Talk:Monty Hall problem/Archive 38

The question (Whitaker/vS) does not unambiguously specify that the player initially picks door 1 and the host opens door 3.
There are sources that question the assumption that the Whitaker/vS statement does specify door numbers. The question says 'say door 1' not 'door 1'. It can therefore be interpreted not to mean that the player does actually pick door 1 but that the player just picks a(unspecified) door.

Of course we know that the latter interpretation was exactly what vos Savant did mean, as she has stated that she added the door numbers to Whitaker's original question only to clarify the game process and that they were not intended to specify which doors were actually involved.

I have amended the article to show this fact. Martin Hogbin (talk) 17:35, 19 June 2014 (UTC)
 * As I've noted on the argument page, even if it is a matter of having chosen a door, the player then knows which door this is. And only if the door chosen and the door opened are known the problem becomes the form of the MHP. Nijdam (talk) 18:54, 19 June 2014 (UTC)
 * What you say is not universally agreed amongst sources. Martin Hogbin (talk) 19:00, 19 June 2014 (UTC)
 * I have removed the 'therefore' from the article text to avoid contention about your point. Martin Hogbin (talk) 19:50, 19 June 2014 (UTC)
 * I am not sure [//en.wikipedia.org/w/index.php?title=Monty_Hall_problem&&diff=613584597 your edit] really clarifies why ("and therefore") a conditional probability approach is preferred by many. It is not a matter of the doors being distinguished by their numbers: when Monty opens a door it becomes definitely and unambiguously distinguished as the one that was opened, whether it is numbered or not. Examining the source cited in that sentence (Grinstead and Snell 2006): They assert that the essential difference between their scenarios is whether one takes account of the particular door Monty opens. It is not. The essential difference is whether one takes account of bias on the part of the host in choosing a particular door, or whether, in the words of Seymann ("Comment" on Morgan et al., 1991, The American Statistician, 45:4, p. 287) "the host is to be viewed as nothing more than an agent of chance" rather than "an alternate game in which the host has an ulterior motive". I think the best explanation of why most sources in the field of probability and statistics use the conditional approach is given in Gill (February 2011) ("The Monty Hall Problem is not a probability puzzle (it's a challenge in mathematical modelling)". Statistica Neerlandica 65 (1): 58–71.) To wit, using the frequentist interpretation of statistics one cannot calculate the probability of winning at all without knowing the frequency with which Monty chooses between goats. Once that frequency is stipulated, the revealed goat is treated as a random event rather than a selective disclosure, and conditional probability is the most natural approach. Without that stipulation, a probability of winning can only be calculated by using the Bayesian interpretation of probability or by using first principles (such as that one ought not trust a card shark who selectively shows he is not hiding cards in one of his sleeves). Imagine you are on a game show and the goat is not a statistical sample but is a selective revelation. Unfortunately, I despair of finding an explanation in reliable (academic) sources of why academic sources tend to prefer the frequentist approach that is both understandable for our general readership and that relates specifically to the Monty Hall problem. ~ Ningauble (talk) 22:13, 19 June 2014 (UTC)
 * Due to the edit conflict you missed the fact that, after Nijdam's comment, I decided to remove the word 'therefore' from the article, for the reasons you have just given. There are no sources showing a causative relationship between giving door numbers and the requirement to use a particular form of conditional probability.


 * On the other hand, the are sources which say that saying 'say door 1' does not necessarily mean the same as thing as 'door 1'. In other words that the (W/vS) question does not specify door numbers.  As that is all that my addition now says I would ask Rick to explain why he has removed it.


 * Like you, I agree with Richard's statement "The Monty Hall Problem is not a probability puzzle (it's a challenge in mathematical modelling)". If you are going to use a frequentist approach you should treat the unspecified distributions consistently, as I have been saying for years.  For some reason many seem to find that hard to do. Martin Hogbin (talk) 08:31, 20 June 2014 (UTC)

