Talk:Monty Hall problem/Archive 23

Mathematical formulation section
The section with the mathematical version using Bayes theorem was wrong, the symbols C, S, H standing both for random variables and for possible values taken by them, and consequentlly a number of the wordings were totally garbage. For instance, "the probability P(C)" is nonsense, and writing Bayes theorem with P(H|S,C) etc is nonsensical - Bayes theorem is about events, not about random variables. @Glopk changed it back again but I have undone his undoings. Maybe some mathematicians would like to take a look. If there's to be a mathematical section showing students of probability theory how to do it by routine (automatic, brainless) application of Bayes' theorem then it should be done properly. Richard Gill (talk) 08:12, 14 February 2011 (UTC)


 * At last. Nijdam (talk) 09:07, 14 February 2011 (UTC)


 * Fie. If P(C) is nonsense, then so are the thousands of referred papers and textbooks using this notation (including the ones referred to in the article). Yes, it is a shorthand, and yes, it is common in the literature (and on Wikipedia). On the other hand, please explain how you can keep in your head a the same time "I want to write P(C=c | H=h)" and "Writing P(... | I ) is rendundant because the background information is always assumed". glopk (talk) 21:19, 14 February 2011 (UTC)


 * @Glopk, I have a question and an answer.


 * Question: Please give me some references (papers, textbooks, or wikipedia articles) where $$P(X)$$ is used as notation for the probability that the random variable $$X$$ takes on some value $$x$$. Such wikipedia articles certainly need correction. Especially if the same text goes on to talk also about $$P(Y)$$ and $$P(3)$$. Students who need to see an explicit computation of the text-book (but rarely used) form of Bayes' theorem also need to learn correct notation. What people actually use in practice is Bayes' rule.


 * Answer (to your question about the consistency of my thought processes): Of course all good mathematical notations "suppress" information which is understood to be present but need not be mentioned specifically, since it never changes throughout a whole text, or because it is crystal-clear obvious from the context. Tell me, what is the point of writing $$P( ... |I)$$ *throughout* a whole text, where I stands for background information which is never specified and never changes? I think it would be wise to agree in advance that $$P( ... )$$ will be used as shorthand for $$ P( ... |I)$$. That would be a service for reader and writer. Richard Gill (talk) 16:29, 15 February 2011 (UTC)

I have added mathematical solutions using Bayes rule, and symmetry in two different forms. Richard Gill (talk) 11:16, 14 February 2011 (UTC)

I would now propose to delete the formal computation with Bayes' theorem, replacing it with a link to Bayes theorem where it already is an example. Richard Gill (talk) 11:23, 14 February 2011 (UTC)


 * Oppose The solution using the so-called "odd form" of the Bayes Theorem is much less referenced in the MHP literature than the full Bayes expansion, therefore placing it prominently in the article would give it undue weight. Further, all three "math formulation" paragraphs added by Richard Gill are unsourced, and the last one appears to be OR. I propose to delete (and will boldly do so soon unless I hear a convincing argument to keep them).glopk (talk) 21:26, 14 February 2011 (UTC)
 * "Odds" not odd, @Glopk. Richard Gill (talk) 08:26, 15 February 2011 (UTC)
 * All three proofs are sourced and all three can be sourced even more if that is desired. Richard Gill (talk) 08:46, 15 February 2011 (UTC)
 * Oppose. I personally like the 3 step modelling and I see no problem with that in the article. Moreover it is bugging me, that this is opening up another completely needless edit conflict. In particular if people said above they are essentially happy with article, why do they have to start another round of edit conflicts rather than leaving it alone.--Kmhkmh (talk) 22:57, 14 February 2011 (UTC)
 * I personally also like the three step modelling. I also like alternative approaches. The article will give a very biased picture of the literature if it is written as if there is only one way a mathematician can do a formal proof of the important result. It would be "undue weight" and a great disservice to all students of Statistics 101 out there, only to show the least attractive and least useful mathematical treatment. Richard Gill (talk) 08:41, 15 February 2011 (UTC)
 * Personally I also am quite happy with the present article and I think it's time we move on from the conflicts, and get back to constructive editing. The discussions of two years have actually opened up a lot more literature and knowledge to all of us, and generated several peer reviewed publications. MHP does not stand still. Richard Gill (talk) 08:46, 15 February 2011 (UTC)
 * Oppose. This is Wikipedia, source of info for all kind of readers. This application of the (now) correct form of Bayes' formula is widely accepted as a way of calculating the (necessary) conditional probability. Nijdam (talk) 23:25, 14 February 2011 (UTC)
 * Yes, and it is already there on the Bayes theorem page. So a reference to that article would be good enough. MHP does not stand alone. It is connected to the rest of the world, to the rest of wikipedia. Let's work on cleaning up the Bayes theorem page, it is quite a mess. But in the mean time, a decent version can of course be kept here on the MHP page, if that's what everyone wants. Richard Gill (talk) 08:46, 15 February 2011 (UTC)
 * You say calculating the (necessary) conditional probability. Of course, a smart mathematician realizes that it is not necessary to compute at all, because of independence. The task of mathematics is to replace calculations by ideas (Riemann). Let's not forget that. The task of mathematics teachers is to teach students ideas and how to use them, not teach them to be calculators. Richard Gill (talk) 08:50, 15 February 2011 (UTC)

Not much fun, is it, Richard? Glkanter (talk) 23:51, 14 February 2011 (UTC)


 * On the contrary, great fun! This is what collaborative editing is about. Discussions and controversies, learning and teaching. The sum is more than its parts. Wikipedia MHP has to be accessible to a huge and varied readership. The present editors are representative of the future readers.


 * To business. Of course a conditional probability can be calculated using Bayes' theorem. It can also be calculated using Bayes' rule. The article on Bayes theorem contains this very calculation as an example. So in my opinion it is unnecessary to have it here, but of course I bow to any concensus (how could I do otherwise?).


 * Saying that doing it by Bayes rule is "undue weight" is in my opinion rather silly. Jeff Rosenthal in his article and his book does it this way, others do it this way too, and it is a method which gives insight into "why". This is the "modern" way to do it; via Bayes' theorem is a rather dull old-fashioned way to do it. It merely translates the numerical calculation already done back into formulas! What's the point of that. The formal derivation using Bayes' theorem just shows that it can be done by an automatic proof computer, but it does not give insight. It is an exercise for a class on Bayes' theorem, not a contribution to the understanding of MHP (in my opinion, that is).


 * My third solution is sourced. It is Vos Savant's solution! The door numbers are irrelevant by symmetry and can be ignored. You do not need to compute a conditional probability because there is no need to condition on anything, by independence. I will add some more references. I know that Persi Diaconis has done MHP in this way and others too. I think it adds insight and moreover builds a bridge between simplists and conditionalists. It therefore serves to unify the wikipedia page. Obviously editors who are still embattled in old positions and not ready for reconciliation and synthesis, will object. Then this proof will only be added to wikipedia in ten years time after it has been around in more easily accessible reliable sources for a longer time. Too bad for the wikipedia readers of the intervening decade.


 * Now about notation in the present formal proof. $$C$$, $$H$$, $$S$$ were introduced as random variables. It would be quite standard notation to use small letters for possible values of those variables, and moreover, it would be quite standard notation to use a small $$p$$ to stand for probability mass function, though many authors also use $$f$$, a probability mass function can be seen as a density function, and we then get the same "theorems" for mass functions as for densities. Properly one should attach to the mass function some indication of which variable we are talking about. So you could write for instance $$p(c)$$ as shorthand for $$P(C=c)$$ but $$p(3)$$ would be ambiguous and one should write something like $$p_C(3)$$. Anyway: if the section with the formal proof via Bayes' theorem is to stay, as some kind of help to students of probability and statistics classes, it had better well use correct notation, otherwise it will be no use for them at all.


 * $$P(\cdot)$$ - note the capital letter - is only ever used, in standard texts, as far as I know, for "Probability of". You can have "probability of an event" but you can't have "probability of a random variable". Richard Gill (talk) 08:23, 15 February 2011 (UTC)

Mathematical formulation
As the lemma Bayes' Theorem - The Monty Hall problem already shows the full Monty Hall problem in mathematical formulation, as talked to students of conditional probability in textbooks, and in order to avoid unnecessary redundancy, I hope that there will be no objection to delete the redundant formal computation with Bayes' theorem here, and to replace it with the proper link to Bayes' theorem - The Monty Hall problem. It's a matter of avoiding unnecessary redundancy. Gerhardvalentin (talk) 18:14, 11 March 2011 (UTC)
 * This article stands on its own. IMO, the full Bayes expansion is useful here.  It's not overly long.  Why delete it? -- Rick Block (talk) 18:39, 11 March 2011 (UTC)


 * Rick, in Bayes' Theorem, the "MHP" is a welcome example to show how to use various assumptions in conditional probability theory and -calculus. And there it does not need to question the sense of licentious assumptions and licentious presuppositions. No need. It's shown there for the sake of understanding conditional probability. And there it belongs, and it is needed there. I repeat: It belongs there.  Period.


 * But, for the reader of the MHP, to understand the paradox, never needed flawed assumptions nor "conditioning" on any completely irrelevant "(im)possible might be-s". The numbers of the doors are irrelevant. No need to condition on their irrelevant "numbers". Please pay respect to what all reliable sources say. Serious "conditional probability", even intentionally pretending to "know exactly" what no one can ever know, and with intentional pretentiousness considering absurd additional information received on a silver platter, never was able to show that staying could ever be better than to switch. Even intentionally "using" the absurd illusion of additional silver platter information. So all of that illusion is of no avail for the MHP.  But, in contrast, it's a matter of "conditional probability". Please notice that this is "two quite different pairs of shoes".


 * All of that illusion, that conditional probability – even in gathering never to be given "silver platter information" – could ever advise another decision than to always switch, is of no avail whatsoever for the MHP. Please understand what the lemma is about. It is about a paradox that more than 90 % of people get wrong. So the lemma should help to clearly see the obvious paradox and how you can match for it. So: No more confusing mind games, please.


 * Once more, Rick: The value of the "conditional probability" is totally irrelevant for the paradox called "MHP". So you should show all of that absurd mockery later in the article, for people interested in. And there should be the link to the Baye's Theorem, also. You are free to present all of that academic odds and ends there. But not in the first place where the secret of the paradox should be accessible for the interested reader. I strictly oppose to shoo interested readers with uninteresting mockery, but that gimcrack mockery should be broadly shown later, for the delectation of readers interested in. So I will be going to do as I proposed and set a link to Bayes' Theorem. Please help the article to be accessible and intelligible for the readers. Thank you. Gerhardvalentin (talk) 21:37, 12 March 2011 (UTC)


 * There are 4 editors here who have opposed deleting this from the article, which at least IMO would make this a change for which there is not consensus - i.e. would make this simply disruptive. -- Rick Block (talk) 22:15, 12 March 2011 (UTC)


 * Thank you. Date of your accredited warrant of attorney? Rick: It is for honesty, it's for the readers, it's for what the sources say. Please help the lemma be what it should be: The unbelievable paradox, and its dazzling history. I strongly hope you will help to "solve the problem". It's for the sake of Wikipedia and for the sake of the readers, to get all of that in a clear manner, not disintegrated, nor misty convoluted, anymore. Hope we can focus on the clearness of the lemma. Thank you. Gerhardvalentin (talk) 22:38, 12 March 2011 (UTC)

Request
@Richard: please do not save every time you change one or more letters. And wait with your next changes until the other parties had the opportunity to react. Nijdam (talk) 08:36, 16 February 2011 (UTC)


 * Those are "minor edits". Richard Gill (talk) 08:43, 16 February 2011 (UTC)


 * By the way, you recently reverted some text near the start of the article so as to promote your Point of View. This was "solutions are almost always based on the assumptions ...". Lots of solutions are indeed based on the usual assumptions of random location of car and random choice of host when he has one. However lots of solutions are also based on the *only* assumption that the player's initial choice is random. See the last section started by an anonymous editor. I think "almost always" is a gross exageration. Richard Gill (talk) 09:11, 16 February 2011 (UTC)
 * Richard, W.Nijdam just only did revert an IP edit - back to your version. Gerhardvalentin (talk) 11:05, 16 February 2011 (UTC)


 * There are assumptions, and there are assumptions. The assumption that the host will behave the same way whether you choose a car or a goat is essential, without this assumption the problem is ill posed. The assumption that the car is randomly placed, or that the host has to open a goat-door (as opposed to opening a car door sometimes, thereby giving away the answer), or that the host has to choose the two goats at random are irrelevant distractions, which do not change the analysis of the problem or the answer. One must never conflate irrelevant assumptions with relevant ones.69.86.66.128 (talk) 11:07, 16 February 2011 (UTC)


 * Exactly, @69.86.66.128. The assumption is essential that the host is always going to open a door and reveal a goat. Most of Vos Savant's readers understood her to mean this, she also later said that she meant that, and Selvin from whom the problem originated (and before him, Gardner) have the same assumption, explicitly. All sources thereafter, as far as I know, also make this assumption. Other assumptions are up to the reader - there is not a concensus though there might be said to be a fairly clear majority opinion. My personal *opinion* is that if you use probability in a subjectivist sense, as I think do most ordinary people - thus probability is a measure of *your* personal (un)certainty - then the assumptions that all doors are equally likely or that either of the host's choices, when he has one, are equally likely, are automatic (logical) consequences of Vos Savant's problem statement  (cf. Laplace (1814), founding subjectivist probability as a rigorous mathematical science - everything is defined in terms of "equally likely", in terms of symmetry, in terms of knowledge and lack thereof). All we have to go on are Vos Savant's words. If however you use probability in the  frequentist sense, as many but by no means all scientists do, and many but by no means all statisticians and probabilists do, then my personal opinion is that the problem is ill posed, unless of course you allow the player the option of introducing randomness himself by choosing his door initially at random. —Preceding unsigned comment added by Richard Gill (talk) 14:12, 16 February 2011 (UTC)

How do the words, 'Suppose you're on a game show..." affect the above response? I read that as equivalent to 'a fair die is thrown in a fair manner'. I would say every American who has watched 3 - 5 hours of game shows on TV every day since birth would have the same interpretation as me. Glkanter (talk) 14:18, 16 February 2011 (UTC)


 * The wording in the lead is a summary of referenced wording in the "Problem" section, i.e. we're not talking about what editors think but what sources say. For example, the sentence "The resulting set of assumptions gives what is called "the standard problem" by many sources" is referenced to Barbeau (2000), which says: "The standard analysis of problem M is based on the assumption that after the contestant makes the first choice, the host will always open an unselected door and reveal a goat (choosing the door randomly if both conceal goats) and then always offer the contestant the opportunity to switch." -- Rick Block (talk) 15:35, 16 February 2011 (UTC)


 * What a sad, valueless rebuke, Rick Block. My comments are wholly consistent with the sources you mention, as well as Selvin, vos Savant, and K& W. All of whom are Americans (U.S) referring to a puzzle made famous in American (U.S.) periodicals about an American (U.S.) game show. Unless you were making those comments to some other editor, which isn't clear at all. Glkanter (talk) 16:29, 16 February 2011 (UTC)


 * Yes Rick, I agree that *many* sources call these assumptions the standard assumptions. Now, here's a specific case: I objected recently to @Nijdam's choice of wording "almost all sources". Do you support my recent alteration of "almost all" into "many"? On the arbitration page I am being viciously attacked for COI and supposedly promoting my own research and for again becoming an active editor of the page: but I think I am not promoting own research at all: I'm doing my best to collaboratively edit the article according to wikipedia principles. (Editors who find my own publications in this area totally unimportant ought to remove them from the reference list as soon as possible, I wish somebody would do that. They used to be on the talk pages, in order to share information.)


 * Amusingly, Barbeau (2000) presents only a simple solution, one which makes no use of the "random choice of the host" assumption at all. Richard Gill (talk) 17:24, 16 February 2011 (UTC)


 * My point, Richard, is that a puzzle about a game show, and Barbeau's problem statement includes the words 'game show', the "random choice of the host" premise exists simply by the common understanding of the term 'game show'. You, Rick, glopk, kmhlmh or anybody else can continue to argue that point. For whatever reasons you so choose. I will no longer do so. No more than I would argue that '6 is a prime number'. Glkanter (talk) 17:38, 16 February 2011 (UTC)


 * I am not arguing, @Glkanter, I think we agree. Is the wikipedia article only for the layperson who learnt about the problem in a discussion at a bar, or is it also for the student of Statistics 101 who learnt about it in his class? I think it's for both. Richard Gill (talk) 16:28, 17 February 2011 (UTC)

Rick Block says above: Barbeau (2000) [...] says: "The standard analysis of problem M is based on the assumption that after the contestant makes the first choice, the host will always open an unselected door and reveal a goat (choosing the door randomly if both conceal goats) and then always offer the contestant the opportunity to switch."

And I repeat: (choosing the door randomly if both conceal goats). –  Exactly as Marilyn vos Savant was underlining of having been her "starting point". Evidently Barbeau knew what he is commenting about. And then I read above: Barbeau (2000) presents only a simple solution, one which makes no use of the "random choice of the host" assumption at all. And we surely can take Barbeau to be knowing what he says, can't we? Being aware of the obvious requirement of the elementary supposition of "randomness", that he emphasized explicitly, as an elementary rule for any approach to give an answer to the famous question, he nevertheless presents a "simple solution".

Am I right in concluding that this obvious requirement of "randomness" that he emphasized explicitly, also for him is the elementary basis for the simple solution he presents? What, imho, for Barbeau like for vos Savant evidently implies the premise of an unbiased host for the "simple solution" they present.

Rick Block, am I correct in reading it this way? Regards, Gerhardvalentin (talk) 06:33, 18 February 2011 (UTC)


 * The solution Barbeau presents is this: "The contestant initially selects a door concealing a goat with probability 2/3. With a policy of always switching, she will win a car with this probability."  This is the same sort of wording Grinstead and Snell use for what they call their simplified version (that analyzes the probability of always switching vs. always staying), and the same sort of wording Carlton uses for his "intuitive solution" ("imagine you plan to play ... and employ the switching strategy").  This solution makes no mention of and has no dependency on how the host chooses between two goat doors.  It is not saying the (conditional) probability of winning by switching if you've picked door 1 and have seen the host open door 3 is 2/3.  It is instead saying if you decide ahead of time to switch and switch regardless of which door the host opens ("with a policy of always switching"), then your probability of winning is 2/3.  If you want to know the conditional probability given you've picked door 1 and have seen the host open door 3 you have to figure it out (somehow).  This is the point Morgan et al., and Gillman, and Rosenthal, etc etc make.  Hypothetically, adding "everything is symmetrical with respect to the doors, so the probability in any individual case must be the same as the 'always switching' probability" would be enough (although one might wonder how you know everything is symmetrical, even given that the host must pick randomly between two goats) - but the sources presenting simple solutions typically say nothing like this and make no mention of the critical assumption.  As Rosenthal puts it "This assumption, callously ignored by the Shaky Solution, is in fact crucial to the conclusion" (where the conclusion is that the conditional probability of winning by switching is 2/3).


 * The distinction between the probability of winning "with a policy of always switching" and the (conditional) probability of winning given you've picked door 1 and have seen the host open door 3 is what Morgan et al. are referring to when they say "The distinction between the conditional and unconditional situations here seems to confound many." -- Rick Block (talk) 18:28, 19 February 2011 (UTC)


 * This is the same 'reasoning' you've been monopolizing this article with for years. In the 2/3 & 1/3 MHP paradox, the 50/50 host bias is a stated premise (it also comes from the definition of a game show). There is no requirement that any solution to a problem uses every premise (piece of information provided) to solve it. You're just making that BS up, sadly. The simple solutions *are* conditional. As I've shown with the 100% conditional decision tree derived from Carlton/Morgan/vos Savant/Selvin/Hall. You make grand leaps of assumptions, and cannot provide a reference that gives a conditional solution that says any of that crap above. Glkanter (talk) 18:46, 19 February 2011 (UTC)


 * Please @Glkanter try to concentrate on the real issue: how the wikipedia article should take account of the hard fact that the sources do not agree how MHP should be solved. You can't alter the fact (except by destroying internet and burning down all the university libraries in the world) that a load of sources state loud and clear that more work has to be done to deduce that the odds are 2:1 on winning by switching given you chose door 1 and the host opened door 3, than to deduce that the odds are 2:1 on winning by switching given only that you chose door 1.


