Talk:Monty Hall problem/Archive 24

Canonical assumptions
Instead of the text from K & W, I suggest using the more structured assumptions from Rosenhouse p. 156:
 * The doors were initially equiprobable.
 * Monty never opens the door you initially chose, and he reveals a goat with probability 1.
 * Monty chooses his door randomly whenever he has more than one option.

(It's an actual quote). Tijfo098 (talk) 13:48, 10 April 2011 (UTC)
 * I think that's a good idea, moreover as long as we focus on the mathematical aspects/ambiguity of the problem citing a math source is more appropriate.--Kmhkmh (talk) 15:36, 10 April 2011 (UTC)
 * I think 1 is rather vague, we all know what it means but it is not clear. In this subject it is best to make things crystal clear.  There is no reason that we cannot state the assumptions in our own words provided that they are all unequivocally supported by reliable sources. Martin Hogbin (talk) 15:34, 10 April 2011 (UTC)
 * An equivalent formulation of a source content would be fine as long as the editors agree on it.--Kmhkmh (talk) 15:39, 10 April 2011 (UTC)
 * Yes, 1 could use some clarification. I'll see if I can dig a better source (possibly some other page in Rosenhouse). My WP:OR is that (1) means (1b) the player initially picks the door hiding the car with [equal] probability 1/3. Neither the car placement nor the player's first door choice need to be actually random, as long as (1b) is met. Tijfo098 (talk) 16:46, 10 April 2011 (UTC)
 * Why not take 1 literally? To wit: every door may hide the car with equal probability. This is a neutral statement that applies regardless of whether it is read as (a) a rule of the game (i.e. a constraint on where the car may be placed before the game is played), (b) the state of knowledge of the player prior to making a choice, (c) the strategy used by the player to make his initial choice (d) (perversely, IMHO) as a long-run frequency of car placements over many games, (e) ...? I may be misinterpreting Rosenhouse here, but I suspect that he chose that form of the statement exactly because it elegantly avoids unnecessary considerations of strategy or, worse, interpretations of probability. glopk (talk) 18:57, 10 April 2011 (UTC)
 * If you say that the doors are intially equally likely to hide the car, and that Monty is equally likely to open either door when he has a choice, you are being strictly neutral with regard to probability interpretation, but you are already biased to solutions which require full probability assumptions "host side". Many people solve MHP without these assumptions at all, they only assume that their own initial choice was random. Please note that "chooses randomly" is ambiguous, I suppose Rosenhouse (who is not a probabilist or a statistician) means "completely randomly". Anyway "chooses (completely) randomly" biases the reader to the frequentist interpretation. For the subjectivist, the uncertainty is in the mind of the player, not in the brain or action of the host. Richard Gill (talk) 07:47, 11 April 2011 (UTC)
 * PS, stating assumptions is fine, but do not call them THE assumptions. They are a commonly made set of assumptions, but not unversally made. The simple solutions don't even use all of these assumptions, which means that they are stronger (in the sense of being more widely applicable). Richard Gill (talk) 07:51, 11 April 2011 (UTC)

Analyses of Glopk and Rick Block's arguments against Martin Hogbin's proposed structure
Rick Block and Glopk listed their policy-based concerns in their straw poll oppose statements. The reader is invited to go there now so as to see the arguments in context, and to consider voicing an opinion in the straw poll. The current count is five support Martin Hogbin's proposed structure, two oppose, one abstains.

The arguments are as as follows (comments mine):


 * Responses inline in italics. -- Rick Block (talk) 15:25, 12 April 2011 (UTC)

Argument: Martin Hogbin's proposed structure violates WP:STRUCTURE:

Argument fails because it either requires the reader to pretend the words "based solely on the apparent POV of the content itself" are not contained in WP:STRUCTURE or it requires an assumption that both Martin Hogbin and Guy Macon are lying when they say that their motive is not based solely on POV of the content, but is instead based upon a desire to conform to WP:TECHNICAL.


 * Using "pretend" here is pejorative and this entire comment is simply a false dichotomy. The proposed structure does separate content based solely on the apparent POV of the content - meaning that only, i.e. solely, content promoting "simple" solutions and, thus, the POV that these solutions are adequate to address the problem, appears first.  What you effectively are arguing is that WP:TECHNICAL supersedes WP:NPOV, or that it's OK to violate WP:NPOV as long as your intentions are pure, and this is manifestly not the case (from WP:NPOV): "The principles upon which this policy is based cannot be superseded by other policies or guidelines, or by editors' consensus." -- Rick Block (talk) 15:25, 12 April 2011 (UTC)

Argument: Martin Hogbin's proposed structure violates WP:WEIGHT:

Argument fails because WP:WEIGHT specifically covers giving some views a larger or more detailed description, and says nothing about structure (which WP:STRUCTURE, being directly above WP:WEIGHT, already covered.)


 * This is a separate objection from the previous one about the section ordering. This one is based on the proposed structure devoting an entire section with several subsections to "simple" solutions (promoting the POV that these solutions are adequate to address the problem), while the predominant approach used in the relevant academic (math) sources, i.e. a solution based on conditional probability, is relegated to a single (later) subsection ("More detailed and comprehensive solutions") of a section titled "Academic criticism of the simple solutions" (!!??)  which is then followed by yet another section titled "Criticism of the criticism".  Not only is this overweighting the "simple" solutions based on sheer amount of text, it is clearly suggesting ALL conditional probability based solutions are disputed which is the exact opposite of the actual situation. -- Rick Block (talk) 15:25, 12 April 2011 (UTC)

Argument: Martin Hogbin's proposed structure violates WP:TECHNICAL (oversimplifying clause):

Argument fails because it is not falsifiable. One could respond to any and every application of WP:TECHNICAL by saying that putting the simpler material near the top is oversimplifying.


 * See the following comment -- Rick Block (talk) 15:25, 12 April 2011 (UTC)

Argument: WP:TECHNICAL does not apply because Krauss & Wang's conditional probability solution is as easy to understand as the simple solutions. Any disagreement with this assertion is simply an editor's opinion and not supported by any reliable source:

Argument fails because it is not falsifiable and works both ways; those supporting Martin Hogbin's proposed structure could just as easily claim that Krauss & Wang's conditional probability solution is harder to understand than the simple solutions and that any disagreement with this assertion is simply an editor's opinion and not supported by any reliable source.


 * The argument is NOT that K&W's conditional probability solution is as easy to understand as the simple solutions, but that K&W provide empirical data suggesting that "simple" solutions are not so simple to understand as opposed to the assertion of various editors that they are simple to understand which is based only on these editor's opinions (not emprical data). -- Rick Block (talk) 15:25, 12 April 2011 (UTC)

Argument: WP:TECHNICAL does not apply because Krauss & Wang's conditional probability solution is as easy to understand as the simple solutions and that this is supported by a reliable source: Krauss & Wang say that the "simple" solutions are "inaccessible" (quotation marks in Rick Block's original comment):

Argument cannot be verified. A search of the Krauss & Wang paper shows no use of the word "inaccessible", and a careful reading of every instance of the words "simple" and "Savant" show no such claim. Given recent history (example; misinterpreting Ohiostandard saying that a table should be included somewhere as saying the table should be included in the simple section), I would have to see an exact quote from Krauss & Wang so I can analyze this claim in detail.

Krauss & Wang describe an experiment where they tested university students who agreed to participate in a study (which alone makes it problemetical to apply the conclusions to Wkipedia readers), but the experiment was not a comparison between a simple solution and a conditional probability solution. It was a comparison between asking the participant whether to switch vs. first asking some leading questions which the authors call "guided intuition." The experiment does mention the word "accessible", so perhaps that is what Rick was referring to.


 * The claim is not that conditional solutions are necessarily easier to understand (indeed, as Tijo098 points out, in experiment 3 they show data suggesting that a conditional solution using Bayes Rule cannot be applied to similar problems - but this in no way means all explanations rooted in conditional probability are hard to understand), but rather that the cited experiment shows that understanding the simple solutions requires a different mental model of the problem than most people (97% in their study) initially create - and that changing mental models is difficult. Indeed, keeping all mention of conditional probability far, far away from the initial "simple" solution seems to be the precise intent of the proposed structure - i.e. the proposed structure seems to be intended to convince the reader that their original mental model is wrong so that they can then comprehend the simple solution.  By doing this the article would be explicitly endorsing the POV expressed by one editor above - "The common mental picture is misleading, if not wrong."   The text of a solution section I've drafted (above) presents both "simple" solutions and what is intended to be an accessible conditional solution, and deliberately relates both types of solutions to the mental model K&W's data says most people construct (where the player has picked door 1 and has then seen the host open door 3).  I don't have empirical data that says this specific text is more accessible to most people, but I strongly suspect that this approach is.  It is a minor variation on the approach used in Grinstead and Snell, but without the pejorative comments about the "simple" solutions not actually addressing the problem.  Since it is also obviously a much more NPOV approach I fail to comprehend why there is resistance to this sort of approach. -- Rick Block (talk) 15:25, 12 April 2011 (UTC)

Guy Macon (talk) 10:02, 11 April 2011 (UTC)


 * I agree that understanding MHP requires changing the mental model. People see before them (in their mind) two doors closed, one opened. Very, very subsidiary to this, they do know that they chose one of the two closed doors, and that the host opened the door which was opened... but they forget the past history, and have in their mind only what their eyes see in front of them at the end of the history. This mental picture makes them think that they needn't switch. In order to be convinced that they ought to switch, they need to change their mental picture. The so-called simple solutions are ways to make you revise your mental picture. For instance the very simple idea that two-thirds of the time your initial choice is wrong, requires you to change your mental picture to the picture that you are standing at a door and behind the door is a goat!!! The host opens the other goat door and you switch in order to pick up the car. To strengthen this change in mental picture, imagine there are a hundred doors. You are pretty certainly not standing in front of the car-door. The door which the host leaves unopened almost certainly is the car-door. This is how you convince people that switching is smarter than staying. Which it is: switching wins you the car 2/3 of the time, staying wins you the car 1/3 of the time (overall, ie not split out over possible door numbers). All this true, assuming *only* that your initial choice is right 1/3 of the time. I go to a frequentistic picture ("1/3 of the time") in order to make it more concrete in your mind.


 * Learning formal probablitity calculus and Bayes' theorem doesn't change mental pictures. It's a dull way to avoid making mental pictures. It provides a mechanical way whereby we can find a solution which is guaranteed to be correct independently of any mental pictures we might have.


 * The simple solutions, on the other hand, provide an alternative mental picture which jolts us to realising that switching is a smart strategy.


 * Once we have realised that always switching is far far superior to always staying, further analysis is for most people a waste of time. However, the conditionalists have a point: how do we know that switching is better than staying not just "on average" but also for every one of the six possible situations we might be confronted with? (door initally chosen, door opened by host). If you are interested to know this, then I can tell you four or more quite distinct arguments. All of them require a small level of abstraction or sophistication. Most people are not interested in any of them. If you are a student in a probability class then you need to learn how to get the answer by routine operations which will satifsy the teacher and give you full marks. There is one way which is foolproof, but unenlightening: it is, use Bayes' theorem. However you need to learn some elementary probability calculus for this.


 * Fortunately, there is a clever way to combine both the insights of the simple solutions and the mechanics of Bayes theorem. That is by thinking of information as coming to us in stages.


 * A: We choose a door. To be specific: let's suppose it was door 1. Nothing changes in our beliefs about the location of the car, when we make a choice of a door. Our subjective evaluation of the chances are that the car is behind our door, number 1, with probability 1/3, just as they were before we chose that door.


 * B: The host opens "a" door revealing a goat (but not telling us which door he opened). *Nothing* changes!!!! (He could do this anyway and he was going to do it anyway. He did do it. No chances have changed!)


 * C: The host tells us it was actually door 3 he opened! This tells us nothing about door 1's chances to hide the car, by symmetry! (Here I am using probability in the subjectivist sense: it's in my own (lack of) knowledge of the situation, not in the physical devices used to make choices). So door 1 must still hide the car with probability 1/3. On the other hand, door 3's chance to hide the car has collapsed to zero, so door 2's chance must have jumped from 1/3 to 2/3.


 * I just don't see a big deal. Two years fighting have gone by because some guys wrote in a low-status statistics teachers' magazine that the intuitive solutions of people like Vos Savant were wrong. They rewrote Vos Savant's question in order to do so. That made her bloody angry, and several smart people defended her. A lot of other folk were delighted to at last have a fun example of application of Bayes' theorem, and copied those guys' rewriting of Vos Savant's question together with those guys' rather boring solution in their elementary textbooks. Yet other people said that the first guys were arrogant bastards and that it was all much easier than they suggested. Just a few other people said that there were a whole load of other ways to approach the problem, all of which actually give insight into the paradox.


 * Additional remark: thinking of information as coming in piece by piece is also a change to the mental model of the problem. We solve a problem by simplifying. The immediate (intuitive) reaction of most people to MHP is to simplify it (ignore the history, just look at the stage) and this leads to the wrong answer. The simple solutions simplify it in a different way (ignore the detail of the specific door numbers). One can marry the "full" conditional probability Bayes theorem solution to the simple solutions in a number of ways, for instance as I did above, by adding the detail of the door number (host opened 3 not 2), at the very last step. One can also mention explicitly in the simplifying stage of forming the "good" mental picture, that by symmetry the numbers are irrelevant. Why make a big thing out of it? Richard Gill (talk) 05:27, 13 April 2011 (UTC)


 * Guy asked if I might explain the difference between the simple solutions and the conditional solutions. The difference is whether you stop at stage B or go on to incorporate also stage C. Most people have no interest in going to the bitter end. For those who do want to, it is an easy final step. That's all there is to say. Richard Gill (talk) 20:38, 12 April 2011 (UTC)


 * I hope you are aware that this is "nice" polemic that barely manages to avoid openly telling false things (politely put). Starting with "rewriting vos Savants question" (in reality it is Whitaker's question and who was allegedly rewriting/rereading what and why it matters or not is actually at the core of a part of the debate) to "Bayes (or conditional probabilities) not changing the mental picture". Of course they do, depending on how you make the Bayesian argument you place the contestant behind the doors as well (just performing a different calculation from that view) and you shift the focus towards the host's behaviour which is another change of the mental picture. The criticism of vos Savant's approach as incomplete (rather than false) stems not only from Morgan's article in that "low level magazine" way back but it fact it ranges over a variety of different publications all the way to the recent most comprehensive publication on the subject yet (Rosenhouse's book).
 * I don't want to comment in detail on the rather subjective (and imho just distracting, not helping and to some degree ridiculous) framing of the issue ("boring", "arrogant bastards". "low level", "smart guys"). Such descriptions might be fine and possibly entertaining for polemic or tongue in cheek essay on MHP, however for encyclopedic articles and their discussion it is inappropriate (as far as the article is concerned) and usually not helping (as far as the discussion is concerned, in particular if it is a heavily contested one anyhow). It really irks me to read something like that, in particular if it comes from a smart mathematician, who should know better.--Kmhkmh (talk) 21:49, 12 April 2011 (UTC)
 * Sorry, we should delete the polemical next to last paragraph I wrote, Preparatory to that, I put it in small print. It was my personal interpretion of the history. But please recall that Whitaker's original question doesn't mention any door numbers at all. I think it supports Vos Savant's intended meaning of her question. Richard Gill (talk) 05:11, 13 April 2011 (UTC)
 * If you want remove the last postings (including mine) that's fine by me, they are not bringing the discussion forward anyhow. Nevertheless I'd like to point out that Whitaker's original question (as quoted in our article and as described on vos Savant's page) does contain the door numbers (and always had). You can of course still argue that the door numbers might not matter and that they did not matter for Whitaker who was using them merely for illustrative purposes, but then again it is only a "can" and not a "must" and how appropriate the "can" is, was again at the core of some parts of the "academic" debate.--Kmhkmh (talk) 06:11, 13 April 2011 (UTC)
 * Sorry Kmhkmh, the last (2010) article by Morgan et al quotes from Whitaker's actual letter to Vos Savant. No mention of any door numbers. Whitaker is comparing Monty Hall to Monty Fall and asks if Marilyn agrees that in the one case you should switch the other not. Vos Savant later insisted that her side remarks "say Door 1" and "say Door 3" were not part of the question, this is mentioned by Rosenhouse in his book. English language semantic ambiguity. Possibly non-native speakers might not be aware of the two possible readings of her words. Anyway, this too is a side issue. Vos Savant asks a question in English with all its ambiguities and vagueness. Different people have converted this into different formal mathematical questions. Fortunately the difference between the two main interpretations is slight. It's merely whether or not you mention explicitly that the actual numbers on the doors are irrelevant to deciding whether to switch or stay. I am assuming here a subjectivist understanding of probability so that the two "equally likely" probability specifications follow from the problem description and do not need to be added as a separate auxiliary assumption. The frequentist cannot solve MHP without adding additional information. Vos Savant doesn't use the word probability or make probability assumptions so she forces the reader to solve the problem with the natural subjectivist (Laplacian) assignent of probabilities. Richard Gill (talk) 06:59, 13 April 2011 (UTC)
 * Ah ok, you were referring to the until recently unpublished "original" letter being different from the version published in vos Savant's column, I misunderstood that, in that context "vos Savant's question" (=vos Savant's formulation of Whitaker's problem) has different meaning too I guess. I don't think though that this in Rousenhouse's book though, as that was published in 2009. Or if it is nevertheless could you tell me the page please? This version has indeed no door numbers, but is not published version that started the debate and hence I agree somewhat of a side issue.--Kmhkmh (talk) 11:10, 13 April 2011 (UTC)
 * Yes. Rosenhouse doesn't know about the original Whitaker letter, indeed. But he does refer to Vos Savant's later writing what her intention was with the words "..., say Door 1" and "..., say Door 3". Maybe this is in her own book, I haven't read it yet (and don't have it). Of course her defence of herself came after the attack by Morgan et al., so we don't know what she really did mean at the moment she published her story. Nijdam thinks she was caught out by Morgan and mates. Who can tell?


 * But, I'm afraid I can't tell you right now where Rosenhouse said that. I'll have to read his whoe book all over again to find the reference, since at the time I did read it, I forgot to note it down.