Rick, I have reverted what you seemed to consider a clarification, because it looses the point being made. The point that I was making was that saying that the Whitaker/vos Savant wording 'say door No. 1' means exactly the same thing as 'door No. 1' is questioned by some sources. It is only one interpretation of the words used. I am happy for you or anyone else to clarify this fact, so long as it is still made in the article. Martin Hogbin (talk) 08:50, 20 June 2014 (UTC)
 * The sources that are being discussed here ("most sources in the field of probability") generally say nothing remotely like "deem the question to specify that the player initially picks door 1 and the host opens door 3". What they actually do is show what the conditional probabilities are "given the contestant initially picks door 1 and the host opens door 3".  Rather than attempt to explain why these sources compute the conditional probabilities in this case, my edit has the article simply say what these sources do.  IMO, they are using the door 1/door 3 combination as a specific example of one of the 6 possible scenarios, which is precisely the meaning of "say door 1".  You apparently disagree with this.  We should keep our opinions out of it and have the article say what the sources actually say.  My edit attempts to do this.  -- Rick Block (talk) 11:25, 20 June 2014 (UTC)
 * Your edit summary gave the impression that you were clarifying not disagreeing.


 * I am perfectly happy to say that most sources in the field of probabilityanswer the question "given the contestant initially picks door 1 and the host opens door 3" but we then need to make clear that this is not the only possible interpretation of the most common statement (W/vS) of the problem. Morgan were the first to make this assumption and, for some reason that is still unexplained, they misquoted the original question to make it unambiguously refer to specific door numbers.  There are reputable sources that criticised them for that interpretation and that criticism should me mentioned in the article.  The original (W/vS) problem statement was unclear as to whether it asked for a solution based on specific door numbers or not, however, Morgan and others chose one particular interpretation of the problem and answered that.  So long as that fact is made clear, we can use whatever form of words you like.  Martin Hogbin (talk) 13:07, 20 June 2014 (UTC)


 * Actually, Selvin was the first to make this assumption (as you often seem to forget). The discussion in "Criticism of the simple solutions" already includes what you seem to be looking for, doesn't it?  Why must it be repeated in the section presenting the conditional solution?  We don't include any criticism of the simple solutions in the section presenting those solutions.  Why the difference? -- Rick Block (talk) 13:31, 20 June 2014 (UTC)
 * You must be talking about a different assumption from me. The assumption I am talking about refers specifically to the Whitaker/vos Savant statement. The ambiguity is about the meaning of the words 'You pick a door, say No. 1'.   Does this mean 'You pick door No. 1' as Morgan assumed or does it mean 'You pick a door', which it could also mean?  Clearly Sevin did not answer this question ten years before it was asked. Martin Hogbin (talk) 14:36, 20 June 2014 (UTC)
 * This question is already addressed in the "Criticism of the simple solutions" section. Why must it be reiterated here?  -- Rick Block (talk) 14:54, 20 June 2014 (UTC)
 * I have looked through that section and I do not see the point that I am making is clearly addressed there. I do not see why when considering the 'conditional' solution we should not make clear reading of the problem statement on which the solution is based. If we do not, those who understand 'You pick a door, say No. 1' to mean 'You pick a door' will be rather puzzled by the solution.


 * 'Deem' is a bit heavyweight, would you prefer 'take'. Martin Hogbin (talk) 16:39, 20 June 2014 (UTC)