 * Yes, Richard, I *will* 'try to concentrate on the real issue'. Can you please clarify what you mean about '...a load of sources...'? I understand there are many different solutions, from many disciplines. That's *not* anybody's problem. Except Nijdam's, perhaps. The problem is that the simple conditional solutions are being prominently and repeatedly bad mouthed with UNDUE WEIGHT and NPOV violations and OR. I get really tired of having to repeat this. Look at the Conditional solution section and all that simple-solution-bashing variants crap in paragraphs 1, 2 & 4, please. We've been over this before, countless times. Glkanter (talk) 20:48, 19 February 2011 (UTC)


 * A load of sources: Carlton, Morgan, Rosenthal; then also just about every introductory probability or statitics text I have ever looked into (and that's a lot). The book by Grinstead and Snell is quite typical. Read it. It's free, it's on internet, it's promoted by the American Mathematical Society. It's quite well written except in my opinion for the Monty Hall section, where they too engage in the rewriting of history which Morgan and his friends went in for. Try working your way through it, so you learn to understand the mind of your opponents. That's how to win a fight. Richard Gill (talk) 10:09, 20 February 2011 (UTC)
 * Yes, there are various solutions. We all know that. And agree they belong in the article. And 1 problem statement. And no critics of the simple solutions. Except in the minds of certain editors. Nowhere else, though. Glkanter (talk) 10:17, 20 February 2011 (UTC)
 * I recall you never ever answered my question whether *you* thought these two matters were the same or different. But I do remember that you were delighted when Boris Tsirelson told you that the single word "symmetry" (which is implied by indifference for a subjectivist like you and like most ordinary folks) was enough to bridge the gap. Later Rick pointed out to me that Boris' approach was already in the literature: William Bell (1992) criticised Morgan et al. for making a mountain out of a molehill. Altogether I found three pretty symmetry-based proofs in the literature and put them in the conditional section, as an alternative to the lengthy formula manipulation which is sometimes used as an exercise or an illustration for students learning formal probability calculus (you can find it where it belongs, as an illustration in the wikipedia article on Bayes theorem!). Richard Gill (talk) 20:19, 19 February 2011 (UTC)


 * Yes, that symmetry *should* satisfy Nijdam, as I understand it. To the best of my understanding, Nijdam still considers the simple conditional solutions wrong, however. Hence, it is of little value to me in these stalemated discussions. In any case, what you describe above should change the argument from 'wrong' to 'not written very well by this particular High Priest' (What?! How can that be!?) or ignorant commoner. But that doesn't seem to have happened, either.
 * Rather, I prefer to argue logic. Cutting out the High Priests. The only way the solution, from the contestant's SoK, is 2/3 & 1/3 is if he is told *before* he makes his door choice all those things that the host is certain to do (that is, the contestant is *not* playing a variant problem with different results). So the contestant, being a thinking, sentient being, just like us Wikipedia editors, can start analyzing and deciding right then, indifferent to the door #s. Or he can wait until later. Lastly, the 'Carlton' decision tree shows that simple solutions *are* conditional, and that the contestant *will* be facing 2 closed doors and 1 open door. And he's indifferent to the door #s. I've said all this thousands of times. All that stuff about before/after, etc is make up and/or grossly over-interpreted and/or grossly over-exaggerated BS, not derived from or representative of a significant minority of reliable sources. Glkanter (talk) 20:48, 19 February 2011 (UTC)


 * I too prefer to argue logic. Remember, Laplace builds probability calculus on logic. It is nothing more than logic, the logic of the concept "equally likely". And "equally likely" is usually invoked because of symmetry - physical symmetry of dice, coins, or symmetry of our knowledge, e.g. indifference. That's why I like my third alernative conditional solution best. It replaces the technical concept of "conditional probability" with the logical concept of "independence". Independence coming again from symmetry, coming from indifference. The door numbers are irrelevant. All that matters for the player is their role (both visible and hidden to the player). He's two times more likely to hit a goat than a car first time, so he should switch. End of story. Richard Gill (talk) 08:38, 20 February 2011 (UTC)


 * It really is all the fight between frequentists and subjectivists. The present intro is pure frequentist. The host's choice is random (the host uses a fair coin toss). No! The player is indifferent to the host's choice. For the player, the host might as well have used a fair coin toss. The intro to the article already is enforcing a particular POV. Richard Gill (talk) 08:38, 20 February 2011 (UTC)

Is there any chance you'll respond directly to the issue I raised above, rather than taking us on another long and useless trip through Richardworld?
 * My first point is that there is only 1 MHP problem statement, the one that has an outcome of 2/3 & 1/3 [please, show me in the literature any others that are *not* variants], and that the contestant has been told the rules before he watches them played out. So from the single 'standard' problem statement, any of the solutions is acceptable, and none is unacceptable.
 * Until this issue is resolved, everything else is wasted energy. And it sounds like, despite it being OR from each and every one of you other editors, this is what you guys want to argue about. Except it's made up. It doesn't exist in the literature. All that F0 and S0 and stuff from Nijdam is bogus. It's phony. A contrivance.
 * I entirely agree with your first point. (Of course, there are examples of solutions whose argumentation is faulty. Eg Devlin, who wanted to do the conditional version of the problem, but overlooked a crucial step. We shouldn't refer to solutions where the logic is patently wrong. Devlin himself retracted his wrong argument.) Richard Gill (talk) 10:12, 20 February 2011 (UTC)
 * Well, then help me put the kibosh on all that wasteful rhetoric that Nijdam, Rick and Martin go on endlessly about. Any source that has renounced a paper in public I would want to know about. I'm not aware of Devlin's retraction, I have read Morgan's, and I disagree with your interpretations of Rosenthal. Glkanter (talk) 11:00, 20 February 2011 (UTC)
 * Tell me why Nijdam ignores what you and Boris have come up with, that the literature had all along?
 * How can I tell you how Nijdam's mind works? It's incomprehensible to me. (Signed HP = High Priest Richard the Omniscient).
 * Tell me why the Conditional solution section in the article is 90% about variants.
 * I don't know why, they don't belong there. (HP)
 * Tell me why you and the others ignore the 100% conditional decision tree that shows the simple solutions *are* conditional? Meeting Nijdam's 'requirement' that the contestant is looking at 2 doors?
 * Because everyone else uses the word "conditional" in a different (and more technical sense) than you. (HP)
 * It's not more technical, they do that to make their argument seem stronger than it is. It's a confusion tactic, using vague words as meaningless 'jargon'. My terminology, with greater precision, shows the canard. I'm disappointed you don't agree with me, and seem to defend their tactics. Glkanter (talk) 11:00, 20 February 2011 (UTC)
 * Tell me why the unconditional interpretation of the problem, and simple solutions are included in the Causes of confusion section.
 * Tell me why. It's a mystery. Could it be part of Rick Block - Nijdam - Glopk attempt to hijack MHP from the people, to whom it belongs, and hide it in a Statistics 101 class? (HP)
 * Tell me why Nijdam and the others continue to insist that *Probability* is the only tool that may be used to solve the puzzle. Despite the facts, and despite the literature.
 * Tell me why. I imagine they didn't know that subjective probability is just logic, anyway. They are slaves of the very calculus which was invented to serve us, they don't see any other way to do these problems except through the "official calculus". But it's not always the best tool for the job. (HP)
 * Tell me how Selvin and vos Savant are both equally too stupid to realize they can't solve the puzzle they each created with a simple table of all possible outcomes. But Selvin's a High Priest. How can he be a High Priest and be wrong? Oh nos!
 * Selvin is not a high priest. He was a guy having fun telling his mates about a stimulating brain teaser. Which can be looked at in all kinds of different ways. I think Vos Savant is so smart that she didn't realise that some people would need to have the symmetry spelled out for them. Also, she is using probability in a subjectivist sense while the Morgan's and others are clearly thinking in a frequentist frame. (HP)
 * The guy is a math PhD writing in a PEER REVIEWED PROFESSIONAL JOURNAL, right?. Saying that he is not a 'High Priest', just like you, or Nijdam, or Rosenthal or Morgan, et al is silly. Any difference that *may* exist is meaningless to us commoner dumbasses.
 * Which misses the point, anyways. They wrote the problem. They know what they meant, fer crissakes! Glkanter (talk) 11:00, 20 February 2011 (UTC)

But please, stop talking in High Priest code that doesn't address these issues. Hopefully, they'll topic ban you along with those hustlers. Then, maybe some old editors will return. Then all the babbling can come to an end, and the article can become what it should have been all along. Glkanter (talk) 09:24, 20 February 2011 (UTC)


 * @Glkanter, please try to learn some of what you call High Priest Code. Did you read Laplace (1814) yet? He was writing for ordinary intelligent people, not for mathematicians! Did you study the wikipedia page on the interpretation of probability? Have you studied the first chapters of a decent introduction to probability theory, so that you finally know what people mean when they write conditional probability, and so that you come to learn and love Bayes' theorem? I suggest this not to make you change your opinion, but so that you can fight your opponents better. I'll answer your questions "in situ" in a moment. Richard Gill (talk) 09:44, 20 February 2011 (UTC)

Intelligence: the ability to make finer distinctions. How to increase intelligence: by reading and thinking. 


 * I've read plenty. I read your papers, even the one that contained the folklore about the dumb Americans that couldn't figure out the game in real life. And I've read all your contradictions, and vagueness, and all sorts of things. I've read the so-called critics. So, stop trying to belittle me. Now, are you done editing, so I can post my responses without getting another edit conflict? The 'preview' button works wonders. As do the browsers with the in-line spell check. They're all the rage, you know! Glkanter (talk) 11:00, 20 February 2011 (UTC)

Too bad you can't focus on the above items, as I have begged you to do forever. We could make the article so much better if we we worked together. But, I insist on not learning, and you insist on teaching. And never the twain shall meet. Of course, you also insist on talking about optimal solutions, game theory, minmax, dense jargon, your papers, and all kinds of stuff that will never appear in the front of the article, where we need to focus our attention. Glkanter (talk) 11:00, 20 February 2011 (UTC)

You see, Richard, right off the top of my head I came up with half a dozen issues that are negatively affecting the article. And improving the article, for many of us, is why we're editing on Wikipedia. And it looks like you agree with some of my ideas as stated above. So, no, I don't need some greater level of knowledge in order to meet my goals. I need less interference and obstructions from editors with bogus goals and agendas. Maybe after we take care of the low hanging fruit I would be interested in pursuing other aspects of the MHP. But not yet. That's not what I've invested 2+ years for. Any interest in being part of the solution, Richard?


 * @Glkanter: That's exactly the perfect throw, showing how the lemma should be : –  Without bloodcurdling inconsistence and without absurdities. Please help to get it that way. Regards, Gerhardvalentin (talk) 11:50, 20 February 2011 (UTC)


 * There big difference there, StatProb a size restriction, that we do not have, so we could (and imho should) treat the problem in greater detail (properly structured of course, separating a fast route with the most important aspects from the more detailed treatments and distinctions).--Kmhkmh (talk) 13:55, 20 February 2011 (UTC)


 * Sure, Gerhardvalentin, I'll "help to get it that way". I'll tell you what I'll do. I make those exact edits I describe above. Then those other guys will all revert my edits, no matter how many times I put them back in the article. Then I'll get reported for edit warring, and I'll get blocked. Meanwhile, you, and Richard, and Martin will pretty much just stand around doing nothing while I get my ass kicked. Sounds good, no?
 * Oh, wait, we all already did that last summer. Glkanter (talk) 15:11, 20 February 2011 (UTC)


 * @Glkanter: Please make a new section down here on the talk page, titled "Glkanters proposal", showing your intended changes, and paying regard to others comment (say you what, I'll be very critical), and please try first to convince. –  And as to me, I will do so alike. Because I'm not as experienced in editing like others. Regards, Gerhardvalentin (talk) 15:29, 20 February 2011 (UTC)


 * The minute I know that the other editors are on board, I will do that in a heartbeat. In the meantime, I want to delete paragraphs 1,2 & 4 from the Conditional solutions section, and that 2nd image in that section. Replace Carlton's solution in the Simple solution section with his actual quote, and add the decision tree inspired by his solution. Remove all mentions of simple solutions form the Sources of confusion section. Just have 1 solution section with no subheadings or biasing narratives - simple, formal decision tree, Bayes. That's a start, anyways. Glkanter (talk) 15:50, 20 February 2011 (UTC)


 * @Kmhkmh, I like your words: "properly structured of course, separating a fast route with the most important aspects from the more detailed treatments and distinctions", and also I would prefer to enforce visibility and to avoid messy confusing jumping around within the convolute, repeating the same several times, and leaving out what all is about. But structured in separated sections, from important to less important (details and variants). And noting that not all sources are equally important to fully understand and appreciate the paradox. Gerhardvalentin (talk) 16:35, 20 February 2011 (UTC)

Getting the Answer to the Question – Solving the Problem
Imo the problem (not to make it a bad-joke-question) is almost always solved using general assumptions, at least using the most general assumption (said by vos Savant to be implicit in her question and her own answers) that the host is certain to open a door showing a goat (which he can always do, because he knows the location of the car). The contestant has no knowledge on the location of the car, so usually it is either assumed that the car is equally likely to be behind each of the three doors, or that the player's own choice is completely random. Some solutions add to this the assumption that, if the host has two goats to show, having a choice which goat door to open, he is equally likely to open either. It's a question about just "one game show". Having no better knowledge at all, and just to stay serious you have no other choice, as you never can expect that show will or "must" be repeated, just to suit your special requirements.

Note that "must randomly choose" is putting randomness into the host's actions. This is obviously a frequentist idea of probability. But most people look at MHP subjectivistically, so chance comes from their lack of knowledge about givens, their lack of knowledge about the world, it is not in the physical world itself.

So, for a subjectivist, the two "equally likely" things (location of car, door opened by host) are not "extra assumptions". No, they are deductions or consequences of the problem statement.

And that's the basis of a reasonable answer to the question whether to switch or to stay. Gerhardvalentin (talk) 09:02, 20 February 2011 (UTC)


 * Very good, Gerhard! I frequently tried to change the present intro "Although not explicitly stated in this version, solutions are often based on the additional assumptions that the car is initially equally likely to be behind each door and that the host must open a door showing a goat, must randomly choose which door to open if both hide goats, and must make the offer to switch" into something like the following:


 * Though not explicitly stated, Vos Savant intended, as indeed most readers interpret her question, that we are to suppose that the host must open a door showing a goat -- something he always can do, since he does know the location of the car. Almost all solutions are furthermore based either on the assumption that the car was hidden at random, or that the player chose his door at random. Alternatively, since we have no information beyond what Vos Savant gives us, for us the car is initially equally likely to be behind any of the three doors (see Probability Interpretations). Some solutions furthermore assume that if the host has a choice of door to open, he determines his choice at random. Alternatively, in this situation, for us either choice is equally likely, since again we have no information beyond Vos Savant's words.


 * The present text is already biased to a frequentist view of probability and towards the conditional solutions. Of course, in introductory statistics texts, in their chapters on Bayes' theorem, such a bias is quite natural. But it is not a natural bias to the ordinary person hearing about MHP in a discussion at a bar or at a party. The problem has to be solved by logical analysis, not by the calculus of probability. Laplace (1814) built the calculus of probability explicitly on the logic of the concept "equally likely", the logic of symmetry, of indifference. And Laplace's probability is the sort used by ordinary people in their day to day lives. Back to basics! Richard Gill (talk) 09:37, 20 February 2011 (UTC)


 * Recall Martin Hogbin's words: no one can think deeply about MHP without pondering on the meaning of probability. Well, wikipedia has an article on that. MHP is not isolated from the rest of the world, it is not isolated from the rest of wikipedia. Good articles have plenty of cross-links. Nijdam should be finding wikipedia articles which explain his point of view, and if he can't find them, he should source the ideas and write the article.


 * Now, how about deleting that passage in the conditional solution with all the formulas? It is totally superfluous since already on wikipedia where it belongs (and where of course it came from!) as an illustration of Bayes theorem. Seems no one ever reads it either, last time I looked it was packed full of gobbledygook (even Nijdam approved of my corrections that time). Richard Gill (talk) 10:02, 20 February 2011 (UTC)
 * +1. Yes, because frequentist is just "one" aspect, dishonestly overstressed in the article. Unnecessarily filling and dominating the whole lemma. Just based on forever unknown (but firmly said to be given) repetitions. Farcical. Yes, the frequentist's aspect has to be arranged and integrated as it belongs. To straighten the hitherto absurdness of the lemma.
 * And to link accordingly to where it is legitimate. Gerhardvalentin (talk) 10:17, 20 February 2011 (UTC)
 * @Richard: I'd wish you'd give that issue a rest, the objections from above haven't changed and as i said before is just (mostly needlessly) created another edit conflict. You should simply accept that some people prefer this treatment - period.


 * We immediately saw "objections" from three objectors, but no comments by anybody else, that's why I repeat this point here. Moreover, I find the objections very poor indeed. I suspect that they are more to do with conservatism than with caring for the interests of the readers of the article. So naturally, those who object will go on objecting till kingdom come, but I think we're all agreed the article is bloated with seondary detail. That derivation is a detail. And: it is elsewhere on wikipedia, so it can be replaced by a link! It is located where it belongs and where it came from: illustrating the workings of Bayes' theorem. It does not illustrate MHP. On the other hand, various proofs with symmetry or independence or the odds form of Bayes' rule give insight into Monty Hall Problem which you could share with your grandma. And insight into the difference between simple and conditional. And all of them used in the past by authoritative, clever, insightful writers to give complete and elegant analyses of the standard problem. The article as a whole doesn't have enough links to the outside world inside wikipedia. It is a little microcosm for editors who only care about MHP and don't know about anything else. That's not good for an important encyclopedia article. And it needs to be linked to, from elsewhere, for the same reason. Richard Gill (talk) 20:06, 20 February 2011 (UTC)
 * I know that you think that. The issue here is that you learn to live with others differing here and let it go.--Kmhkmh (talk) 21:26, 20 February 2011 (UTC)
 * I can live with others differing in their opinion from me, and I do let that go. I am angry that a minority is holding on to their minority point of view which has become engrained into the present article, blocking a majority from allowing the article to progress back to a balanced reflection of the literature. Richard Gill (talk) 06:42, 25 February 2011 (UTC)
 * As far as problems frequentist arguments are concerned, one should keep in mind although apparently favoured by some here, it is not without problems eithe. The assigning of priors but not be as obvious/justified as one might think. Another thing is that the concept of various implicit rules (don't show a car, don't open the candidates door, follow some rule at all) are somewhat associated with the notion of repeatability (the very nature of law or rule is repeated application in a way). Now if one insists one one time event nature, one might argue as well that you lose the rules and even symmetry (some publication(s) actually make that argument) and as consequence you loose 2/3 and simple approach altogether. --Kmhkmh (talk) 13:44, 20 February 2011 (UTC)


 * I disagree, Kmhkmh. I'm talking about the standard MHP, not some academic variation. Given only the words of Vos Savant (including her suggestive and parenthetical "say, Door 1", and "say, Door 3") together with the uniformly agreed addition that the host will certainly open a door becaues he always can, the subjectivist's prior is completely determined. I'm talking about the basic standard MHP, the one for ordinary people talking in a pub or at a party. Who are told vos Savant's question and the universally agreed clarification. And nothing else. Its a one time question with no other information, no previous experience. (Credit to Glkanter for hammering away at this point). Of course one can change the problem and things will become problematic. That's fine for some section for academic readers about variants.


 * BTW I agree with Martin Hogbin that for the frequentist, the same basic MHP is insoluble, precisely because we have no information to go on. The smart frequentist will therefore randomize his choice in advance of the show and switch whatever. His unconditional probability is 2/3, he doesn't know and doesn't care what his conditional probability is. He knows he's doing the best he can do, anyway (game theory) so why bother. Richard Gill (talk) 20:15, 20 February 2011 (UTC)
 * Well the problem everybody is talking about is first and foremost Whitaker's (and not vos Savant's) question. And the "uniformly agreed addition" is not uniformly agreed in literature either. It is just the most common assumption (or simplification) generally considered as reasonable to get a handle on the problem. And the real Monty Hall for instance did not always open a door nor behave strictly regular in anyway.--Kmhkmh (talk) 21:23, 20 February 2011 (UTC)
 * Whitaker's question was I’ve worked out two different situations based on whether or not Monty knows what’s behind the doors. In one situation it is to your advantage to switch, in the other there is no advantage to switch. What do you think?


 * Marilyn explained regarding her own question that the words "say, Door 1", and "say, Door 3" were added to help you visualise the situation but are not part of the question. They are parenthetical comments. They can be deleted.


 * Morgan et al. and later authors changed the problem and then told Marilyn off for giving the wrong solution. Richard Gill (talk) 06:36, 25 February 2011 (UTC)


 * What you say above is consistent (derived from?) my insistence that there is only 1 MHP, *not* as you have said in the past, [paraphrasing] "an ever evolving MHP that no longer belongs to Selvin or vos Savant". I'm trying to make this very point in the arbitration to support my contention that Rick's & Nijdam's POV about the 'critics' is *not* supported by the reliable sources (they [except Morgan, who just flat out lies] change the problem in order to teach conditional probability, Rick incorrectly reads this as criticisms), and I put together that table in my evidence that some arbitrator decided to collapse. I think you'd be doing the arbitration, the article and me a lot more good if you would write something in your evidence section in support of what I have been saying all along, and which you seem to now agree with. Glkanter (talk) 11:23, 25 February 2011 (UTC)


 * My friend, please learn to make distinctions. This is the mark of intelligence.


 * There are Vos Savant's words, they are fixed, and will never change. How they are interpreted is not fixed. Morgan et al. interpret Vos Savant differently from how you do. Personally, I think that there are many legitimate formalizations of her question. Some more popular than others. My personal opinion is unimportant. Which formalization is most popular, can change and does change, in time. The meta-MHP is the problem to formalize Vos Savant's words responsibly into a problem which allows logical analysis. The answer is the answer. The path between formalization and answer has to be a logically correct deduction from starting point to end point.


 * So we have to distinguish three things: The exact problem which we are going to solve. A logical deduction from that problem. An answer which is the endpoint of the logical deduction. Richard Gill (talk) 14:49, 13 March 2011 (UTC)


 * Please provide examples of sources that give different problem statements (and hence, different premises than K & W or Selvin) that have a result of 2/3 & 1/3. The only ones are the so-called critics, not the mainstream. I have a table that shows this in the evidence section. You'll need to un-collapse it. Glkanter (talk) 15:13, 13 March 2011 (UTC)
 * Morgan, et al, but not the other authors, say they are quoting vos Savant, use quotation marks, then change vos Savant's problem by improperly and deceptively eliminating the 'say' part of 'say door #1' and 'say door #3'. They're bums. And their journal's peer-reviewers aren't much better. Glkanter (talk) 06:50, 25 February 2011 (UTC)

Citing accurately
It is just a minor thing, but still in the symmetry solution we have:
 * In Tierney (1991), the mathemagician and Stanford professor Persi Diaconis stands up for vos Savant,

However in Tierney's cited article Diaconis does no such thing. What he in fact does is being sympathetic towards his colleague Sachs (not vos Savant) for getting it wrong and later in the article he even explicitly argues that strictly speaking the problem cannot be solved without knowledge about the host's behaviour rather than siding with vos Savant. Now Diaconis might have explicitly supported vos Savants approach elsewhere, but he certainly does no such thing in Tierney's article. In short either the sentence regarding Diaconis needs to be changed or another source would be required.--Kmhkmh (talk) 21:56, 20 February 2011 (UTC)
 * Sorry. That was careless of me. I was reading an article from New York Times, 1991 or so, and thought it was the same as the one referenced in the article (Google search on "Diaconis Monty Hall"). I took the wording "Diaconis stands up for Vos Savant" right from the article. I know Diaconis was also interviewed in another article. By the way, the point about the host's behaviour, Diaconis' point, is that we need to be told that the host will always open a different door and reveal a goat. And if we are frequentists we need explicit symmetry (uniform-random) assumptionsm (but Diaconis is usually Laplacian, and then symmetry follows from lack of information to the contrary and/or Vos Savant's indication that the specific door numbers are not meaningful. Which she also wrote somewhere, later). Diaconis says he had written on the MHP problem before Vos Savant had made it famous, he knew the Selvin version from years earlier. More literature search needed. Richard Gill (talk) 14:28, 21 February 2011 (UTC)
 * PS, one could also refer to Yours Truly (2011), but I am not going to promote that. COI, OR. Question is, is it useful? Since we're both mathematicians, Kmhkmh, we might privately agree on the Mathematical Truth. You work in geometry. Symmetry in geometry is as old as geometry, just as symmetry in probability is as old as probability. But I'm about to take a hopefully restful WP:WIKIBREAK. Richard Gill (talk) 14:41, 21 February 2011 (UTC)
 * I corresponded with Persi about all this. He agrees with me about the content. Of course a private communication is not a reliable source. But I just mention it, anyway. Richard Gill (talk) 06:32, 25 February 2011 (UTC)
 * See below for why non-uniform choice (by the host) should not affect the players' game expectation overall, although it does still affect their expectation after the door is opened. Rich Farmbrough, 11:35, 27 February 2011 (UTC).