 * Anyway, does it matter? The important thing is perhaps how the average reader understands Vos Savant's question, not what her own intention was or what she later claimed it to be. But you'll notice how many authors jump to read into Marilyn's question things which are not there. For instance, Grinstead and Snell write "Other readers complained that Marilyn had not described the problem completely. In particular, the way in which certain decisions were made during a play of the game were not specified. This aspect of the problem will be discussed in Section 4.3. We will assume that the car was put behind a door by rolling a three-sided die which made all three choices equally likely. Monty knows where the car is, and always opens a door with a goat behind it. Finally, we assume that if Monty has a choice of doors (i.e., the contestant has picked the door with the car behind it), he chooses each door with probability 1/2. Marilyn clearly expected her readers to assume that the game was played in this manner." I disagree that Marilyn expected her readers to assume anything like this at all. Even Monty Hall himself writing to Selvin 10 years earlier did not mathematize the problem in this way. And how do Grinstead and Snell know that Marilyn is a frequentist at heart? And if she was, why didn't she tell her readers her frequentist assumptions? (A frequentist can't do anything without making assumptions, while a subjectivist (Laplacian) can and does). Then they go on and write "We begin by describing a simpler, related question" as introduction to the problem which many editors call the simple problem. Then they write "This very simple analysis, though correct, does not quite solve the problem that Craig posed. Craig asked for the conditional probability that you win if you switch, given that you have chosen door 1 and that Monty has chosen door 3." But Craig does no such that thing! He does not ask for a probability, nor does Marilyn! Richard Gill (talk) 12:37, 13 April 2011 (UTC)
 * Well I agree such comments are annoying and somewhat misleading. On the other you can't quite fault the authors for it, since at least until 2010 (and probably even now) most people did not know Whitaker's original wording but simply based their (maybe somewhat pretentious) judgement on the publication in vos Savant's column (Parade Magazin). Another thing is that even into Whitaker's original wording one can read ambiguities and pursue different solution strategies. Personally I'd agree that the "canonical" MHP and Monty Fall is most likely what he had in mind, but then again we don't know for sure. Morgan's intepretation of Whitaker's words is at least possible though indeed it might appear a bit contrived, Georgii's approach (bayesian angle) however still looks appealling in particular since Whitaker explicity refers to actual Monty rather than formulating an "abstracted" game show riddle. More importantly before ultimately judging Whitaker's original words, I think we would need to see his complete letter rather than having only this snippet by Morgan without knowing whatever else might have been written before or after. I agree that there is no reason to peg vos Savant as frequentist, but there might be none to peg her as a Laplacian or Bayesian either. I agree that solving a problem with no (additional) assumptions (or better minimal assumptions as we still need to exclude Monty Fall) has a certain elegance, but then again making "reasonable" assumptions has always been a basic tool of solving or modelling "practical" problems.--Kmhkmh (talk) 13:29, 13 April 2011 (UTC)

My theory:
I dont beleive everyone else, This is my theory: alright, now lets say each O is a door-

door #'s:    1    2    3

doors:       O    O    O

probability: 1/3 1/3  1/3

okay each door has a 1/3 chance; but the two on the left have 66% chance.correct, So now say we pick one for example, door 3; awesome possum, so we say that it has a 33% chance of it being a car. So now we open door 2 and it's a goat. Now some think that door one still has a 66% chance and door 3 has 33% chance, but thats not true because now we only have 2 numbers; therefore it's 50/50 chance between the two and I was looking at this and they were using larger numbers to try and explain it, this is what I think:

door #'s: 1  2  3  4  5  6  7  8  9  10

doors:     O  O  O  O  O  O  O  O  O  O

So each door has a 10% chance now say we predict door 9; at this point, you only have a 1 in 10 chance of getting it right. now next step: lets say we open all the doors except for door 7 and door 9. Now you would think it's in door 7, BUT thats not true now we just have two doors and TWO doors only not 10 but Two:

door #'s: 7  9

Doors:    O  O

okay and now they both have a 50/50 chance. PROBLEM? Well then,This is what I think, my friend tells me I'm crazy but I just don't know. —Preceding unsigned comment added by 68.225.14.69 (talk) 06:06, 13 April 2011 (UTC)


 * You are supposing just any door got opened and it just happened by chance to show a goat. Then the chances of the other doors jump to 50%, 50%. But the point of MHP is that the guy who opens a door knows where the car is hidden and always opens a door different from your and with a goat behind it. He deliberately shows you a goat, not coincidentally. The chance your initial door hides a car doesn't change since the host is certain to open a door and show a goat, whether or not your door hides a car. And "which" of the two doors he opens is 50-50 whether or not your door hides a car, so this little piece of news doesn't tell anything about the chances your initially chosen door hides the car, either. Of course, the chance that the car is behind the opened door collapses to to zero, and the the chance that the car is behind the other closed door jumps up to 2/3. The host didn't open it. Maybe, because there is a car behind it. In fact 2/3 of the time your initial choice of door was hiding a goat and so it is twice as likely that the host's choice is forced than that it is free. Richard Gill (talk) 06:47, 13 April 2011 (UTC)


 * BTW there is an "arguments" page and a "FAQ" where this point is mentioned. It's a common misunderstanding of the MHP. Richard Gill (talk) 07:04, 13 April 2011 (UTC)


 * And BTW, the couple of sentences I have just written solve MHP in plain words, *completely*, incorporating the difference between the simple and the conditional solutions. That's the parenthetical addition "and "which" of the two doors he opens is 50-50 whether or not your door hides a car, so this little piece of news doesn't tell anything about the chances your initially chosen door hides the car, either", which is just as parenthetical as Vos Savant's "say, Door 1", and "say, Door 3". Richard Gill (talk) 07:15, 13 April 2011 (UTC)


 * Yet another editor who needs to have it explained to them why the answer is 2/3 and not 1/2. This is what nearly everyone wants to know and what the article should make clear at the start. Martin Hogbin (talk) 17:54, 13 April 2011 (UTC)


 * Indeed. Another editor thinking of a specific (conditional) example, who has clearly read the "simple" solution and finds it completely and totally unconvincing.


 * Many people are only convinced by doing some kind of simulation themselves. They'll simulate the simple solution. And quickly become convinced. I'm not sure if anyone is convinced by formula manipulations. These showed to that famous pure mathematician Erdos that the naive answer is wrong but he still was not convinced. Of course, he wasn't a probabilist or a statistician, he was a pure mathematician and number theorist. Some people will never be convinced. Richard Gill (talk) 17:02, 14 April 2011 (UTC)


 * If the original poster is checking in here, can you try reading the "Solution" section suggested above (repeated here so you don't have to go looking for it) and let us know what you think? Thank you very much. -- Rick Block (talk) 18:51, 13 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 )

Quick poll on structure
To get some idea of what the general feeling is here can ask for a quick poll on my proposed structure for the article. Please sign in the appropriate section below.

Support

 * Martin Hogbin (talk) 22:28, 8 April 2011 (UTC)


 * Guy Macon (talk) 05:51, 9 April 2011 (UTC) This structure (or one like it) is required by WP:TECHNICAL. After this poll was taken, Rick Block presented a version with a different structure that meets the requirements of WP:TECHNICAL. My position on this is now neutral. Guy Macon (talk) 20:40, 13 May 2011 (UTC)


 * Ohiostandard (talk) 09:16, 9 April 2011 (UTC) I really would like to see Rick's table included somewhere, though, along with some mention of a see it from Monty's perspective argument and especially of Monty's (indirectly) choosing the "car" door when the contestant doesn't, at first.


 * Gerhardvalentin (talk) 19:55, 9 April 2011 (UTC)   New attempt, a chance not least for the readers.


 * Richard Gill Simple first, to reach the largest possible number of readers. I don't believe this equates with choosing a point of view or giving an unbalanced picture. A poll is not a discussion, but we have been already been discussing this for years, it's not a bad idea to roughly count heads now. We need a sensible framework within which fresh editors can contribute with fresh editorial ideas, and who can capitalize on the insights (regarding simplication and reconciliation of different approaches) which were uncovered in the literature in recent years. The work of past editors has been a tremendous achievement, creating a magnificent resource and overview of the problem. Let's hand this over to new folk who rightly see possibilities for improvement (in particular under the philosophy of "less is more".). The article got "politicized". We need to remember that Monty Hall problem is fun for young and old, "educated" and not, it stimulates the imagination, it's found its way into so many different environments, both popular and academic. Richard Gill (talk) 10:45, 10 April 2011 (UTC)


 * Lambiam 16:29, 17 April 2011 (UTC): Support as better than most of the alternatives (and rather similar to the structure I proposed before). I think we also need to present the unstated assumptions that some authors think make the problem statement unambiguous, but this is something best postponed until after the simple solution has been presented. P.S. I see that Richard Gill made the same point already above.

Oppose

 * glopk (talk) 22:55, 8 April 2011 (UTC) -- Violates both WP:STRUCTURE by "hiding" one whole class of treatments in a late subsection, and WP:TECHNICAL by oversimplifying the presentation until that section.


 * Rick Block (talk) 07:08, 10 April 2011 (UTC) -- Violates WP:STRUCTURE (per glopk) by elevating "simple" solutions to a position of primacy. Violates WP:WEIGHT by over-emphasizing "simple" solutions (published by popular sources) and under-weighting conditional probability approaches, which are at least as prominent (arguably more so since they are the dominant approach in the relevant academic literature).  Relegating any mention of the predominant academic approach (i.e. using conditional probability) to a section titled "Criticism of the simple solutions" is (IMO) blatantly POV.  These objections relate to sections of WP:NPOV which is a fundamental policy that "cannot be superseded by other policies or guidelines, or by editors' consensus".  Violates WP:TECHNICAL (per Glopk) - the opinion that "simple" solutions are easier for most people to understand (and, therefore, a better way to satisfy wp:technical) is simply the opinion of various editors here.  A published wp:reliable source (Krauss & Wang) provides experimental data indicating most people have trouble connecting these solutions to the mental model they create of the problem.

Abstain
I don't see why we have to vote en bloc for your proposal, Martin. I see nothing wrong with a better specified version of the "standard problem" for instance. Many RSes do give it in more detail than Whitaker. At the same time, the structure you propose doesn't plan to include some of the topics I've raised in my FAR review. Also "3 Academic criticism of the simple solutions" with that many subsections seems blown out of proportion. There is more substantive stuff to be said about the interpretation of the vague natural language statement. Also "3.1a More detailed and comprehensive solutions" is unclear to me what is going to include; see point (3) in my FAR for possible issues here, if you plan to include solutions that most naturally apply (per multiple RSes) to other variants before presenting them at "4 Variants". Tijfo098 (talk) 01:05, 10 April 2011 (UTC)

Comments
Ohiostandard, I think we can better decide which tables and arguments should be included in the article after we have decided on its structure. Bear in mind that all material in the article must be supported by reliable sources. Martin Hogbin (talk) 16:38, 9 April 2011 (UTC)


 * The proposal is for a structure (what goes where). The table you mention can be added (or we can decide not to add it) no matter what the structure is Guy Macon (talk) 20:06, 9 April 2011 (UTC)


 * I interpret Ohiostandard to be saying he would like the aforementioned table to be in the "simple" section - which means to me he is actually opposing Martin's proposed structure. Issues like this are one of the reasons polling is not a substitute for discussion, which says "On Wikipedia, we generally do not line up simply to cast ballots, without some sort of discussion alongside of voting. In some cases, editors decide to use straw polls during discussions of what material to include in various Wikipedia articles. Although such polls are occasionally used and sometimes helpful, their use is often controversial and never binding. Where used, article straw polls should be developed in a way which assists in reaching consensus, rather than in an attempt to silence an opposing opinion."


 * Insisting on a particular structure despite other editors' (policy based) objections to that structure, and then initiating a "poll" to gauge consensus is simply not the right way to go about things. What we should be doing is trying to address the concerns raised by all editors.  Complying with WP:TECHNICAL is a valid concern.  But so is the concern that the "simple" solutions may not be the most accessible way for most readers to understand a solution, and so is complying with WP:NPOV.  None of these concerns can be dismissed by any poll. -- Rick Block (talk) 00:29, 10 April 2011 (UTC)


 * You are entirely correct that a straw poll is not a vote and that some sort of discussion alongside of voting is required. Ohiostandard and Richard Gill, could you please edit your comments to include a reason for supporting the proposed structure? Also, make sure that you really do support the proposed structure and not something else, and move your vote to oppose or abstain if you don't support the proposed structure. Thanks!


 * Re "I interpret Ohiostandard to be saying...", You are ignoring Ohiostandard's plain words ("included somewhere") and are claiming without evidence that he really meant "included in the simple section."


 * Martin Hogbin could just as easily claim that it is you who are "Insisting on a particular structure despite other editors' (policy based) objections to that structure," but wouldn't it be better if you assumed that Martin Hogbin is doing the best he can to follow Wikipedia policies and he assumed that you are doing the best you can to follow Wikipedia policies? See Begging the question. When the issue being discussed is whether Rick Block or Martin Hogbin is correctly interpreting Wikipedia policy, you cannot make an argument that assumes that only Rick Block is correctly interpreting Wikipedia policy. That is a classic example of the petitio principii fallacy. Guy Macon (talk) 07:39, 10 April 2011 (UTC)
 * Rick, Your main objection to my proposal is that you think it promotes the simple solutions as 'the correct solutions to the MHP'. Could we not work together within my proposed structure to avoid giving this impression. I would say that we should also avoid giving the opposite impression, that the simple solutions are 'wrong', or making off-putting remarks such as,  'these solutions are incomplete' or 'answer the wrong question' but there must be other ways to present the simple solutions first in a way that would be acceptable to you.  Why not suggest some? Martin Hogbin (talk) 09:27, 10 April 2011 (UTC)
 * Martin - I have suggested ways to present simple solutions first - see my suggested text above. Can we not work together to address your wp:technical-based concerns about accessibility of this text? -- Rick Block (talk) 16:55, 10 April 2011 (UTC)
 * Your structure does not work. 'Aids to understanding' refers only to the simple solutions, so should come immediately after them. What you call the 'conditional solutions' are presented in isolation, with no discussion of why or in what circumstances they are required. The 'conditional' solutions should surely come only after a discussion of why some sources consider them necessary. Martin Hogbin (talk) 20:22, 10 April 2011 (UTC)
 * Re: "'Aids to understanding' refers only to the simple solutions, so should come immediately after them". Why? It can come after an integrated solution section just as well. In Rick Block's proposed structure the "conditional solutions" are not presented in isolation. They are presented immediately after the "simple" ones, each with a clarification of what question is being answered (player decides "a priori", player decides before the opening of the door, player decides after). The finer point that differentiate the "simple" and "conditional" interpretations of the problem, and their associated solution, can come much later. But both kind of interpretations and solutions can be clearly stated one after another without loss of clarity in an initial integrated Solutions section. glopk (talk) 07:00, 11 April 2011 (UTC)
 * Dear Glopk, your assessment of which problem is being solved by which solution is (IMHO) your own personal opinion, together with that of some sources, but by no means all. (It seems to me that Rick Block has the same opinion as you do). You seem to me to be repeating poor Nijdam's insistence that there was a certain Truth and that the Truth could be used to differentiate Reliable Sources from Unreliable Sources. You say that the difference between simple and conditional solutions corresponds to whether the player has to decide on his actions before or after seeing a door being opened. I say that this is not correct, and I have many times explained why, and I am happy to do so again, if you don't object to reading yet another of my Professorial Pamphlets. There are reliable sources a-plenty which support me (and I refer here to academic mathematical statistical and probabilistic sources as well as academic game theoretic sources). Richard Gill (talk) 07:40, 11 April 2011 (UTC)
 * I haven't followed all the arguments regading the simple solution solving the conditional problem (that's what you are referring to right?) too closely, but I think there is still a difference. There is no disagreement that the simple solution "can be extended" to solve the conditional problem, there is disagreement about the exact description of this "extension" in the article. There was also a disagreement about the proper sourcing of the extension, but that's probably resolved by now. Still we should not have the situation where the simple solution is given as apparently computing to the overall probability (solving the unconditional problem) und not really telling whether and why this may also solves the conditional problem, such approach would be rather obfuscating to readers (and was partially a problem of the article in the past). Latest after reading the whole article (with a short mentioning in the lead) the reader should be aware that there were (at least) 2 promiment readings or versions of the problem (unconditional, conditional) and how and why the solution of the unconditional one solves the conditional one as well. --Kmhkmh (talk) 12:25, 11 April 2011 (UTC)


 * Gill110951, I would very much like to see "another of your Professorial Pamphlets." What I would very much like a second opinion on is this: [1] In words an engineer can understand, what, exactly is the core difference between "simple" and "conditional" in the context of MHP? [2] Is there any truth to the claim that "simple" and "conditional" are equally easy for the average reader to understand and thus I am in error when I try to apply WP:TECHNICAL?  [3] Does any reliable source claim that "simple" is actually harder for the average reader to understand than "conditional"?  I think I know the answers, but would like a second opinion. Guy Macon (talk) 13:26, 11 April 2011 (UTC)

Tijfo098, as you know, there has been a very long running dispute on the validity of the simple solutions which even arbitration has not resolved (although I believe that they did give us a few pointers). My proposal, which essentially involves putting the 'simple mathematical puzzle' aspect of the MHP first was intended to circumvent this issue to allow us all to work together in improving the article. Martin Hogbin (talk) 09:16, 10 April 2011 (UTC)


 * Unfortunately (for you guys) settling disputes between editors is not the purpose of Wikipedia. This is what Rosenhouse says at p.29 about the Morgan et al. vs. vos Savant, after citing their critique:

Rather strongly worded, wouldn't you say? And largely unfair, for reasons I have already discussed. Indeed, continuing with their lengthy essay makes clear that their primary issue with vos Savant is her shift from what they call the "conditional problem," as posed by her correspondent (in which it is stipulated that the contestant always chooses door one and the host always opens door three), to the "unconditional problem," in which we stipulate only that after the contestant chooses a door, the host opens one of the goat-concealing doors. She did, indeed, make this shift, but this was hardly the point at issue between vos Savant and her angry letter writers.
 * -- Tijfo098 (talk) 12:24, 10 April 2011 (UTC)
 * And R. Isaac also makes this distinction between the two variants, although he calls Morgan's "conditional problem" "another version, analyzed by Gillman" (ISBN 038794415X, p. 27). So framing the dispute today in the terms that Morgan did it (instead of problem variants) violates quite a few policies and guidelines WP:PRIMARY, WP:UNDUE, etc. Tijfo098 (talk) 12:44, 10 April 2011 (UTC)
 * Tijfo098, I agree with you on both points. I too see the Morgan solution as addressing what I would call an academic extension of the MHP, however, I am not proposing to push that POV in this article, I am proposing a structure that specifically avoids WP becoming a battleground between those who support that POV and those who oppose it.