 * I think Rick's version of the sentence is marginally better, but in the immediately preceding sentence the article is still saying the same thing that he removed: that the difference between methods lies in whether or not one considers which door was opened. I have problems with the whole paragraph: Firstly, I find it odd that the opening sentence of the section on solutions using conditional probability is not about those solutions at all, but is an allegation that other solutions are not taking something into account. I appreciate that this may be intended to motivate the conditional approach, bit it strikes me as argumentative and overtly pejorative. Secondly, it is misleading to suggest the simple solutions ignore the fact that some definite door is opened. Just because, from a simple perspective, it does not matter which door it is, that does not mean it is being disregarded. Symmetry inherent in the problem is accounted for by symmetry inherent in the solutions. Thirdly, as stated above, the essential difference between approaches lies not in whether Monty opens a particular door but whether, in doing so, Monty's action is regarded as given by some particular probability distribution. Nota bene: The very first thing conditional approaches do, before even getting to solutions, is to restate the original problem by stipulating that probability distribution (or at least representing it by a variable). From a Bayesian perspective or a perspective using symmetry (explicitly or implicitly), this is irrelevant to the original question at hand. From a frequentist perspective, unless one uses symmetry, this is essential to making a definite, closed form solution possible. Therein lies the essential difference. Fourthly, with specific reference to sources cited in the first sentence: (1) Grinstead and Snell (2006) do assert something like the first clause, but not the second. What they demonstrate is that the simple solution does not apply if different frequency distribution is specified in the problem. (2) Carlton (2005) does not say this, or anything remotely like it. Period. Finally, I agree with Rick that "rather than attempt to explain why these sources compute the conditional probabilities" we should "simply say what these sources do"; but with a caveat. One of the things many of them do is to criticize simple solutions as inadequate to the problem as they see it, but I do not think this section is the place for it. We already have entire sections devoted to this under "A second controversy" and "Criticism of the simple solutions", as well as associated discussion insinuated in multiple other sections. This section should focus entirely on the conditional solutions themselves. Therefore, to introduce what is essentially distinctive about the conditional approach, I suggest replacing the opening paragraph of the section on Solutions using conditional probability with something like the following language:
 * Most academic sources in the field of probability show the conditional probability that the car is behind door 1 or 2 given that the host has opened door 3, based on the likelihood for the host to open door 3 after the contestant chooses door 1. Solutions in this section assume the host is equally likely to open any door other than door 1 that contains a goat.
 * I think this introductory paragraph does not really need citations, since the solutions below are (or should be) fully cited, unless we need a source expressly stating that "most" academic sources use conditional probability. I have taken some liberty in using terminology that might be considered specific to Bayes' theorem, but I don't think the vernacular usage misrepresents what is going on with conditional probability generally, and it's a lot more succinct than talking about the frequency distribution for the observed event. I have attempted to indicate, in a neutral and non-prejudicial manner, what is essentially distinctive about the conditional approach. Would this language, or something like it, be acceptable? ~ Ningauble (talk) 18:03, 20 June 2014 (UTC)
 * I think you have missed my point entirely. I was not trying to restart a an old argument but just to clarify the particular interpretation of the Whitaker/vos savant problem statement that is used as the basis for the conditional solutions.  It is simple the point that  'You pick a door, say No. 1'.  could mean 'You pick door No. 1', as all sources who give 'conditional' solution do, or just, 'You pick a door' without specifying a door number.  No sources use this interpretation as the start of a 'conditional' solution. Martin Hogbin (talk) 18:18, 20 June 2014 (UTC)
 * Because thes sources understand that the specific number of the chosen door is of no importance. Any of the three numbers leads to the 'same' problem and the 'same' solution. Nijdam (talk) 18:38, 20 June 2014 (UTC)
 * Martin, by saying "another door, say No. 3", vos Savant indicates that it is a distinguishable door, conveniently labels it, and hints at the symmetry of the problem. (If they were not distinguishable somehow, by number or color or position or something, then Monty could not know which one has the car!) As Gill (February 2011) says, "The wording ' say, Door 1' and ' "say, Door 3' (my [his] italics) emphasize that we know nothing about the behavior of the host, whether in hiding cars or in opening doors." Not everyone will read the same emphasis in this wording, which is subtle, but the point about knowing host behavior is the essential difference in adapting the Whitaker/vos Savant problem for solution by conditional probability: the conditional probability cannot be calculated without stipulating probabilities for this behavior. My proposed wording above removes reference to the simple solutions, which are not what the section is about, and adds reference to the likelihood of host behavior, which is what makes conditional probability work. It is not how the doors are specified, but specifying host behavior with respect to the doors that really matters. Do you find either of these specific changes problematic? Is there something you think ought to be added? ~ Ningauble (talk) 21:45, 20 June 2014 (UTC)
 * You have all still missed my point completely and seem to be trying to re-start old arguments. I am talking only about the wording of the W/vS statement and what those words mean.  'You pick a door, say No. 1' could mean be taken to mean 'You pick door No. 1', or just, 'You pick a door' without specifying a door number.   I am simply pointing out that the w/vS statement does not necessarily specify door numbers, that is all.  What it means to specify or not specify door numbers is another question. Martin Hogbin (talk) 22:00, 20 June 2014 (UTC)
 * What does that have to do with the section entitled "Solutions using conditional probability"? Have I misunderstood what edit, to what section, you were referring in opening this thread? I thought it was [//en.wikipedia.org/w/index.php?title=Monty_Hall_problem&&diff=613584597 this one] (as subsequently amended). ~ Ningauble (talk) 23:41, 20 June 2014 (UTC)
 * Yes, in response to comments here, it now reads, 'take the question to specify that the player initially picks door 1 and the host opens door 3 and use ...'. It is a simple introductory statement of fact showing the starting point for the 'conditional' solutions. It states how the sources that give conditional solutions have understood the W/vS problem statement.  As is not the only possible interpretation of the words vS used (as some sources have made clear) it is sensible and informative to state the basis on which the solutions proceed.  Not to do so would leave confusion in the minds of readers who naturally understood the original W/vS problem not to be specifying door numbers.  I do not think it should be particularly contentious as it is now worded. Martin Hogbin (talk) 13:53, 21 June 2014 (UTC)
 * The difference you seem to be emphasizing is that your wording says "take the question to specify" rather than, or in addition to "show the conditional probability that". Three users have objected to this, and have given cogent reasons. The meaning of "say #3" is indeed open to interpretation, but this section is not about the semantics of whether the door said to be #3 is taken to be door #3, it is about solutions. I agree with the proposition that the conditional solutions are not quite addressing the original Whitaker/vos Savant statement of the problem, but the difference is more substantive than the semantics of what saying "say" means. Indeed, they expressly add a material stipulation about the probability for Monty to open one door or another. I have added this stipulation in my proposed wording bulleted above. I believe it is important to mention this in introducing the conditional solutions because in the [//en.wikipedia.org/w/index.php?title=Monty_Hall_problem&oldid=613810771 current article] it is not part of the problem statement in the lede or, even as a variant, in the section "Origin and full assumptions". The only place it is even alluded to prior to this section is under "A second controversy", yet this restatement of the problem is essential to applying the method of conditional probability. ~ Ningauble (talk) 18:03, 22 June 2014 (UTC)
 * I cannot see what the problem is now. Any solution to a problem must start with the way the problem statement is interpreted.  The conditional solutions all start by interpreting the W/vS statement to say that the player chooses door 1 and the host opens door 3.  I now state this simple and uncontroversial fact in plain language at the start of the section.  Without this, the following solutions may make no sense to readers who interpret the problem statement differently.