Useful resource
Jason Rosenhouse's book's first chapter is on internet:. (I don't agree with him on many issues, but still, this must be considered a highly "reliable source"). Richard Gill (talk) 11:30, 23 February 2011 (UTC)

N doors and biased choice

 * I changed the N door section to reflect that N/p is the important number, not N or (number of doors -1). This is a factual correction (quite a risk saying that on Monty Hall talk.. if I'm wrong - revert obviously).
 * The biased choice - if the host has a 100% bias to door 2 (i.e. he will open it whenever he gets the opportunity) then the "value" of his disclosure is split unevenly depending on the door he opens. Door 2 - no information probabilities are .5 .5, door 3 full information probabilities 0, 1 (always switch, always win).  The net expectation is unchanged at 2/3, and indeed (sensibly enough) the exception contribution of each door opening event stays at 1/3. We should include in the table "Other host behaviors" enough information to show that the biased choice does not a priori give the player an advantage, in some cases it works out as a disadvantage, in others as an advantage, on average makes no difference.
 * (Detail: Bayes theorem confirms 1/(1+p) as the expectation (from switching) given door 2 is chosen. The probability of the host choosing door 2 is 1/3+p/3+0/3=(1+p)/3, the expectation contribution form door 2 is therefore E2= (1+p)/3 . 1/(1+p) = 1/3, by symmetry E3 =1/3, the total expectation E= E2+ E3 = 2/3. RF.)


 * "proves that they form the minimax solution." should this read "proves that they form a minimax solution." ? Given the above I would say yes - there are other strategies that work equally well for the TV company.
 * Rich Farmbrough, 11:27, 27 February 2011 (UTC).


 * What the cited source (Granberg's appendix to vos Savant's book) says about the N-door variant is that if the host opens one door there is always an advantage, but this one door advantage approaches 0 as the number of doors grows, and also that if the host opens all incorrect doors except 1 (p=N-2), the advantage increases as the number of doors increases and approaches 1. It's clearly true that if you hold p constant, as N/p grows the advantage drops to 0, and if p=N-2 then as N increases N/p approaches 1 (not 0) and the advantage approaches 1 - but this isn't what the source says.  I think essentially everybody agrees (see Arbitration/Requests/Case/Monty Hall problem) one of the main issues we've had with this article is editors injecting their own conclusions.  How about if we either change this back to what it said, or attempt to clarify the wording while sticking to what the source says (and not adding any essentially WP:OR conclusions)?


 * Regarding the Nash equilibrium, I suspect when the current ruckus has died down the bit about this will be expanded into an entire section on "game theoretic" approaches. It's probably not worth worrying too much about "the" vs. "a" until then (although the specific variant currently described in the table is Richard Gill's rather than either of the variants discussed in the Mueser and Granberg paper, and for this variant "the" might be more appropriate than "a").  -- Rick Block (talk) 00:34, 28 February 2011 (UTC)


 * Clarifying is good. What was there before, though doubtless it was supposed to mean what you said, seemed to me wrong as it stood.  Given the nature of the article linguistic precision reflecting the mathematical precision seems a good idea.  I am sure you can clarify the wording without introducing OR. I was reluctant to actually go and read the sources, although I may look at Gill's if I find time, I'm sure you know what you are talking about.
 * Again as far as the biased door opener is concerned, I'm sure that my little bit of OR is not very O - and all I am suggesting is that we don't give the wrong impression - if RS say or imply something demonstrably false (which I am not assuming they do) we do not  have to reflect it, if they don't say or imply it we certainly shouldn't.
 * None of these are big deals, and can wait happily for the page regulars to resolve, unless I find time weighing on my hands, which is unlikely. Rich Farmbrough, 01:32, 1 March 2011 (UTC).


 * I read most of Gill's 1 March 2010 paper, he does say "the" in "proves that these are the respective minimax strategies" but does not support uniqueness. He also mentions the extremum where the expectation from switching is a half, so he has considered at least the edge cases, and that also can be cited into the article. maybe he also states the expectation calculation I gave above elsewhere, which would be cool. Of course he cites these (talk) pages as a source so prepare for self-ref arguments! Rich Farmbrough, 23:53, 4 March 2011 (UTC).


 * Are you OK with the clarification I made ? This stretches the source a tiny bit (the source says "... if all of the incorrect doors except one are shown, the advantage of switching increases as the number of doors increases and approaches 1.0 when the number of doors is very large" - clearly what is meant is that the probability of winning by switching approaches 1.0 rather than the "advantage" of switching).


 * Regarding the effect of a biased host, Morgan et al. (in their conclusion) show that the unconditional probability of winning by switching is 2/3 regardless of the host's bias. This same paper shows the probability of winning by switching regardless of the host's bias is never less than 1/2 - so there's no particular reason to reference Richard's paper for the extreme cases.  BTW - a fully general Bayesian solution, with a discussion of numerous variants involving host bias and other variables, is presented in a paper by Puza, Pitt, and O'Neill (Teaching Statistics, v27(1), Spring 2005). -- Rick Block (talk) 06:37, 5 March 2011 (UTC)


 * "The minimax solution" indeed should, of course, read "a minimax solution" unless we have also proved uniqueness. We agree that there are strategies of host which guarantee him maximally risk of 2/3 to lose the car, and strategies of player which guarantee him minimally chance of 2/3 to win the car. This proves that the 2/3 is "the" value of the game, by von Neumann's minimax theorem. Now, to any *other* strategy of the player it is clear that the host can figure out a strategy which decreases the host's risk from 2/3, but to do that, he must use a different strategy from "random, random". Conversely, for any other strategy of the host, the host can figure out a strategy which does better than 2/3, but it has to be different from "random, switch". This proves that the minimax solutions of both host and player are unique. So: yes, one may talk about THE minimax strategies, but one has to do a bit more work to check that indeed this is legitimate. Richard Gill (talk) 14:42, 13 March 2011 (UTC)

Morgan, "...the answer is 2/3, period.", and "False-ness"
In their 1995 paper, Morgan concludes:
 * "2. CONCLUSIONS"
 * "In general, we cannot answer the question "What is the probability of winning if I switch, given that I have been shown a goat behind door 3?" unless we either know the host's strategy or are Bayesians with a specified prior. Nevertheless, in the vos Savant scenario we can state that it is always better to switch. The fact that Pr(W | D3) ≥ 1/2, regardless of the host's strategy, is the key to the solution."

In their 2010 response to Martin and Nijdam, Morgan writes:
 * "To wit, had we adopted conditions implicit in the problem, the answer is 2/3, period."

Which can only mean that they recognize that from the contestant's SoK, doors 2 and 3 are equally likely to be opened. Which leads to the question, how many of the various claims of 'false' that Morgan makes in their paper do they no longer consider false?
 * vos Savant's proof of her original solution
 * vos Savant's simulation
 * All 6 solutions that they reject

Posted by Glkanter (talk) 02:50, 12 March 2011 (UTC)

Subjectivist view missing

 * Perhaps Morgan (unlike Rosenthal) actually had a point in 1991. But his back and forth on the issue, failure to notice a highly questionable result due to an error in calculation (corrected in 2010!) and frankly muddled writing, makes them a questionable source. I think that this (WP:RS, possibly wp:secondary too, but this debatable) explains best the issue with the interpretation of probability as applied to this problem (please read all three pages in the source)--and, yes, under the subjectivist interpretation, the answer can even be 1/2 (because, it's argued, the player may not know they're playing MHP, so they may think they are playing Monty Fall). This should be covered in the article in the variations section, preferably under a "subjectivist interpretation" sub-section. However, the horribly complex proofs in this "FA" (which are still entirely frequentist) are of questionable value. I'm not at all surprised that most of the vituperation on this talk page has been on the completely wrong issue(s). Tijfo098 (talk) 10:09, 13 March 2011 (UTC)

Game-theory solutions WP:RS (viewpoint mostly missing)
Chun 1999 also in American Statistician, covers this correctly (by solving a much more general case than even Morgan, and particularizing it to that):

Thus, the answer is clear in the vos Savant scenario; switching doubles the player's chances of winning from 1/3 to 2/3 regardless of p and q.

(p is the preference of the host for one of the doors when he has a choice; p+q=1) And I bothered to prove that myself, duh. In other words, the host's strategy does not matter at all; no need to assume he's equally likely to open either goat as said in the lead for instance. So much for this being a "comprehensive FA". Tijfo098 (talk) 09:36, 14 March 2011 (UTC)


 * You should quote a bit more of this source:

In the vos Savant scenario, if follows from (11) that the player's unconditional probability of winning is z=1/3 + (qx + py)/3 and the player's optimal strategy is (x* = 1, y* = 1). Thus, the answer is clear in the vos Savant scenario; switching doubles the player's chances of winning from 1/3 to 2/3 regardless of p and q. When the host specifically opens door No. 3, which has no car behind it, the conditional probability of winning the car is shown to be ...


 * The fact that the unconditional probability is 2/3 regardless of p and q (BTW, Morgan et al. show this result) is what the following sentence and subsequent text in the Conditional solution section means:


 * The popular solutions correctly show that the probability of winning for a player who always switches is 2/3, but without additional reasoning this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens.


 * Making this somewhat more clear has been one of the issues in mediation. -- Rick Block (talk) 14:26, 14 March 2011 (UTC)
 * [if I'm not clear enough, please ask and I'll elaborate, although I'd rather discuss some actual changes rather than the problem & solution, which we both clearly understand well enough already]: the conditional probability (zc in Chun) can indeed be 1/2 in some cases (2/3 of them overall) if the host prefers a certain goat door (aka Monty Crawl in Rosenthal). But "[...] in this situation, the player cannot gain (or lose) by switching." (Chun, 1999, p. 47) So it obviously doesn't change the player's strategy, because in the remaining cases of Crawl (which happen 1/3 of the time overall), switching wins with probability zc=1. In more mechanistic terms: if the host has a preference for a goat door, there's a redistribution of the unconditoinal probability of winning z between two groups of cases (that will have different zc), but this redistribution is subject to the constraint that the sum is always 2/3 and the conditional probability in either group cannot be less than 1/2. This is indeed fairly messy to explain in plain English, and it's also not something that is assumed to be "standard" MHP by plenty of authors: Rosenthal,, ; I can surely find more, but surely others think this is what Savant meant for some reason. There's also the question of whether the player knows what the host's strategy is. I've not seen Savant state that. Since none of this actually affects the player's strategy, for didactic reasons (assuming the Wikipedia audience is a general audience), I'd prefer we leave the more complex discussion (Monty Crawl/Monty Small) as a variant rather than making it the standard problem. Perhaps we should mention in the lead that not assuming that Monty "must randomly choose which door to open if both hide goats" assumption changes the conditional probabilities for cases (some go up some down), but ultimately does not alter the player's optimal strategy, because the conditional probabilities never drop below 1/2 for switching. I deleted part of the "aid to explanation" section that actually solved the Monty Small problem (to use Rosenthal's terms). Using that as aid to understanding what the lead sets up as a simpler problem is a case of: "let me explain to you Galois theory so you better understand how to solve the quadratic equation" (Yes, I'm exaggerating a bit, but I hope you get the point.) Tijfo098 (talk) 15:42, 14 March 2011 (UTC)

Succinctly: Morgan's "vos Savant scenario" (as cited by Chun 1999) is actually Monty Small in Rosenthal 2005/2008 terminology. Tijfo098 (talk) 16:29, 14 March 2011 (UTC)


 * The plain English way to describe the difference between the unconditional and conditional probabilities (per Gillman) is whether the player's decision point is before the host opens a door, or after the host opens a door (with knowledge of which door the player picked and which door the host opened). The point that Morgan et al. make (and Gillman, and Grinstead and Snell, and many others) is that since the decision point is clearly after the host opens a door, the probability of interest is the conditional probability - for example, the probability of winning by switching for a player who has picked door 1 and has seen the host open door 3.  This probability is 2/3 only if the host picks evenly between two goat doors, or if you change the problem to be restricted to the player's state of knowledge (and assume the player is ignorant of any host's preference - which is in effect moving the decision point to before the host opens a door), or if you (less realistically) assume the doors are indistinguishable (which, again, effectively moves the decision point to before the host opens a door).


 * I have suggested unifying the two different solution sections several times, e.g. (in the show/hide box).  Would this change help address the point you're bringing up?  If not, can you propose some other change(s) that would? -- Rick Block (talk) 17:01, 14 March 2011 (UTC)


 * Well, I'm going to make some baby changes in that direction. At least all the back and forth on the symmetry argument (gap) in informal proofs should be in one place; it's in at least 3 different places now! Also, how do you feel about renaming "simple proofs" -> "informal proofs". The other are not formal proofs, but are more formal in the sense of mathematical proof. Tijfo098 (talk) 17:49, 14 March 2011 (UTC)

Disputed edits

 * I don't think these edits quite capture the point which is not whether the host uses a uniform selection criteria but whether the probability of interest is the conditional probability.  Rather than get into this subtlety in the lead, I think it might be better to defer it to the body of the article.  -- Rick Block (talk) 03:12, 15 March 2011 (UTC)

No, they do capture the point. The bickering whether in the symmetrical/uniform host strategy problem (p=q=1/2 in Morgan) the symmetry argument needs to be spelled out (to say that conditional and unconditional probabilities are the same) is absolutely silly for the lead. That is a completely unintuitive and (some but not other RSes say) obvious technicality unless we discuss it in a case where the distinction could actually matter by the numbers, which is what Morgan (and followers) do discuss. There's no reason to bring it up in the lead otherwise; do you really want the following in the lead instead?

Morgan et al. state that many popular solutions are incomplete because they do not explicitly address their interpretation of vos Savant's rewording of Whitaker's original question. In contrast Bell writes that "I will leave it to readers as to whether this equivalence of the conditional and unconditional problems is intuitively obvious."

Feel free to ask at WP:WPM, but you'll likely hear the following principle: Wikipedia mathematics articles should discuss concepts not words. When authors call the same mathematical thing by different names, or call different things by the same name, the Wikipedia article needs to make some arbitrary choice of terminology (NPOV-based if what is a majority/minority can be established) and unambiguously discuss the concepts, while make a note of the terminology variations. Like said above, Morgan's "vos Savant scenario" (as cited by Chun 1999) is actually Monty Small in Rosenthal 2005/2008 terminology (i.e. arbitrary p for host strategy, not necessarily 1/2). Tijfo098 (talk) 03:55, 15 March 2011 (UTC)


 * @Tijfo098 - I'm not sure, but I think we're agreeing here, i.e. that the conditional/unconditional issue is not appropriate to bring up in the lead. What you've added is the reasoning used by some sources that insist a conditional approach is the only way to go - but in the context of the whole article this is a fairly minor point, IMO not worthy of being in the lead (which I think is what I said above).  As far as I know most math sources do approach the problem conditionally (either defining or assuming p=1/2), but only some go the trouble of spelling out why they approach the problem this way.  My very strong preference is for the article first to present one or more unconditional solutions (like most popular sources do) as well as one or more conditional solutions using the common assumption that p=1/2 (as most math sources do), and somewhere either just before or just after the conditional solutions explain the difference (presumably with a forward reference to the "Variant" section, where the Monty Small variant is already discussed).  Please don't think I'm a rabid foaming at the mouth conditionalist POV-pusher - I'm really not.  -- Rick Block (talk) 05:04, 15 March 2011 (UTC)

Selecting proofs (and which variants to give proofs for)

 * I do not insist on that being in the lead, although it is/was a significant part of the real-world controversy on MHP. But it needs to clearer when we're solving a different problem, not introduce the variants in the 2nd half of some proof. Although G & Sneel (from whom that section appears to have been largely lifted) have a couple of free-form pages of text covering both the proof for uniform-Monty-choice and then discussing the p ≠ q variant, a more structured approach, where it's made more clear what problem is solved and why you need more "proof firepower" is practiced by quite a few other RSes also written at introductory level:
 * Rosenthal 2005/8 (except for totally unclear argument on the "shakiness" of the simple proof, he has the progressive-difficulty didactic approach)
 * math textbook - formalizes the "simple" argument, only uses more complicates formulas (Bayes') for the generalization.
 * crypto textbook - no proofs, but variants given as exercises in order of difficulty
 * monte carlo textbook - ibid
 * solves p=1 (Monty Crawl) first to stress one's need to consider the conditional probabilities, then p=1/2, then the general case. He fist gives proof without Bayes' first even in the general case, then with Bayes'. Even in this approach clearly directed at impressing the conditional probability as a concept, an easier by insightful numeric case comes first, and the switching between problems is quite clearly indicated.


 * The didactic/editorial point to consider is: what is the point in providing a more calculation-intensive approach for the uniform-Monty-choice variant (p=1/2) if the result is the same as much simpler solution? It's like applying the formula for the quadratic equation to solve a linear equation. It's absolutely not insightful. Proofs are not supposed to be calculation for the sake of calculating. That's why many RSes introduce some other variant before using more "proof firepower". Tijfo098 (talk) 15:13, 15 March 2011 (UTC)


 * I don't know if you've read all the archives, but we've been here repeatedly. Is the problem "is always switching better than always staying" or "is switching better given that you've picked door 1 and the host has opened door 3"?  Which of these questions do the popular (simple)  solutions answer?  Are these the same question if the host chooses uniformly between goats? 20 archives later and none of these questions have definitive answers.


 * Our task here is to represent what RSes say, fairly, proportionately, and as far as possible without bias. The point in providing a more calculation-intensive approach for the uniform-Monty-choice variant is because it is what a significant number of sources do (way more than just the ones that then go on to show how varying the host preference changes things).  Whether you or I personally think it's worthwhile is a POV-based decision.  We don't get to weight RSes based on whether we think their approach is unnecessarily complicated.  Personally, from an editorial standpoint, I think it improves the article as well.  History showed vos Savant's "simple" explanation was incredibly unconvincing.  Eisenhauer mentions this: "Consequently, what could and should have been a correct and enlightening answer to the problem was made unconvincing and misleading".  Krauss and Wang reflect this as well: "Note that once formed, this assumption [that the specific case of interest is player chooses door 1 and host opens door 3, i.e. the conditional, not unconditional case] prevents the problem solver from gaining access to the intuitive solution illustrated in Figure 1. [an unconditional solution]"  The fact that the "simple" solutions are mathematically simpler does not at all imply they are easier to comprehend or more convincing - per K&W the mental model of the problem that most people create on reading the standard version requires a conditional solution.  IMO, providing one is not just required by NPOV but a good idea.  -- Rick Block (talk) 19:27, 15 March 2011 (UTC)

If, as you seem to imply, there's empirical evidence in K & W as to which proofs people find more convincing, we should definitely consider that issue in deciding which proof to emphasize. I'll reply in more detail after reading their paper (I see it's 20-page long, and I lack the time right now.) Tijfo098 (talk) 23:04, 15 March 2011 (UTC)

Quick quote from the abstract: In a training study (Experiment 3) frequency formulation and mental models, but not Bayes’s rule training, showed significant positive transfer in solving related problems.

So Bayes' seems out of favor. Tijfo098 (talk) 23:07, 15 March 2011 (UTC)


 * The focus of this paper is not on what solution is most convincing, but on varying the presentation of the problem statement to better lead people to the "correct" (2/3 chance by switching) solution. The point about the intuitive solution being inaccessible reflects the empirical observation that 97% of their test subjects reading the standard version of the problem drew a picture with 1) labels on the doors, 2) door 3 open showing a goat, 3) the player's pick being door 1 - exactly like the image at the beginning of the article.  The chance that the car is behind door 3 in this mental model of the problem is 0 (not 1/3) - this is a conditional probability (given the player has picked door 1 and the host has opened door 3).  The probabilities of the other two doors (that the subjects are trying to determine) must also therefore be conditional (given the player has picked door 1 and the host has opened door 3). This conditional mental model already exists by the time the standard version of the problem has been read, and (as K&W say) once this model is formed it makes the "intuitive" simple solution inaccessible.  The car is NOT behind door 3.  Any solution that then continues to consider the chance that the car is behind door 3 with probability 1/3 (like vos Savant's) becomes hard swallow ("I already picked door 1, I already saw the host open door 3, why are you saying the probability the car is behind door 3 is 1/3?, what are you, stupid? it's obviously 0").