 * True, settling disputes between editors is not the purpose of Wikipedia. Settling disputes between editors is just a happy side effect of the editors embracing the letter and the spirit of Wikipeedia's policies and guidelines. Guy Macon (talk) 13:26, 11 April 2011 (UTC)


 * Could I ask you to re-read my original reasons for proposing my structure and see if you do not agree with them. Martin Hogbin (talk) 15:30, 10 April 2011 (UTC)


 * Similarly, I am proposing actual text (not just structure) that is intended to be editorially neutral with regard to ALL solutions - whether they are "simple" or conditional or (we don't really have any of this form yet) based on game theory. -- Rick Block (talk) 16:55, 10 April 2011 (UTC)
 * I do not unerstand your point about editorial neutrality. I want to put the simple solutions, with associated discussion, first because they are simple, and because they are by far the most well known, not because they are 'right'.  Martin Hogbin (talk) 20:25, 10 April 2011 (UTC)
 * Martin, can you please state exactly what part you do not understand of what has already been explained above, by Rick and myself, in the "Oppose" section of your straw poll? After these many repetitions it's becoming harder for me to tell whether you have actually read what I and Rick have written and the WP policies we have quoted, and if so, what communication problem there may be for you to still not understand. Because, please believe me, it's not at all about you and what you personally believe to be right (or I, or Rick, for that matter).  We are saying that the article needs be editorially neutral with respect to the different POV's of the sources.  You may not agree that there is a variety of sources' POV (so you have stated above), but please have at least the courtesy to acknowledge that there is a good-faith disagreement on this very point, and it is not a reflection on you personally as an editor. Do you seriously believe that I or Rick are questioning your own editorial good faith? If you do, then this discussion is pointless, and you and I will have to start another round of dispute resolution. Otherwise, let's please debate on the issues, rather than on ourselves. glopk (talk) 07:00, 11 April 2011 (UTC)
 * It may be that Martin has read and understands the arguments you made in your oppose comment, and is using a common verbal shorthand where "I don't understand how you can claim X" or "X does not make any sense to me" actually means "I comprehend X, but it is an invalid argument." Please see the section titled "Analyses of Glopk and Rick Block's arguments against Martin Hogbin's proposed structure" for some reasons why he may have come to that conclusion. Guy Macon (talk) 13:26, 11 April 2011 (UTC)
 * And let me repeat there there simply are not two opposing POVs, to which we must give equal weight, in the literature. By this I mean that we do not see two opposing groups each saying that the other is wrong, as you would in  theist/atheist, protestant/catholic, conservative/liberal debates.  What we see is most sources giving simple solution (presumably believing them to be correct), some sources which give more complicated solutions, some sources which criticise the simple solutions on the basis that the host may not choose evenly, and maybe one or two sources who criticise the simple solutions to the conditional symmetrical formulation of the problem.  In addition we see a number of good secondary/tertiary sources that do discuss the complicated relationship between the simple and complicated solutions in a scholarly and comprehensive manner and from which the later parts of this article should take their lead.  Martin Hogbin (talk) 15:02, 11 April 2011 (UTC)

(Tijfo098's Response to Rick Block moved here from "Oppose" section -Guy)

The same source also says that most people can't apply the Bayes' formula solution to another problem, so while they "buy" that kind of solution, the don't really comprehend it (Experiment 3). You always cite only what suits you from K & W. Tijfo098 (talk) 12:51, 10 April 2011 (UTC)
 * The difference is I'm not promoting an article structure that prominently features a Bayes formula solution (in my suggested text above, Bayes formula is not even mentioned), let alone one that puts all other approaches in a subservient "Approaches that the academic community criticizes" section. My point is NOT that we should eliminate "simple" solutions - but that we must not endorse the POV that the "simple" solutions are the best way to approach the problem in an accessible manner. -- Rick Block (talk) 16:42, 10 April 2011 (UTC)


 * Rather than imposing a kind of apartheid I would like to see a kind of synthesis. Let's focus on a different distinction from the technical distinction between solutions which use conditional probabiility more or less explicitly and those that don't. Let's consider the distinction between solutions which use *only* plain English and plain logic, and those which use anything than the most basic concepts from probability theory. For instance: "equally likely" is plain English, but "conditional probability given this or that" is sophisticated.


 * With this in mind, quite a few solutions can be given in the early, plain English parts of the article. For instance:


 * You choose door 1. The host opens a door showing a goat. He deliberately shows you a goat, not coincidentally. The chance your initial door hides a car doesn't change since the host is certain to open a door and show a goat, whether or not your door hides a car. And "which" of the two doors he opens is 50-50 whether or not your door hides a car, so this little piece of news doesn't tell anything about the chances your initially chosen door hides the car, either. Of course, the chance that the car is behind the opened door collapses to to zero, and the the chance that the car is behind the other closed door jumps up to 2/3. The host didn't open it. Maybe, because there is a car behind it. In fact 2/3 of the time your initial choice of door was hiding a goat and so it is twice as likely that the host's choice is forced than that it is free.


 * This is a couple of sentences of plain English. Other editors will be able to make them even more simple. It gives a simple solution and a conditional solution in one. And I think it does it in such a way that that distinction is not forced up anyone's throat. It is a synthesis of both solutions, not an opposition. Richard Gill (talk) 07:22, 13 April 2011 (UTC)


 * It is difficult for me to get an overview of what is at stake in this whole discussion, but as I said about Devlin, I'm convinced an argument like: The chance your initial door hides a car doesn't change after the host opened a door with a goat, is not correct. And I notice Richard Gill saying the same. There is a probability law before the game starts, and one after the host has opened a door, and they are different laws. And for the chosen door 1 they share the same value on hiding the car, but it is misleading to suggest it is the same probability. Handy2000 (talk) 17:42, 15 April 2011 (UTC)

Synthesis, not conflict (Richard Gill's theory)
Rather than imposing a kind of apartheid I would like to see a kind of synthesis. Let's focus on a different distinction from the technical distinction between solutions which use conditional probabiility more or less explicitly and those that don't. Let's consider the distinction between solutions which use *only* plain English and plain logic, and those which use anything other than the absolutely most basic (pre-formal) concepts from probability theory. For instance: "equally likely" is plain English, but "conditional probability given this or that" is sophisticated. I know you can teach it to primary school children but still ordinary people find it confusing. That's why there are all those articles about how to teach conditional probability. Because it is a major challenge time and time again both to teachers and to students.

With this in mind, some technically speaking "full conditional solutions" can be given in the early, plain English parts of the article. They just avoid technical language. For instance:


 * You choose door 1. The host opens a door showing a goat. He deliberately shows you a goat, not coincidentally. The chance your initial door hides a car doesn't change since the host is certain to open a door and show a goat, whether or not your door hides a car. And "which" of the two doors he opens is 50-50 whether or not your door hides a car, so this little piece of news doesn't tell you anything about the chances your initially chosen door hides the car, either. Of course, the chance that the car is behind the opened door collapses to zero, and the chance that the car is behind the other closed door jumps up to 2/3. The host didn't open it. Maybe, because there is a car behind it. In fact there is a 2 out of 3 chance that your initial choice of door was hiding a goat and so it is twice as likely that the host's choice is forced than that it is free.


 * His actions don't tell you anything about what's behind the door you have chosen. But they do tell you a lot about what's behind the other doors.

This is a couple of sentences of plain English. Other editors will be able to make them even more simple. It gives a simple solution and a conditional solution in one. And I think it does it in such a way that that distinction is not forced up anyone's throat. It is a synthesis of both kinds of solution, not an opposition. (And you can find it in reliable soures, too). Richard Gill (talk) 07:24, 13 April 2011 (UTC)


 * I would have no objections to having the above (or something like it) in the lead. Very understandable, so no violation of WP:TECHNICAL.


 * Rather than "collapses to zero" perhaps something like "is now zero" or "changes to zero" would be better? Guy Macon (talk) 07:39, 13 April 2011 (UTC)


 * Sure. Changes to zero is fine, too. People find it strange that chances jump about - they think the closed doors were equally likely to hide a car before, so should stay equally likely after one door is elemintated by the host. Yet they don't find it strange that a door which first hid a car with probability 1/3 has zero probabilty of hiding a car after it's been opened and seen to hide a goat. And the idea that door 1 versus door 2 stays 50-50 would mean that the chance door 1 hides the car has jumped from 1/3 to 1/2, which is counter-intuitive when we emphasize that the host was going to show you a goat behind a different door anyway. True, the specificity of "which door" helps them overlook this. He wasn't certainly going to open door 3.


 * One can get people "off" their wrong initial solution both by giving an alternative apparently equally logical route to a different solution, and by showing some internal inconsistencies in their own. A paradox means an apparent contradiction. You have to get people re-thinking, you have to get them uncover hidden assumptions or patterns of thought. You do have to do it many ways since different people are pushed over the edge - a phase transition from 50-50 to 1/3-2/3 - in different ways.


 * I think it's good to anticipate the conditional solutions in the preliminary material. Then it is not as if we editors are hiding something from readers which only the initiated will learn later. And I'm convinced it can be done without technical developments which will alienate some readers. And it does not mean that the wikipedia foundation (through its editors) is taking some kind of stand on The Truth concerning how MHP should be solved. Richard Gill (talk) 12:55, 13 April 2011 (UTC)
 * And maybe it could help to "understand" if you could show that - as soon as the contestant has selected her door, ie even long before the host will open one door, you already know that there's only one car in the game, and so the chances of those two host's doors from now on are clearly  "0+2/3=2/3"  resp.  "2/3+0=2/3".
 * As soon as the guest has made her first choice, at least one of the two host's doors has zero chance. Only one. The "other" door is 2/3 from now on.
 * You just don't know which one hides a goat "for sure".  –   In 1 out of 3 there will be two goats, but only one goat is "for sure" anyway.
 * So, when the host opens one door, the chances need not to jump anymore. They "collapsed" and "jumped" long before. Gerhardvalentin (talk) 13:07, 13 April 2011 (UTC)


 * Re: "I'm convinced it can be done without technical developments which will alienate some readers.", If all can agree on it, that would be a very happy result. Re: "And it does not mean that the wikipedia foundation (through its editors) is taking some kind of stand on The Truth concerning how MHP should be solved." That also would be a relief. I know that the editors who accused me of same meant well, and that exploring possible bias/POV is not the same as assuming bad faith, but it's annoying just the same. Guy Macon (talk) 16:13, 13 April 2011 (UTC)


 * I agree in principle with this sort of approach, although the specific wording Richard has suggested is precisely the kind of wording that is problematic. The sentence "The chance your initial door hides a car doesn't change since the host is certain to open a door and show a goat, whether or not your door hides a car." is mathematically at least misleading - unless you read this very carefully it sounds like it is saying (and I suspect many, many people misinterpret it this way) the host opening door 3 doesn't change the chance the player's door hides the car ..." which it definitely is NOT saying (unless you are [somewhat unnaturally] numbering the doors only after the player has picked one and the host has opened one).  I'd suggest something more like "The player's overall odds of winning by staying with the original choice are not changed since the host is certain to open one of the two unchosen doors and show a goat, whether or not the player's door hides the car.  Since the overall odds of staying with the original choice are 1/3, the overall odds of switching must be 2/3.  The overall odds are the same as the odds in the specific case of door 1 and door 3 as long as all the probabilities are symmetrical, which they are with the usual assumptions."  This might not be the best wording, but I suspect the point I'm attempting to make is clear. -- Rick Block (talk) 18:51, 14 April 2011 (UTC)


 * Rick's wording is equally acceptable to me with two small exceptions. I would like to see "probabilities" and "symmetrical" replaced with simpler words or phrases (perhaps using "chances" and "the same"? Easy to dislike a word, hard to suggest a good replacement). Guy Macon (talk) 19:05, 14 April 2011 (UTC)


 * One has to be very precise, and one has to emphasize the important point. Indeed there is a major ambiguity in the English phrase "the host opens a door". One has to tell the story explicitly as "the host opens a door without telling the player which door he opened (the player is looking the other way)". And then at a later stage "the host tells the player which door was opened". Better still, the player had better be blind, in order to emphasize the separation of the two pieces of information. Step 1: the player hears the host opening another door and hears a goat bleating. Step 2: the host tells the player that it was actually door 3 he opened and asks if the player wants to switch to door 2. I like it! Richard Gill (talk) 21:16, 14 April 2011 (UTC)

Conflict?
I do rather object to my proposed structure being referred to as conflict and as a kind of apartheid. My proposal was just to put the simple solutions first and the more complex later, after some degree of explanation of the simple solutions. This was specifically to avoid conflict over which solutions are 'correct'. It would seem that there is a consensus for that structure although I am disappointed that more editors have not expressed an opinion on the subject so that the consensus is clearer. We now seem to have moved on now to the question of what simple solutions to have.

Can we all accept the principle of 'easy first', or to put it another way that the first part of the article should explain as clearly as possible to the new reader what the answer is and why? Martin Hogbin (talk) 09:06, 15 April 2011 (UTC)


 * OK, sorry for my use of divisive words like "conflict" and "apartheid". My proposal was intended to remove the merest suspicion that this might be how the article was being organised. I'm all in favour of simple first, complicated later. Always have been. I am just noting that there also exist simple solutions, simple in the sense of using only plain English and plain logic, and not needing any technical knowledge, which could be included early in the article (alongside many others of course) and which at the same time make a bridge to the more technical "conditional solutions".


 * Recall that Devlin's solution was wrong, as he himself admitted. He missed a small step and didn't realise it could be fixed very easily, in a way which is both obvious and intuitive. No surprise to me, Devlin's background is not in probability. "My" solution is Devlin's simple solution. It's also the simplest possible way to present Rosenthal's solution. It goes back to William Bell's comments to Morgan et al, namely that whether or not the car is behind door 1 is independent of whether the host opens door 2 or door 3. A fact which Bell wrote is so obvious it is hardly worth remarking on. As one of the editors on the arbitration remarked, this solution is so simple nobody is ever going to write a research paper on it. It goes back to Seymann's remarks on symmetry. Fortunately it has been published, so one can cite a reliable source for it too, if that is felt necessary.


 * The solution works by a clever switch in focus. Instead of seeing the two closed doors and not "seeing" their history, focus on the door which the player chose first, and consider the history of information coming to the player. Don't focus on the other door. Ask yourself if the actions of the host have changed anything, regarding the hidden secrets of your own door. Initially, the chance is 1 in 3 you are standing in front of a car-door. Imagine you're blindfolded and hear the host open another door and you hear a goat bleating. This was certain to happen whether or not there is a car behind your door, it tells you nothing about that issue. You remove your blindfold and note that it was actually door 3 which was opened. The chance was 50% that that door would be opened, whether or not the car is behind your door. So again it tells you nothing about the question you are focussing on. That's it! Richard Gill (talk) 08:08, 22 April 2011 (UTC)

Which simple solutions?
Richard Gill's proposal takes us away from what seemed to be a requirement previously, which was that all solutions presented here must be taken directly from reliable sources. Personally I think that this is too strong a requirement and not actually part of by WP policy; I have never supported a cut-and-paste approach to sources. There is always a balance in WP between sourcing and plagiarism.

If we do make this move, however, we must ensure that what we write is supported by reliable sources, otherwise it will be too easy for all new editors to add their pet solutions to the article making it cumbersome and confusing.

We also need to remember what it is that makes this problem so famous, it is the fact that most people get it wrong and do not believe the right answer even when it is clearly explained to them. We need to try to look at our simple solutions through the eyes of people who have never seen the problem before and then, supported by reliable sources, come up with the most easily understood explanations, ignoring, if necessary mathematical complications. Mathematical sophistry can come later, for those who are interested. Martin Hogbin (talk) 09:06, 15 April 2011 (UTC)


 * The solution I gave you is in reliable sources written by one Richard D. Gill. It is a minor variation on Rosenthal's Bayes' rule solution and a fix to Devlin's botched solution. It's a version of Bell's solution in plain words instead of in formulas. It's fully intuitive and it's fully correct; it's based on the common Bayesian interpretation of probability in the problem, which I think is the way plain people understand probability in this context, and which is the tradition of the elementary probability puzzles of the 19th and earlier centuries. Richard Gill (talk) 07:47, 22 April 2011 (UTC)


 * Within the constraints of what the WP policies allow, we should give as the first solution the one that is the most convincing to the largest number of people . For the solution to be convincing, it has to be understandable. Therefore it must be simple, straightforward, and expressed in plain language. Doesn't MvS's solution as originally presented fit these requirements quite well? Of course, some people will never be convinced (just like some people will never be convinced that the gambler's fallacy is indeed a fallacy), and we should not attempt to convince the inconvincibles by making the initial solution presented more complicated and thereby losing more than we win. --Lambiam 18:22, 15 April 2011 (UTC)


 * I think the MvS description will stay in the lead paragraph no matter what, being the most notable example. I don't see it being retained in the problem description section because I think we can do better, having the advantage of a wealth of research done since the MvS published it. In my opinion, with  the structure proposed by Martin Hogbin and the wording proposed by Richard Gill and Rick Block, we are there. Yes some tweaking and negotiating will make it even better, but it seems to me that the long-running content dispute is essentially solved though good-faith attempts to achieve consensus.


 * I can look at something and tell if it is understandable, simple, straightforward, and expressed in plain language. I think the latest proposals are doing quite well on those aspects and expect a final tweaked version to be even better. As for determining what is convincing to the largest number of people, I cannot tell be looking at it. Someone has to cite a reliable source showing some experiments establishing what is convincing to the largest number of people (the ones cited so far don't compare the same things and the test subjects were all college students). I personally cannot even tell you what convinced me because I got the right answer withing seconds of first encountering the Vos Savant column.