 * I am not sure what you are getting at when you say, 'they expressly add a material stipulation about the probability for Monty to open one door or another'. Could you explain please.Martin Hogbin (talk) 23:28, 22 June 2014 (UTC)
 * Regarding your first point, on the principle that "Any solution to a problem must start with the way the problem statement is interpreted" [emphasis added] you would presumably also add prefatory remarks to the simple solutions about how the problem statement is being interpreted. I would oppose that as well. It is best for the section on solutions to just state how the problem is solved, and not insinuate that any of them involve idiosyncratic interpretations of what the problem is asking. Regarding your second point, I refer to stipulating that hiding of the car and showing of the goat are done at random with equal distribution. I just realized the 'section on Origin and full assumptions' does mentioned this is "typically presumed". Some solutions presume this implicitly, or explicitly presume it as an uninformative prior, or even derive it from symmetry, and some do not; but conditional solutions expressly stipulate that it is part of the problem definition. ~ Ningauble (talk) 21:17, 23 June 2014 (UTC)
 * I understand your answer to the second point now and generally agree with you. I think, though, that it is reasonable to assume that symmetry is an implicit part of the simple solutions because they are simple solutions and most people can intuitively see the symmetry with respect to door number (I am not sure if there is a source which says this).  The 'conditional' solutions, on the other hand, generally claim to be more rigorous and complete and therefore I think they have a duty to explicitly state all assumptions or deductions made. I think the 'conditional' solutions presented in most sources fail in this and other respects.


 * Regarding the original point of this section, I cannot see how any solution can be presented without first stating what is considered to be the problem to be solved. I changed my wording from 'deem' to 'take' specifically to avoid insinuating that the 'conditional' solutions involve an idiosyncratic interpretation of the problem. If you can think of a way to make that clearer it would be fine with me, so long as it is clear that it is not the only reasonable interpretation. Martin Hogbin (talk) 22:05, 23 June 2014 (UTC)
 * Your proposed wording is not really introducing rigor that one may feel the sources lack (which would be original research). Rather, it is raising a caveat that the solution may not address the question at hand as some understand it. You, yourself, have strenuously opposed prefacing simple solutions with caveats to the effect that they may not address the question at hand as some understand it; and your proposal for "no disclaimers that they do not solve the right problem" was adopted in a mediated request for comments with wide participation. I think symmetry is called for, adopting of neutral point of view by not doing unto conditional solutions what you would not have done to simple solutions. ~ Ningauble (talk) 14:57, 24 June 2014 (UTC)