 * All of this is mostly irrelevant anyway, since appropriate wp:weight depends on prominence of views, not "understandability" of views. The view that the solution to the problem is conditional is (as far as I know) far and away the prevalent view in math sources.  We should feature a conditional solution because of this, not because of how understandable it might be (not that we shouldn't try real hard to make it understandable as well).  -- Rick Block (talk) 00:01, 16 March 2011 (UTC)

You are ignoring WP:MTAA (which ArbCom chose to quote for this issue in the case's principles), but please elaborate how do you propose we determine wp:weight in this case? Which of the myriad of proofs out there should be in, and which not? Weight the number of sources giving a proof by their notoriety? Just as two sampling points: What notoriety would you assign to Morgan's, and what notoriety to Ken Binmore's proof (from his Game Theory--A very short introduction ISBN 0199218463)? And how do you propose we determine when two proofs are the sufficiently similar that we need to put them in the same bin for wp:weighting purposes? Tijfo098 (talk) 07:21, 16 March 2011 (UTC)


 * Actually, no, I'm not ignoring MTAA - did you miss "try real hard to make it understandable as well"? We need to satisfy both NPOV and MTAA.  Determining WEIGHT is tricky, and (IMO) it deserves a reasoned, in depth discussion that it has never had (on this page or even during mediation).  As to how, it should be based on secondary sources like Rosenhouse's book or Barbeau's book (or earlier paper).  A coarse measure of prominence is number of cites - one source (but probably not the best source in the world) for this is google scholar.  For example, according to google scholar Morgan et al. is cited by 59 other other publications.  Binmore's book is cited by 34 others, and presumably only a very small subset refer specifically to what he says about the MHP.  So, by this measure I think we'd have to say Morgan et al. is much more prominent.  Determining whether two sources share the same POV (and, in this case, apparently even what POV a source is advancing) is ultimately a judgment call.  Fortunately, we have local experts we can rely on here like Dr. Gill, and Dr. Prefers-to-remain-anonymous (user:Woonpton), and Dr. Tsirel (and many other participants in WP:WPM) if it should come up that editors here disagree (and it has).  All in all, the point is that we should be WAY more focused on what the preponderance of references say and WAY less focused on what we each individually think about the problem. -- Rick Block (talk) 15:19, 16 March 2011 (UTC)
 * Not correct. Sources about conditional probability theory, using the MHP as an example, are sources of prominence on conditional probability theory, and never "automatically" sources on the MHP. Especially as there are enough of sources saying and showing that conditioning on irrelevant door numbers is of no avail whatsoever for "solving" the MHP-paradox. They are just only of interest in applying conditional probability theory. Their "weight" and prominence regarding conditional probability theory is irrespective for their "weight" in finding the correct answer to the MHP-question. So we have to focus on relevant sources here that have enough "weight" to finding the correct decision. The main focus here should be to show that conditioning on irrelevant door numbers is of no avail to find the correct decision asked for. And conditional probability should be shown in the variants section for pupils and students interested in conditional probability theory. With a link to Bayes' Theorem. Gerhardvalentin (talk) 15:50, 16 March 2011 (UTC)
 * It can but it doesn't have to. Moreover you seem to ignore that fact that many sources that focus directly on the MHP are using conditional probabilities (in particular the 2 books about the MHP (Randow, Rosenhouse) use conditional probabilities promimently), i.e. on that probability textbook in their treatment of conditional probabilities as ypou seem to suggest. Also conditional probabilities are at the core of some aspect of the MHP and the disputes about it. So it very well makes sense to have conditional treatment here in detail rather than in article for Bayes' theorem, where it scould be mention as an example as well in less detail. Last but not least "Wikipedia is not paper", i.e. there is nothing wrong having more detailed description of conditional or other "advanced" further down in the article. I also don't like the notion to treat non random host behaviour as variant, since it was at the core the core of the MHP dispute as well. Generalization of the original question are variants, but not different approaches to the original question and the original question did not specify random behaviour of the host (nor symmetry). You cannot simply pick your favoured solution (and its legitimate additional or implicit assumptions) and declare other approaches (with differing legitimate assumptions) as variants. That's nothing but a subtle POV pushing.--Kmhkmh (talk) 16:13, 16 March 2011 (UTC)


 * Since I read this as a direct accusation leveled at me, please provide a list of references agreeing that MHP is preferentially Morgan's "vos Savant scenario" (or whatever other variant you think the article should be discussing first). I have given a bulled-ed list above of refs that first solve (or ask the reader to solve) the uniform-Monty-choice case ( p = q = 1/2 in Morgan's notation). We need to have some clarity as to what mathematical problem-object has what name, even if various sources do not use consistent terminology, otherwise the article is an unintelligible dog pile. And bear in mind that when I arrived here, the article already had most if not all of its proofs written for the p = q = 1/2 case, and the p ≠ q introduced in the last part of a proof as a variant ! I merely made that clear in the lead, assuming this shiny FA is at least NPOV in that respect. Tijfo098 (talk) 20:44, 16 March 2011 (UTC)
 * First of all it was reply to GehardValentin not you and I did not make any comment regarding your edits or lead suggestion (note i use a newsgroup style/tree structure for my replies and my ident is to read be as reply to Gehardvalentin not you, otherwise my ident would have been on the same level as Gerhardvalentins). And second I don't quite get your question, I simply stated any answer/publication explicitly dealing with the original problem posed to vos savant is not variant. It is variant if it modifies the original problem (for instance n doors instead of 3). As far as the different approaches to the original problem go I did not suggest any preference and imho the article in doubt should cover them all. I didn't support treating Morgan's p!=q case as variant either - on the contrary. Imho the word variant is/was used to sideline arguments contradicting the 1/3 vs 2/3 solution and to represent that one as the only reasonable way to address the problem's ambiguities. By classifying other valid solutions to the ambiguities as variants we (falsely) signal the reader that they do not deal with the original problem but something else. This is the POV I was talking about in my posting above. The link to Gerogii's textbook you've recently supplied in another posting is an illustration of my point, it like Morgan's p!=q should not be treated as a variant as it addresses the original problem. They should however be described later in the article after the simple solution since they require more background knowledge (some familiarity with conditional probabilities that is). The lead however should at least shortly mention the ambiguity.--Kmhkmh (talk) 21:31, 16 March 2011 (UTC)
 * Right. The impressive history of the MHP-literature should be shown in the article, yes. Especially the prominence of papers that try to give formally "correct theorems", should be mentioned. That never could give any better advice for the famous decision asked for, however. Nevertheless pretending to be indispensable for the only correct decision to be made (always YES and never NO). Disregarding and neglecting the target course. Not noticing that correct conditional probability theorems and the decision asked for are two different pairs of shoes, when unable to give any better advice for that decision. Just interesting within the field of studying conditional probability theory, when using the MHP as a favored and popular example.


 * Sources on conditional probability theory showing that whole a lot of additional assumptions can be handled also, should be in the article, yes. No question. Sources that condition on conditions that are irrelevant for the decision asked for, and never are able to give better advice than always to switch. All of that should be shown.  But never as "to be the MHP", but that they are quite an interesting side aspect within that impressive history of the MHP. Sources are important. But important is also their actual impact and their actual relevance. The article should stop to be retarded.  Regards, Gerhardvalentin (talk) 19:19, 16 March 2011 (UTC)
 * Anything an author publishes as explicitly dealing with "the MHP" needs to be treated as such, we do not get to pick what "the MHP" is, even if some here seemingly would like to do so. By the way the whole Monty Python affair was never about "you should always switch", since almost everybody in particular including vos Savants  misguided early critics in the letter to Parade agree, that switching doesn't harm (you can apply a variant of Pascal's wager here). All the ruckus was and is about the exact and correct reasoning for why you should always switch. In other words what you seem to dismiss as "irrelevant analysis" producing the same advice is actually the real MHP debate and what has caused the controversy. The resulting advice "always switch" itself was never really controversial to begin with.--Kmhkmh (talk) 21:55, 16 March 2011 (UTC)

This Is What Passes for 'Consensus' and 'Discussion'?
[refactored for clarity]
 * Why are such large scale edits to the article taking place during arbitration, and without talk page discussions or consensus? Glkanter (talk) 04:01, 15 March 2011 (UTC)
 * We are discussing them right above your rhetorical question, aren't we? If you expect arbitrators to pass down a solution to the content issues, you are seriously misunderstanding the arbitration process. The only thing that will come out of arbitration are WP:discretionary sanctions. Then we can try to have each other blocked by filing reports at WP:AE for "large-scale changes without discussion" vs. "filibustering that has been going on for years". Feel free to revert if you have a substantive, content- rather than process- based reason to do so. This page is choke-full of meta-meta-meta... discussion of how discussion/consensus should happen, but this is an article talk page, not a wikipolicy talk page. Tijfo098 (talk) 04:31, 15 March 2011 (UTC)


 * My question was in no way 'rhetorical'.
 * Perhaps you could define 'we', 'discuss', and 'consensus' as you understand them vis a vis your near-monologue, above.
 * My reference to arbitration did not say I expected 'direction' from them. Rather, I thought it was obvious to all that we ought to wait until all editors can give this talk page & article their undistracted attention, and wait for any editor participation remedies that are handed out. Glkanter (talk) 04:46, 15 March 2011 (UTC)

Let's see, here are your timestamps for the article:
 * (cur | prev) 15:59, 14 March 2011 Tijfo098 (talk | contribs) m (67,737 bytes) (undo)
 * (cur | prev) 15:57, 14 March 2011 Tijfo098 (talk | contribs) m (67,736 bytes) (undo)
 * (cur | prev) 15:56, 14 March 2011 Tijfo098 (talk | contribs) (67,735 bytes)
 * (cur | prev) 09:31, 14 March 2011 Tijfo098 (talk | contribs) (66,637 bytes)
 * (cur | prev) 04:53, 14 March 2011 Tijfo098 (talk | contribs) m (68,827 bytes)
 * (cur | prev) 04:37, 12 March 2011 Taro James (talk | contribs) (68,823 bytes) (undo)
 * (cur | prev) 19:53, 11 March 2011 Tijfo098 (talk | contribs) (68,838 bytes)

...and here are your timestamps for the talk page:
 * (cur | prev) 00:46, 15 March 2011 Glkanter (talk | contribs) (164,808 bytes) (→Edits) (undo)
 * (cur | prev) 00:31, 15 March 2011 Tijfo098 (talk | contribs) (164,275 bytes) (→Edits) (undo)
 * (cur | prev) 00:01, 15 March 2011 Glkanter (talk | contribs) (163,398 bytes) (→Edits) (undo)
 * (cur | prev) 23:59, 14 March 2011 Tijfo098 (talk | contribs) m (163,185 bytes) (→Edits) (undo)
 * (cur | prev) 23:58, 14 March 2011 Tijfo098 (talk | contribs) m (163,186 bytes) (→Edits) (undo)
 * (cur | prev) 23:55, 14 March 2011 Tijfo098 (talk | contribs) (163,193 bytes)
 * (cur | prev) 23:13, 14 March 2011 Rick Block (talk | contribs) (161,422 bytes)
 * (cur | prev) 13:50, 14 March 2011 Tijfo098 (talk | contribs) (160,892 bytes)
 * (cur | prev) 13:49, 14 March 2011 Tijfo098 (talk | contribs) (160,858 bytes)
 * (cur | prev) 13:02, 14 March 2011 Rick Block (talk | contribs) (160,162 bytes)
 * (cur | prev) 12:59, 14 March 2011 Tijfo098 (talk | contribs) m (158,597 bytes)
 * (cur | prev) 12:29, 14 March 2011 Tijfo098 (talk | contribs) (158,592 bytes)
 * (cur | prev) 11:42, 14 March 2011 Tijfo098 (talk | contribs) (158,379 bytes)
 * (cur | prev) 10:26, 14 March 2011 Rick Block (talk | contribs) (155,756 bytes)
 * (cur | prev) 05:56, 14 March 2011 Tijfo098 (talk | contribs) m (154,534 bytes)
 * (cur | prev) 05:40, 14 March 2011 Tijfo098 (talk | contribs) (154,507 bytes)
 * (cur | prev) 05:36, 14 March 2011 Tijfo098 (talk | contribs) (154,272 bytes)
 * (cur | prev) 11:13, 13 March 2011 Glkanter (talk | contribs) (153,581 bytes)
 * (cur | prev) 10:49, 13 March 2011 Gill110951 (talk | contribs) (153,180 bytes)
 * (cur | prev) 10:42, 13 March 2011 Gill110951 (talk | contribs) (152,136 bytes)
 * (cur | prev) 06:09, 13 March 2011 Tijfo098 (talk | contribs) (151,076 bytes)

You began editing the page well before you began posting on the talk page. I would hardly describe three after-the-fact/catch-up responses from a single editor in a known-to-be-contentious article representative of a 'discussion' or a 'consensus'. I don't think you believe that, either, Tijfo098. Glkanter (talk) 05:20, 15 March 2011 (UTC)

Picture edit request
Can someone change "total probability" to "case probability" (or perhaps "branch probability") in File:Monty_tree_door1.svg? Tijfo098 (talk) 17:49, 14 March 2011 (UTC)
 * Do you have a copy of Chun's 1991 letter to OR/MS Today? I can't seem to find my copy but I'm pretty sure the labels are his and, if so, I'd prefer not to change them. -- Rick Block (talk) 05:18, 15 March 2011 (UTC)
 * Meh, let's not bring the debate on "WP:OR in math articles" into this trivial issue. Technically they are joint probability (you are intersecting/and-ing all the events on a branch), but that will probably read too technical at so early in the article. Grinstead & Snell have an almost identical picture (p. 138) using "path probability" as label, probably for the same ease-of-comprehension reason. Tijfo098 (talk) 13:42, 15 March 2011 (UTC)
 * See also "path probability" here. Tijfo098 (talk) 13:57, 15 March 2011 (UTC)
 * I agree that independent of what Chun might have written (I don't have access to his publications) the term "total probility" is misleading as it is usually used for (total) probability of an event being computed from its "subevents" (formula of total probability). So in the case of a probability tree you would use it for events spread over several branches. Using it to describe the probability of an "elementary event/outcome" at the end of a branch is not false strictly speaking but it is definitely not meaningful. In particular since in the MHP scenario "real" total probabilities are used as well. I've uploaded a revised picture that can be found here: File:Monty_tree_door1_revised.svg
 * --Kmhkmh (talk) 15:08, 17 March 2011 (UTC)
 * OR/MS Today is a magazine aimed at OR professionals, so it might not have used the most mathly terminology. I can't be bothered to check out that article because it's only available in print, and I won't learn much from it anyway. (Chun has published a more comprehensive article on MHP later in American Statistician). Tijfo098 (talk) 12:43, 19 March 2011 (UTC)

Something else
Proposal - and what about that table from citizendium, together with those few lines of text:

"For some readers, numbers speak louder than words. The following table should be self-explanatory. We observe that the player who switches wins the car 2/3 of the time. We also see that Door 3 is opened by the host 1/2 = 1/6+1/3 of the time (row 2 plus row 3), as must also be the case by the symmetry of the problem with regard to the door numbers &mdash; either Door 2 or Door 3 must be opened and the chance of each must be the same, by symmetry.

Winning by switching in combination with Door 3 being opened occurs 1/3 of the time (row 3). The conditional probability of winning by switching, given Door 3 is opened, is therefore (1/3)/(1/2)=2/3. Since this is the same as the overall chance 2/3 of winning by switching, we see that knowing the identity of the opened door doesn't change the chance of winning by switching. Not only does the switcher win 2/3 of the time, he also wins 2/3 of the time when Door 3 is opened by the host, and 2/3 of the time when Door 2 is opened by the host.

In other words, the combined chance of winning by switching and Door 3 (rather than Door 2) being opened, 1/3, equals the product of the separate chances of "the car being behind the other door", 2/3, and "host opens Door 3", 1/2. Whether or not the car is behind the door not opened by the host is statistically independent of whether the host opens Door 2 or Door 3.

This last fact could have predicted in advance, by the symmetry of the problem. The contestant may simply ignore the door numbers: they do not change his chances of winning by staying or by switching."

See also here and there

Regards, Gerhardvalentin (talk) 16:02, 17 March 2011 (UTC)

Asking explanation for revert
Glkanter, can you explain what do you mean by this edit summary? It makes no sense to me. Clearly explicit calculation of conditional probabilities takes place there, unlike the simple solution. Would the word "explicit" satisfy you? Tijfo098 (talk) 23:30, 15 March 2011 (UTC)

Glkanter (talk) 23:35, 15 March 2011 (UTC)
 * There have been no recent discussions or consensus for this change.
 * The change you made has been a very contentious issue for some years now.
 * The article is currently the subject of arbitration. Editors' attention is elsewhere. Editor remedies may affect future discussions and consensus.
 * I mentioned and supported my concerns as stated above, earlier today.
 * Thanks for the complete lack of content-based rationale. I'll wait until they well-deservedly topic ban you, which should be in a day or so now. Tijfo098 (talk) 23:43, 15 March 2011 (UTC)

Perfect solution. Which, of course, was all I was suggesting all along, isn't it? Glkanter (talk) 23:51, 15 March 2011 (UTC)

Initiating featured article review
This article has been tagged for featured article review again. I strongly suggest waiting until the current Arbitration case is resolved before initiating the review. Some contributors who are involved in the case, and very interested in the article, may consider it imprudent to participate until the dust has settled. ~ Ningauble (talk) 22:21, 16 March 2011 (UTC)
 * Don't worry, it will take me a while to finish writing my review given the complexity of the issues; it's already 5Kb long. The Arb case will be closed in a day or so, but FARs last much longer. Tijfo098 (talk) 09:12, 17 March 2011 (UTC)
 * You should be aware that WP:Discretionary sanctions will make editing in this topic area a different affair, even after the arbitration is over. WP:AE is always open. Tijfo098 (talk) 09:15, 17 March 2011 (UTC)
 * Yes, exactly. It is not just that people may be walking on eggshells during the proceedings, but also that Arbcom will effectively put the article on probation by authorizing discretionary sanctions. ~ Ningauble (talk) 12:10, 17 March 2011 (UTC)
 * @Tijfo098: The issue is less your review, but that now since it is tagged others might start reviewing it before the ArbCom case is settled. Reviewing a "moving target" may if of little use, because the review might be outdated right away. First the Arbcom case should settle, then remaining active editors agree on a somewhat stable version (assuming the can do so within a reasonable time frame) and then we should have a review not in the middle of that all.--Kmhkmh (talk) 14:20, 17 March 2011 (UTC)

Ideas for a rewrite to make the problem and solution more understandable.
Looking over the current article, I see some areas where we can improve it to make the problem and solution more understandable to non mathematicians. We still want to retain the more technical sections, but lower in the article.

I think we should lead with material based upon [ http://www.marilynvossavant.com/articles/gameshow.html ] Right now there is no mention in the lead of the firestorm the Vos Savant column ignited, the thousands of letters saying she was wrong, The Research Mathematical Statistician from the National Institutes of Health who said she was wrong, etc. That, properly referenced, should be our lead.

Next we should list arguments why she was right, and one stands out as being an obvious top of the list; the many elementary schools that performed the experiment and found that switching wins on average two out of three times. It's the strongest argument and the most understandable argument, and nobody can argue that it is a "wrong solution." We need to search out citations and make that the first argument listed.

Lets discuss the above proposed changes and other needed changes so we can reach consensus on what to do. I especially want to hear from editors who have never edited this article in the past, but of course everybody is welcome. Guy Macon (talk) 11:25, 25 March 2011 (UTC)


 * I generally agree with the above. The 'Aids to understanding' section is rather weak and unclear.  It address the question of why it matters that the host know where the car is but does not explain it very well.  Many people are puzzled by this aspect. Martin Hogbin (talk) 10:41, 26 March 2011 (UTC)
 * I don't quite agree the lead is supposed a shortsummary/overview of the article and is after 40 years of MHP that's not just the media storm created by vos savant's column. So the media storm should be mentioned there (actually rereading it I'm a bit surprised that it wasn't), however more detailed examples or arguments from the media storm don't belong the lead but a separate section at the top.
 * A reputable (secondary) source on the media storm would be Rosenhouse's book on the MHP (p. 22-32)--Kmhkmh (talk) 12:11, 26 March 2011 (UTC)
 * Agree. Martin Hogbin (talk) 12:39, 26 March 2011 (UTC)
 * Agree that detailed examples or arguments from the media storm don't belong the lead. We do need to briefly describe the media storm in the lead, simply because it is so notable. Guy Macon (talk) 14:40, 26 March 2011 (UTC)

The key here is to closely follow the advice found in WP:TECHNICAL and WP:MOSINTRO. Given the history of this article, I am going to ask a personal favor of everyone reading this. Please click on the links above and read the guidelines, even if you have done so in the past. Arguments for or against proposed changes to the lead of this article should be in the form "proposed change / existing text is a better match with what paragraph X of WP:TECHNICAL says."

In my opinion, the paragraph starting with the words "A common variant of the problem, assumed by several academic authors as the canonical problem, does not make the simplifying assumption..." does not follow the guideline found in WP:TECHNICAL, which says:

"A general technique for increasing accessibility is to consider the typical level where the topic is studied (for example, high school, college, or graduate school) and write the article for readers who are at the previous level. Thus articles on college topics can be aimed at a reader with a high school background, and articles on graduate topics can be aimed at readers with some undergraduate background."

We know that the Monty Hall problem is studied in some elementary schools and that some readers come to the page as a result of a bar bet. Neither audience is likely to understand what a "canonical problem" is, and they may not understand "simplifying assumption." Thus that material belongs lower in the article.

Likewise in my opinion, the first paragraph not mentioning the huge volume of mail Vos Savant got or the mathematicians who said she was wrong arguably does not follow the guideline found in WP:MOSINTRO, which says "Consideration should be given to creating interest in reading the whole article." It's a story that grabs the reader. When I read the Vos Savant column and see the following quote:

"As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and in the future being more careful."

It makes me want to read further. "A common variant of the problem, assumed by several academic authors as the canonical problem" simply does not create the same kind of interest in reading the whole article. Its important information, but in the wrong place. Guy Macon (talk) 14:33, 26 March 2011 (UTC)
 * Maybe I'm misreading you a bit, but at first glance I strongly disagree.
 * I agree that lead should focus on general accessibility and refrain unnecessary "academic" vocabular such as "canonical form", I also agree that it helps if people take a look on the cited guidelines.
 * However I strongly disagree on your guideline based discussion &editing style, this is almost preparing the stage for wikilawyering par excellence. Halfway capable authors should have no problem a proper lead on their own compiling a lead - common sense should be good enough for that. You may refer to the guideline in case of disagreements or to double check the results.
 * I also have an issue with your suggested/insinuated target audience and the literal WP:TECHNICAL application here ('elementary school kids nowadays learn about MHP, hence we write a lead in kindergarden english'). For that we have the Simple English WP and for this article it is a imho a no-go.
 * I don't quite agree with your interpretation of that one line in WP:MOSINTRO. We are writing an encyclopedic articles not a human interest story. Such technique of "grabbing" the reader's interest might be fine for news paper articles, blogs, particular essays or even a book on subject, however they are largely inappropriate for encyclopedic articles. What that guideline essentially means, is that the lead should already outline why the topic is notable and answering the basic W-Questions. Therefore the media storm needs to mentioned prominently in the lead (btw it currently already is just somewhat hidden at very end of the lead). However the heart felt concerns of some letter writing math teacher have no place there.--Kmhkmh (talk) 18:05, 26 March 2011 (UTC)
 * Pretty much agree with all of the above except the wikilawyering comment. I didn't mean to imply that we should write at an elementary school level, just at a lower level than we now have. Also you make a good point about grabbing the reader's interest. I am not married to the ideas I proposed above, but rather i threw them out there as a springboard for discussion. I have no doubt that when a consensus is formed it will have little resemblance to my original suggestion, which is fine by me - If I wanted to own an article I would run my own Wiki.