 * It appears from the limited data we have that no argument is convincing. Every argument appears to fail to convince well over 50% of the population. So we should go with what is most understandable, simple, straightforward, and expressed in plain language Guy Macon (talk) 20:17, 15 April 2011 (UTC)


 * Well, I'm in favour of a different structure, one in which the MvS description is moved out of the lead into the body, and other formulations are only discussed in later sections. While no argument can convince everybody, I do expect that what I think of as the simple solution, if carefully and clearly presented, will convince a large number of people. But I have no experimental data on that. --Lambiam 20:44, 15 April 2011 (UTC)
 * I agree, the simple solutions are essentially those solutions presented by sources with the specific intention of convincing people of the correct answer. The more complex solutions are generally intended for a different audience.  That is how we should treat them here.  Martin Hogbin (talk) 09:34, 16 April 2011 (UTC)
 * Lambiam, from what you have said it would seem that you support my proposed structure for the article. If this is so, perhaps you could add your name under 'support' above.  If not perhaps you could tell me what you do not like about it.  Martin Hogbin (talk) 00:02, 17 April 2011 (UTC)
 * Done, with a remark. --Lambiam 17:19, 17 April 2011 (UTC)

Recent changes
I am trying to get a feel for how the well the consensus is working out in real life. Is everybody at least somewhat happy with the way things are going. or at least does everyone have a wait and see attitude? If anyone thinks their preferred structure / content is being walked on, please discuss it. Thanks! Guy Macon (talk) 09:30, 23 April 2011 (UTC)

We Won an Award!


This section is a humorous essay and is not intended to directly show a serious opinion.

Monty Hall problem is on the prestigious Lamest edit wars list!

Alas, the edit wars over "Cauliflower" and "Cute" (search for them on the lamest edit wars list) are still ahead of us.

For those who wish to amp up our lameness, here are some links:

The Most Important Thing Possible

The Truth

The Last Word

Beware of the tigers

No angry mastodons

Civil POV pushing

Megalomaniacal point of view

Requests for medication

How to win an argument

Please be a giant dick, so we can ban you

Why Wikipedia cannot claim the earth is not flat

Don't be a fanatic

Don't be a dick

How to lose

Get over it

Defend Against Passion

I hope this helps... :)

Guy Macon (talk) 18:05, 28 April 2011 (UTC)


 * Great list. Can we have a show of hands on taking Monty to the list of common misconceptions?  Tkuvho (talk) 03:48, 29 April 2011 (UTC)


 * I say go for it. The worst that can happen is that someone shoots it down. The 50%/50% guess is clearly a misconception, and is clearly commonly held by those who have heard of the MHP. The only question is whether the MHP is widely known enough.


 * BTW, I especially recommend searching for the word "Cute" on Lamest edit wars. Fascinating what gets called NPOV, isn't it? Guy Macon (talk) 05:00, 29 April 2011 (UTC)


 * An editor at WPM expressed concern that there is still disagreement as to which solution is the right one and which the wrong one. For this reason, it would be preferable to gracefully come to an agreement, without lamely triggering nuclear holocaust, as to the text of a brief addition at list of common misconceptions before we actually take it there.  I would suggest the following text:  In a gameshow, there are three closed doors, one hiding a car, and two hiding a goat.  The player wishing to win a car guesses a door, which remains closed.  The host proceeds to reveal a goat behind one of the remaining doors, and offers the player a chance to switch his choice of door to the remaining door.  Should the player switch?  The correct answer, contrary to a common misconception, is affirmative.  I am running for the bunker.  Tkuvho (talk) 07:45, 29 April 2011 (UTC)


 * I suggest the following insertion: ... The host  , knowing where the car is hidden,  proceeds to reveal a goat ... Richard Gill (talk) 09:54, 1 May 2011 (UTC)


 * Looks good to me, that is what this puzzle is all about. Should we add that the player doubles their chance of winning the car by doing so? [Also runs for bunker].  Martin Hogbin (talk) 09:22, 29 April 2011 (UTC)


 * No Martin! You shamelessly exhibit a biased point of view! Does MHP belong to decision theory, or to probability theory? Your saying doubles their chance means that you are triply biased. You think (A) MHP is a probability puzzle, to be solved (B) using subjectivist probability, from the point of view of (C) the player. Richard Gill (talk)


 * Great! Let's go for it. Tkuvho (talk) 11:21, 29 April 2011 (UTC)


 * See if you can help, I am down to my last 3R already at list of common misconceptions. Tkuvho (talk) 13:36, 29 April 2011 (UTC)


 * (Start humor) Bad Tkuvho! Bad Tkuvho! No donuts for Tkuvho!! (End humor) (feel free to delete my attempt at lighthearted humor if it offends rather than amuses.)


 * Seriously, though, one revert was enough to tell you that it wasn't an uncontroversial change and that at least one editor opposed it. In such cases, you need to go to the talk page and seek consensus, leaving the article in its previous state while you discuss things.


 * Speaking of seeking consensus, Martin's suggestion that we come to an agreement as to the text of a brief addition at list of common misconceptions before we actually take it there. is just the sort of seeking consensus we need to be doing. Guy Macon (talk) 16:23, 29 April 2011 (UTC)


 * May I suggest instead, in the interests of having a consolidated and inclusive discussion, to continue this consensus-building at Talk:List of common misconceptions, where there is already a lengthy discussion underway? (Just my 2c.)  Sławomir Biały  (talk) 17:15, 29 April 2011 (UTC)


 * Sounds good to me. Guy Macon (talk) 17:21, 29 April 2011 (UTC)

Any proposed new entries to list of common misconceptions must at least fulfill the following:

The common misconception's main topic has an article of its own.

Done. Monty Hall problem is the main topic..

The item is reliably sourced, both with respect to the factual contents of the item and the fact that it is a common misconception.

Done. Monty Hall problem has extensive references establishing both.

The common misconception is mentioned in its topic article with sources.

Done. Monty Hall problem has multiple sources about the common misconception

The common misconception is current, as opposed to ancient or obsolete.

Done. Vos Savant is current

Looking at Talk:List of common misconceptions (header labeled Please read before proposing new entries) can anyone see a reason why I should not propose the following text?

''In a game show, there are three closed doors, one hiding a car, and two hiding a goat. The player wishing to win a car guesses a door, which remains closed. The host, knowing where the car is hidden, proceeds to reveal a goat behind one of the remaining doors, and offers the player a chance to switch his choice of door to the remaining door. Should the player switch? The correct answer, contrary to a common misconception, is that he should.''

Guy Macon (talk) 10:22, 1 May 2011 (UTC)


 * I don't see why we need to mention the fact that the host knows where the car is hidden. This clause merely complicates an already long sentence.  Since he is the host, it is understood that he knows where the car is.  Also, I think the editor who opposed inclusion did so on merely technical grounds, namely that there is no footnote giving a source where this misconception is described as being common.  Furthermore, while there may be debate among editors in this space as to whether the solution is true or not, I would like to point out that wiki's goal is not the search for elusive truth.  The goal is simply to reflect what's in the literature, and the literature certainly says that he doubles his chances.  There may be a systemic error in the literature, but that's not our concern.  Tkuvho (talk) 10:33, 1 May 2011 (UTC)
 * I have to disagree rather strongly. Mentioning details about the host explicitly is important, as much of the "academic" debate of the topic hinges exactly on such details and the ambiguities of what can/should be implicitly assumed and what not. There are plenty of publications dealing exactly with those details. Also I'd like to point out that the literature certainly does not say switching doubles his chances, but the literatures says that for the "most common"/"canonical" interpretation of the (ambiguous) problem he doubles his chances by switching. However there is literatures dealing "less common" interpretations and having a differing notions on the exact nature of the "canonical" problem. These subtle differences are exactly one of the reasons for the ongoing debate after the media hype caused by vos Savant's column. In short your assessment above seems to based on a false assumption of what is actually written on the subject in literature. It is true that WP cannot embark on quest for the elusive truth, however that quest is in already contained in the iterature, so mirroring/summarizing it accordingly will mean we have to deal with it here as well.--Kmhkmh (talk) 10:59, 1 May 2011 (UTC)


 * Thanks for your comment. You are obviously more familiar with the literature than I am.  Now we seem to agree that "the most common/canonical interpretation is that he doubles his chances".  That's all we need to mention at list of common misconceptions, I think.  The quest for truth in the literature is appropriate at this page.  I have no problem with mentioning the host's knowledge if you prefer.  Tkuvho (talk) 11:13, 1 May 2011 (UTC)
 * Yes, I have no objection regarding the list. I think, I misunderstood the context above, I thought your comment was referring to this article rather the entry in that list.--Kmhkmh (talk) 11:48, 1 May 2011 (UTC)
 * By necessity, List of common misconceptions has short summaries rather than exhaustive descriptions. If we do our job and Monty Hall problem covers those details (You *are* all monitoring the latest changes and making improvements, right?), a "See: Monty Hall problem" is all the List of common misconceptions entry needs. I do think we should retain the mention of what the host knows so as to reduce the number of readers who assume that the host could have randomly chosen the car. Guy Macon (talk)


 * It's great to have a consensus. Can one of you experts come up with a suitable footnote citing an established source to the effect that students have a hard time relating to this? Tkuvho (talk) 14:07, 1 May 2011 (UTC)


 * I listed two references at Talk:List of common misconceptions that you might stick in just after the word "misconceptions". (I think only the first reference is any good, really.)  It's likely Monty Hall regulars have something better, though.  It seems from the attitude at Talk:List of common misconceptions that anything without a citation supporting "misconception" is likely to be an automatic no-go.   Sławomir Biały  (talk) 15:03, 1 May 2011 (UTC)

Huck reference
In Statistical Misconceptions, By Schuyler W. Huck, on page 100: The Monty Hall problem (or three door problem) is one of the most famous examples of a "cognitive illusion", often used by psychologists, economists, and even law scientists to demonstrate people's resistant deficiency in dealing with uncertainty. This was published by Psychology Press, Taylor & Francis group in 2009. Tkuvho (talk) 16:34, 1 May 2011 (UTC)


 * I propose the following addition to common misconceptions: In a game show, there are three closed doors, one hiding a car, and two hiding a goat. The player wishing to win a car guesses a door, which remains closed. The host, knowing where the car is hidden, proceeds to reveal a goat behind one of the remaining doors, and offers the player a chance to switch his choice of door to the remaining door. Should the player switch? The correct answer, contrary to a common misconception, is that he should. The Monty Hall problem (or three door problem) is one of the most famous examples of a "cognitive illusion", often used by psychologists, economists, and even law scientists to demonstrate people's resistant deficiency in dealing with uncertainty. Schuyler W. Huck, Statistical Misconceptions. Psychology Press, Taylor & Francis group, 2009, page 100. Tkuvho (talk) 16:37, 1 May 2011 (UTC)


 * I suggest adding after the sentence ending he should., Indeed, switching his choice doubles his chance of winning, just to make clear it is not a marginal gain. Martin Hogbin (talk) 18:03, 1 May 2011 (UTC)

OK, let's try this: ''In a game show, there are three closed doors, one hiding a car, and two hiding a goat. The player wishing to win a car guesses a door, which remains closed. The host, knowing where the car is hidden, proceeds to reveal a goat behind one of the remaining doors, and offers the player a chance to switch his choice of door to the remaining door. Should the player switch? The correct answer, contrary to a common misconception, is that he should. Indeed, doing so doubles his chances of winning. The Monty Hall problem (or three door problem) is one of the most famous examples of a "cognitive illusion", often used by psychologists, economists, and even law scientists to demonstrate people's resistant deficiency in dealing with uncertainty.'' Schuyler W. Huck, Statistical Misconceptions. Psychology Press, Taylor & Francis group, 2009, page 100. Tkuvho (talk) 18:59, 1 May 2011 (UTC)


 * Very nice, IMO. Guy Macon (talk) 21:13, 1 May 2011 (UTC)

Last structural changes
I have now moved most of 'Sources of confusion' to follow 'Aids to understanding'. This section can now describe the basic puzzle, have the most well known solutions, help readers to understand why the answer is 2/3 and cover the media furore and the reasons that people get the answer wrong.

There is then a major section on why some sources think the simple solutions are wrong/incomplete/answer the wrong question, where we can discuss the subject properly, calmly, and according to what the sources say. The relevant section of 'Causes of confusion' has been moved here. There is no attempt whatever to hide this view or suggest that the simple solutions are 'correct' but the structure does recognise that the conditional/unconditional issue is discussed in a minority of sources and will interest a minority of readers.

I know Rick and Glopk will not like this but I really would welcome the opinion of other editors. I think this structure, properly used (and without 'health warnings') is the only way to permanently end the long standing argument. Martin Hogbin (talk) 09:18, 29 April 2011 (UTC)


 * I am confident that Rick (and possibly Glopk if he chooses to participate) will work to make the article the best it can possibly be even if it doesn't agree with their vision concerning the ongoing content dispute. Later, when I Martin to step aside for a while and I ask Rick to make the best possible version that incorporates his vision, I am confident that Martin and (indirectly) Richard Gill will also work to make the article the best it can possibly be even if it doesn't agree with their vision concerning the ongoing content dispute.


 * I am hoping that this article becomes a shining example of how editors with deep disagreements can work together in a friendly and collegial atmosphere on the many areas where they agree. After that, we will get together and agree how to settle the remaining differences - hopefully we will at that time be looking at a difference of a couple of short sections that are different and/or a minor re-ordering of the structure - small changes with a major impact. Guy Macon (talk) 11:03, 29 April 2011 (UTC)


 * I would like to encourage everyone reading this to be bold and make any edits that you think will improve the article. Also, revert any edits that you think didn't improve the article and discuss the edits here. This has been proven to result in a better article than being cautious and wary of someone not liking your edit does. Guy Macon (talk) 11:28, 1 May 2011 (UTC)


 * Would you kind co-editors object to me also editing? I would only try to make uncontroversial and evident improvements, e.g., removing some of the obvious duplications. Anyone can revert me, and I promise not to undo a revert. Or is that the slippery slope like the Brits sending in military advisors to Libya, and the US sending in their drones? I'm avoiding at present even reading the article, in order not to fall into temptation. Richard Gill (talk) 13:39, 1 May 2011 (UTC)


 * That is what I was hoping people would do. I hope that you agree that the present structure makes cooperative editing easier as there should be nothing to argue about now. Martin Hogbin (talk) 18:07, 1 May 2011 (UTC)


 * I would like to see Richard Gill make the edits he describes. Guy Macon (talk) 06:01, 5 May 2011 (UTC)

Comment
In two places early(ish) in the article, it says "solutions are often based on the additional assumptions [...] that the host [...] must uniformly choose which door to open if both hide goats". Also, in similar vein, "Many sources add to this the assumption that the host chooses at random which door to open if both hide goats [...]"

It is far from obvious why this should matter. From what comes later, I'm guessing that it only matters if the contestant knows the host's preference and can adjust his behaviour accordingly. However, from the statement of the problem up to the point at which one reads these things, there is no reason to imagine that the contestant would know any such thing. If anyone can think of a good brief way to cover this point at first mention, it might allay some confusions. 109.153.234.58 (talk) 02:51, 5 May 2011 (UTC)


 * This gets at the question of what probability we're interested in and what we mean by "probability". Concerning what probability, is it:
 * The probability of winning for a fixed strategy of switching (as opposed to a fixed strategy of staying with the player's initial choice)? Imagine 900 shows.  This would be the percentage of all 900 players we would expect to win if they all switch, regardless of which door they initially choose and which door the host opens.
 * The probability of winning for all players who initially pick Door 1 and switch (as opposed to players who pick Door 1 and stay with their initial choice)? Again, 900 shows.  This would only consider the subset who have initially picked Door 1 - if players are picking the initial door randomly then this would be about 300 players (not 900).
 * The probability of winning by switching for just those players who initially pick Door 1 and then see the host open Door 3 (as opposed to players who initially pick Door 1, see the host open Door 3, and stay with their initial choice). Again, 900 shows.  This would only consider the subset who have initially picked Door 1 and have then seen the host open Door 3 - with an initial random choice of door, and a host who chooses randomly between two "goat doors" this would be a population of about 150 players.
 * These are all different probabilities which can have different values depending on what assumptions are made. Assuming the car is randomly placed (or that the player's initial choice is random) is enough to make #1 and #2 be 2/3 (but #3 can be anything between 1/2 and 1).  The full set of "standard" assumptions ensure all of these are the same (2/3 chance of winning by switching for all players, 2/3 chance of winning by switching for just those players who pick Door 1, and 2/3 chance of winning for just those players who pick Door 1 and see the host open Door 3).  If what we're interested in is #3, then according to numerous sources how the host chooses between two "goat doors" affects the player's chances (whether the player knows how the host makes this choice or not).  According to these sources, if the host has an unknown preference the player's probability of winning by switching is not 2/3 but rather an unknown value between 1/2 and 1 [which can be expressed as a function of the host's preference for the door he did open - 1/(1+p) ].


 * The question of what we mean by probability relates to whether we're talking about a frequentist view (the relative frequency we'd observe in a large number of trials) or a subjectivist view (the probability given a particular state of knowledge). As far as I know, this difference isn't explicitly addressed in many of the sources although there are editors here who think this is a vitally important difference.  IMO (based on numerous sources), if what we're interested in is #3, then in either a frequentist or subjectivist view the player's probability is a function of the host's preference between two "goat doors", and if the player doesn't know this preference the best we can say is the probability of winning by switching is an unknown value between 1/2 and 1.  This is analogous to the "Monty Forgets" version, where we're told (but not necessarily the player) that the host forgets where the car is, opens a door randomly, but happens to avoid showing the car.  In this case, whether the player knows it or not, the player only has a 50/50 chance of winning by switching.  The question is (at least typically) NOT what is the probability limited only to what the player might or might not know, but rather what IS the probability.


 * Saying all this succinctly would be challenging, particularly in an early part of the article where some editors are attempting to avoid "technical" language in an attempt to satisfy WP:TECHNICAL. -- Rick Block (talk) 06:33, 5 May 2011 (UTC)


 * 109.153.234.58 you have walked into the aftermath of an arbcom case on this very subject. Some editors agree, to varying degrees, with you and others hold opinions similar to those expressed by Rick Block above.  You further comments would be welcome. Martin Hogbin (talk) 14:27, 5 May 2011 (UTC)


 * @Martin - you imply the arbcom case was about this content issue which is not true. The arbcom case was about user behavior.


 * @109.153.234.58 - further comments would indeed be welcome, but comments rooted in sources. If you're interested I would be happy to provide references to the "numerous sources" I mention above (they are mostly in the article already). -- Rick Block (talk) 15:05, 5 May 2011 (UTC)


 * Yes you are quite right Rick, my point was simply that this is a very contentious topic. Martin Hogbin (talk) 16:01, 5 May 2011 (UTC)


 * Not wanting to pile on, but I would like to emphasize for 109.153.234.58's sake that, while there indeed have been deep disagreements on this subject and some editors were banned for their behavior (the arbitration committee does not rule on topic disputes, only on user behavior) I am not seeing even the slightest hint of misbehavior by anyone currently editing or discussing this page. All I see are good-faith efforts to improve the article, and a plan for moving forward and resolving the content dispute.