 * Whereas the proposed wording bulleted in my post of 18:03, 20 June 2014 above has not received any endorsements, the proposal is hereby withdrawn. Please do not speak of it again. I apologize if my alternative wording involved too many changes not directly related to the subject of this thread. I may raise individual points (such as bogus citations in the first sentence) separately, after the dust has settled. ~ Ningauble (talk) 14:59, 24 June 2014 (UTC)

Martin's revision has not received any endorsements, and has been opposed by three contributors. Is it time to lay this to rest by reverting to the previous version of the sentence, or should additional opinions be solicited with a "request for comments"? ~ Ningauble (talk) 15:02, 24 June 2014 (UTC)
 * I really do not see this as a continuation of the old battle between the simple and the 'conditional' solutions and I am not sure that three editors do oppose my clarification as it is now.


 * Sources that give the simple solutions present them as solutions to the problem as stated, regardless of whether specific doors are considered to be involved. The disclaimers that we agreed not to have were based on sources that gave other solutions or editors' opinions here.


 * Sources which propose the 'conditional' solutions themselves say that the problem being addresses is one in which door 1 is initially chosen etc. Including that fact in the article is not a disclaimer but part of the solution as it is stated  in the sources that give those solutions. Why should we miss it out? To do so misrepresents what the sources say.


 * I suggest that we ask for opinions of those watching this page or maybe ask a few regular editors before going to a formal RfC. Martin Hogbin (talk) 15:35, 24 June 2014 (UTC)


 * Referring to [//en.wikipedia.org/w/index.php?title=Monty_Hall_problem&diff=613810771&oldid=613633700 this diff] specifically:
 * The sentence already says "given the contestant initially picks door 1 and the host opens door 3"
 * You added "take the question to specify that the player initially picks door 1 and the host opens door 3"
 * It was already perfectly clear, by saying they solve it given this information, that they take it as given. Your addition suggests doubt that they take a meaning which is, in your words, "not the only reasonable interpretation", and is otherwise completely redundant. If you want to cite sources that say what is given is open to interpretation (not just sources that take it as given) then please do it in another section discussing ambiguities or criticizing solutions, not in the section presenting solutions. This is how debatable points about the simple solutions are being handled, and it accords well with the policy at WP:MNA: "there is probably not a good reason to discuss some assumption on a given page [section], if that assumption is best discussed in depth on some other page [section]." [italics and interpolations are mine] ~ Ningauble (talk) 19:54, 24 June 2014 (UTC)
 * The problem with 'given', as it was, is that it is in Wikipedia's voice; it appears to say that WP takes the door numbers as a given not the sources.


 * I accept that my original wording was critical of the sources but I am not so sure that is true now. Surely, to state the correct basis (as stated in the sources themselves) on which the sources base their solutions shows them in a better light than giving a basis on which it could be argued that the sources are wrong. This is not an attempt by me to cast doubt on the correctness of the sources, it was meant to be an essential 'brief, unobtrusive pointer', without which the 'conditional' sources might be accused of making unjustified claims.  Martin Hogbin (talk) 13:03, 25 June 2014 (UTC)


 * Solving "the probability of A given B", written "P(A|B)", is what conditional probability does, in the voice of those who do it. Your "pointer" explains nothing: Saying that B is a matter of choice ("they take") or opinion ("they deem") only raises doubt, in Wikipedia's voice. We already describe ambiguities in the problem statement elsewhere. It is not neutral to single out one method of solution as being particularly susceptible to vagaries of interpretation. Given that your change was reverted and you re-reverted (notwithstanding minor alteration) without obtaining consensus, would you at least be willing, in the spirit of WP:BRD, to undo [//en.wikipedia.org/w/index.php?title=Monty_Hall_problem&diff=613810771&oldid=613633700 this edit] pending endorsement by anybody of your addition? ~ Ningauble (talk) 17:58, 25 June 2014 (UTC)
 * I find it a great pity that you cannot assume a little more good faith on my part rather than assuming that everything I do is aimed at pushing my own POV. I am not singling out a method as being particularly susceptible to vagaries of interpretation, even the sources themselves are not doing that, they are simply explaining how they arrived at their conclusions.