 * As for wikilawyering (I encourage the reader to read WP:WL again even if he/she thinks they have it down cold) I strongly believe that the solution to someone quoting the letter of a policy or guideline while violating its spirit or underlying principles is to focus more closely on those underlying principles. The common sense of halfway capable editors has led to several failed meditations and finally to arbitration committee sanctions. In general the common sense of the editors is enough, but in this case I would have to say that we tried it that way and it didn't work. During the arbcom it became clear that some of the involved editors had zero familiarity with the policies that they should have been following (confusing bans with blocks and not knowing that mediation is privileged for example).  Based upon that history, I really do think we should start out with a strong emphasis on policies, at least for a while. This, of course, is also subject to consensus, so please keep disagreeing when what I write doesn't seem right. I love being corrected; that's how we learn. Guy Macon (talk) 22:04, 26 March 2011 (UTC)


 * It occurred to me after I wrote the above that I mentioned consensus without encouraging the reader to read WP:CON again even if he/she thinks they have it down cold. If they really do have it down cold, I wasted a few minutes of their time. If they only think they have it down cold, perhaps reading the actual policy will enlighten them. Remember that users tend to falsely believe that Wikipedia's policies conform to their point of view, so strive to understand the spirit or underlying principles even if especially if you don't agree with them. Guy Macon (talk) 22:16, 26 March 2011 (UTC)
 * If we are still talking about the lead here can I make the suggestion that we leave it until we have sorted out the body. If the body is well written, the lead should almost write itself. Martin Hogbin (talk) 22:30, 26 March 2011 (UTC)

I made a concrete proposal for the structure of the article almost three years ago (archived here; search for "Here is the TOC"). The basic approach outlined there still seems valid to me. --Lambiam 23:48, 26 March 2011 (UTC)
 * I think the current structure is fine. Do you have any objection to it? Martin Hogbin (talk) 09:39, 27 March 2011 (UTC)


 * The current structure has resulted in an article which does not follow the guideline found in WP:LEAD. The proposed alternate structure (especially "Use a condensed formulation of the problem statement in the lede") appears to address that issue.  Something has to be done; the lead of the present article fails to follow the guidelines found in WP:LEAD.  If anyone want to claim that the article is fine the way it is, I can make a list of specific areas where the lead does not conform to WP:LEAD and WP:TECHNICAL, but I shouldn't have to.  Anyone reading those guidelines and examining the current article should be able to see what needs improving. Guy Macon (talk) 00:04, 29 March 2011 (UTC)


 * Lambiam and I were referring to the structure of the article not the lead. You will see above that I suggest that we leave the lead until last, when it should be easy for any editor to write it on the basis that you suggest.  Martin Hogbin (talk) 22:09, 29 March 2011 (UTC)


 * I know of no Wikipedia policy or guideline that allows the retention of a lead paragraph that does not conform to Wikipedia standards simply because it would be more convenient to change it later. rather, the lead paragraph should be brought up to Wikipedia standards at once and if necessary edited a second time if needed to reflect improvements made elsewhere in the article.


 * In addition, editors like myself who are not experts in statistics would be wise to defer to those with more expertise in those later sections, but it could be argued that a non-expert has an advantage in crafting a lead paragraph that is understandable to all readers. Guy Macon (talk) 15:26, 30 March 2011 (UTC)


 * From wikipedia to citizendium.org to statprob.com has also seen an evolution in the lead of an article on MHP. And that text can be freely used (creative commons license). Anyway, I just mention it as an example which might inspire an active editor here. Even if it inspires him or her to do something completely different. (One small paragraph is intended for a specialist reader).  Richard Gill (talk) 16:51, 30 March 2011 (UTC)

Probability notation and MHP
I was asked by an editor following the arbitration if I'd like to write an essay on mathematical notation in probability and statistics. You might like to take a look at a first draft essay on probability notation, you can talk about it at  essay-talk. It includes both Bayes Theorem, Bayes Rule, and MHP, as examples. Richard Gill (talk) 17:55, 27 March 2011 (UTC)

Two years focussing on the MHP page also produced some publications which might or might not be useful resources...

Some are of survey nature, others more about my personal (professional) opinion. The intended readership is teachers and students of probability and statistics. The articles are intended to supply a resource of Indisputable Mathematical Truths, including relationships between different proofs, and giving special care to show what mathematical assumptions are needed to get which results. This work started life as working notes on my home page, useful since I often include MHP in talks to lawyers and doctors and students about statistics and society. Later I was asked to contribute to the Springer international encyclopedia of statistical science and its open source companion StatProb.com, and it was a welcome opportunity to write the article about MHP, which *I* would have liked to read, when first I learnt about the problem.

All are peer-reviewed and published now (except for the second on the list - extended version of the top one). Comments are welcome. The top two (the most recent) are IMHO the most potentially useful for editors here.


 * The Monty Hall Problem, Article in StatProb.com, the statistical societies' internet encyclopedia. (2011)


 * The Monty Hall Problem, extended version (includes technical appendices to previous). (written in 2011)


 * MHP is not a probability puzzle: it's a challenge in mathematical modelling, Statistica Neerlandica (written in 2010)


 * The three doors problems, Springer International Encyclopaedia of Statistical Science (written in 2009)

Richard Gill (talk) 18:03, 27 March 2011 (UTC)


 * Those appear to be notable and reliable sources (feel free to disagree, as an engineer I may not be the best person to judge). If nobody objects in the next seven days, I will add them to the external links section. If somebody does object, we will discuss it further (insert usual encouragement to read WP:CON here). Guy Macon (talk) 18:50, 27 March 2011 (UTC)


 * I offered those links as resources rather than sources. Reliable, on the whole, not necessarily notable. I simply hope they are useful to (some) wikipedia editors, as containing a compilation of different proofs of the different results which were already out there in the literature (with references to who and where). They emphasize synthesis: showing bridges between the simple and conditional solutions. They contain alternative derivations of the conditional solution: by Bayes' rule instead of Bayes theorem, as well as two derivations which use symmetry to argue that recourse to Bayes is unnecessary. There is also a derivation of the Morgan et al. result for the biased host case, showing by a simple argument that "Monty Fall" is the most favourable to the player, but that even in this situation, always switching is the best strategy (@Lambiam found this one).


 * My aim was to search for derivations which make sense at an intuitive and informal level as well as being mathematically sound, so that "ordinary folk" can understand the argument; students can convert the verbal reasoning into the formal language of the probability calculus, if they feel the need.


 * On my talk page, @Tijfo points out some good references to MHP in the game theory and decision theory literature. He also pointed out a mistake in one of my papers: I talk about "the" minimax solution, but only exhibited "a" minimax solution. One can easily check uniqueness but I forgot to mention that.


 * (The above unsigned text was written by Richard Gill on 11:50, 28 March 2011)


 * I am going to open this one up to discussion so as to seek consensus. I think the references are notable, but am open to arguments that they are not.


 * I would also like to commend Dr. Gill for following good practice in respect to him being a Wikipedia editor who is himself a reliable source.Guy Macon (talk) 00:54, 29 March 2011 (UTC)


 * Notable or not, I think these links fail #1 under WP:ELNO and do not satisfy #3 under WP:ELYES - so should not be included as external links. Whether they should appear as references is another matter. -- Rick Block (talk) 04:13, 29 March 2011 (UTC)
 * WP:ELNO #1 only works because it is usually ignored in practice, it is meant to block spam and linkfarms (where it serves as "killer rule"), which we are not creating by adding those links, I agree however that there is a danger that people start linking all articles on MHP which might not be desired and getting us close to a link farm. However the real underlying question is here, whether we should create something like a "further reading" or "alternative reading" section. In which we list other comprehensive/recommendable treatments of the subject, that we'd recommend to readers (beyond their function as a source). Such would be books on the subject, treatments in other (encyclopedic) online sources, particularly extensive/comprehensive Artikels, etc.--Kmhkmh (talk) 13:38, 29 March 2011 (UTC)
 * Fair enough. I am throwing the above idea on the No Consensus Scrapheap. Thanks! Guy Macon (talk) 13:45, 29 March 2011 (UTC)
 * I don't mind such an additional section as long as it is not leading to new tug of war about every entry listed in there.--Kmhkmh (talk) 13:49, 29 March 2011 (UTC)
 * I agree with @Kmhkmh and @Rick. If editors ever wanted to use material from one of these sources it would become a regular reference. See also WP:ELMAYBE #1: "The recommendation to consider professional reviews as external links was repealed. The reviews should instead be used as sources in a "Reception" section". (But maybe "should" should be "could".) Richard Gill (talk) 05:34, 30 March 2011 (UTC)

A Fresh Start
I just archived this page so everybody can make a fresh start after the Arbitration. I encourage all involved editors to put away any hard feelings from the past and to make a fresh start. Please remember that this article is now on probation and that editors making disruptive edits may be blocked by an administrator. I would strongly encourage all editors to stop and review WP:NPOV, WP:OR, WP:V, WP:OWN, WP:CIVIL, WP:NPA, and WP:EW. Even if you have read them before, read them again, and treat them as your road map for editing this article, its talk page, any any discussions about this article on your user page. Thanks! Guy Macon (talk) 03:59, 25 March 2011 (UTC)


 * The latest edition of the Wikipedia Signpost has an article about this page titled "Lessons from Monty Hall problem." It includes the following:


 * What is the effect of the decision and what does it tell us?


 * Article talk pages should not be used by editors for proposing unpublished solutions, forwarding original ideas, redefining terms, or so forth. Although more general discussion may be permissible in some circumstances, it will not be tolerated when it becomes tendentious, overwhelms the page, impedes productive work, or is otherwise disruptive.


 * Users who disrupt the editing of articles by engaging in sustained attacks on other editors may be banned from the affected articles, and in extreme cases, may be banned from the site.


 * Articles should be understandable to the widest possible audience. For most articles, this means understandable to a general audience. Every reasonable attempt should be made to ensure that material is presented in the most widely understandable manner possible.


 * If editors disagree on how to express a problem and/or solution in mathematics articles, citations to reliable published sources that are directly related to the topic of the article and directly support the material as presented must be supplied by the editor(s) who wishes to include the material. Novel derivations, applications or conclusions that cannot be supported by sources are likely to constitute original research within the definition used by the English Wikipedia.


 * Guy Macon (talk) 00:39, 29 March 2011 (UTC)

While we are making a fresh start, let's all get on the same page concerning indenting. Here are some rules that not only predate Wikipedia, but predate the World Wide Web, having been established to facilitate commenting on USENET and LISTSERV.

Keep your entire comment on the same level (same number of colons).

The number of colons should be the number found in the comment you are replying to plus one.

Do not assume that a comment is a reply to whatever is directly above it. If they are at the same indent level, assume that they are two replies to the same thing.

Newer replies go after older replies on the same indent level.

Breaking in to a comment with a reply is allowed, but please consider placing the comment on the bottom using "Re 'quote from above comment':" format.

If the indentation gets too deep, insert on a line by itself.

If you see an outdent, treat the thread as if there were an invisible multi-level indent before everything that follows the outdent.

Comments / suggestions welcome; the important thing is that we all follow the same indenting rules, not the exact ruleset.

Guy Macon (talk) 21:39, 5 April 2011 (UTC)

Improvement suggestions from a casual reader
I see that Gerhard just moved the section I initiated here named "Increasing the number of doors rewrite?" to the Arguments page. I'm not thrilled about that, because it was about a specific proposal, but I'm not going to restore it, either. I will however repeat here that I think it would be of great benefit if the article were more focused on giving non-technical readers the nudge they need to recognize that Monty, too, makes a choice, and that his choice reveals relevant information to the contestant. I'll also reiterate that of all the presentations I've seen here so far, the first immediately convincing one that worked for me, for the simple version, is the one ( permalink ) that Rick Block posted to the Arguments page under the heading, "A different table". Best, –  OhioStandard  (talk) 20:35, 4 April 2011 (UTC)


 * What is the choice that Monty makes, and what information does it give to the contestant? Martin Hogbin (talk) 21:21, 4 April 2011 (UTC)


 * He chooses one particular door to leave closed. To see that, just imagine that after the contestant makes his initial door selection, Carol Merrill were then obliged to call out from the stage, "Monty, which one of the remaining doors do you want me to leave closed?" Monty makes a choice, alright. It's just that in the game's usual verbal protocol he identifies it in a negative or indirect way.
 * And in the likely case where the contestant misses choosing the "car" door at first, Monty (indirectly) identifies it for him! I don't know how to mathematically express the information value he communicates in doing so, but it's much easier to recognize that value intuitively when the case that the contestant first chooses a "goat" door becomes very likely, i.e. when the number of doors is increased so outlandishly that the contestant's chance of choosing the car at random becomes almost nil.
 * In such a case where there's only one car, and a gazillion goats behind a gazillion doors, the contestant could get a glimpse of the information value in Monty's choice pretty easily by asking, (1) "How likely is it that I chose correctly in the first place?", and (2) "Making the reasonable assumption that I chose a goat, why did Monty choose to leave one particular door closed, from among all those he could (negatively) choose?" I only assert that the foregoing applies to the "simple" version of the problem, btw. I have to admit that I've barely registered the more complex versions so far. –  OhioStandard  (talk) 01:28, 5 April 2011 (UTC)


 * The above is a good example of the sort of good-faith effort that has in the past, through no fault of the person doing it, led to enough problems that the arbitration committee had to ban some users restrict others, and remind everyone of Wikipedia policies. The above is an example of what has come to be knows as a "simple solution" and there are math experts who argue that all such simple solutions are flawed - and they are correct. And yet, to the non-technical reader, the only chance he has of understanding the solution is through one of the simple solutions. So, do we repeat the arguments that this has caused yet again?  No. The archives contain plenty of examples of that not accomplishing anything.  Do we discount or ignore your comments? No. They are pretty good and we don't want an editor editing the simple solutions sections to not have access to them.  So the solution is to put them on the arguments sub-page. Yes, they are a specific proposal, but at this point (and through no fault of yours) the consensus is to list simple solutions that have been published in reliable sources, and to move discussions about simple solutions that may be examples of Truth - discussions that led to stalemates and frustration in the past - to the arguments page, and to have the talk page focus on the sort of Verifiability, not truth discussion that leads to Consensus.  Please don't think the arguments page is somehow inferior or that your comments will only be noticed if they are on the main talk page.  Actually, it's a better choice for this sort of thing, as you can see by reading it.  You may wish to copy the above comment there. I strongly suspect that you will be happy with the discussion that results. MHP has attracted some of the nicest and most helpful editors on Wikipedia. Guy Macon (talk) 20:01, 5 April 2011 (UTC)


 * Thanks for your reply, Guy. I wasn't trying to make an argument, just a brief, four-sentence suggestion for improving the article from the perspective of someone who (frankly) is mostly interested in helping the casual reader to quickly grasp why the "simple problem's" seemingly counter-intuitive solution is correct. I imagine that's what over 95% of readers of this article want from it, and ( without the least offence intended ) I don't believe it "delivers the goods" that way in its current form, focusing as it does so heavily on variations of the problem. I understand why you're saying simple solutions are flawed, btw, and won't argue the point except to say that those 95% of readers probably won't care about variant versions of the problem in which conditional and unconditional solutions differ. I also understand the need in this context for WP:RS and I imagine I could go find some for my initial suggestion if I were motivated to do so. Anyway, I only responded to Martin's question because he asked; I didn't really want to, actually, and perhaps I shouldn't have. Tell you what: Leave this thread here for a week, and if I don't move it to the arguments page myself, then, feel free. Let's not make it any longer here by debating, though, okay? –  OhioStandard  (talk) 21:15, 5 April 2011 (UTC)


 * Sounds good. Again I would like to emphasize that there is no hint of you doing anything wrong, just that this post-arbcom page is a special case. Guy Macon (talk) 21:43, 5 April 2011 (UTC)

Alternative derivations
I have removed the last two "alternative" derivations. The reason I have done so is that they are not supported by the cited references, hence I consider them as unverified content until proven otherwise. I have carefully read the Bell 1992 letter in the American Statistician, and an explicit appeal to symmetry is nowhere to be found there. The argument therein is one of conditional independence. I quote from the letter:
 * The point to make is that AGG and D3  turn out to be independent events,  thus  P(AGGID3) =  P(AGG),  and  the  answer  to  the  conditional  and  unconditional  problems  is  the  same.  This  is  formally demonstrated  by  observing  that P(D31AGG) =  1/2 by  assumption, and that P(D3)  =  P(D31AGG)P(AGG)  +  P(D31GAG)P(GAG)  + P(D31GGA)P(GGA) =  (1/2)(1/3)  +  1(1/3)  +  0(1/3)  =  1/2.  If  the independence  of AGG  and D3 is judged intuitively obvious,  then the use  of  a  solution  to  the  unconditional  problem  (e.g.  the  solution offered by vos Savant that the authors label F2) is valid. In this regard it is interesting to note  that while (AGG,D3)  are an independent  pair of  events,  obviously  (GAG,D3)  and (GGA,D3)  are not.  I will leave it  to  readers  as  to  whether  this  equivalence  of  the  conditional  and unconditional  problems is intuitively  obvious.


 * The independence of AGG and D3 is intuitively obvious by symmetry, and easily proved by symmetry. Seymann also refers to symmetry in his discussion, by the way. The point is that these arguments are so obvious that no-one bothers to write papers on them. Fortunately however Richard Gill has recently filled in the gaps for those not able to supply them, themselves. Richard Gill (talk) 17:35, 31 March 2011 (UTC)

The third derivation suffer from an even worse case of the same disease, as the only cited references are a secondhand journalistic account and a generic statement of support.


 * This derivation is written out fully in a recent reliable source by one Richard Gill. That person also consulted with Persi Diaconis who agreed 100% with the reasoning, and agreed that his book was a good source. I suggest, @Glopk, you also study more recent reliable sources, e.g. good secondary sources, as well as the ancient primary ones. Richard Gill (talk) 17:35, 31 March 2011 (UTC)

NOTE CAREFULLY. I am not questioning the mathematical truth of the removed alternative derivations, only their lack of documented support in reputable sources. glopk (talk) 06:08, 31 March 2011 (UTC)


 * There is now documented support in reliable sources written recently by one Richard Gill. So you could put those alternative derivations back, if you think they are interesting and useful, and adapt the references.


 * If you find those sources insufficient (e.g., insufficiently explicit), I'm sure Richard Gill would be happy to expand them. After all, this is his job.


 * I thought that no changes would be made to the page unless there is concensus. Yet, @Glopk, you are proposing changes without seeking concensus. This is not a good example of collaborative editing. Richard Gill (talk) 17:27, 31 March 2011 (UTC)


 * I'll be happy to collaborate on editing in a side page the paragraphs I removed, provided that sources can be provided for them. As for using your (Gill's) material, I do not have exceptions, but I'll leave to others to decide whether it counts as a primary source (hence off-limits).
 * However, I don't believe that my removal should be controversial: I showed above that the text for the removed second derivation (simple solution + symmetry) did not fairly represent its purported source. Symmetry is simply not mentioned there, not even indirectly. You may think that it is obviously implied, but your thinking as a WP editor does not a source make - see WP:SYN . Further, the Bell's derivation does use algebraic manipulation of formulae, so it is a misreprentation to cite it as source for a section of "derivations can also be given which avoid explicit computations or formula manipulation".
 * As for the third derivation I removed, I am afraid that stating "Persi thinks so, and told me too" likewise does not a reputable source make. Lastly, in regard to your comment on my alleged unwillingness to read sources, pray look at the evidence: I did read the one source YOU gave for that paragraph (the Bell letter to Am. Stat.), and found it sorely wanting. glopk (talk) 20:14, 31 March 2011 (UTC)

Glopk's edits were no improvement of the article, therefore I had to revert. Please use this page first, to make propositions. Btw, these alternative proofs are interesting and illuminating for the reader, there are proofs in the reliable (and maybe even notable) sources: http://statprob.com/encyclopedia/MontyHallProblem2.html, http://www.math.leidenuniv.nl/~gill/#MHP and   http://www.math.leidenuniv.nl/~gill/mhp-statprob.pdf. Gerhardvalentin (talk) 10:07, 2 April 2011 (UTC)
 * PS: and please read the section above Probability notation and MHP, esp. its "Extended content" (show). Gerhardvalentin (talk) 10:23, 2 April 2011 (UTC)
 * If you re-add the section then instead of giving reading recommendations, you could at least added proper citations, which would have removed glopk's concern above.
 * Overall I'm getting the impression that beside glkkanter and rick being out of the picture or slightly reduced the rest of the old gang seems to bedetermined to continue as before.--Kmhkmh (talk) 10:37, 2 April 2011 (UTC)
 * Yes @Kmhkmh. Notice Rick Block's extensive argumentation for his point of view on Guy Macon's talk page. Personally, I do not wish to edit the article for the coming months. But I'm available for discussions here or elsewhere on the content of specialist (mathematical) literature if that is needed. Note the discussions on the MHP arbitration pages concerning the proper interpretation of the "No O.R." rule as far as the exposition of mathematics is concerned, and the proper role of experts. Richard Gill (talk) 13:08, 3 April 2011 (UTC)
 * Kmhkmh, yes. The proper citations are the links I just gave. They are not reading recommendations for everyone, they are reading recommendations for those interested in the "conditional" approach.
 * If one rejects to read it, he should not edit the "advanced" part. Those who are happy with "popular" can ignore the links I gave (but such people should also not edit the "advanced" part.) Regards, Gerhardvalentin (talk) 13:32, 2 April 2011 (UTC)
 * I was talking about adding proper citations to the article, where they belong rather than just talking about them here. Because the criticism of glopk above essentially still stands, the section on symmetry has no proper citations (worse it nevertheless "pretends" to have some which are however not really valid as glopk pointed out). Now, if you want that section in the article it would be primarily your responsibility to add the proper citations rather than just mentioning them here and suggesting glopk to read them. --Kmhkmh (talk) 13:47, 2 April 2011 (UTC)
 * Please Kmhkmh could you check whether I did it right, I'm no expert. Thank you. Regards, Gerhardvalentin (talk) 16:10, 2 April 2011 (UTC)

[Outindent] Gerhardvalentin, do you have a source for Alternative Derivation N. 3? If not, do you think that a secondhand journalistic account (Tierney) and "Persi Diaconis is OK with Vos Savants" counts for WP:V ? Also, can you please explain how a formula that is a mile wide, and does not render properly on handheld devices, is better than the equivalent one I entered and you reverted? Please note that the formula I entered is essentially identical to the one that passed two FA reviews. Thanks. glopk (talk) 23:58, 2 April 2011 (UTC)


 * @Glopk, if you want to use abbreviated notation for your proof of Bayes' theorem, then use a small p to denote a probability mass function or discrete density and not P to denote a probability measure, and explain your notation, rather than using unexplained and non-standard notation which is not consistent with earlier notation used in the article. See my essay on probability notation. Regarding the procedure for passing FA status I doubt that any professional mathematicians were involved in that procedure so it doesn't surprise me that this kind of blunder passes the procedure. Moreover, the section is superfluous and no doubt no-one has read it carefully. It is superfluous since a) contained in the article on Bayes theorem a useful illustration of the derivation of that theorem, b) it not add any useful insight into MHP whatsoever. Richard Gill (talk) 08:15, 3 April 2011 (UTC)

Here is a reliable source speaking, not an editor of the MHP page. Reliable sources for both symmetry proofs are and [. These are not primary sources, but secondary source, they are of review nature (a commissioned and peer reviewed encyclopeadia article), standard classroom background material.