 * The plan, in case anyone missed it, is that Martin, with Rick and everyone else helping, makes it the best article he can following his vision, then Rick steps in and, with Martin and everyone else helping, makes it the best article he can following his vision, We then all work toward making the two versions as close as possible, differing only where the editors actually disagree, and we leave the Rick Block version up while we follow the normal steps for resolving a good-faith disagreement about content. Guy Macon (talk) 19:07, 5 May 2011 (UTC)

Move to arguments?

 * Arguing based on conditional probabilities/Bayes' formula is not the same as having a "bayesian view". Bayes formula holds for all (sane) probability inpterpretations (=views) might they be frequentist, bayesian, classic, axiomatic, whatever. Morgan and Eisenhower use conditional probabilities but with frequentist interpretation iirc, to my knowledge that's also the "mainstream" view still as in (most) introductionary courses into probability & statistics usually implicitly or explicitly teach a combination of axiomatic and frequentist views. A real bayesian approach to the problem however can be found in Gillman and Georgii and it is also covered somewhat in Rosenhouse's book.--Kmhkmh (talk) 02:35, 6 May 2011 (UTC)


 * Rosenhouse discusses the whole shebang in the context of MHP, so those interested are probably better off reading that book than making/attending a wikicourse on it. However, I'm hopeful that some article improvements will come from editors' better understanding of these issues. Tijfo098 (talk) 21:55, 5 May 2011 (UTC)
 * I agree however with the thrust of Rick Block's comments above that this is a fairly technical discussion, so it's best left for the latter part of the article. Ronsehouse only addresses it in the last (substantive) chapter of his book "Philosophical Monty". The distinction Guymacon speaks of above is generally seen a core issue in the philosophy of probability, namely probability interpretations; sadly the Wikipedia articles on those topics aren't great. The book of Donald A. Gillies is a pretty good one on this topic (doesn't discuss MHP though). I'll avoid theorizing here, but still note that there's more than one epistemic/epistemological interpretation of probability: logical, subjective and intersubjective being the main ones. (Nowadays when one says Bayesian interpretation, one means usually means subjective Bayesian as this is the most popular variant.) Similarly, the frequency and the propensity interpretations are sometimes grouped under the objective sobriquet. Tijfo098 (talk) 01:01, 6 May 2011 (UTC)


 * If you want the nutshell variant of that, Paul Bartha's lecture notes are here, but they may be too terse (he uses a sightly different categorization in "objective" vs "subjective" than Gillies.) Tijfo098 (talk) 04:24, 6 May 2011 (UTC)
 * Btw there is also an online overview/summary article of all of that at Stanford Encyclopedia of Philosophy [Interpretations of Probability--[[User:Kmhkmh|Kmhkmh]] (talk) 09:24, 6 May 2011 (UTC)

Consensus
It looks to me that we have reached a consensus on the rough form that the content should take. So let's talk about the next step. Let me throw out an idea here: Martin Hogbin writes up a version in userspace that he believes reflects the consensus, paying careful attention to the recent suggestions made by Rick Block and Richard Gill, all the other editors look it over and make suggestions, we all seek consensus on the details, and finally we all agree that it should be moved to the main article. What do you thing? Too soon? --Guy Macon (talk) 02:23, 18 April 2011 (UTC)


 * How can one reach consensus about nonsense? I visited the article and discussion page hoping to find an easier way to explain the problem, rater then with the Bayes calculation, and learned about the use of odds, but I did this because the so often heard simple ways of explanation are not correct, in the sense they do not address the conditional probability. And it seems naive, not to say a lack of logic, to me, to belief that the problem could be solved with the unconditional probability as well as with the conditional probability. It's either the one or the other. And it is in my opinion the other, i.e. on the base of the conditional probability given the door chosen by the player, and the door showing a goat. The simple solutions may convince people about the chances being 1/3 vs. 2/3, instead of 50-50, but do this on false grounds. That should not be the purpose of the article. —Preceding unsigned comment added by 81.167.87.82 (talk) 08:54, 18 April 2011 (UTC) — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.


 * It is the consensus among the current editors of this page (some of whom are recognized experts in the field) that both solutions should be included. The disagreement was over where in the article to put them. If you wish to make an argument for some other content, please do so in a civil manner and seek consensus. If your arguments are sound, they are likely to prevail. Please note that you won't get far by calling the hard work of other editors "nonsense" or accusing them of a a lack of logic. Guy Macon (talk) 14:11, 18 April 2011 (UTC)


 * Sorry, if I sounded impolite, but I really would like to know, which of the experts in the field disagrees with me. 81.167.87.82 (talk) 18:12, 18 April 2011 (UTC) — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.


 * You asked "How can one reach consensus about nonsense"? The answer is easy: do not write any nonsense. Wikipedia is an encyclopedia so it just reports, selectively, what other people have written, and organises this material in a sensible way. Marilyn asked such and such a question, and gave this as an answer. Later someone else did something else. Later someone else criticised some other people. All we have to do is to present the facts in an intelligible way, so that intelligent people can think it through for themselves. The literature contains disputes about what is the actual question which has to be answered. That too has to be reported. Wikipedia isn't going to take sides. Fortunately, 20 years on, the clouds of dust raised by earlier fights have settled somewhat. The article shouldn't be poisoned by old wounds from old wars. The world has moved on. MHP is a moving target. Richard Gill (talk) 08:40, 22 April 2011 (UTC)


 * But a lot of nonsense has been written and still is written. I do not quite follow what you're trying to point at. Wikipedia doesn't takes side, you say, but also Wikipedia reports selectively. Well, Wikipedia should not take side in a controversial matter, but referring to nonsense as nonsense can hardly be considered as taking side, but better be called being selective. It is of course the duty of Wikipedia to make clear to its readers why switching to the remaining closed door is advantageous, But it also is its duty to point from the start to the fact that several sources tells you that the simple explanations do not show the correct arguments why you should switch. Let the reader judge for himself. BTW, do you Richard Gill, as an expert, agree that the simple solution is not solving the problem?81.167.87.82 (talk) 09:19, 22 April 2011 (UTC) — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.
 * I looked again at the simple solution section, and noticed that the picture along side, showing the doors 2 and 3 combined, is absolutely misleading, and should not be presented as a solution. It should not just be left out, but instead presented as an example of incorrect arguing . 81.167.87.82 (talk) 09:33, 22 April 2011 (UTC) — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.


 * Would it help if we were to move the more precise (K&W) problem definition until after the simple solution section. It is not clear exactly what Whitaker's original question is asking.  You could therefore assume (but we will not say so) that the simple solutions answer a slightly different question from that presented by K&W and thus you could sleep easy in your bed. Martin Hogbin (talk) 10:11, 22 April 2011 (UTC)


 * Well, being a reader, I surely would like know what the problem is, and I think the version of Kraus and Wang is picturing exactly what people will have in mind. But the simple solutions do not solve this problem. On the other hand there has to be a simple formulation of the solution in order to avoid the use of the words conditional probability and Bayes' theorem. I'm not well known with all the sources about the problem, but doesn't one of the sources gives such a (correct) simple explanation? 81.167.87.82 (talk) 10:34, 22 April 2011 (UTC) — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic. — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.  — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.
 * The consensus is that the simple solutions are fine for the start of the article although the complicating details will, of course, be explained fully later. There was a very long argument about this subject ending in an arbitration decision in which an editor with views almost identical to yours was banned for a year for tendentious editing. Best go with the consensus and add your views later in the article.  Martin Hogbin (talk) 19:56, 22 April 2011 (UTC)
 * It sounds like a threat, but do not all the editors share the same view? Do you have a different view than me? Could you please explain? 81.167.87.82 (talk) 21:55, 22 April 2011 (UTC) — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic. — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.
 * It is not a threat, I was merely noting the similarity between your views and those of a previous editor who was recently banned. I was also pointing out that we do not want to go through the same long arguments again. There is a consensus and that is what we should go with. Martin Hogbin (talk) 22:03, 22 April 2011 (UTC)
 * As I said before, consensus about ... what? Look again at the pictures next to the simple solution section. Now take door 1 and door 3 together in a box. They have 2/3 chance on the car. Because door 3 shows a goat after opening, its chance collapses to 0. Does the chance on the car for door 1, the first chosen door, now goes up to 2/3? 81.167.87.82 (talk) 22:10, 23 April 2011 (UTC) — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.
 * I still did not get any comment on my concern about the pictures next to the simple solution section. 81.167.87.82 (talk) 22:51, 8 May 2011 (UTC) — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.
 * Hi! Would you be so kind as to make a new section at the bottom of the talk page and explain your concerns? The above was a bit confusing to me. Guy Macon (talk) 23:12, 8 May 2011 (UTC)
 * K&W conditions are not part of Vos Savant's words and are added themselves by many solvers, especially those who want to exhibit MHP as an example to liven up their lecture on Bayes theorem, but not all. For instance the mathematician Georgii presents a solution in which door numbers 1, 2 and 3 are not even observed by the player. In fact, for him, door number 1 is defined as the door hiding the car. Georgii's solution is Vos Savant's solution. Maybe Georgii understood Vos Savant better than many readers of her article. Vos Savant's English is ambiguous and she later told Morgan et al. off for misreading her obvious intended reading (obvious when you see her solutions). They were so cock-sure of themselves that they didn't see that she was totally consistent in question and answer, as long as your read her question properly. Georgii assumes only that the player has one third chance to have picked door number 1, initially. From this he shows that switching gives the car with probability 2/3. Gill shows that the simple solution corresponds to a solution where the solver ignores door numbers, the conditional solution corresponds to a solution where the information coming from the (observable) identity of the doors is pasted on afterwards. Bell showed that simple and conditional solutions are connected through the obvious independence between what is behind door 1, and the number of the door opened by the host. This is so obvious it goes without saying that the door numbers are irrelevant.
 * The game theorists and mathematical economists, who tend to be frequentists not subjectivists, all give simple solutions where it's the player's initial choice which is random. The only thing which the player himself can guarantee. The frequentist can't solve the problem without assuming that the player has been wise enough to choose his door at random. The conditional probability that the car is behind the other door is unknown and uninteresting. Richard Gill (talk) 13:49, 24 April 2011 (UTC)
 * Not everyone is a professor in statistics. Common people automatically assume equal chances when they are shown 3 doors with a car behind one of them, from which they may choose. An over precise mind is needed to state the car is randomly hidden, an extra over precise mind to say completely random, etc. Ordinary people do see the doors, the one chosen, the one opened and the one remaining closed. It shows them the Monty Hall problem, and makes them think the chances for the still closed doors are equal. Nothing there to promote Bayes' theorem. On the contrary, people pushing forward the "simple version", to which the simple explanation applies, presumably do this for specific reasons. Because they do not understand the conditional nature, because they once used the simple explanations themselves, or whatever other reason. Maybe Georgii, like Vos Savant, initially also was mistaken, who knows? The K&W version is quite natural for most people, much more acceptable as the intended problem than the artificial simplified version. 81.167.87.82 (talk) 17:34, 24 April 2011 (UTC) — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.

Proposed action

 * If Martin is OK with this proposal and feels up to it, it's fine with me if he gives it a first shot. --Lambiam 17:08, 18 April 2011 (UTC)


 * I agree that there is a consensus but please note that my suggestion was only about the structure of the article, I have no special desire to rewrite the content. Again, my point is that, if we change the structure to have the simple solutions first, without a precise definition of the question, we can all work cooperatively on the rest of the article to present the issues concerning the so called conditional solution.


 * By giving no more that Whitaker's ambiguous version of the question at the start we allow those who wish, such as 81.167.87.82, to view the simple solutions as solutions to the so called unconditional interpretation of Whitaker's question.


 * Rather than a sandbox version, I suggest that we start by moving sections into the suggested order, which I am happy to do if it is wished. After that we should all be able to work together.  Martin Hogbin (talk) 22:15, 18 April 2011 (UTC)
 * As no one has objected, I will move a few things around as per my suggestion. I will do it slowly to give people time to object or revert if they do not like what I am doing. Martin Hogbin (talk) 10:27, 21 April 2011 (UTC)
 * Strongly approve. Guy Macon (talk) 12:59, 21 April 2011 (UTC)


 * Good plan. Richard Gill (talk) 08:14, 22 April 2011 (UTC)

A have the solution to your problem right here
It's not that the problem is smart, it's that the presentation is misleading and easy to miss the obvious that the host is picking explicitly a goat door, purposely not opening a car door. If this is stressed, "scientists and nobelists" would not have such a problem. It's pretty certain this 'problem' is used by idiots that want to say "see, those smart people are idiots" when the only thing they do is not present the premises of the problem clearly or at least without any doubt. --195.74.255.127 (talk) 23:26, 10 May 2011 (UTC)


 * It sure does look like Marilyn Vos Savant stressed that exact thing on here follow-up columns, and the "scientists and nobelists" continued telling her she was wrong. BTW, using bold in discussions about probability theory do not make your argument more compelling. Quite the opposite, actually. Emphatic language conveys strong feelings about a statement, but in this case the audience is unlikely to be swayed by strong feelings. They are looking for rational and logical arguments. Guy Macon (talk) 08:28, 11 May 2011 (UTC)


 * 195.74.255.127, you are quite right in saying the fact that the host knows where the car is and always picks a goat-hiding door is critical. If you have any suggestions as to how we might explain this more clearly to our readers they would be most welcome. Martin Hogbin (talk) 09:35, 11 May 2011 (UTC)

Trigger table of contents
Empty section to keep ToC out of collapsed section. Glrx (talk) 17:42, 12 May 2011 (UTC)

Other solutions
I have renamed the 'Conditional solutions' 'Other solutions' because the section covers a wide range of approaches. I think using the terms 'conditional' and 'unconditional' is a relic of the old argument. I do not particularly like 'other solutions' but it is at least neutral.

I have made 'Conditional solutions' a sub heading under 'Other solutions'. Martin Hogbin (talk) 08:38, 23 April 2011 (UTC)

Conditional intro section
I have moved the intro that Rick restored to a more appropriate place, as the introduction to other solutions. I think it is another relic of the old argument and should be replaced by something based on one of the better-known secondary/tertiary sources.

It its new position we have no need to present a veneer of neutrality, we can boldly say that some sources consider the simple solutions to be wrong, or that they answer a different question. We can give reasons why the sources say this and arguments for and against the claim. Martin Hogbin (talk) 08:38, 23 April 2011 (UTC)

I must have forgotten to save the above changes, but I have made them now. Martin Hogbin (talk) 12:44, 23 April 2011 (UTC)


 * The point of the previous placement was to provide a forward reference to the section presenting the very commonly presented conditional probability solutions. As discussed above, without this forward reference and given the current headings IMO the article is endorsing the POV that the simple solutions are correct and complete in violation of WP:NPOV - specifically WP:STRUCTURE: "It [segregating content based on the POV of the content] 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."  The current structure exhibits this exact problem.  The "Simple solutions" section is presented first, with no mention of other approaches or that these approaches are criticized by numerous sources (as if the simple solutions are complete, correct, and undisputed) - with the approach that is unquestioned by any source (i.e. showing the conditional probability of winning by switching for a player who has picked door 1 and has seen the host open door 3 is 2/3) relegated to a subservient section - implying this approach is somehow less common than the "simple solutions" or even controversial.


 * No one has adequately addressed this concern, which has been raised by multiple editors. In particular, the claim that the content is segregated to satisfy WP:TECHNICAL is not responsive to the concern. I will tag the article with POV unless this concern is addressed. -- Rick Block (talk) 20:19, 23 April 2011 (UTC)


 * Everyone is being civil so far, but I would like to remind everybody to be especially nice to each other while we work together to resolve this. Agreed?


 * Rick, no need to threaten to tag the article at this point. That just encourages the digging in of heels on both sides. Just say, in the gentlest language possible, that such and such a content/structure doesn't address your concerns. Assume that Martin meant to address your concerns and help him to do so.


 * I have not looked at the page in detail (I am on a job site and occasionally looking at Wikipedia while waiting for the technicians to set up a test), but I thought that we had pretty much agreed to mostly put things where Rick wants them (and that Rick agreed that the structure that Richard G. and Martin were talking about did that) as long as Rick could supply text that isn't too technical for the average reader. If anyone thinks that the text is still too technical. we need to give Rick a chance to make it simpler while still retaining his meaning. Only if he cannot do that should we invoke WP:TECHNICAL. If it doesn't violate WP:TECHNICAL, it stays as per WP:STRUCTURE. This is non-negotiable, because it is based upon the consensus of all the editors who worked together to create WP:TECHNICAL and WP:STRUCTURE. Too technical? Simplify it or move it lower. Not too technical?  Keep it near the top - the policies and guidelines list no other reason why it should be moved other than being too technical.


 * Note that the use of phrases like "probabilistic" or "conditional probability" are too technical for the early sections. If someone can explain Albert Einstein's Theory of Relativity in words of four letters or less we can find a way to avoid "probabilistic."


 * So, lets all sit down and talk this over. Let;'s agree that we are not going to let it blow up in our faces again. OK? Guy Macon (talk) 22:13, 23 April 2011 (UTC)


 * Rick, what exactly is the POV that you believe the current structure promotes? Martin Hogbin (talk) 22:48, 23 April 2011 (UTC)


 * The POV the current structure promotes is that the "simple solutions" are undisputed and entirely sufficient to address the problem, and are the easiest and "right" way for mathematically unsophisticated readers to understand the solution. The POV this structure does not present (as required by WP:NPOV) "fairly, proportionately, and as far as possible without bias" is the POV that the problem is asking about the chance of winning by switching for a player deciding after picking a door and after seeing which door the host opens - and that this is (by definition) a conditional probability (mathematically different from the probability of winning by always switching) and the problem should therefore be addressed by a solution that considers this probability.