 * However, I am willing to revert if you will agree to consider changing the wording to address the points that I have made but without the negative connotations that you see. Martin Hogbin (talk) 18:10, 25 June 2014 (UTC)


 * I have reverted anyway and then made a small change elsewhere. Is that OK? Martin Hogbin (talk) 15:39, 26 June 2014 (UTC)
 * Thank you for reverting. I do not doubt your good intentions for making the article clear, accurate, and informative. I just think you sometimes go about it the wrong way. I too sometimes go down the wrong track. I have also undone an intermediate edit[//en.wikipedia.org/w/index.php?title=Monty_Hall_problem&diff=614494742], apparently intended to reduce redundancy, that ended up with no referent for "those two". It is exceedingly difficult to find suitable words to address the underlying assumptions of the conditional probability approach. Don't be too disappointed about not resolving what years of discussion here have not been able to resolve. ~ Ningauble (talk) 14:25, 27 June 2014 (UTC)
 * The small change that I made, from 'use' to 'calculate' does resolve the issue to some degree in that 'calculate', to me at least, suggests some degree of choice by those doing the calculation ('choose to calculate' would have been better for me but I have refrained from starting a new battle), while 'use' suggests a greater degree of pre-existence. As you, or any others, have not commented on that change I presume that it is OK so we can leave it at that. Martin Hogbin (talk) 15:46, 27 June 2014 (UTC)

Simulations are not Solutions
What is the purpose of the Simulations section? Why is it in solutions? Surely it can be pulled out or moved and trimmed to slim things down and make this article a little easier to read? SPACKlick (talk) 11:36, 26 June 2014 (UTC)
 * I agree 110% that simulations are not solutions. The point of having the section, somewhere, is that many people remain unconvinced by any explanation of the counterintuitive result until they see it demonstrated empirically. In the past, contributors felt it was important to provide this information immediately after the simple solutions, before getting into more technical analysis, in the hope that readers would "get it" before their eyes glaze over. However, the simple and conditional approaches were moved under a single heading in order to avoid giving the impression that they do not have equal legitimacy. There ought to be a better resolution than this awkwardly illogical compromise . ~ Ningauble (talk) 13:49, 26 June 2014 (UTC)
 * I also agree that simulations are not solutions and should be put where they best fit.


 * Ninguable, are you referring to the RfC conclusion above. I do not see why sticking to that should result in an awkwardly illogical compromise. Martin Hogbin (talk) 09:14, 27 June 2014 (UTC)


 * A better form for the article might be


 * 1) Lede
 * 2) The Paradox
 * 3) Origins and History [Combining two sections]
 * 4) Assumptions (where the need for assumptions is explained)
 * 5) Solutions
 * 6) Simple Solutions
 * 7) Solutions using conditional probability
 * 8) Criticisms of the simple solutions [Combine refining the simple solution]
 * 9) Conditional Probability
 * 10) Bayes Theorm
 * 11) Direct Calculation
 * 12) Strategic Dominance
 * 13) Solutions by Simulation
 * 14) Sources of Confusion (I'm not overly sure this section needs to exist, it could be put in assumptions and criticisms without losing much)
 * 15) Variants
 * 16) Host Protocols
 * 17) N-Doors
 * 18) Quantum Version
 * 19) Recent Discussion
 * 20) See Also
 * 21) References
 * 22) External Links

What do people think? SPACKlick (talk) 13:37, 1 July 2014 (UTC)

The collapse of the Monty Hall Paradox
The following may be cobbled together referring reliable sources, but until now it is not explicitly stated there. Nevertheless it may not harm if we mention it meanwhile:

A, explaining the 2/3 solution of the original MHP of vos Savant to B:

Now the host must open another door with a goat.

B: Why must he do that?

But on top of collapsing MHP in this way:

The "standard" members of the 2/3 fraction didn't ever say, that the host "must" open another door with a goat. They only said (say?), that the host opens another door with a goat. (B: So what?) So concealing the reason for the 2/3 solution. But missing the crucial rule makes it not only difficult, but impossible to explain the 2/3 solution, because it is simply wrong. That's the main reason for the fog publicists produced around the problem, and for the alleged "great paradox".