Here's the mathematics. (For the notation, see my draft essay on notation in elementary probability and statistics, invited by wikipedia mathematics editors, User:Gill110951/Probability_notation). Let S, C, and G be the usual doors selected by player, hiding car, and opened by host revealing a goat.

(A) Consider the situation with the door chosen by the player fixed. Note that Pr(S=C|S=1,G=3)=P(S=C|S=1,G=2) by symmetry. With p denoting a generic probability mass function, recall that p(x|y,z) does not depend on y is equivalent to X is independent of Y given Z. Thus the event S=C is independent of G, the door opened by the host, given that S=1. Whether or not the player should switch is independent of the goat-door opened by the host, and the conditional probability equals the unconditional probability 2/3, when we consider the door selected by the player as fixed.

(B) Consider the situation with the door chosen by the player uniform random. Note that Pr(S=C|S=s,G=g) by symmetry does not depend on s and g. Thus the event S=C is independent of the pair S,G. Thus whether or not the player should switch is independent of both his own selected door and of the goat-door opened by the host, and the conditional probability is equal to the unconditional, 2/3.

These two proofs are so simple and obvious that it is not worth writing an article about them. Unfortunately therefore perhaps only professionals will recognise that other professionals are referring to these arguments when they mention (conditional) independence or symmetry. However I hope there can be no misunderstanding of this anymore. If the editors of the Monty Hall page find the proofs interesting and useful for the readers, and do not find present references in the literature explicit enough, this reliable source would be happy to oblige. Richard Gill (talk) 07:53, 3 April 2011 (UTC)


 * I just tried to enter: "Monty Hall Problem" (version 5). StatProb: The Encyclopedia  as a source to the article. Please could someone check whether I did it right? Thank you. Gerhardvalentin (talk) 21:56, 3 April 2011 (UTC)

@Glopk, I wrote out the proof of Bayes' theorem in the context of MHP using the more abbreviated notation of probability mass functions. You can find it at.


 * You know, I think I have a better idea for using your time and mine on WP: rather than writing your own proof of the MHP solution using the Bayes theorem, and asking me to read it, how about we find a reputable source with one? I think this method is called WP:V and WP:NOR? By the by, the proof currently in the article follows closely Henze's, but at the time of the last FA review it was criticized for not being close enough, and because the source was in German. A good one that has been recently mentioned here is the one at page 64 of DJC MacKay' book: is adequate, accessible and reputable. glopk (talk) 00:08, 6 April 2011 (UTC)


 * Yes this is also fine. But he is using different symbols and notation from the rest of our article, so you would have to rewrite MacKay's formulas substantially to bring them into line with the rest of the article, or rewrite the rest of the mathematics in the article in line with MacKay. I have no problem with his approach. He is careless with his notation but he says so in advance. He's very careful that nothing he writes is ambiguous. He's from engineering and phyiscs, not from mathematics.


 * I wanted for a long time to get hold of Henze's writing on MHP, do you have a pdf or a scan of the relevant pages for me? Go ahead and do what you like. MacKay does (but not very carefully) distinguish probabilies from probability mass functions, and random variables from possible values which they take. I'm sure my old friend Norbertus Henzius is much more careful - he is a very precise German mathematician. Richard Gill (talk) 10:04, 6 April 2011 (UTC)
 * The 8th edition (2009) of Henze's rather popular 1997 book Stochastik für Einsteiger is partially available at Google books, the MHP problem is treated (in a frequentist manner) at p.52 (unconditional) and pp. 104 (conditional). You may also want to take a look at another probabilty textbook by another German mathematician posted by math portal member earlier, which considers the Bayesian angle with an interesting twist (leading to p=1/2 rather than p=2/3 as the "best answer" to ascertain the probability of winning by switching). It can also be found at Google Books pp. 54-56--Kmhkmh (talk) 11:34, 6 April 2011 (UTC)


 * Henze is indeed very careful with his notation which follows mainstream mathematical conventions, no short cuts. (MacKay follows conventions often seen in computer science). Georgii too uses mainstream notation, no short cuts. Unfortunately Google Books wouldn't let me see the interesting page 55 of Georgii. It looks to me that the issue which Georgii is studying is what Rosenthal calls Monty Hall versus Monty Fall, ie depends on whether or not the host is in advance certain to open a non-goat door different from ours, or merely certain to open a door (everything else being settled by assuming "equally likely"). Most of Vos Savant's readers did correctly divine her intention to be the former. Henze and Georgii are both writing for newcomers to probability. MacKay is writing for pretty serious scientists who are going to get into very deep and technical material. I certainly wouldn't recommend his book to an amateur or beginner. By the way, MacKay presents the simple solutions as well as the conditional with no comment at all on the conceptual difference! My opinion remains that the elementary line by line going through the proof of Bayes theorem is only useful (if at all) for beginners, and for those readers, one should be careful with notation. MacKay wants his sophisticated engineering and physics and hard computer science readers to get used to a highly short-hand notation. OK for them, but it creates a lot of dangers in unskilled hands. Richard Gill (talk) 14:46, 6 April 2011 (UTC)


 * Georgii takes the door numbers as unknown to the player. Door number 1 is the door hiding the car. Doors 2 and 3 are the two goat doors. His proof is essentially Glkanter's proof since he conditions on the certain event "host opens different (unspecified) door revealing a goat". It corresponds to my/Diaconis/Vos Savant's "symmetry in advance" proof where we take no notice of the visible numbers written on the outside of the doors at all (we ignore them safely, by symmetry). The player's choice is completely random, relative to the hidden numbers on the doors.


 * He also considers the alternative problem, Helpful Host: Monty opens one of the two goat doors, chosen at random, taking no account of the door chosen by the player. We condition on the *not certain* event that that door happens not to be door chosen by the player. This event has probability 2/3. Again, he does not condition on specific (visible) door numbers!


 * This reference shows that also in the academic literature the point of view is taken by some reliable sources that the door numbers (the ones which the player and the audience can see) are irrelevant. He follows Vos Savant literally in ignoring her side remarks "say Door 1", "say Door 3". He completely approves of her solution, and he completely approves of the experiments done by all the school children. Richard Gill (talk) 16:26, 6 April 2011 (UTC)
 * Well, you do however just tell half of the story since Georgii gives 2 solutions one along the line of vos Savant with p=2/3 and another one with p=1/2 and he doesn't really prefer one over the other and considers the problem and the debate in the community as yet "unsettled". So simply stating Georgii completely approves vos Savant is accurate but nevertheless misleading since you've neglected to mention that he approves her critics (the p=1/2 crowd) as well (as odd as that might appear at first glance).--Kmhkmh (talk) 00:51, 7 April 2011 (UTC)

I repeat that I don't see the point of writing out a proof of Bayes' theorem as part of an article on Monty Hall problem, when Monty Hall problem is already used as an example in the article on Bayes theorem itself. Especially when Bayes' rule gives a more insightful derivation with less computations, more ideas.
 * And I repeat that I don't think it matters much whether you personally see the point or not. Many many reputable sources on the MHP present a derivation of the solution using the Bayes expansion similar to the way it is done currently in the article. We can't just ignore them, and they didn't write: "The problem can also be solved using Bayes Theorem, go find a reference to read about it". Further, it definitely does not matter whether you personally find the derivation using the Bayes rule more insightful. I personally find it annoying instead, but that may be because I prefer to think in terms of probabilities rather than odds, and prefer to calculate rather than trust intuition. glopk (talk) 00:08, 6 April 2011 (UTC)

I agree with you that my references to Bell and to Diaconis were inadequate as exact sources for these elementary derivations. I was researching the literature and composing these sections simultaneously just before the arbitration started in ernest. These passages were certainly not finished.

I know that Diaconis was interviewed twice on the Vos Savant controversy (different interviewers, different newspapers) but I have not yet found the reference for the second interview. I also know (since Rosenhouse says so, with reference, in his book), that Vos Savant clarified her intended meaning of "say Door 1", "say Door 3", namely that these words should not be considered part of the question. But I didn't yet re-read Rosenhouse's book to track this one down and I didn't read Vos Savant's book yet, where it probably can be found, either.
 * But that's irrelevant, isn't it? One cannot combine sources on WP to express a view (WP:SYN is a form of WP:OR). Either you editor have a reputable source for the whole of the alternative derivation you edited in, or you don't, in which case it does not belong. glopk (talk) 00:08, 6 April 2011 (UTC)

As to Bell, please focus your mind on his words "I will leave it to readers as to whether this equivalence of the conditional and unconditional problems is intuitively obvious". He wrote out a proof in full, using Morgan et al's notation, because his contribution is a short letter commenting on that article. That is not the best way to bring across the idea, the intuition. I can understand that he can be thought to be "preaching to the converted". His remark makes clear that he finds writing out the proof more or less superfluous. You were bothered that he didn't explicitly use the word "symmetry" here. However symmetry is what he is using, whether he names it or not. And the word is used by other writers in the same context and is enough hint for any professional. For a student, writing out these proofs constitutes a kind of warming up exercises before proceeding to serious problems in conditional probability. Richard Gill (talk) 08:23, 5 April 2011 (UTC)
 * You seem to infer a lot from the silence of the sources. I (and WP) prefer to read and summarize the non-silent portions instead (see WP:V). What is obvious is that Bell is making an argument that requires proving conditional independence, and he proves it. That's it. You as a WP editor simply cannot put in his pen that which he did not write. On WP you are not even allowed to write "and here Bell is using symmetry", unless you have a reputable source for that statement. glopk (talk) 00:08, 6 April 2011 (UTC)


 * I do have a reputable source for that last statement, and it's in Professor Richardius van der Gille's most recent pamphlets. He states that Bell is using symmetry. Which he is. Symmetry means "having the same measurements". He is exchanging symbols in formulas and claiming equality. Why are they equal? By sym-metry. He doesn't have to say it out loud because it's in front of your very eyes. But it doesn't matter, you can remove the attribution to Bell if you like. That's an interpretation by a secondary source (said pamphlet-writing Professor from the Nether Lands).


 * You didn't respond to Bell's "I will leave it to readers as to whether this equivalence of the conditional and unconditional problems is intuitively obvious". What do you suppose that was supposed to mean? And maybe you missed some of the maths editors comments during the arbitration, in my own words "put like this it is so bloody simple there is no point in writing a paper in a mathematics journal about it". It is indeed hard to find reliable sources for some things which are so obvious that no-one bothers to write it out. Fortunately I'm happy to help out, I see it as a service to the community, not as pursuing some kind of hidden agenda of my own. Let's have the Mathematical Truth's out of the way, listed in some uncontroversial compendium, so that we can concentrate on writing a good article. Richard Gill (talk) 09:43, 6 April 2011 (UTC)


 * By the way, of course Bell is proving conditional independence. He proves it by using symmetry. He states that it's so obvious that a proof is hardly necessary. It seems to me that we have no disagreement. The other proof using symmetry works by proving full independence rather than conditional independence. It is also so obvious that a proof is hardly necessary. The key word is "independence". Independence means that conditional and unconditional probabilities are equal. It means that you don't need to sit down and compute a conditional probability the long way. It means that you don't need to take any notice of the condition. What's the problem? Richard Gill (talk) 10:00, 6 April 2011 (UTC)

Course on probability
Moved to Arguments. Handy2000 (talk) 21:40, 7 April 2011 (UTC)

Article structure
Now that the dust has settled from the arbitration, I am going to put forward my proposal for the structure of the article. The structure that I propose has, in my opinion, a number of advantages.

It is encyclopedic, it goes from simple to complicated like most good text books and encyclopedia articles. It does not put up any solution as 'The Truth', enabling us to put that argument behind us and get on with improving the article. It approaches the subject from the POV of our readers. The vast majority will want to know if it is better to swap, by how much, and why.

The section titles are indicative of what should go in the sections. If they are considered too POV I would be happy to change them.

1 The problem

Just Whitaker's statement. It is by far the most well-known problem statement. Although it is very vague, most people seem to understand what it is about

2 Vos Savant's and other simple solutions

2.1 Vos Savants solutions 2.2 Other simple solutions 2.3 Media furore 2.4 Aids to understanding Mainly why it matters what the host knows and helping people to understand the solution 2.5 Sources of confusion and the psychological aspects Why people find this problem so hard.

3 Academic criticism of the simple solutions 3.1 Morgan's paper 3.1.1 K&W formulation 3.1a More detailed and comprehensive solutionsMartin Hogbin (talk) 17:02, 5 April 2011 (UTC) 3.2 Other 'probability' sources 3.3 Criticism of the criticism 3.4 Summary of 'The Truth' (essentially that there is no such thing in this case, as per reliable secondary sources)

4 Variants

4.1 Other host behaviors 4.2 N doors 4.3 Quantum version

5 History Martin Hogbin (talk) 17:53, 4 April 2011 (UTC)


 * That's quite a good structure IMO. Is there any way we could fit in some properly referenced experimental results (everything from elementary school classes playing the game again and again to various software simulations)?
 * Guy Macon (talk) 23:12, 4 April 2011 (UTC)


 * And, where would the first solution using conditional probability appear in this structure - in the "academic criticism of the simple solutions" section? Given that there are many (nearly all) popular sources presenting "simple" solutions, and many (nearly all) math sources presenting solutions using conditional probability, I think a far more NPOV approach is for the first few sections to discuss the "standard" interpretation (e.g. the K&W version), presenting both "simple" solutions and solutions using conditional probability - without expressing an editorial preference for one or the other - followed by sections discussing variants (such as Morgan's "the host preference between goats is a variable" version).  Certainly within the initial "solution" section, one or more "simple" solutions would be presented first, but I think omitting any mention of solutions using conditional probability in the initial section would be POV.  IMO a structure that would be more NPOV (and better as well) would be


 * 1 The problem


 * More or less like the current "problem" section, i.e. Whitaker's version, plus the "standard" clarifications (including host chooses evenly between two goats if it comes up)


 * 2 Solutions


 * 2.1 Vos Savant's solution
 * 2.2 Other simple solutions
 * 2.3 Solution using conditional probability


 * 3 Aids to understanding
 * Mainly why it matters what the host knows and helping people to understand the solution
 * 4 Sources of confusion and the psychological aspects
 * Why people find this problem so hard.


 * 5 Academic criticism of the simple solutions
 * Discussion of Morgan's paper and difference between unconditional and conditional solutions, including the "leftmost host" variant most of these sources use to illustrate this difference. Criticism of the criticism would go in this section as well.


 * 6 Variants


 * 6.1 Other host behaviors
 * 6.2 N doors
 * 6.3 Quantum version


 * 7 History


 * Through section 4 this would be discussing the "standard" problem, but with both "simple" and "conditional" solutions presented (per NPOV). -- Rick Block (talk) 02:15, 5 April 2011 (UTC)
 * I have just one comment on the proposals above and that is that I think that one should *not* have the list with the so-called "standard" K and W assumptions early on in the article. Introducing this list biases the presentation to a textbook probability or statistics approach. The popular solutions do not make all the assumptions of K&W, they do not need them all! The popular solutions are more widely applicable than the "full" conditional solution. The full conditional solution needs *all* the assumptions. In fact there is a spectrum of solutions in the literature - as you heap on further assumptions, you get stronger conclusions, but you simultaneously restrict the scope of your argument. (If a statement like this needs a reliable source to back it up, I can tell you some references, but I hope everyone would agree it is plain uncontroversial logic). The corollary of this is that I prefer Martin's proposal to Rick's. Moreover I find it weird to suppose that one is imposing a POV by saving material which many readers cannot appreciate till a bit later in the article. Whether or not any individual editor has the POV that the simple solutions are wrong and the conditional ones are right, or vice versa, one can make sure that what is written in the initial sections simultaneously reflects the sources which are being discussed at that point and respects the Mathematical Truth. Similarly in the later sections. Bear in mind is that Mathematical Truth consists of tautologies, it does not have a POV. A mathematical theorem does not make a value judgement. It does not tell you what you *must* do. It only aids you in using wisdom to make your own choice.


 * People who write down lists of assumptions ought to attempt to justify them. But justifying those assumptions necessitates taking a stance on the interpretation of probability. This too is a more advanced topic. See on my talk page.


 * * If*, despite these reasons, the list *is* presented early on, it should be presented in a non-committal way. Some sources make some of these assumptions, some don't. Many solutions don't need them. Some sources see randomness in the choice of door of the player. Some sources see randomness in the location of the car. For some sources randomness means real physical random number generation, for other sources it means subjective uncertainty. (Ontological versus epistemological notions of probability). See the just cited essay on my talk page.
 * Richard Gill (talk) 07:28, 5 April 2011 (UTC)
 * I took the liberty of correcting the indentation in the thread above. In a threaded discussion where indentation signifies who is saying what, it causes confusion if a participant in the conversation either fails to add a level of indent (one more colon before each paragraph) or adds extra colons to indent part of his text. Also, be sure to use the "show preview" button; it shows you the formatting so you can notice things like the fact that surrounding a word with asterisks doesn't do what you expect it to do if it's the first word on a line.
 * Do you therefore support my structure, which does not present the K&W formulation early? Martin Hogbin (talk) 17:02, 5 April 2011 (UTC)
 * Re: "omitting any mention of solutions using conditional probability in the initial section would be POV", it would be helpful if you were to say exactly what policy (I am guessing WP:NPOV) is not being followed and quote that policy. WP:NPOV says "Segregation of text or other content into different regions or subsections, based solely on the apparent POV of the content itself, may result in an unencyclopedic structure", but the proposed structure does not move the solutions using conditional probability lower solely on the apparent POV of the content. It moves them lower because WP:TECHNICAL says to do so: "Put the most understandable parts of the article up front. It's perfectly fine for later sections to be highly technical, if necessary."  Not following Wikipedia standards would be a violation of WP:NPOV. Following them would not. BTW, If you or anyone else thinks I am misinterpreting Wikipedia policy, please make that argument. If we need to do so we can get clarification.
 * I cannot see why you consider my suggestion to be POV. What POVs are you referring to? Martin Hogbin (talk) 16:57, 5 April 2011 (UTC)
 * Refusing to do what WP:TECHNICAL says to do (see above) appears to be POV pushing. You may have some other motive, but it really doesn't matter why you want to not do do what WP:TECHNICAL says to do. Guy Macon (talk) 20:07, 5 April 2011 (UTC)
 * Looking over the proposed structure, I think that experimental results (especially the results of elementary school classes playing the game again and again) should be early in the "Vos Savants solutions" section. First, because Vos Savant used the experimental results argument early in her original column. Second, because it is an especially convincing argument to someone with a little math knowledge that has concluded that Vos Savant is wrong, and third because there can be no academic criticism of experimental results; either switching wins roughly 67% percent of the time or it doesn't.
 * I agree about the xperimental results be presented early, I suggest as part of the vS furore. In fact they are no nearly as decisive as you suggest, it all depend on ''exactly' what experiment you do. However as the seem to convince people let us have them.Martin Hogbin (talk) 17:02, 5 April 2011 (UTC)
 * I am also thinking that "Academic criticism of the simple solutions" might be better if it followed a section called "Academic solutions" or perhaps "Statistical solutions." I would like the article to convey the fact that rigorous mathmatical arguments exist outside of criticizing the simple solutions. Guy Macon (talk) 13:40, 5 April 2011 (UTC)


 * My response above was indented to indicate it was a response to Martin's, not Guy's post (and, BTW, I am a highly experienced editor [an wp:admin in fact], so please don't lecture me about Wikipedia conventions). Re "would be POV" - yes, I am referring to WP:NPOV here, specifically WP:STRUCTURE.  WP:TECHNICAL is only a guideline while WP:NPOV is a fundamental policy, so if there is a conflict between these two WP:NPOV wins - i.e. the article MUST be NPOV first and foremost, and (I'm not saying this is the case here) if we can satisfy only one of WP:NPOV or WP:TECHNICAL we have to satisfy WP:NPOV.  If anyone disagrees with this, let's get clarification from another source.  -- Rick Block (talk) 15:27, 5 April 2011 (UTC)
 * Not lecture, but imho correcting your level of indent was okay, wasn't it.
 * +Guy, I like your wording    "... because there can be no academic criticism of experimental results  [...]  I would like the article to convey the fact that rigorous mathmatical arguments exist outside of criticizing the simple solutions" Gerhardvalentin (talk) 16:13, 5 April 2011 (UTC)
 * It is self-evident that, in an indented threaded discussion, using additional indents to format your comments is harmful to following the thread. This basic principle goes back to USENET and was established long before the World Wide Web was invented. I will leave it to the reader to determine whether they have made this error; I did not mention anyone by name. Regarding "lecturing," and I am bound by WP:GOODFAITH to assume that any confusing indentation was a simple oversight, and I must explain why one should stay at one level of indentation because some here may not know that. Guy Macon (talk) 20:34, 5 April 2011 (UTC)
 * Guy, I don't think you are misinterpreting WP:TECHNICAL, but I do think that WP:TECHNICAL cannot trump WP:NPOV where the reputable sources (as opposed to opinionated WP editors) give clear guidance. It is a fact that there are important primary and secondary sources that criticize the simple solution as incomplete or false. I believe that relegating their contribution - in fact, the very mention that such criticism exists - to the bottom of the article in order to make the article more understandable, is a disservice to the readers.
 * I am pretty sure that I am correct on this, and I note that you have not quoted the section of WP:NPOV that you think requires a clear violation of WP:TECHNICAL. If you wish, I would be glad to enter into dispute resolution (note that this is not an aggressive act but rather the proper way for Wikipedia editors to resolve disagreements about policy) and get a clarification of Wikipedia policy on this. Guy Macon (talk) 20:44, 5 April 2011 (UTC)
 * To make a bold analogy, consider the problem of us editors (scholares) trying to decide in about 1640 AD the proper structure of the Sublimis Wikia Pedica article on Planetary Orbits. Kepler's work is now old enough (20 years, same as Morgan's article) to be in secondary academic sources, but it is still barely registering in the popular press (a.k.a. textbooks for the Dauphin) . If we were to follow the dictate of "thou shalt put the most understandable parts of the article up front" in the way a Proff. Martinus de Hogbines forcefully proposes, then the first few sections must not contain any mention of Galileo's Dialogue because, hei, that's criticizing the geocentric system, which is the simple and intuitive model for the common reader (a.k.a. unwashed masses). Further, with heaps of added assumptions, epicycles, deferentes etc. (that the unwashed masses never even imagine because, well, they just follow intuition, but nevermind that), the geocentric system is correct too, as Proff Doctor. Ricardus Van der Gilles never ceases to prove and reconfirm in new essay after newer pamphlet - the good Doctor tells us that it is just a problem of mathematical modeling after all. Therefore we ought to start with the article with Anaximander, continue with Ptolemaic "aids to understanding", and relegate Copernicus, Galileo and Kepler at the bottom, in a section entitled "Academic nothingies" that the common reader never notices.
 * Do you see now Rick's point (i.e. what has been driving he and I and Nijdam insane for all this time, and led straight to mediation and arb)? Martin Hogbin (and GerhardValentin, and Glkanter) have consistently regarded even a simple mention, within the Solution section, that some things may be wrong with the simple solutions, as simply too hard to contemplate for the readers. Myself, I'd be happy to have the finer point of the conditional vs. unconditional difference at the bottom, if there was a common Solution section where the differences between the two solutions and their underlying assumptions were at least introduced. glopk (talk) 16:43, 5 April 2011 (UTC)


 * Glopk's comment makes sense. If the solution is criticized, why should this be banned to the end of the article? Handy2000 (talk) 20:12, 5 April 2011 (UTC)
 * 'Banned to the end of the article'?? Martin Hogbin (talk) 09:14, 7 April 2011 (UTC)
 * The solution to the above hypothetical is simple. If after the present violations of policy are fixed you find the article lacking, go ahead an add a reference in the simple solutions referencing the Academic criticism. If your edit violate WP:TECHNICAL it will be removed. If it doesn't and someone tries to take it out without referencing a policy it violates, they will get reverted and it will stay in. There is there no need to anticipate future wrongdoing, and in fact WPGOODFAITH requires that you assume that everyone here has taken up my suggestion to make a fresh start. Glkanter was banned, so mentioning past wrongdoing by him is irrelevant. IIRC,  Martin Hogbin was not banned, sanctioned, or warned, so predicting future bad-faith misbehavior on his part based upon alleged past bad-faith misbehavior is especially inappropriate in his case. We will deal with misbehavior when it actually happens, We are not to assume that it will occur. Guy Macon (talk) 21:16, 5 April 2011 (UTC)

I think some are still seeing monsters where there are none.