 * The POV that the solution involves considering the conditional probability is the POV of most introductory probability textbooks, as evidenced by their use of this problem as an example of conditional probability, as well as the POV of the fewer, but significant number of papers and books that explicitly discuss this very issue. Structuring the article so that a mathematically unsophisticated reader reads content presenting "simple solutions" (with no mention whatsoever that other solutions exist or that these solutions are considered deficient by some sources) violates both NPOV and WP:TECHNICAL (specifically the section "Don't oversimplify" which says "It is important not to oversimplify material in the effort to make it more accessible. Encyclopedia articles should not "tell lies 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 understanding that by the host opening, say, door 3 (after the player has picked door 1) the chances the car is behind door 1 must remain unchanged is (according to many sources) wrong.  This is the (mis)understanding promoted by the "simple solutions" which has been commented on even by vos Savant, unfortunately now a dead link, but available on the Internet Archive at  (italics in the original): "Back in 1990, everyone was convinced that it didn’t help to switch, whether the host opened a losing door on purpose or not. ... Now everyone is convinced that it always helps to switch, regardless of what the host knows. But this is just as incorrect!"


 * Vos Savant refers here, I believe, to the Monty Hall versus Monty Fall interpretations of the problem. For Monty Hall, biased host ( an academic diversion), Morgan et al show it is never wrong, always to switch (as long as the car is hidden completely at random and as long as Monty always must open a (different) goat door. @Lambiam, reproduced in a recent publication of Gill, shows that this result can be got by an elementary argument without using conditional probability: argue that maximally biased host (Monty Crawl) is most favorable to the player, yet even then, by consideration of the only two possible cases, always switching gives you the best you can hope for. Since you can't beat the overall 2/3 win chance of always switching, it's a waste of time to figure out conditional probabilities. This is a nice example of how game-theoretic ideas - consider the extreme cases - can simplify a probability problem, making probability calculations superfluous. What we want on the MHP are the ideas, not the calculations or formula manipulations. It's a page for humans to read. Also math challenged humans.  Richard Gill (talk) 12:18, 25 April 2011 (UTC)


 * Regarding what words to use in the early sections to distinguish the "simple solutions" from the solutions based on conditional probability (at the point of providing a forward reference to a later section), I'm very open to suggestions. We previously worked on this and the consensus at the time was for the wording Martin initially deleted and has now moved to below the "Simple solutions" section (which is pointless, since the intent is to alert the reader that other kinds of solutions exist and what kinds of sources present what kinds of solutions and provide a forward reference to the section it now appears in).  If "probabilistic reasoning" and "conditional probability" are too technical, how about the following (to appear immediately after the header "Simple solutions"):


 * There are several approaches to solving the Monty Hall problem all giving the same result—that a player who swaps has a 2/3 chance of winning the car. Most popular sources present solutions analyzing the overall chances of winning by switching as opposed to staying. Another approach to solving the problem, commonly used in mathematical sources, is to examine the probabilities in a single specific case such as the one given in the problem statement where the player has picked door 1 and the host has opened door 3, see Conditional probability solutions, below.


 * If we want to be more inclusive, we could also reference a section (still to be written) presenting a game theory approach, e.g. "Yet another approach, commonly used in economics sources, is to treat the problem as a 2-player game where the host may be trying either to prevent the player from winning the car or to help the player win the car, see Game theory approach, below." -- Rick Block (talk) 06:52, 24 April 2011 (UTC)


 * I hope that others here will be able to see that this is the same old argument that seems to exist only on the minds of some editors of WP. In the outside world, there is no split of opinion of the sources into those that assert 'the "simple solutions" are undisputed and entirely sufficient to address the problem' and those that assert the reverse.


 * What we actually see is a large number of sources that give simple solutions and other sources that criticise the simple solution to varying degrees and which present alternative and generally more complex solutions. That is what we should reflect here.  I can only hope that someone else will see this elephant in the room and bring this pointless argument about a nonexistent split of opinion to a close. Martin Hogbin (talk) 09:08, 24 April 2011 (UTC)


 * We also see sources which present both solutions in peaceful coexistence, and other sources which present solutions of different nature again. BTW I don't like "all giving the same result—that a player who swaps has a 2/3 chance of winning the car". This is again divisive, not synthesizing, since it begs the question "what probability". Moreover "probability" might mean something conceptually completely different from one reader or solver to another.


 * All solutions tell you that you should switch. They do that in different ways and with different degrees of persuasiveness. ie they give weaker or stronger reasons to switch. They make different assumptions so they are not even comparable. The important distinction we need to make as editors is between solutions (arguments for switching) which can be explained in words of one syllable and without assuming prior technical knowledge, and those which require more.


 * If the conditionalists want to have some conditional probability solutions up front, they should concentrate on making some of such solutions intelligible to people with a phobia for mathematics and with a beginner's English vocabulary. While keeping them interesting and brief at the same time. That's what I tried to do with my patch to Devlin's failed argument. It's published now in reliable sources so it can be used. If it's intelligible to ordinary folk (I think that Glkanter recognised its sanity) then it's a simple solution. And it's a simple solution which provides a lead to a later section using Conditional Probability. Richard Gill (talk)


 * Do either or both of you object to including a paragraph such as the one I suggested above before the solutions? If so, on what grounds? -- Rick Block (talk) 16:18, 24 April 2011 (UTC)
 * There is a consensus to have the simple solutions first without complicating issues. Martin Hogbin (talk) 19:12, 24 April 2011 (UTC)


 * I think such a paragraph is at best superfluous. And I already said what I was unhappy about, in it. Superfluous: Because there are now easy (non technical, one syllable words only) solutions (in reliable sources) which *do* take explicit account of the non-informativeness of the identity of the door opened by the host. (Do I need to spell one out for everyone, yet again?). From the mathematical point of view, such solutions are verbal forms of a conditional probability solution. The article should start simple (=non-technical, accessible for all), should concentrate first on helping the reader experience the mind-shift of realising that switching is smart. The first solutions should be about jolting the reader's mind to look at the problem from different persectives which enable the mind shift: e.g. from the host's perspective; or from the perspective of "information reaching you about what is behind the door you are standing a, (not what is going on at other doors)". Richard Gill (talk) 06:39, 25 April 2011 (UTC)

I am not part of that consensus as I understand the phrase "simple solutions." My position is to have solutions written to be easily understandable near the top. If Rick can write up a conditional probability solution that is simple and easy to understand (this may not be possible), it belongs with the other solutions near the top. We might have to come up with a different name in that case. Perhaps just "solutions."

As for whether Rick's proposed text should appear immediately after the header "Simple solutions" and before the list of simple solutions, in principle adding a short amount of carefully-written introductory text at that spot does not complicate matters and there is no consensus that text should not be placed there.

Alas, the proposed text as it stands does complicate matters where a simpler text would not. Let me take a shot a simplifying it (for discussion only - consider the following a starting point, not a finished proposal).

Replace this:

''There are several approaches to solving the Monty Hall problem all giving the same result—that a player who swaps has a 2/3 chance of winning the car. Most popular sources present solutions analyzing the overall chances of winning by switching as opposed to staying. Another approach to solving the problem, commonly used in mathematical sources, is to examine the probabilities in a single specific case such as the one given in the problem statement where the player has picked door 1 and the host has opened door 3, see Conditional probability solutions, below.''

With this:

''There are several approaches to solving the Monty Hall problem. The approaches below analyze the chances of winning by switching as opposed to staying. see Conditional Probability for approaches that analyze a single specific case.''

(With "Conditional Probability" a link)

Again, I expect someone else to rewrite the above with an improved version, but it is the sort of short introductory text we need there. Guy Macon (talk) 21:02, 24 April 2011 (UTC)
 * I think your statement is extremely complicating. Most people are going to puzzled as to what the difference is between the two cases and will wonder if they have different answers.


 * The whole purpose of my proposal was that editors here should not make a decision about which solutions solve which problem. That is what we had two years argument about and what I am trying to avoid. The subject should be discussed later in a scholarly fashion and according to secondary/tertiary sources.  Martin Hogbin (talk) 21:49, 24 April 2011 (UTC)


 * It sounds like you're saying you want to present a section intended to convince mathematically unsophisticated readers that the answer is 2/3, but deliberately avoid making it clear what the solutions in this section are talking about (!!??). Wikipedia is an encyclopedia, not "Parade Magazine".  We don't have to (and, per WP:TECHNICAL shouldn't) "dumb down" anything.  There are numerous secondary sources that discuss what specific problem the "simple solutions" address, and it is they (not "editors here") who say these solutions address the overall chances of winning by switching vs. staying (note to Guy - this is what is meant here by "simple solutions" - not solutions that avoid "probability jargon").  It is certainly a fact that the overall chances must be the same as the chances in any specific case if the problem is assumed or constrained to be symmetrical (it is also a fact that this is not mentioned by any of the popular sources).  There is a POV (popular among some editors here) that this is obvious and no more than technical nit-picking, so the "simple solutions" are just fine.  However, multiple papers have been published addressing this very issue which means we don't have to resort to asking editors what they think about this.  Given the number of papers published about this that express the POV that the simple solutions address the overall chances (as opposed to, say, the chances if the player picks door 1 and has seen the host open door 3) and that this is more than simple nit-picking, IMO we are obligated (per WP:NPOV) to present this POV fairly and without bias.  I think it is certainly possible to describe in an accessible manner the difference between addressing the overall chances of winning by switching as opposed to the chances of winning by switching in a specific case.  The simplest way (provided by Gillman) is differentiating the chances of switching knowing the host will open a door but before the host opens a door as opposed to the chances after the host has opened a specific door.  I really don't think anyone would fail to understand this.  Please explain why you are so reluctant to do this.  Note that what I'm after here is NOT a disclaimer that the "simple solutions" are wrong, but some introductory text describing what it is these solutions address with forward references to other approaches.  All the approaches (for the problem with standard assumptions) say the chances of winning by switching are 2/3.  I think this should also be mentioned.  Are you thinking saying the simple solutions address the "overall chances" is tantamount to calling them wrong?


 * The text I'd prefer would be something like:


 * There are several approaches to solving the Monty Hall problem. With the standard assumptions these all give the same result—that a player who swaps has a 2/3 chance of winning the car.  The solutions immediately below analyze the overall chances of winning by switching as opposed to staying. Another approach is to examine the probabilities in a single specific case such as the case where the player has picked door 1 and the host has opened door 3, see Conditional probability solutions, below.  Yet another approach is to treat the problem as a 2-player game where the host may be trying either to prevent the player from winning the car or to help the player win the car, see Game theory approach, below.


 * This text is not saying the "simple solutions" are wrong, but it does say what these solutions do (in the opinion of numerous reliable sources). If you object to this text, can you suggest revisions (short of deleting it)?  -- Rick Block (talk) 03:47, 25 April 2011 (UTC)


 * I can see your point. I also see Rick's point. I am (for now) going to take a neutral position on whether the added text, even after I edited it, adds complication.  Would anyone else like to weigh in with an opinion?


 * On a related point, are there really only four editors interested in this, or are there others following but not commenting? If so, is there anything we can do to make it easier for you to contribute?  I would especially like opinions from folks who aren't experts in statistics. Guy Macon (talk) 03:01, 25 April 2011 (UTC)


 * I'm trying to grasp the discussion here, and I do understand the analysis of Rick Block. The comments of Martin Hogbin: elephant in the room, and pointless discussion, doesn't have any conceptual content. Richard Gill, although saying: we have to very precise, speaks about "peaceful coexistence" (whatever that may have to do with the discussion) of solutions, without mentioning the different solution solving different problems. Then he speaks about solutions of different nature, without telling what this difference is. Then Gill seems to object the use of probability, although  the problem typically is a probability puzzle. He continues: all solutions tell you to switch. I do not see any relevance in such a statement, any proper solution should tell so, and something telling you to switch doesn't have to be a solution. He speaks about different ways and with different degrees of persuasiveness, and weaker or stronger reasons. The implied solutions however only consider the relevant probability, and apply to different versions of the problem. He is right about the different assumptions, but then it should made clear in the article which assumptions goes with which solution. Anyway, if something cannot be explained in simple one syllable terms, it is not meant for readers who do not understand otherwise. On the other hand it should not be to difficult to explain about probability before and after.  81.167.87.82 (talk) 10:16, 25 April 2011 (UTC) — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.
 * This has all been discussed here before. If you want to talk more about it I would be happy to do so on the Arguments page. Martin Hogbin (talk) 11:39, 25 April 2011 (UTC)
 * Alas, dear Anon-apparently-from-Stavanger, you are making exactly the same mistake our poor former friend Nijdam, May Jimbo Bless His Soul, used to make. You think that The Truth exists, and can be used to decide what is a Reliable Source. Yet on Wikipedia, just as in a Court of Law, it is the other way round. If you believe you are in possession of The Truth then please generate Reliable Sources which promote and explain your point of view. On Wikipedia, what counts is Consensus, and Collaborative Editing. Anyway, Wikipedia doesn't say what is the MHP and what is the solution. It merely reports what others have put forward as "the problem" and what others have offered as "a solution". Hopefully it does this in a clear and efficient and informative way, so that any reader can form their own opinion, if they so wish, concerning The Truth. Statistics textbooks contain what the authors consider to be "the" solution in the context of the version of the question which they present to their students of their courses. Games theory and optimization textbooks do that too, and their solutions are different. Authors of collections of brain teasers do that too, and their solutions are different yet again. Wikipedia editors must try to represent the multi-cultural phenomenon of MHP in as transparent a way as possible, so anyone can find out what's out there, in all it's diversity. Richard Gill (talk) 11:59, 25 April 2011 (UTC)
 * I just reviewed all of the edits made by 81.167.87.82 (And SPA tagged them; see User talk:81.167.87.82 ) and agree with Richard Gill's assessment. Guy Macon (talk) 13:28, 25 April 2011 (UTC)
 * Well, professor Gill-apparantly-from-Leiden-Holland, you're right about Stavanger (you could have asked me), but the rest of your text I don't understand. I didn't go through all I've written, but I'm prety sure the truth is I didn't speak about The Truth. What's your problem apart from Monty Hall. Above I analyzed that although you say "we have to be precise", you yourself were not very precise. Now you say Wikipedia does not tell what MHP is; doesn't look very precise to me. Wikipedia should be precise in reporting what others consider the MHP. and not be vague about this. 81.167.87.82 (talk) 12:54, 28 April 2011 (UTC) — 81.167.87.82 (talk • contribs) has made few or no other edits outside this topic.
 * Re "apparently-from-Stavanger" and "(you could have asked me)", OK, I am asking. What are yoou two talking about? You don't have to answer - I would not have asked if not for the "you could have asked me" invitation, but is there something that the rest of us might want to know about?  Also, you might want to read WP:SPA for a discussion about single purpose accounts.


 * I stand by my conclusion regarding your comments and WP:TRUTH. You would do well to read Richard Gill's comments about Wikipedia and The Truth again. He is correct. Guy Macon (talk) 16:43, 28 April 2011 (UTC)

What is neutral?
There are and have been for a very long time two POVs presented by editors here about the simple solutions to the MHP. Everyone agrees that they are perfectly good solutions to the unconditional formulation of the problem. One group of editors thinks (to varying degrees) that they are good solutions to the conditional formulation and another group thinks (to varying degrees) that they are bad solutions to the conditional problem. I am in the former group and am perfectly happy to discus the subject in a civil manner on the arguments page with anyone who disagrees with me.

However, my personal opinion and Rick's personal opinion (as stated above) have no place in WP. The issues are complicated and discussed at length in various secondary sources. There is no easy way to summarise the whole subject and we will never agree how to do that, thus my proposal, which is of itself completely neutral and in accordance with WP policy, is to say nothing about the subject until it can be presented in a proper scholarly manner according to good secondary sources.

What we must not do, because it will go on forever and will be OR in any case, is try to find a simple statement that summarises neutrally the views of editors here. It simply cannot be done. Martin Hogbin (talk) 09:33, 25 April 2011 (UTC)


 * In other words being neutral means here leaving out the criticism you dislike.--Kmhkmh (talk) 09:53, 25 April 2011 (UTC)


 * Your comment is unnecessarily hostile and violates WP:GOODFAITH. I have placed a reminder on your talk page concerning Wikipedia's policy of assuming good faith. There is zero evidence that anyone here has any motive other than improving the article. Guy Macon (talk) 11:07, 25 April 2011 (UTC)


 * Not at all Kmhkmh. I am just saying that we cannot put words to the effect that the simple solutions are a bad, or that the simple solutions are good in the article.  What we should put is what secondary sources say about the subject.  This cannot be done in one simple paragraph, it needs to be done properly. Martin Hogbin (talk) 11:33, 25 April 2011 (UTC)


 * I've also noticed that from "just reorganizing the (unchanged) content in a new structure" we apparently moved on to changing the content.--Kmhkmh (talk) 09:53, 25 April 2011 (UTC)


 * Nobody promised to not change the content. The reason we are here is to come up with the best structure and the best content we can. Rick was totally in line when he suggested adding the text above, I was totally in line when I suggested an alternate text, and Martin was totally in line when he suggested that no text is better. That's what building consensus is all about - listening to each others' views and reaching a compromise.


 * Your opinion on the content dispute is welcome. Attacking other editors is not. Do you have a civil comment about Rick's proposed text you would like to share? Guy Macon (talk) 11:07, 25 April 2011 (UTC)


 * I have not removed Rick's paragraph but moved it to a more appropriate place where we can all discuss its content based on what the secondary sources say on the subject rather than our personal opinions. Martin Hogbin (talk) 11:33, 25 April 2011 (UTC)


 * Martin - you have moved it to a place where it's entire function, i.e. alerting the reader that the "simple solutions" that immediately follow are not the only approach to the problem with forward references to other approaches, has been eliminated. You are at least implying you think the wording is not neutral.  What about this wording is NOT neutral?


 * There are several approaches to solving the Monty Hall problem. With the standard assumptions these all give the same result—that a player who swaps has a 2/3 chance of winning the car. The solutions immediately below analyze the overall chances of winning by switching as opposed to staying. Another approach is to examine the probabilities in a single specific case such as the case where the player has picked door 1 and the host has opened door 3, see Conditional probability solutions, below. Yet another approach is to treat the problem as a 2-player game where the host may be trying either to prevent the player from winning the car or to help the player win the car, see Game theory approach, below.