Because of the confusion in their own heads they are not able to give the following simple explanation for the problem when formulated with the crucial rule:

If the contestant requests the host to open door 2 or door 3 with a goat, he wins in two of three cases: If the host opens door 2, the contestant selects door 3, and if the host opens door 3, he selects door 2.--Albtal (talk) 11:35, 1 July 2014 (UTC)


 * Not to be too blunt ut have you read this policy?SPACKlick (talk) 13:57, 1 July 2014 (UTC)


 * I only wished to mention it here on the talk page. Therefore no harm.--Albtal (talk) 19:44, 1 July 2014 (UTC)


 * The famous paradox (switching doors doubles the chance to win the prize): who did ever read the "original" paradox presented by Selvin (three "boxes" instead of curtains or doors, just one box contains the prize, the other two boxes are empty). It's exactly the same intended paradox. What do the sources explicitly say about this famous clean paradox? Leonard Mlodinow says that it is the "role of the Host" that generates the paradox, in intentionally showing a nut (avoiding to ever show the prize), in order to offer a swap to the second closed door. This extremely biased role of the host – who never will show the car – initiates and secures the striking paradox. In the first line the article must present the clean famous paradox. Sources in heaps. All other "variants" that do not generate the famous paradox can be shown later on in the article. Gerhardvalentin (talk) 16:37, 1 July 2014 (UTC)

Another Reference
''Jan C. Schuller, Journal of Applied Statistics '' "ABSTRACT The classic solution of the Monty Hall problem tacitly assumes that, after the candidate made his/her first choice, the host always allows the candidate to switch doors after he/she showed to the candidate a losing door, not initially chosen by the candidate. In view of actual TV shows, it seems a more credible assumption that the host will or will not allow switching. Under this assumption, possible strategies for the candidate are discussed, with respect to a minimax solution of the problem. In conclusion, the classic solution does not necessarily provide a good guidance for a candidate on a game show. It is discussed that the popularity of the problem is due to its incompleteness." IVLeeg (talk) 19:34, 1 July 2014 (UTC)


 * Concerning actual TV shows, also Morgan et al. and Georgii argue in the same manner (see German Wikipedia ("Ziegenproblem", section "Frequentistische Sicht")). There is also an article in DIE ZEIT which refers to Schuller's paper (online).--Albtal (talk) 20:52, 1 July 2014 (UTC)

''Cognition and Chance: The Psychology of Probabilistic Reasoning Raymond Nickerson, Psychology Press, 2004. '' "Problems like those described by Vos Savant illustrate in a particularly compelling way that probabilistic reasoning can be tricky, even for people well versed in probability theory. What makes these problems, and many others that could be considered, difficult may have a complex answer.  However, I want to argue that a major contributing factor is the fact that statements of the problems often are incomplete or ambiguous in the sense that they admit of more than one interpretation, depending on assumptions that the reader may make about the situation described, possibly without realizing he or she is making them, in the process of deriving an answer.  [We should] insist that the assumptions on which probabilities are calculated be made explicit.  Two probabilistically knowledgeable persons working on the same assumptions should not produce different answers to the same problem"IVLeeg (talk) 20:40, 1 July 2014 (UTC)

Reliable Sourcing - Econ Working Papers?
Some of the references in this article are to things that seem questionable for meeting the Wikipedia standards for Reliable Sources. For example, there are numerous references to this item:


 * Mueser, Peter R. & Granberg, Donald (May 1999). "The Monty Hall Dilemma Revisited - Understanding the Interaction of Problem Definition and Decision Making". University of Missouri. Working Paper 99-06. Retrieved 10 June 2010.

What exactly is a "Working Paper"? Is it peer reviewed? The link is to a web hosting site called Econ papers, which doesn't mention anything about peer review, etc. It seems to be just an archive where people can upload materials. Am I under-estimating the quality of this reference? In the interests of shortening and streamlining the article, would it be worthwhile to review the entire list of references, and delete all those that don't meet Wikipedia standards for Reliable Sources?IVLeeg (talk) 04:35, 28 June 2014 (UTC)
 * Reliable sources do not necessarily have to be formally peer reviewed (which only exists journal papers anyhow). The reliability can also be derived from general reviews, the reputation of the publisher and most important the reputation of the author. The latter case seems to apply here as Mueser and Granberg are established academics.
 * Before you attempt to "shorten & streamline" the article, I recommend to check with other editors whether they agree and see a need for that. The optimal presentation of content has a long and contentious history, which already went all the way up to the arbitration committee in the past (see archives for that as well).--Kmhkmh (talk) 12:56, 28 June 2014 (UTC)


 * 1984--Albtal (talk) 20:25, 28 June 2014 (UTC)


 * Yes, I will certainly not attempt to shorten and streamline (or make any other significant changes to) the article on my own. My purpose in starting this discussion section was just to prompt a re-examination of the article from the standpoint of Wikipedia policies against synthesis of primary sources, especially in a field where one can find primary sources that say every possible thing.  It seems to me this article makes rather heavy use of primary sources (doesn't it?), such as numerous psychology papers in which individual "researchers" express opinions (or interpretations of highly dubious psychological "experiments") on a subject in which they are arguably not really experts.