My proposal is not a dastardly plot to push my POV (that I do not actually have) that the simple solutions are correct and completely beyond criticism.

It is still being viewed from the very limited perspective of teaching undergraduate probability. Most of our readers will not be probability students and, and those students that do read the article should learn to read it all the way through.Martin Hogbin (talk) 18:10, 5 April 2011 (UTC)


 * glopk, I disapprove your making others ridicule. Spurning other editors as feeble is a PA.
 * Besides that, I must strictly reject the flat simplification of the facts. Dolorous again, as all the past years. Gerhardvalentin (talk) 18:39, 5 April 2011 (UTC)


 * Martin, the only perspective I am viewing it is WP:V and WP:NPOV, and - subject to them - WP:TECHNICAL. I don't have a view of the readers other than they are intelligent people that need not be kept away from the "hard stuff" till the very end.
 * Gerhardvalentin, if my lighthearted attempt at summarizing the past catfights with a little humor has pained you, I do humbly apologize - spurning brother editors was not my intent in the least, and I rejoice at noticing that the none of the other spurned editors have taken offense. However, how is what I wrote an utter simplification of the facts? Is there or not a group of editors (you and Martin Hogbin most notably now) who still reject the idea of inserting in the up-front solution a balanced, NPOV mention of the existence of sources criticizing the simple solutions as inadequate? And remember, we are talking of the POV of the sources here, everything else is irrelevant, please do not answer trying to teach me your own solution/interpretation/view of the MHP. glopk (talk) 19:36, 5 April 2011 (UTC)
 * Please quote the exact section of WP:V or WP:NPOV that you think requires a clear violation of WP:TECHNICAL. Guy Macon (talk) 21:19, 5 April 2011 (UTC)
 * I wrote that WP:TECHNICAL is to be followed subject to the requirements of WP:V and WP:NPOV. Note, however (1) that my humorous analogy was an illustration of Don't oversimplify and (2) that WP:STRUCTURE addresses specifically what's wrong with both the current state of the article and with Martin Hogbin's proposed structure above. glopk (talk) 03:26, 6 April 2011 (UTC)


 * Re "...as all the past years...": we are making a fresh start here. Please set aside the past, assume that all editors have reviewed WP:NPOV, WP:OR, WP:V, WP:OWN, WP:CIVIL, WP:NPA, WP:EW and the ArbCon ruling, and that all concerned want to start over. Referencing old pre-ArbCom disputes is toxic. Please stop doing that. Guy Macon (talk) 21:59, 5 April 2011 (UTC)


 * Re "I must strictly reject the flat simplification of the facts" please read Make technical articles understandable. Nobody is advocating simplification in thhe later, more technical sections, but tt is required in the earlier sections. Guy Macon (talk) 22:08, 5 April 2011 (UTC)


 * In what way does including a solution based on conditional probability violate WP:TECHNICAL? How NOT including a solution based on conditional probability until a section titled "Academic criticism of the simple solutions" or even simply until pages and pages of text fully elaborating only "simple" solutions has been presented violates WP:NPOV is that it creates an apparent hierarchy of fact.  Per WP:STRUCTURE (section of WP:NPOV):


 * It may also create an apparent hierarchy of fact where details in the main passage appear "true" and "undisputed", whereas other, segregated material is deemed "controversial", and therefore more likely to be false. Try to achieve a more neutral text by folding debates into the narrative, rather than isolating them into sections that ignore or fight against each other.


 * Approaching the MHP with a solution based on conditional probability is neither controversial or disputed - it is the dominant way the problem is approached in math sources (there are other approaches as well, but explicitly solving for the conditional probability is completely and absolutely standard).


 * The approach I favor is an integrated approach, where multiple solutions (simple and conditional) to the "standard" problem are presented - for example per the following suggested "Solution" section. These proceed from simpler to more complex (in accordance with WP:TECHNICAL) - so if anyone thinks this sort of approach is violating wp:technical, please state exactly how. -- Rick Block (talk) 23:00, 5 April 2011 (UTC)


 * Solution


 * Different sources present solutions to the problem using a variety of approaches.


 * Simplest approach


 * The player initially has a 1/3 chance of picking the car. The host always opens a door revealing a goat, so if the player doesn't switch the player has a 1/3 chance of winning the car.  Similarly, the player has a 2/3 chance of initially picking a goat and if the player switches after the host has revealed the other goat the player has a 2/3 chance of winning the car. (some appropriate reference, perhaps Grinstead and Snell)


 * What this solution is saying is that if 900 contestants all switch, regardless of which door they initially pick and which door the host opens about 600 would win the car. Assuming each specific case is like any other, this means a player who initially picks Door 1 and sees the host open Door 3 wins the car with a 1/3 chance by not switching and with a 2/3 chance by switching.


 * Enumeration of all cases where the player picks Door 1


 * If the player has picked, say, Door 1, there are three equally likely cases.


 * A player who switches ends up with a goat in only one of these cases but ends up with the car in two, so the probability of winning the car by switching is 2/3. (some appropriate reference, perhaps vos Savant)


 * What this solution is saying is that if 900 contestants are on the show and roughly 1/3 pick Door 1 and they all switch, of these 300 players about 200 would win the car. Assuming the cases where the host opens Door 2 or Door 3 when the player picks Door 1 are the same, this means a player who initially picks Door 1 and sees the host open Door 3 wins the car with a 1/3 chance by not switching and with a 2/3 chance by switching.


 * The probability of winning by switching given the player picks Door 1 and the host opens Door 3


 * This is a more complicated type of solution involving conditional probability. The difference between this approach and the previous one can be expressed as whether the player must decide to switch before the host opens a door or is allowed to decide after seeing which door the host opens (Gillman 1992).


 * The probabilities in all cases where the player has initially picked Door 1 can be determined by referring to the figure below or to an equivalent decision tree as shown to the right (Chun 1991; Grinstead and Snell 2006:137-138 presents an expanded tree showing all initial player picks). Given the player has picked Door 1 but before the host opens a door, the player has a 1/3 chance of having selected the car.  Referring to either the figure or the tree, in the cases the host then opens Door 3, switching wins with probability 1/3 if the car is behind Door 2 but loses only with probability 1/6 if the car is behind Door 1. The sum of these probabilities is 1/2, meaning the host opens Door 3 only 1/2 of the time. The conditional probability of winning by switching for players who pick Door 1 and see the host open Door 3 is computed by dividing the total probability of winning in the case the host opens Door 3 (1/3) by the probability of all cases where the host opens Door 3 (1/2), therefore this probability is (1/3)/(1/2)=2/3.


 * Although this is the same answer as the simpler solutions for the unambiguous problem statement as presented above, in some variations of the problem the conditional probability may differ from the average probability and the probability given only that the player initially picks Door 1, see Variants below. Some proponents of solutions using conditional probability consider the simpler solutions to be incomplete, since the simpler solutions do not explicitly use the constraint in the problem statement that the host must choose which door to open randomly if both hide goats (multiple references, e.g. Morgan et al., Gillman, ...).


 * What this type of solution is saying is that if 900 contestants are on the show and roughly 1/3 pick Door 1, of these 300 players about 150 will see the host open Door 3. If they all switch, about 100 would win the car.


 * A formal proof that the conditional probability of winning by switching is 2/3 is presented below, see Bayesian analysis.

( Preceding offered by Rick Block at 23:00, 5 April 2011 UTC )

Introduction has to serve intelligibility first
The following phrases [EDIT: which are in the present article] are too technical for inclusion in a section near the top of the article:

"canonical problem"

"formalization is authoritative"

"altering the optimality"

"simplifying assumption"

"uniformly choose"

It may not be possible to write a presentation of a solution based on conditional probability that meets Wikipedia standards for being near the top of the article. Some things are innately technical. Guy Macon (talk) 01:59, 6 April 2011 (UTC)


 * None of those phrases appear in the text I'm suggesting, so I'm not sure what your point is. -- Rick Block (talk) 06:18, 6 April 2011 (UTC)
 * But you are still trying to make some kind of point about the way the problem should be tackled in the article, mainly by insisting on giving a more complex solution than is given by the vast majority of sources before dealing with the problems most readers will have in understanding why the answer is 2/3.


 * Guy is absolutely right, especially as, on first seeing the problem, most readers will not only get the answer wrong but will not believe the correct solution when presented to them. We need to make the start as easy as possible. Martin Hogbin (talk) 07:42, 6 April 2011 (UTC)


 * Sorry for being unclear. The phrases I listed as being too technical for inclusion in a section near the top of the article are currently included in a section near the top of the article.


 * The problem with the text you are suggesting can be found in the phrase "...whether the player must decide to switch before the host opens a door or is allowed to decide after seeing which door the host opens..." That describes an alternate problem that is not what Vos Savant described, and thus discussion of it belongs in the variants section. It appears to me that you wish to introduce it earlier in order to justify introducing a more complex, more accurate and less understandable solution near the top of the page (everyone agrees that it belongs in the article; the disagreement is where) Guy Macon (talk) 16:56, 6 April 2011 (UTC)


 * There is a little problem in how people understand the word "random". The door hiding the car is random. Do we mean that each door is equally likely? Or do we just mean that each door has some different probability to hide the car, but the three probabilities are not necessarily equal? I would suggest to use the wording "completely at random" if you mean random with equal probabilities, and avoid the word "random" without qualification. Or alternatively, to explain (parenthetically) that by "random" we mean completely at random, till further notice (in the advanced sections). Richard Gill (talk) 09:30, 6 April 2011 (UTC)


 * I don't see a problem with a "simplifying assumption". After all, we are talking about a brain teaser. A paradox. People who don't understand what an assumption is, or a simplifiying assumption, are probably not going to see any difference between the right answer and the wrong answer to the problem.


 * By the way, there is "simple wikipedia" for people who have difficulties with long words and complicated sentences (for whatever reason). Still, if they can be avoided, they certainly should be avoided, in the initial parts of the article. The parts which everyone ought to be able to read and understand (even journalists and lawyers). Richard Gill (talk) 09:30, 6 April 2011 (UTC)


 * On my talk page Rick Block claimed that "How to interpret the problem vos Savant describes is a question that should be resolved by secondary sources." I replied saying "I will respond on the MHP talk page. My user talk page is for discussions about my behavior, not for discussions about the content of the MHP article."


 * The fact that forcing the player to decide to switch before the host opens a door is not what Vos Savant described is already referenced. Read what Vos Savant described. It isn't there. Guy Macon (talk) 16:28, 7 April 2011 (UTC)


 * Pardon, I wasn't logged in: 62.47.230.105. I just added the headline "Introduction". Gerhardvalentin (talk) 10:06, 6 April 2011 (UTC)


 * @RichardGill: Please do not invoke the existence of the simple Wikipedia as justification for what is a clear violation of WP:TECHNICAL. You are free to try to change the policy so that WP:TECHNICAL is satisfied by a reference to the simple wikipedia, but until you get consensus to make that change, we must do what WP:TECHNICAL tells us to do.


 * @RichardGill: I am concerned by your statement that if long words and complicated sentences in the early sections can be avoided they should be avoided. That is not Wikipedia policy. Wikipedia policy is that they must be avoided near the top of the article but are allowed in later, more technical sections.


 * @RickBlock: It appears that you are dead set on introducing highly technical information near the top. Please remember the Arbitration Committee Findings of fact: "Rick Block has displayed ownership of the article, and has been excessively controlling of both content and presentation."  Try to reach a compromise where you don't get everything you want and the editors who disagree with you don't get everything they want. You need to avoid any hint that you are repeating the behavior described in the arbcom finding of fact. You need to avoid even the appearance of ownership or controlling behavior. And, if I may inject my personal opinion, you really should fully embrace the letter and the spirit of WP:TECHNICAL rather than looking for reasons why it does not apply in this case. Guy Macon (talk) 16:56, 6 April 2011 (UTC)

Are we talking about the lead now, or the overall structure, or something else? @Martin - my point is that for an extended (arguably "main") section of the article to present and discuss only the "simple" solutions has the effect (regardless of the intent) of endorsing the POV that these solutions are correct. IMO, this means the article would not be NPOV, as both simple and conditional solutions are prevalent in the literature and a not insignificant number of sources are critical of the simple solutions. I think we can satisfy both wp:technical and wp:npov by presenting simple solutions first, but immediately after presenting one or more simple solutions also present a conditional solution (per my suggested text above). Your claim that we can convincingly explain to the satisfaction of most readers why the answer is 2/3 rather than 1/2 without mentioning anything about conditional probability is (as far as I can tell) based only on your opinion about this. The sources we have that discuss this (feel free to add any you think are relevant) are


 * Krauss and Wang, who say the "simple" solutions are "inaccessible" once the mental image most people create on reading the problem is formed (the conditional one where the player has picked door 1 and the host has opened door 3) - "Although, semantically, Door 3 in the standard version is named merely as an example ("Monty Hall opens another door, say, number 3"), most participants take the opening of Door 3 for granted and base their reasoning on this fact. In a pretest we gave participants (N = 40) the standard version [the original Parade version], asking them to illustrate their view of the situation described by drawing a sketch. After excluding four uninterpretable drawings, we saw that 35 out of the remaining 36 (97%) indeed drew an open Door 3, and only a single participant (3%) indicated other constellations also remain possible according to the wording of the standard version. The assumption that only Door 3 will open is further reinforced by the question that follows: "Do you want to switch to Door Number 2?" Note that once formed, this assumption prevents the problem solver from gaining access to the intuitive solution illustrated in Figure 1."


 * Eisenhauer, who says (speaking of vos Savant's solution) "Consequently, what could and should have been a correct and enlightening answer to the problem was made unconvincing and misleading."

We know for a fact that many people are not convinced by the simple solutions (10,000 letters to vos Savant). IMO, K&W explain why. If we present both simple solutions and an accessible conditional solution, we will be helping both people who are able to change their mental model to one that the simple solution fits as well as people who cannot do this. The table that user:Ohiostandard likes (below) fits this as well - it allows both views to be accessed, without requiring one view to dominate the other. If you're already thinking about the (unconditional) chances of winning by switching vs. staying (or you're able to change to this view of the problem) you can see what they are. If you're thinking about only the case where the host opens door 3, you can see that as well.

IMO, not only is this style of presentation (multiple approaches) much more NPOV, but it better serves our readers as well because we're not forcing anyone to change their mental model. -- Rick Block (talk) 16:45, 6 April 2011 (UTC)


 * I see no evidence that people who are not convinced by Vos Savant's simple solutions will be convinced by K&W's more technical solution. Guy Macon (talk) 16:28, 7 April 2011 (UTC)


 * {| class="wikitable" style="margin:auto; text-align: center;"

! colspan=4 | Situation BEFORE the host opens a door !colspan=4 | Situation AFTER the host opens a door ! Door 1 || Door 2 || Door 3 || total cases !colspan=2 | host opens Door 2 !colspan=2 | host opens Door 3 ! || || || || cases || result if switching || cases || result if switching
 * Car || Goat || Goat || 100 || 50 || Goat || 50 || Goat
 * Goat || Car || Goat || 100 || 0 || N/A || 100 || Car
 * Goat || Goat || Car || 100 || 100 || Car || 0 || N/A
 * }
 * Goat || Goat || Car || 100 || 100 || Car || 0 || N/A
 * }
 * }


 * In my experience ordinary people *are* convinced by the simple solutions. Especially the "Monty is essentially offering you to exchange one door for two", and also the "hundred doors variant". I have not had much succes trying to get ordinary people even interested in the conditional solutions. The simple solution is needed to raise their interest in the problem, anyway. It is true that people get stuck in their thinking by knowing only what they see: a door open with a goat behind it, two doors shut, (themselves happening to be at one of them). What people have to realise is that the process which led to that goat door being open depended crucially on their initial choice and determined the host's choices; the process is as important as the end result.


 * Teaching them conditional probability and doing formula manipulation or numerical calculations might appeal to some people, but not to most, I think.


 * There is a way to have your cake and eat it. Note that the fact of Monty opening "a goat door" (unspecified) does not change the chance that your door has the car behind it (cf. Georgii). Then add to this, the remark that *which* of the two doors he opens can't change that probability. Why do we have to make a big deal out of this? Give the reader the information and let them form their own opinion. Richard Gill (talk) 23:41, 6 April 2011 (UTC)


 * I think you're spot on, Richard, when you say that lay people aren't likely to have much interest in conditional solutions. But if I understand what you've said correctly, I also have to respectfully say that I think you're straight out in the assertion that people are at all likely to accept that the probability doesn't change after Monty opens a "goat" door, probably because you're so very familiar with the problem. I wasn't at all convinced by that assertion, myself, when I first saw it. My reaction was, "Nonsense! Of course the probability changes!" and I dare say most people have the same initial reaction or they wouldn't have any trouble with seeing the solution.
 * It may help you accept this, btw, if I mention that while I did all I could in college to avoid applied math ( I had only one probability and statistics course, and finished with the applied side after real and then complex analysis ), I re-read Russell's pre Principia Mathematica work − I'll get to PM one day − every few years with great pleasure, and continue to amuse myself reading in logic and set theory. All this is very much as an amateur, but my expectation would be that if I can follow Russell and couldn't see immediately that the probability doesn't change when Monty opens a door, that the average lay person isn't likely to accept that assertion either, without he has some previous "flash of insight" motivation to do so. But I suspect I may have misunderstood your intention? –  OhioStandard  (talk) 02:12, 7 April 2011 (UTC)
 * Yes, you're right, straight out people won't believe you. It all depends on how the conversation goes. You pose them the question. Instinct (they focus on what they "see" and forget the history which led to it: two doors closed, one open with a goat) tells people that the two closed doors are equally likely to hide the car. So they answer that they won't switch (but they're suspicious, it must be a trick question, but they don't see the trickiness yet). You tell them that their answer implies that the initial probability of 1/3 that their initially chosen door hides the car has jumped from 1/3 to 1/2! How can that be?! When you tell this to people they realise there is a contradiction but still don't know what to believe. Next you offer them the concept "Monty is offering you the choice between your one door and the other two doors, and helping you making up your mind by showing that one of the other two can be neglected". This, and only this, clinches the matter. Or you offer them the picture of one hundred doors, only one with a car hidden behind it. The player chooses door number 11 because he was born on 11th September. The host opens door after door revealing goat after goat, except for your door and door 42, and asks you to reconsider. Richard Gill (talk) 20:15, 7 April 2011 (UTC)


 * There is no POV problem with placing K&W in the (lower in the article) "academic criticism of the simple solutions" section and placing a brief note that such criticisms exist (with an internal link) in the (higher in the article) "simple solutions" section. Again nobody is advocating removing the material you want to put near the top. All that is being asked is to follow WP:TECHNICAL and put it lower on the basis of it being more technical.


 * So how do we proceed? Would you like a straw poll showing what the consensus is? Would you like to seek official word clarifying the policies? Guy Macon (talk) 17:08, 6 April 2011 (UTC)


 * Just to be clear, are you saying you think there is a wp:technical problem with the suggested text above? What, exactly, is the issue?  The more detailed, more technical, section IS lower (follows the easier, simpler sections).  This text would follow a "Problem description" section like the current one  (permanent link) that also starts with the simpler description and then proceeds with additional (more technical) detail.  IMO, this structure fully complies with wp:technical.  So, before proceeding, I'd appreciate it if you could explain your objections. -- Rick Block (talk) 17:48, 6 April 2011 (UTC)


 * Multiple conversations seem to be going on simultaneously within this thread. Can I suggest folks stop inserting additional comments mid-thread and only append (on the bottom)?  Thanks. -- Rick Block (talk) 19:06, 6 April 2011 (UTC)


 * Sorry, Rick, but I can't comply with the request. It's the norm to insert comments immediately after the post you're replying to, unless someone else has previously replied to that same post, in which case your reply should be positioned below that previous one, but at the same indentation level. I suppose you know that, but it would be still more confusing, imo, to use a strictly temporal posting order. And if you'll pardon me for saying so, the main difficulty I see in this thread is that you posted your 16:45 (UTC) contribution at flush-left, in contravention of that norm. No offence intended (or taken) but your having done that means that subsequent contributors can no longer use simple indentation to indicate who they're replying to. Anyway, I need to reply to something Richard wrote above, and there's now no good way to do that except to post after his comment. Again, sorry; I'm not trying to be contrary here.   –  OhioStandard  (talk) 01:43, 7 April 2011 (UTC)
 * Sigh. There's a definite trade-off here between keeping responses together vs. making it possible to follow the thread without stepping through all the diffs.  For example, can you tell who "you" refers to in Guy's post of 16:56, 6 April above?  Positionally by indentation he seems to be addressing Richard Gill.  By content his first paragraph seems to be referring to Richard, but his second paragraph seems to be referring to me (perhaps in response to my [earlier] post of 16:45 which is positioned physically further down in this thread - or perhaps not).  I outindented my 16:45 post because I quite literally could no longer follow who was responding to whom about what (particularly with the addition of the new heading smack in the middle of what seemed to be an ongoing conversation!).  I've asked a couple of times for Guy and Martin to say how they think including a conditional solution (more or less like the draft text above) violates wp:technical.  Should I read every diff of this page from now to eternity looking for a response?  I completely missed the top half of Guy's post here until just now, rereading the entire thread.  If folks don't want to append on the end to make the thread easier to follow, that's fine.  But I'll bet if we do this we'll continue to have discussions that go nowhere and leave everyone frustrated.  -- Rick Block (talk) 05:29, 7 April 2011 (UTC)
 * Yes, the reader can can you tell who "you" refers to in my post of 16:56, 6 April. It is completely clear from the context. Otherwise I would have written "You (Rick Block)" Because the second sentence refers to you by name you can infer from the context that the first sentence does so as well.