 * Is it the one sentence The solutions immediately below analyze the overall chances of winning by switching as opposed to staying? Or is it both this sentence and the following one Another approach is to examine the probabilities in a single specific case such as the case where the player has picked door 1 and the host has opened door 3, see Conditional probability solutions, below?  Please explain what your objection is. -- Rick Block (talk) 14:49, 25 April 2011 (UTC)


 * The statement as it stands is your opinion that the simple solutions do not solve what we often call here the conditional case. This is too strong a statement and not what reliable secondary sources say about the subject.  There is indeed some criticism of the simple solutions when applied to the 'conditional' case but to say that the simple solutions do not apply to the 'conditional' case is not justified; some people think that they do.
 * Perhaps you could tell me what its purpose is in the position you propose. In its current position its purpose will be to introduce the more complex solutions that follow in a manner supported by reliable secondary sources.  Please explain why the reader must be 'alerted' to anything right at the start.  Martin Hogbin (talk) 16:08, 25 April 2011 (UTC)
 * How about leaving it where Rick wants it for now, editing it in place to make it simpler as per my suggestion above, finishing the rest of the restructuring, and then revisiting the issue again when you are done? It might be that the finished revision addresses Rick's concerns. Or it might be that the finished revision makes it obvious that Ricks concerns are valid.  Or it might be that the finished revision makes it obvious that Ricks concerns are without merit.  I would like to see, in detail how well the revision addresses the concerns and then from that point address the current good-faith content dispute. Guy Macon (talk) 02:25, 26 April 2011 (UTC)


 * @Martin - Where does this statement say anything like what you're claiming? It says the simple solutions "analyze the overall chances of winning by switching as opposed to staying".  Period.  Full stop.  Nothing about what they DON'T do.  You're reading something into this that is simply not there.  Whether such an analysis is sufficient to also conclude the probability is the same in a particular case is the dispute you're talking about differing to a later section, which I'm fine with - but this text is not saying anything about that.  The purpose is what I say above - i.e. to alert the reader that the "simple solutions" that immediately follow are not the only approach to the problem with forward references to other approaches.  If you'd like we could also add a forward reference to the "criticism" section, e.g. something like:


 * There are several approaches to solving the Monty Hall problem. With the standard assumptions these all give the same result—that a player who swaps has a 2/3 chance of winning the car. The solutions immediately below analyze the overall chances of winning by switching as opposed to staying. In the literature there are mixed opinions about this sort of approach, see Criticisms of the simple solutions, below.  Another approach is to examine the probabilities in a single specific case such as the case where the player has picked door 1 and the host has opened door 3, see Conditional probability solutions, below. Yet another approach is to treat the problem as a 2-player game where the host may be trying either to prevent the player from winning the car or to help the player win the car, see Game theory approach, below.


 * I'd bet there's at least a 2 out of 3 chance you won't like this either. To some extent, my point here is that presenting a solution based on conditional probability (or game theory) is distinct from criticizing the simple solutions.  The are multiple types of solutions presented in the literature.  Your outline denies this, and instead focuses the entire article on the "simple solutions" - presenting other approaches as if the only reason they exist is to criticize the simple solutions.  IMO, this is highly POV.  If this article were about any other subject matter I think it would be quite obvious.  Mentally replace MHP with religion, "simple solution" with Catholicism, "conditional solution" with "Buddhism", "game theory approach" with "Islam".  See what I mean?  Presenting a section on Buddhism (conditional solutions) doesn't say Catholicism (simple solutions) is wrong even if there are individual Buddhists who might think so. -- Rick Block (talk) 15:48, 26 April 2011 (UTC)
 * You have not explained why the reader needs 'alerting' to anything. The simple solutions are presented first because they are simple, because the vast majority of sources give them, and because it was in response to these that there was a media furore.


 * The 'conditional' solutions do only exist because some people consider the simple solutions inadequate, they would not be needed otherwise.


 * Some editors here obviously find the adequacy of the simple solutions an interesting subject but far more notable is the fact the most people get the answer wrong and vos Savant got it right. That is not to say that we should not discuss the subject in the article but it is something of a specialist interest. Why not join in discussion in the section below about the criticism of the simple solutions? Martin Hogbin (talk) 17:19, 26 April 2011 (UTC)


 * I have explained, repeatedly, why the reader needs to be alerted to the existence of other approaches. You are simply ignoring my reasons rather than responding to them.


 * No one is arguing that the simple solutions should not come first, so why are you bringing that up? The point is not which solutions are presented first, but (in violation of WP:NPOV) presenting one type of solution as if it is the "main" or "undisputed" approach (with all others presented in a subservient manner).  All I'm asking for to address this concern is a very brief section providing a pointer to different types of solutions.  Why you are so dead set against this puzzles me.  If you could explain what you think the issue is that might help.


 * If by "the vast majority of sources give them" you're saying these solutions deserve more WP:WEIGHT, I disagree. WEIGHT is about more than sheer number of sources.  The simple solutions are repeated uncritically by lots of popular sources, so, of course by sheer number there are more sources presenting these sorts of solutions than any others.  However, since WEIGHT speaks to prominence among reliable sources, not to popularity, I think a far better measure is the prevalence of different solutions in the relevant academic literature.  As has been mentioned before, nearly every introductory probability textbook includes a conditional solution.  Some also include "simple" solutions without taking a stance about which kind of solution is better.  However, some also take a stance and express an explicit preference for a conditional approach.  Among papers on the MHP that discuss the differences between "simple" and conditional solutions, I'd say most (nearly all) express a definite preference for conditional.  From a prominence viewpoint, I think this adds up to approaching the problem with a conditional solution is at least as prominent as approaching the problem with a "simple" solution.  And your claim that conditional solutions "do only exist because some people consider the simple solutions inadequate" is completely unfounded speculation on your part (reflecting nothing other than your own POV).  If you can find a single source that says anything like this I'd be extremely surprised.


 * And again, for the umpteenth time, I'm NOT talking about a section that compares and contrasts simple and conditional solutions - but a section presenting a conditional solution as an alternate approach. I agree comparing and contrasting simple and conditional solutions is likely something of a specialist interest.  Why do you insist on conflating this with simply presenting a solution based on conditional probability? -- Rick Block (talk) 18:14, 27 April 2011 (UTC)

'Criticism of the simple solutions' heading
I have added this heading to Rick's text in what I hope will be seen as a sign of good faith. It is not my intention to try to hide the fact that the simple solutions have been criticised. I think this section should be expanded to include the criticism of the simple solutions based on what the sources say. Martin Hogbin (talk) 16:24, 25 April 2011 (UTC)
 * Again I want to emphasize that both of you are obviously editing and discussing in good faith. I know that in the midst of a heated content dispute one can be fooled into thinking that the other side is being difficult on purpose, but it is usually the case that it looks the same way to the other side. I also want to emphasize that sometimes compromise involves getting part of what you want but not all, and to encourage further efforts to find common ground. Guy Macon (talk) 17:06, 25 April 2011 (UTC)
 * That is what I hope will happen now. I we work on expanding this section to include a full description of what reliable sources say about the simple solutions, which is the only way it should be, it will, I hope, become obvious to all where the section should go. Martin Hogbin (talk) 18:03, 25 April 2011 (UTC)
 * I suggest that we give Martin enough time to reorganize the article and then see what changes, if any, Rick suggests to improve the finished product. When I was very young, I pulled some carrots out of the ground to see how they were growing. Sometimes it is better to wait a while... Guy Macon (talk)
 * I do really want to do any reorganising with a consensus of editors. My plan is for this section to show clearly, openly, and in a scholarly manner, who criticised the simple solutions, what the criticism was and why it was raised, and the various responses to the criticism.


 * The only other major, and possibly contentious, change I propose to make is to move the 'Sources of confusion' up under 'Aids to understanding' and rename it to include the media furore and psychological aspects of the problem. Martin Hogbin (talk) 07:54, 26 April 2011 (UTC)

I have restored the above heading, whilst we consider what might go in it. Criticism of the simple solutions has played a major role in discussion of the MHP here and elsewhere. Martin Hogbin (talk) 08:23, 27 April 2011 (UTC)


 * I have now moved the criticism into this section. perhaps it should stand on its own rather than being a subsection of 'Other solutions'.   All the arguments we have had over the years should now be restricted to this section. Martin Hogbin (talk) 09:26, 27 April 2011 (UTC)

Plan to avoid conflict
I hope the above action makes my plan to avoid conflict clear. The sections are now just what they say they are. The simple solutions section gives simple solutions. It does not say they are 'correct' or 'complete', neither does it say they are 'incomplete' or 'answer the wrong question' it just gives them as they are.

The 'Conditional solutions' section just gives the conditional solutions. It does not assert that they are 'better' or 'necessary' or that they are 'pedantic' or 'not needed', or 'answer academic extensions of the question'.

All the disagreement should now be confined to the criticism section, where we should put what the sources say about the subject rather than what we think. Do others see what I am trying to do? Martin Hogbin (talk) 09:37, 27 April 2011 (UTC)
 * I like the plan. Richard Gill (talk) 12:20, 27 April 2011 (UTC)
 * (I notice that right now, the page includes a number of duplications or near duplications of text in the early sections. I suppose there is still some cleaning up to be done after the initial rearrangement.) Richard Gill (talk) 12:20, 27 April 2011 (UTC)
 * The "duplications or near duplications" are because of WP:LEAD. My concerns about POV have still not been addressed, so I've tagged the article as POV. -- Rick Block (talk) 12:47, 27 April 2011 (UTC)
 * I don't really care which way (i.e. in which chapters and order) the article is organized, but the lead has clearly deteriorated over the last 2 weeks and is imho not providing an accurate summary of the article anymore (see WP:LEAD).--Kmhkmh (talk) 13:04, 27 April 2011 (UTC)


 * I cannot see how presenting all the information clearly can be described as POV. It certainly does not promote my POV. We have a section on criticism of the simple solutions, we can hardly be said to be promoting them.
 * I think we all agree that the lead has deteriorated. That may be partly my fault as I suggested that we leave rewritng the lead until the body is substantially finished.
 * Do you agree that the new arrangement will help end the long running dispute? This is now confined to the criticism section, where we should summarise what the sources say. Martin Hogbin (talk) 14:52, 27 April 2011 (UTC)
 * Well the lead has deteriorated because it was rewritten rather than left alone. I was essentially ok with Tijfo098's version. Whether your new arrangement helps to end the long running dispute is something you'd have to ask Rick, Glopk or others having a strong differing opinion regarding the arrangement. As I've repeatedly stated I have no strong preference regarding order and arrangement. As long as all the content is appropriately covered, I'll go with any arrangement the others can agree on.
 * As far as the lead is concerned now. If indeed we are in agreement, then why not changing it back to Tijfo098's version (or a similar one)--Kmhkmh (talk) 16:41, 27 April 2011 (UTC)
 * Just do it. If someone objects, they can revert and we can discuss the issue. Guy Macon (talk) 18:35, 27 April 2011 (UTC)
 * Why not change the lead back to a version during the Great Argument. It was stable during that period.  Martin Hogbin (talk) 19:55, 27 April 2011 (UTC)


 * There is an excellent reason to not go back to before the arbcom: we do not have a current consensus to do so. We cannot rely upon an alleged consensus from a time when we had now-banned editors muddying the waters with misbehavior, because we cannot know how much of the consensus was simple a matter of those who disagreed being shouted down or giving up in disgust. Guy Macon (talk) 20:10, 16 May 2011 (UTC)

POV tag?
Can anyone else explain to me why Rick has added a POV tag, I really cannot see it. Martin Hogbin (talk) 14:57, 27 April 2011 (UTC)
 * Just POV of Rick Block. Remove it. 95.9.89.70 (talk) 15:34, 27 April 2011 (UTC)
 * No, it is not. At least not as long as there's no proper lead.--Kmhkmh (talk) 16:24, 27 April 2011 (UTC)
 * Just revert it on the basis of there being no consensus that there is a POV problem Everyone please read NPOV dispute and discuss whether the tag should be reverted or whether policy calls for it to remain. Guy Macon (talk) 17:16, 27 April 2011 (UTC)

Rick Block is under a WP:1RR restriction imposed by the arbcon. That puts him on an unequal basis in any content dispute - anyone can revert his changes and he cannot put them back. Because of this, it is important that we all pay careful attention to Rick's objections and if at all possible to try to accommodate him. That being said, right now the consensus appears to clearly favor letting Martin finish his restructuring (with others, including Rick, free to rewrite and revise as it progresses). Then, after everyone has made whatever changes they think are needed (with the usual seeking of consensus) we should carefully examine the finished work, ask Rick exactly what changes would address his concerns, look at both versions, and then seek consensus as to which way we should go. I for one am not prepared to either accept or reject Ricks objections without having two versions in hand, both as polished as possible, that I can compare. I am also not prepared to battle over them to the detriment of improving the article. In the short term, Rick simply is not going to get what he wants. In the longer term, his concerns have not been rejected or accepted, but are instead being deferred so we can get on with the work of improving the article.

For those who did not follow the details of the arbcom, there is another restriction in place, but this one is voluntary. Richard Gill is not only a Wikipedia editor, but is himself a recognized and published reliable source on the topic of the Monty Hall Problem. Because of this he has chosen to not edit the article but instead to restrict himself to commenting on the talk page. This means that we all need to pay careful attention to Richard Gill's comments and suggestions so they don't get lost in the clutter.Guy Macon (talk) 17:16, 27 April 2011 (UTC)
 * You cannot remove a POV-tag simply arguing that there's no agreement regarding it, that's almost self contradictionary as setting a POV obviously requires disagreement to begin with. As far as Martin's restructuring is concerned, that's completely unaffected by the POV tag anyhow. He can do it whether the tag is in the article or not. All the tag does for now is probably attracting reviews from additional editors, which is probably a good thing. So rather than starting a fight over the POV tag, leave it in there until Martin has completed his changes and then do a review whether the tag is still needed or not. If that review yields a(n almost) consensus, the tag may be removed then.


 * Also at some point then we still have to review the FA status of the article.--Kmhkmh (talk) 17:32, 27 April 2011 (UTC)


 * Very good point. I struck my comment above and replaced it, as per your argument. I also put a comment on the FA review page about progress as I see it. Guy Macon (talk) 17:43, 27 April 2011 (UTC)


 * Will somebody please articulate what the NPOV issue is? What statements are in dispute. I don't see a POV issue, so the tag should be removed. Glrx (talk) 18:21, 10 May 2011 (UTC)


 * You don't have to see a NPOV dispute. As long as one editor sees one, the tag stays In this case at leas two editors believe that there is a NPOV dispute. Rick thinks that the current structure violates NPOV. If we change it so Rick's objection goes away, Martin will think it violates NPOV. I am neutral on who is right, but certain that a NPOV dispute exists. So the tag stays. Guy Macon (talk) 20:23, 16 May 2011 (UTC)

NPOV policy
Here are a few quotes from the NPOV policy.

''The editor who adds the tag must address the issues on the talk page, pointing to specific issues that are actionable within the content policies, namely Wikipedia:Neutral point of view, Wikipedia:Verifiability, Wikipedia:No original research and Wikipedia:Biographies of living persons. Simply being of the opinion that a page is not neutral is not sufficient to justify the addition of the tag. Tags should be added as a last resort''.

Perhaps Rick would like to say exactly which policy the article is not in accordance with.

''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''.

The overall presentation clearly is broadly neutral. In fact there is not really anything to be neutral about. There simply are not two great schools of thought in the outside world, that the simple solutions are right and that the simple solutions are wrong.

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.

There is no back and forth dialogue.

''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''.

There is no 'main passage' with other material segregated from it. The only 'narrative' is the section on criticism,

Pay attention to headers, footnotes, or other formatting elements that might unduly favor one point of view, and watch out for structural or stylistic aspects that make it difficult for a reader to fairly and equally assess the credibility of all relevant and related viewpoints.

I can see no problems of this nature here.

''However, when reputable sources contradict one another and are relatively equal in prominence, describe both approaches and work for balance. This involves describing the opposing views clearly, drawing on secondary or tertiary sources that describe the disagreement from a disinterested viewpoint''.

That is exactly what I propose we do here.

when discussing a subject, we should report what people have said about it rather than what is so.

That is exactly what this structure does. It is not up to us to alert people to possible deficiencies in solutions, rather we should say what sources say about them.

''Wikipedia describes disputes. Wikipedia does not engage in disputes'' It is not up to us to take sides here. If there is a dispute we describe it.

[Neutrality] involves describing the opposing views clearly, drawing on secondary or tertiary sources that describe the disagreement from a disinterested viewpoint.

Again that is exactly what I propose, not making up our own 'health warnings' to insert where we see fit.

It is hard to see how we could follow the policy better. Is there anyone apart from Rick who thinks that the body of the article pushes any particular POV? Martin Hogbin (talk) 23:16, 27 April 2011 (UTC)


 * Per Glopk clearly agrees with me, but per  has apparently decided further conversations with you about this are pointless.  Per, I'd guess Handy2000 might agree (but I will not ask him/her to confirm this lest anyone accuse me of WP:Votestacking).  Per  (and various other edits), user:81.167.87.82 apparently agrees with me.


 * User 81.167.87.82 (Anon, apparently from Stavanger) is anonymous and his or her comments are strongly similar to those of another editor whose intransigeance led to him getting a ban for a year. Whether or not this is the same person, I feel strongly that we cannot have anonymous editors influencing what is going on here. Remember: the article has suffered from ownership and from battles of persons with entrenched positions. It needs fresh editors who are fresh enough that they are able to change their minds about desired content as well as form of this article. Editors who hide under double anonimity are not reliable partners in the present atmosphere. Richard Gill (talk) 10:54, 28 April 2011 (UTC)


 * Specific responses (Martin's questions in italics):


 * Perhaps Rick would like to say exactly which policy the article is not in accordance with.


 * I think I've been very clear about this. IMO, the article does not adhere to WP:NPOV (specifically WP:STRUCTURE and WP:WEIGHT) or even WP:TECHNICAL (specifically the "Don't oversimplify" section).


 * The overall presentation clearly is broadly neutral. In fact there is not really anything to be neutral about.  There simply are not two great schools of thought in the outside world, that the simple solutions are right and that the simple solutions are wrong.


 * As I have explained repeatedly, the POV issue is not whether the simple solutions are right or wrong (although there are indeed two schools of thought about this, per Morgan et al., Gillman, Grinstead & Snell, Eisenhauer, etc etc) but that approaching the problem conditionally is at least as prominent as approaching the problem with "simple" solutions - which the current article does not convey accurately or without bias. Instead, it overweights the (contested by numerous sources) "simple solutions" placing them in a position of primacy within the article, with all other solutions relegated to a subservient section titled "Other solutions".  Again - a religion analogy.   Imagine an article on "world religions".  It starts with a section on what "religions" are (analogous to the problem statement in the MHP article).  This section is immediately followed by a very long section on Christianity (that goes into the various Christian sects and reasons why people believe in Christianity and etc etc).  All other religions are then clumped together in a section called "Other religions" (none of which have been referenced anywhere in the article to this point), of course including a section on how other religions differ from Christianity.  This is the precise structure currently exhibited by this article.  Denying that this creates a POV problem seems absurd to me.  Although I would strongly prefer a single solution section with multiple solutions presented as equally valid alternatives, I am more than willing to compromise about this so long as the fact that multiple approaches exist is made clear before any extended sections are presented about any solution approach.