 * Take a look at the Alternative derivations section. First Gerhardvalentin used an indent on a letter from American Statistician he was quoting (he should have used quotation marks and stayed on the same indent level) then Richard Gill replied inline (not at the bottom) one level down from the AS letter and two down from the comment he was responding to, then a couple of lines later Richard Gill replied inline one level down from the the comment he was responding to, then glopk responed exactly where he is supposed to - right below the comment he was responding to and at the correct indent level - but because of the earlier decision by Richard Gill to post inline, glopk's comment was inline as well (I don't see what else glopk could have done). The end result was a mess that I was unable to untangle when I fixed the indentation of some of the other sections.


 * Later, in the Article structure section, Rick Block (23:00, 5 April 2011 edit) indents one paragraph 4 levels in the middle of a 3rd level comment for no apparent reason. In the light of his previous comment ("I am a highly experienced editor (an wp:admin in fact), so please don't lecture me about Wikipedia conventions.") I am not going to assume that it was a simple typo, but I also cannot say why it was written that way. I do know that if someone responds inline instead of at the bottom the above confusion will be repeated. Guy Macon (talk) 16:28, 7 April 2011 (UTC)


 * Is "you" in the first two paragraphs Richard Gill, and "you" in the third paragraph me? This is what I'm saying is confusing.


 * The indented paragraph from my 23:00 5 April post is a quote (from WP:NPOV), visually indented and without quotation marks per MOS:QUOTE.


 * IMO, this whole discussion has reached the point of absurdity. This is a TALK page.  Slavish adherence to MOS or indentation guidelines is definitely not required.  What is required is that participants in this discussion and someone reading this later be able to figure out who is saying what to whom (and the temporal aspects, i.e. when, are occasionally important as well).  ANY common sense formatting is just fine (if you need a policy reference for this, it is wp:sense).  As a participant in this discussion, I'm saying what we're doing is not working for me.  Feel free to ignore my feedback, but then don't get upset with me if I miss certain comments you make (that you might expect that I've read).  If I'm missing comments, I'm expecting others are as well.  -- Rick Block (talk) 19:27, 7 April 2011 (UTC)


 * Okay, Rick. Sorry to have caused you to sigh, above. ;-) I see your point, and agree that the top-posting (or whatever you want to call it) might have gone a bit overboard. I've clarified which of Guy's paragraphs refer to Richard and which refer to you, as that also seemed confusing to me. ( I hope you don't mind, Guy. ) I think most of the problem in the above was that everyone could have thought more carefully about where to place his comments per WP:INDENT, but it's also true that normal indentation can break down in certain circumstances. I'm tempted to refactor the section, but that would probably just make people mad in this context. Anyway, I hope it's moot, now that a new section has been started below. –  OhioStandard  (talk) 23:54, 7 April 2011 (UTC)


 * I don't mind at all. The best thing about wikipedia is the ability to easily undo things, so go ahead and be bold if I was unclear (and, of course, notify me so I can double check the change) Thanks!

The POV argument makes no sense at all
I can understand that in an article on, say, two well-known branches of a religion, putting a large volume of text on one branch first might lead a reader to assume that WP considers that branch to be more important in some way. However, that argument simply does not apply to the MHP article. There are not two well-known and well-defined positions on the MHP outside the pointless argument that has been raging here.

Does anyone seriously think that my proposed structure would lead a reader to assume that WP supports the 'simplist' view of the MHP? Can you imagine a new reader coming to the article and thinking to themselves, 'This article is clearly biased towards the 'simplist' view on this subject'? In fact I would say that even if Rick and Glopk with their current knowledge of the MHP but never having seen the article before were to read the article using my proposed structure they would see no problem with it. For an instant they might think to themselves, 'These people have missed and important point regarding the problem', but on looking further down the page they would see that Morgan's contribution was in fact comprehensively covered.

The objection to my structure is a remnant of the old arguments and page ownership and it should stop. Let me repeat again, I do not see my structure as a victory for my POV, I see it as the right way to write an encyclopedia. The MHP is not about a real-life situation it is essentially a contrived simple mathematical puzzle that everybody get wrong. Let us get that, most notable, aspect out of the way first before looking at mathematical subtleties and the problem's value in teaching undergraduate statistics. Martin Hogbin (talk) 08:09, 7 April 2011 (UTC)


 * @Martin, I think it'd be helpful for the discussion if you (and others) avoided repeating your own personal opinions about what the "average MHP article reader" might think or want as if they were somehow obvious, a given. They are not. You show no evidence that your intuition of the reader is factual, and you personally are neither an average nor a median nor a mode in the WP readership. In fact, as an opinionated editor, you are very likely in the long tail of the user population. On the other hand, we have guidelines (WP:TECHNICAL, WP:NPOV and WP:V) to follow and more to the point, for the MHP article, we are blessed to have sources to guide us about how people's mental picture of the problem is formed. The K&W snippet that Rick quoted above is eminently relevant here ("we saw that 35 out of the remaining 36 (97%) indeed drew an open Door 3"). That is strong evidence that the simple solution does not match the common mental model of the problem - and as such it deserves mention right next to where the simple solutions are presented. glopk (talk) 15:29, 7 April 2011 (UTC)
 * The first thing that WP:TECHNICAL does is to tell us to do exactly as I did and consider our audience when writing an article. Please tell me what class of reader you think would draw any kind of adverse inference from my proposed structure; I can think of none.


 * WP:TECHNICAL specifically says, 'Put the most understandable parts of the article up front', and, 'It's perfectly fine for later sections to be highly technical, if necessary. Those who are not interested in details will simply stop reading at some point, which is why the material they are interested in needs to come first', which is exactly what I have been saying.


 * WP:NPOV is irrelevant here because I have not expressed nor proposed that we support any particular POV. What POV exactly do you think I am pushing? Martin Hogbin (talk) 17:00, 7 April 2011 (UTC)
 * For the umpteenth time, we are talking of the POV of the sources here, and how to reflect it in the article in an NPOV manner. Of course noone here cares about whether you personally have a POV to push on not. Please do stop interjecting personal considerations (a.k.a. "preemptively playing the victim"): noone is attacking your own opinions on the MHP, we are just having a legitimate disagreement on how to best present the MHP on wikipedia, and we need to resolve it. glopk (talk) 21:39, 7 April 2011 (UTC)


 * Re: "as such it deserves mention right next to where the simple solutions are presented (glopk)", No Wikipedia policy supports that opinion, and WP:TECHNICAL clearly states that is should be placed lower. I am still waiting for you to quote the specific section of WP:NPOV that you believe calls for a clear violation of WP:TECHNICAL, which says "Put the most understandable parts of the article up front. It's perfectly fine for later sections to be highly technical, if necessary." This is the fourth time I have asked you for a specific quote from WP:NPOV. Guy Macon (talk) 16:57, 7 April 2011 (UTC)
 * Yes, and I did reply here. But with so many conversations occurring at once in too many indents I guess it's getting harder and harder to follow who's talking :-) glopk (talk) 18:28, 7 April 2011 (UTC)


 * Indeed, @Glopk, the simple solution does not match the common mental model of the problem. That's the whole point of MHP. The common mental picture is misleading, if not wrong. That's why the simple solution should be presented first without (emphatic) criticism. Most people are convinced by the simple solution that the answer "it doesn't make a difference, you might as well stick with your original choice" is wrong. The simple solution shows that the common mental model of the problem needs to be reappraised. For many, the simple solution is enough to settle the matter. You should switch, not stay. The conditional solution only refines this, and it refines this in a way which adds absolutely nothing for - say - fifty percent of the readers who come to the page. The other fifty percent will read on and their knowledge and insight will be refined. What's the problem? I don't see that K&W's insightful analysis of why ordinary people mostly initially give the wrong answer,  should necessitate that the article from the start sows doubt into the correct insight which is given by the simple solutions, and which for most people is quiet enough to understand why "stay", paradoxically, must be the wrong answer. People who without thinking use MHP to add some sex appeal to Bayes theorem in an elementary course on probability, can't impose their solution-driven solution as a guideline for how to write an encyclopedia article about a famous brain-teaser, a media uproar, a modern myth. [User:Gill110951|Richard Gill]] (talk) 19:57, 7 April 2011 (UTC)
 * @Richard, afraid I don't follow. The K&W's experiment in the article (Page 5, Item 4 - have you actually read it?) shows that the common mental picture of the problem is that a solution is asked given that a specific door has been opened. It is also stated therein that the simple solution does not address this mental model (as we all know). This common model is neither misleading nor wrong, it just is the model of the problem that, for the overwhelming majority, a solution must address. That experimental result offers strong evidence, from a reputable source, for what the common WP reader may expect from an article about the MHP. As such it is (a) much more relevant than Martin's (or yours) personal opinion the common reader, and (b) it gives guidance to us editors on how to properly structure the article per WP:TECHNICAL, because it indicates what is and isn't simple for the common reader. Martin (and you, I believe) continue to regard the conditional interpretation of the problem as a mathematical contrivance of interest to a tiny minority of the readers only, and too complex to show or (in Martin's case) even hint at in the Solution section. Yet here we have evidence that this is not the case, that in fact the common expectation is that the conditional interpretation of the problem needs addressing. Therefore an article structure like the one Martin is proposing, in which the simple solutions are presented and nothing is said, until the very end of the article, that they do not address the common model, is a violation of both WP:NPOV and WP:TECHNICAL (do not oversimplify). glopk (talk) 23:47, 7 April 2011 (UTC)


 * The link you (Glopk) just provided references a specific section of WP:TECHNICAL, not WP:NPOV So again I ask you to please quote a specific section of WP:NPOV rather than simply asserting that the specific instructions I quoted from WP:TECHNICAL are to be disobeyed based upon some undefined requirements that you claim are somewhere in WP:NPOV.
 * I did answer that already. For the second time, from WP:NPOV, section WP:STRUCTURE: The internal structure of an article may require additional attention, to protect neutrality, and to avoid problems like POV forking and undue weight. Although specific article structures are not, as a rule, prohibited, care must be taken to ensure that the overall presentation is broadly neutral. Segregation of text or other content into different regions or subsections, based solely on the apparent POV of the content itself, may result in an unencyclopedic structure, such as a back-and-forth dialogue between proponents and opponents. It may also create an apparent hierarchy of fact where details in the main passage appear "true" and "undisputed", whereas other, segregated material is deemed "controversial", and therefore more likely to be false. Try to achieve a more neutral text by folding debates into the narrative, rather than isolating them into sections that ignore or fight against each other.
 * As WP:NPOV is a policy, it must trump WP:TECHNICAL, which is a guideline, where the two are in conflict. In this case they are: as the reputable sources present different POVs on the proper interpretaton of the MHP, and therefore on what its proper solution is, we must present them in a NPOV manner (including in our choice of how to structure the article), while doing our best to adhere to the guidelines for technical articles. I think don't think it is that hard to reach a good compromise. I will hammer out my proposal for the structure later tonight. glopk (talk) 02:41, 8 April 2011 (UTC)


 * The quoted section of WP:TECHNICAL says: "Don't oversimplify. It is important not to oversimplify material in the effort to make it more accessible. Encyclopedia articles should not "tell lies to children" (with a link to Lie-to-children) in the sense of giving readers an easy path to the feeling that they understand something at the price that what they then understand is wrong."


 * The quoted section of WP:TECHNICAL appears to concern content, not where in an article the content is placed. In fact, the very first example used in Lie-to-children is Classical mechanics, which we know to be an oversimplification and factually incorrect. So, does Classical mechanics describe special relativity or quantum mechanics near the top of the article to preserve NPOV? No. It does not. The lead mentions that the Newtonian "simple solution" is wrong, but the actual description of special relativity or quantum mechanics are two-thirds of the way down, under "Limits of validity."  I thus conclude from the example given that if we follow the same pattern found in Classical mechanics when we structure MHP, we will have satisfied the "Don't oversimplify" clause, and we will have done so without violating the clear instructions we have been given to put highly technical material in the later sections. Guy Macon (talk) 20:35, 7 April 2011 (UTC)
 * The example does not apply, and your conclusion is wrong. Of course an article on Classical mechanics will not expound in any depth on either special relativity or quantum mechanics, as neither are branches of Classical mechanics. Therefore a health warning at the top, and an explanation of the theoretical limits at the bottom sufices therein. However the conditional interpretation of, and solution to, the MHP are instead part and parcel of the MHP just as much as the simple (unconditional) solutions. As the original MHP controversy indicates (and the K&W experiment shows), the conditional interpretation is the common one, and the so-called simple solutions (vos Savants, etc) require abstracting away from it. This is not my own personal opinion, it's the one of reputable sources: K&W and Falk, for example. Let's face it: the reputable sources on the MHP have different POVs. It is our job as editors to present them in a balanced manner. In Rick Block's and my opinion, an article structure like the one proposed by Martin Hogbin above, in which no mention of the conditional interpretation and solution is made until we are near the bottom of the article, is not balanced. And, am sorry to say, you cannot (a) use WP:TECHNICAL to hide this lack of balance and force that structure on on the article, nor (b) make vaguely threatening references to the arb case (FoF on Rick Block) or the dispute resolution process (to me) to gain an upper hand. I believe we can reach a compromise that is satisfactory for all involved. Let's please stay focused on the article.glopk (talk) 00:24, 8 April 2011 (UTC)


 * Glopk, if anyone is attempting to oversimplify anything it is you. I am proposing to have, in the appropriate place, a full and scholarly discussion of the Morgan solution to the MHP, based on the sources that discuss the subject.  You on the other hand are pushing a simplistic view that the simple solutions are 'wrong' and the Morgan solution is 'right'.  Whichever approach you favour you should surely realise that things are much more complex than this and that the issue deserves a full and proper discussion rather a 'health warning'.
 * If WP:NPOV applies to anyone it applies to you and Rick. You are both trying to push your own personal view on what is right and what is wrong.  I am trying to write an encyclopedia article according to WP policy.   Martin Hogbin (talk) 21:41, 7 April 2011 (UTC)
 * Martin, please tone it down, I will not tolerate any more personal attacks like the above. I am not pushing for any personal POV's, simplistic or otherwise, but I do require that the POV's of the reputable sources be presented in a NPOV and balanced manner. glopk (talk) 00:24, 8 April 2011 (UTC)
 * I have made no personal attack on anyone. You raised the POV argument, which I think is irrelevant to this article anyway so let us leave NPOV out of this.  We both want to represent what the sources say on the subject and I have proposed a structure which lets us present in a proper and encyclopedic manner, and in accordance with the appropriate WP:TECHNICAL policy, what all the sources say on the subject.  Let us now get on and do it.  Martin Hogbin (talk) 11:47, 8 April 2011 (UTC)
 * @Martin Hogbin, Re: "I have made no personal attack on anyone" - of course you have, how else am I to interpret your signature following "You on the other hand are pushing a simplistic view" and "you and Rick. You are both trying to push your own personal view on what is right and what is wrong", with absolutely no evidence being given? Please re-read and internalize WP:AGF, WP:CIV, WP:NPA, and avoid making false accusations in the future, will you?
 * Re: "You raised the POV argument, which I think is irrelevant to this article anyway so let us leave NPOV out of this". We cannot leave it out. The reputable sources have diverging POV's on the subject matter of the article, and it is our job as editor to present them with a NPOV and in a balanced way. Until you accept this simple fact of the matter we will not be able to move forward.
 * The article structure you insist upon does not match this goal, as it violates WP:STRUCTURE, a section of WP:NPOV, and the latter (as a WP policy) has primacy over WP:TECHNICAL, which is a guideline. Most sources with mathematical background present a conditional approach, the very one your proposed structure would relegate to the bottom of the article. You have asked above  (rethorically, I guess) what readers would be hurt by your proposed structure. I answer: all of them, and particularly those without the patience to reach the bottom and learn that there are serious issues with the simple view of the problem they have been given so far. We have guidance from both a WP policy and, most importantly, from the sources, that an article structure in which the two main approaches are presented next to each other is worthwhile. Will you please try to reach a compromise along these lines? glopk (talk) 14:14, 8 April 2011 (UTC)
 * Sources clearly do not give us any guidance on how to structure a WP article but the relevant guideline WP:TECHNICAL does. As I have clearly explained above WP:NPOV is irrelevant since there simply no split of opinion amongst the sources just a spectrum of sources treating the problem with varying degrees of rigour and detail which should, of course, be mentioned at the appropriate place in the article. Martin Hogbin (talk) 16:28, 8 April 2011 (UTC)
 * So, am I to understand that you are not interested in reaching a WP:CONSENSUS on this matter, that it is either strict adherence to the article structure you have proposed or nothing? In particular, that the presence of ambiguity in the statement, the K&W unambiguous statement and the conditional approach and solution will not be mentioned, within your proposed article structure, until the section you named "Academic controversies"? glopk (talk) 17:09, 8 April 2011 (UTC)

Let us see what other editors have to say regarding consensus. I believe that my proposed structure avoids POV issues, allows all aspects of the MHP to be properly presented, and meets the needs of our readers. Martin Hogbin (talk) 22:16, 8 April 2011 (UTC)

Glopk, Please do not assume bad faith.
 * Guy, I am responding to this point in your own talk page, since it relates to your own personal conduct in this conversation. I will only note here that asking an editor for clarity (whether he will accept any changes to his proposal) is not the same as questioning his good faith. glopk (talk) 16:02, 10 April 2011 (UTC)

Martin Hogbin has made it clear that he supports any structure that meets Wikipedia standards / policies in general and WP:TECHNICAL in particular. Feel free to propose another structure if you don't like his. I will support it if it does not violate Wikipedia standards.
 * I may do so, but at the moment the "integrated" structure proposed above by Rick Block above meets my preferences. glopk (talk) 16:02, 10 April 2011 (UTC)

As for consensus, We are not allowed to reach any consensus that violates Wikipedia guidelines or policies, because those guidelines and policies reflect the consensus of Wikipedia as a whole.
 * We perfectly agree here. However, as I have said, where a choice must be made, an applicable policy (WP:STRUCTURE here) takes primacy, and the applicable guideline (WP:TECHNICAL) must be followed subject to the requirements of the former. In my view this is absolutely not controversial, but I'll be happy to go through WP:DR if it needs clarification in your view. glopk (talk) 16:02, 10 April 2011 (UTC)

I have shown you, quoting the exact wording of the relevant policy, why it is that an explanation of the K&W solution must be placed in a lower section (exactly where is an open question; the guideline simply says "not near the top").

I see from an earlier comment that you believe that the WP:NPOV section WP:STRUCTURE applies. It reads:

"The internal structure of an article may require additional attention, to protect neutrality, and to avoid problems like POV forking and undue weight. Although specific article structures are not, as a rule, prohibited, care must be taken to ensure that the overall presentation is broadly neutral. Segregation of text or other content into different regions or subsections, based solely on the apparent POV of the content itself, may result in an unencyclopedic structure, such as a back-and-forth dialogue between proponents and opponents. It may also create an apparent hierarchy of fact where details in the main passage appear "true" and "undisputed", whereas other, segregated material is deemed "controversial", and therefore more likely to be false. Try to achieve a more neutral text by folding debates into the narrative, rather than isolating them into sections that ignore or fight against each other." Please explain why it is that you believe that Martin Hogbin's proposed structure is "Segregation of text or other content into different regions or subsections, based solely on the apparent POV of the content itself" despite his and my repeated statements that we wish to segregate the content according to the clear instructions provided by WP:TECHNICAL, which reads:

"Put the most understandable parts of the article up front. It's perfectly fine for later sections to be highly technical, if necessary."


 * As I have already explained (but I don't mind doing again, it's Sunday), Martin Hogbin's proposed structure is a violation of WP:STRUCTURE, part of WP:POV because sources presenting a fundamental POV on the problem are, in that proposed structure, relegated to a sub-section later in the article, and one entitled "Academic controversies", which further demeans their weight. I note that, historically, the difference in intepretations of the MHP, and the criticism of the "simple" POV, appeared very early (arguably, this difference in interpretations it is at the root of the controversy surrounding the appearance of the puzzle), and continues to be remarked in publications on the subject. This is further indication that there are significantly different POVs on the matter at hand, and care must be taken in presenting them in a balanced manner.
 * The "most understandable" argument you make is highly questionable, when applied to Martin's proposed structure, given evidence from reputable sources that the "simple" interpretation is, in fact, the one more removed from the intuition of the average person. Rather, a presentation that shows up front that there are different ways to interpret the (ambiguous) problem statement, and that logically different reasonings must apply depending on the chosen interpretation is not, in my view, "less understandable": it is just a way to celebrate the intelligence of the readers and avoid telling lies to children. glopk (talk) 16:02, 10 April 2011 (UTC)

I am pretty sure that I am properly interpreting Wikipedia guidelines and policies, and I think it is fair to assume that you also think that you are as well, so I am taking this to the next step found in WP:DRR so we can get clarification on the guidelines and policies. See Editor assistance/Requests to see my request for assistance. Guy Macon (talk) 05:45, 9 April 2011 (UTC)


 * Guy, at this point I am pretty sure you are misinterpreting both the relevant WP policies involved herein. I welcome the assistance. glopk (talk) 16:02, 10 April 2011 (UTC)


 * Editor assistance was unable to help. The next step will be a RFC. Guy Macon (talk) 13:37, 11 April 2011 (UTC)


 * You go right ahead. On further reflection, am calling myself out. Have fun while you can. glopk (talk) 18:38, 11 April 2011 (UTC)


 * You are, of course, free to withdraw, but the above comments appear to assume that an RFC is an aggressive act of some kind rather than what it is, a good-faith attempt to resolve a dispute between editors as to how to properly interpret Wikipedia policy. Guy Macon (talk) 12:02, 12 April 2011 (UTC)