 * There is no 'main passage' with other material segregated from it. The only 'narrative' is the section on criticism,


 * The 'main passage' is clearly the section on "simple solutions", with all other solutions presented in a subservient manner. The advice from WP:NPOV here is "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".  This is exactly what I would prefer.  Again, I'm willing to compromise as mentioned above.


 * I'll repeat, again, that what I'm looking for here is simply a brief section of text before the initial section on "simple solutions" that makes it clear other approaches exist and provides forward references (links) to the appropriate sections. I'm not at all hung up on the specific text, so the claim that I'm insisting the simple solutions be defamed in any way is absolute nonsense.  Anything like the following would be just fine:


 * There are several approaches to solving the Monty Hall problem. With the standard assumptions these all give the same result—that a player who swaps has a 2/3 chance of winning the car. The solutions immediately below [insert absolutely neutral, brief, description of the "simple solutions" here]. [completely optional - include if you want -  In the literature there are mixed opinions about this sort of approach, see Criticisms of the simple solutions, below.] Another approach is [insert absolutely neutral, brief, description of solutions based on conditional probability here], see Conditional probability solutions, below. Yet another approach is [insert absolutely neutral, brief, description of solutions based on game theory here], see Game theory approach, below.


 * Is there anyone apart from Martin who thinks ANY text providing links to the approaches presented in the following sections of the article is "making up our own 'health warnings'"? -- Rick Block (talk) 04:27, 28 April 2011 (UTC)

You are both wrong. Martin Hogbin thinks that he is clearly right and Rick Block clearly wrong. He has some points but has not presented a compelling argument. Rick Block thinks that he is clearly right and Martin Hogbin clearly wrong. He also has some good points but he has not presented a compelling argument either. I am not just acting as if the arguments are equally good/bad in order to seem fair - it really is not clear at this point who is right.

It is clear that both parties think that they are so obviously and clearly right that the other side must agree if only they understand the argument. I think it's also pretty clear by now that that simply is not going to happen. It is also painfully clear that the struggle has over time, decreased the quality of the article.

I am asking you to please stop arguing about this. Martin and Rick, work together to make the article the best it can be while following Martin's vision. The faster we get that done, the faster we get to the point where we start working together to make the article the best it can be while following Rick's vision. I want you to make the two as close as possible in other areas, differing only where you cannot reach agreement. Once we have two versions to evaluate, we will address the underlying content dispute. In case anyone thinks that letting Martin go first is unfair, I remind you that by letting Rick go last it will be his version that stands while we work on resolving the content dispute.

So please drop it for now and spend some time improving the many areas you agree on. OK? Guy Macon (talk) 08:02, 28 April 2011 (UTC)

Proposal for moving forward on resolving longstanding content dispute
As we all know, there is still a longstanding content dispute and a consensus plan for resolving it.

The plan, in case anyone missed it, is this:

See improved version in later section) STEP ONE: Martin, with Rick and everyone else helping, makes this the best article they can following Martin's vision.

STEP TWO: Rick steps in and, with Martin and everyone else helping, makes it the best article he can following his vision, saving Martin's version on a subpage.

STEP THREE: We then all work toward making the two versions as close as possible, differing only where the editors actually disagree about the content.

STEP FOUR: We leave the Rick Block version up while we follow the normal steps for resolving a good-faith disagreement about content.

STEP FIVE: We request a featured article review.

I believe that we have pretty much gone as far as we can go with step one. Please make any final improvements you can, and later this week we will move to step two.

As always, any plan we make as a group is subject to consensus, so if anyone disagrees with the above plan, please feel free to discuss your concerns. Guy Macon (talk) 16:21, 10 May 2011 (UTC)


 * I do not know where you think a consensus for that plan came from. I have not agreed to it and I am not sure that anyone else has. The only thing representing any sort of a consensus was the 'Quick poll on structure' where there was a clear preference for my structure.


 * I have proposed a structure to avoid conflict between Rick and the majority of other editors which I hope most editors will agree to. I think that the great majority of editors agree to the structure as it is, with no 'health warning' (please note that I use this term somewhat lightheartedly to refer to any form of words to the effect that the simple solutions are wrong/incomplete/ answer the wrong question or alerting the reader that there other solutions or ways of tackling the problem.  It is, of course true that there are other solutions but we need not say so because they should all be clearly given later in the article, where they will be clearly visible to all).  I therefore see no need for steps 2-4.


 * If other editors do not see this structure as a way to avoid conflict between Rick and the majority of other editors then please let them say so. It is not a reflection of my POV or that, I suspect, of many others.  We do not need a compromise between this and Rick's POV, this is the compromise. Martin Hogbin (talk) 17:34, 10 May 2011 (UTC)


 * I find it interesting that you did not voice any objection when I laid out the plan - twice - and Rick Block stopped fighting about it and let you edit the page the way you prefer without any criticism or interference. At that time I asked him to give you a fair chance to finish molding the page into your vision. Now I am asking you to do the same thing - give him a chance to show us his finished product without interference.


 * We all know that this is going to go into content dispute resolution and that outside administrators are going to make a binding ruling. Neither of you are going to give up. Each of you is 100% certain that you are not only right, but obviously right, and neither of you can understand how anyone can not see that. Re: "I have proposed a structure to avoid conflict between Rick and the majority of other editors which I hope most editors will agree to", you didn't just propose it.  You implemented it while Rick stepped aside rather than constantly disputing it.  I think it only fair that you give Rick the same opportunity.


 * So we know that this is going to content dispute resolution. Past experience has shown us that nobody involved in the content dispute resolution is willing to wade through lengthy talk page arguments, so we need to show them two versions of the page that differ only in the specific areas where there is a content dispute. That's what my plan does, and I believe that there is a general consensus for doing that, based upon the total lack of objections to it until just now. If I am wrong and there is not support for following the plan, I invite those who oppose to discuss it so we can come up with another plan that will move us forward to content dispute resolution. And of course anyone who thinks I have come up with a good plan is invited to say so as well. Guy Macon (talk) 19:13, 10 May 2011 (UTC)


 * Guy, firstly, I know of no WP process where, ' outside administrators are going to make a binding ruling'. That is not their job.


 * Secondly, this is not a fight between myself and Rick. My proposed structure was supported by a considerable majority. The current version does not represent my POV or that of the majority it is already a compromise in an attempt to satisfy the minority who hold a different POV and who are still asserting ownership over the page.  Rick does not have to approve everything that goes here, that is decided by consensus and we now have one. Martin Hogbin (talk) 09:45, 11 May 2011 (UTC)


 * I never implied that it was a fight between you and Rick, but it is a good-faith content dispute that all efforts so far have failed to resolve, and you and Rick are the most vocal proponents of each side. The current version most certainly does represent your position on the proper way to resolve the longstanding content dispute, and does not represent Ricks position on the proper way to resolve the longstanding content dispute. (My position is to support neither at this time).


 * Here is the mechanism where outside administrators end up making a binding ruling: first we start down the series of content dispute steps (previous efforts focused on editor behavior). A consensus will emerge among the outside editors (editors, not administrators). At this point either you or Rick (depending on who the consensus supports) will either accept defeat, or you will keep arguing and editing against consensus, and we will once again have a behavioral problem that will end up with somebody being topic banned. I would hope that it will not come to that, but all past efforts at mediation have failed.


 * By my count, your claim of being in the majority is not correct. I count two editors (one who is not actively posting) supporting your position, two editors (one who is not actively posting) supporting Rick's position, one editor (myself) who takes a neutral position, and several editors who recently started working on this page and have not expressed an opinion on the longstanding content dispute. So far, that looks like a tie. Do you have another count? I could be wrong.


 * My proposed plan is by my count also a 1:1 tie at the moment, and of course will stay stalled unless more editors weigh in and there is a consensus to follow the plan. Previously it was "proposed by 1, opposed by 0" - a fragile consensus which goes away as soon as one person objects. Now that you have objected, it needs more support before I can consider moving forward with it. I suspect that Rick likes the plan and will say so, which would make it 2:1, but I would not move forward with 2:1. If I see at least 3:1 supporting the plan then I will know I have consensus to move forward with it.  Guy Macon (talk) 17:32, 12 May 2011 (UTC)

Reverted edits
User:Guymacon reverted several of my edits to the lead. The discussion above wants the lead reworked; I reworked it - including removing some comma splices. There are comments that several duplications exist; I deleted one. My edits do not impinge upon the POV debate which seems to revolve about weight to be given to informal and formal proofs. My impression is this article is suffering from people being too scared to make reasonable edits. For example, the lead didn't disclose that the proper strategy should be to switch.

I believe my edits should stand, but there are these ugly statements at the top of this page. Some encourage edits. Others warn of a topic ban. I believe I'm doing the right thing. And now Guymacon blows away all the edits while also stating that they are not all bad. I don't get it. WP:BOLD

May I restore my earlier edits?

Glrx (talk) 20:30, 10 May 2011 (UTC)


 * No need to. I self-reverted 13 minutes before you posted the above with the comment "On second thought, lets leave it as is while we discuss."  You did do the right thing, and I apologize. Guy Macon (talk) 20:50, 10 May 2011 (UTC)


 * OK, thanks. Glrx (talk) 20:56, 10 May 2011 (UTC)
 * I rather still see Tijfo098's lead or similar version reinstated. The current changes didn't really address the deterioration of the lead. In particular the lead needs to mention shortly the disambiguity of the problem and how that allows differing interpretation and an "unsolved academic dispute". Or to put in another way, the current lead is not summarizing the article accurately.--Kmhkmh (talk) 21:31, 10 May 2011 (UTC)


 * I suggest picking one edit that you think was not an improvement and discussing it here, then another, etc. Guy Macon (talk) 08:30, 11 May 2011 (UTC)


 * I dug back in the history to see Tijfo098's lead. It is overlong, buries the obvious, and is too detailed. I'm thinking about adding a part about the assumptions to intro the academic twists. But if somebody else wants to jump in.... Glrx (talk) 14:13, 11 May 2011 (UTC)


 * Looked at Tijfo098's lead. Much prefer Glrx's. Anything can be improved though, so WP:BEBOLD. Guy Macon (talk) 15:56, 11 May 2011 (UTC)


 * Added more to lead to intro acad and variations. It seems overlong.
 * Next step is to move the more detailed problem statement down. I believe the usual reader is more interested in the solution rather than several paragraphs disecting the problem statement.
 * Glrx (talk) 00:57, 12 May 2011 (UTC)

Use of present tense
What appears to be potentially misleading is the use of present tense when we say that the host "opens" the door. One conceivable way of interpreting this is that the host's strategy is to keep mum when the player picks a goat, and to offer to "switch" when he luckily picks a car. In a way this is very similar to language problems over there at drinker paradox. Tkuvho (talk) 09:55, 11 May 2011 (UTC)


 * We cannot change the tense used by Vos Savant. We can only report what she wrote. It the following discussion, which we can change, the sentence "Monty will always open another door and give the contestant an opportunity to switch (the game is played the same way each time)." makes it clear that one of the implied assumptions is that Monty always gives the contestant the choice of switching. Vos Savant's followup columns also made that clear, and yet did not result in overwhelming agreement among academics. Your theory about the problem as stated being misleading and your theory that most academics would get it right if it was described differently are interesting, but lack citations to reliable sources establishing them as being true. Guy Macon (talk) 08:22, 12 May 2011 (UTC)

Is this structure (without a 'health warning') the best way to present this subject?
Please could we have the views of some other editors. My view is that if we do not go with this structure we are doomed forever to argue about the subject and article. What do others think? Martin Hogbin (talk) 17:38, 10 May 2011 (UTC)
 * I think you are presenting a dalse dilemma - the logical fallacy of only considering two alternatives when in fact there are additional options. There are at least four options, not just the two you mention.  We could do things your way. We could do things Rick's way. We could argue indefinitely. Or we could follow the procedures found in Dispute resolution, (resolving content disputes section).  We, of course. already went through the steps in the resolving user conduct disputes section, and in my opinion we no longer have any user misconduct, just a good-faith dispute over content. There is no reason to think that content dispute resolution will not work equally well. Guy Macon (talk) 18:25, 10 May 2011 (UTC)
 * I came to this page from an RfC as part of the dispute resolution process. I would really like to get the views of others on this subject. Martin Hogbin (talk) 18:39, 10 May 2011 (UTC)
 * User conduct dispute resolution process. As far as I can tell, nobody has ever tried going through the steps of content dispute resolution process. Guy Macon (talk) 19:21, 10 May 2011 (UTC)
 * The RfC was really a content dispute, and we have tried mediation. Martin Hogbin (talk) 21:40, 10 May 2011 (UTC)
 * I wasn't going to suggest mediation. Mediation involves guiding the parties into reaching an agreement that can be acceptable to everyone. No such agreement is possible in this case. This isn't a misunderstanding. This isn't a case of user misconduct. This is a pure good-faith content dispute. Nobody has ever tried resolving the underlying content dispute, and instead addressed the now-solved user conduct issues. I am going to try. And the next step is to come up with two versions that differ only in the specific areas where you and Rick have your disagreement. Guy Macon (talk) 08:52, 11 May 2011 (UTC)
 * Guy, I cannot believe that you have written, 'Nobody has ever tried resolving the underlying content dispute'. We have been arguing about this for two years.  It was because absolutely no progress was made on the subject that the dispute became rather heated and went to Arbcom.  If you want to go back to discussing the content that is fine with me but let me stress that my plan was intended to avoid the disagreement by structuring the article in a natural way so that the issues could be dealt with properly and according to sources.


 * I am not sure where you came into this discussion but I would ask you to look through the arbcom workshop and evidence to see all the evidence I put forward about the long standing disagreement. A considerable majority of editors have been against the position asserted by Rick and the others who were sanctioned for page ownership.  I have been trying to find a way forward that does not push my POV but it has been continually blocked by Rick.


 * The majority view is substantially the same as that of myself, Glkanter, and Gerhard, which is that, considering all the circumstances surrounding the puzzle, the simple solutions are perfectly adequate. The Morgan case should be treated as nothing more than an obscure academic extension to the problem.  If you want to restart that argument it is fine with me but please remember that my latest suggestion was intended to circumvent the argument and let us move on.  It does not represent my POV or that of the majority of editors here.   Martin Hogbin (talk) 09:30, 11 May 2011 (UTC)


 * I came in when the arbcom was first featured in Signpost. I have read the entire history of the arbcom case and every diff anyone provided in that case. In many cases I read them several times. I am very familiar with the history. It may well be that the consensus favored your position at that time (it is difficult to determine consensus in the middle of severe user misconduct) but WP:CCC makes it clear that past consensus is not binding. What matters is today's consensus, and I believe that my count on that is accurate.


 * If, as you claim, somebody has tried following the steps listed in Dispute resolution for resolving content disputes, how then do you explain the fact that when I go to Content noticeboard and search for "Monty" there are no results? The steps listed at Dispute resolution for resolving content disputes clearly state that help should be sought at WikiProject Mathematics and Content noticeboard. Instead, everyone followed the steps listed in Dispute resolution for resolving user conduct disputes.  This was the right decision, of course, but now the user misbehavior has been resolved (not a hint of bad behavior by you or anyone else) we can, at last, go through the content dispute resolution steps. As for Arbcom, they could not have made it more clear that they do not rule on content disputes.  They could not have made it more clear that they are on the user conduct dispute path and not on the content dispute path. Perhaps are confused by the fact that some steps (mediation, for example) are on both paths. See diffs provided by Rick Block below.Guy Macon (talk) 17:18, 13 May 2011 (UTC)
 * The last poll on structure came up with 6:2 in favour of the current structure.  This is the consensus structure.  It would be nice if some others would confirm that they are happy with it.  That was why I started this section, to confirm that there is still a consensus, although there is no reason to believe that anyone has changed their mind.


 * As for content dispute resolution, I am happy to participate in any content resolution process although I should point out that the recent failed mediation was about content. Martin Hogbin (talk) 17:50, 13 May 2011 (UTC)


 * Would you be willing to follow my plan for moving forward on resolving the longstanding content dispute if it was modified? Perhaps temporarily holding the two versions on a subpage of this talk page, as I originally suggested?  Content dispute resolution will undoubtedly fail unless I can get you and Rick to agree on a version X and a version Y that are identical except for the longstanding content dispute, with you agreeing everywhere else. That will give everyone a simple and easy to grasp view of exactly what content is disputed so they can comment intelligently. If we do that, then you and Rick can add comments to the two versions that, with a minimum amount of verbiage, present your best arguments for each version. Nobody will be willing to crawl through lengthy talk page discussions; we need to give them something short that everyone agrees describes the longstanding content dispute and the arguments on each side. Guy Macon (talk) 20:42, 13 May 2011 (UTC)
 * There was a 6:2 poll in favour of this structure. Unless any editors say that they have changed their minds, I would say that the consensus it to leave the page as it is.


 * Also let me say again that this structure is an attempt to avoid conflict. It is the compromise.  If you want people to comment on it that is, of course, fine but some may editors want to increase the prominence of the simple solutions, reduce the criticism and describe the Morgan-style solutions as solutions to a different problem, and academic extension of the MHP in which the host is known to choose unevenly.  That would essentially take us straight back to the old arguments. Martin Hogbin (talk) 22:04, 13 May 2011 (UTC)
 * Content disputes have two or more parties who disagree. Does the other party concur that your structure avoids conflict?


 * As has been explained to you before, an old poll taken when the page was in the middle of a firestorm of user misconduct - conduct that led to some users being banned - cannot be used as evidence of current consensus. I believe that my count is accurate. I count two editors (one who is not actively posting) supporting your position, two editors (one who is not actively posting) supporting Rick's position, one editor (myself) who takes a neutral position, and several editors who recently started working on this page and have not expressed an opinion on the longstanding content dispute. Feel free to take a new poll to get a better count, but please stop citing that old poll again and again without addressing my repeated explanations as to why doing so is not valid. Guy Macon (talk) 03:34, 18 May 2011 (UTC)