 * It isn't even clear what academic field represents the experts on the Monty Hall Problem: statistician, mathematician, psychologist, evolutionary biologist, neurologist, sociologist, carnival worker, game show host, game theorist, economist, financial advisor, actuary, newspaper columnist, philosopher... do ALL of these people qualify as experts on the Monty Hall Problem? You say that Mueser and Granberg are established academics, but surely not every "established academic" in any field qualifies as a recognized expert on the Monty Hall Problem.  Wikipedia policy reminds us that expertise in one field doesn't transfer to other fields.  This is especially true in a subject about which the literature contains so much disagreement.  And when we make the article rely on unpublished "working papers" (whatever that is) posted on the web, written by people who are not necessarily qualified as experts on the Monty Hall Problem (regardless of their academic standing in their own field of study), it seems to me we are in danger of violating Wikipedia editorial policies.


 * I'm under no illusions that the article can be much improved, because so much nonsense about the subject has been published, and there's a scarcity of mature secondary literature. Still, I think some of the stupidity of the existing article could be trimmed by the scrupulous application of Wikipedia editorial policies.  Just my two cents.IVLeeg (talk) 14:27, 28 June 2014 (UTC)
 * Well to be a(n academic) expert on Monty Python (in the widest sense), you need be an established academic and have worked/published on the MHP. Since there various aspect to the MHP and different contexts/perspectives from which you can approach the topic, there is no single academic field from which the experts arise. Most of them however are in fields statistics, math, economics and psychology.


 * As far as mature secondary literature is concerned, there is certainly Jason Rosenhouse's book (covering the subject up to 2008). There are also a few other decent secondary or tertiary sources. However the difference between secondary and primary is not always the most important, but the reputation of the publication (journal) and the author matters as well. Meaning a publication in a "high impact" reputable journal might be a better source than a secondary publication (like a book) by hardly known author with a hardly known publisher. The real problem however might be that there is no real consensus in academic literature about how the problem should be treated and which (implicit) assumptions are "obvious", "justified", "likely" or "intended by the original author" or more generally about the way the ambiguities of the original formulations should be adressed


 * Lastly the lack of consensus in literature and some sloppy treatment in it are not the only problem. Another (probably worse) is an never ending amount of WP editors who feel the need to tell the world their take on MHP and what they consider as the "true" solution or "optimal" treatment.--Kmhkmh (talk) 22:37, 28 June 2014 (UTC)


 * Yes, the fact that we have only Rosenhouse and "a few other (unnamed) decent" ones is why I say there is a scarcity of mature secondary literature. Regarding expertise, let's take a specific example: Mueser is an economist, specializing in labor economics, and as far as I know, the only thing he has written about the Monty Hall problem is the unpublished "Working Paper" referenced in the article, that is accessible as a pdf file in an online repository.  Does he have any work actually published on the Monty Hall Problem?  If not, then he doesn't meet your criteria, correct?  (I don't mean to pick on Mueser, I just selected that reference at random and checked to see who he is and what he has published.  The results for this sample of one are not encouraging.)


 * I agree that the lack of clear consensus in the literature is a real problem (closely related to the fact that there are no clear experts on this highly inter-disciplinary subject), which makes the subject problematic for Wikipedia, because Wikipedia articles are supposed to mostly represent the consensus in the literature. Lacking such consensus, the article just becomes a rambling hodgepodge of different views, synthesized to reflect the beliefs of the latest batch of editors to have their way with the article.  And, as you say, there are always new editors coming along with their own views, who would synthesize the presentation differently.  Whether any particular batch of editors have better ideas than any other batch of editors about how the article should be written is obviously subject to debate.  But we all know there is no ownership of Wikipedia articles, so we all have to realize that our edits are only temporary. ("Look on my works, ye mighty, and despair!")IVLeeg (talk) 01:56, 29 June 2014 (UTC)


 * In the section Sources of confusion we can read:
 * Indeed, if a player believes that sticking and switching are equally successful and therefore equally often decides to switch as to stay, they will win 50% of the time, reinforcing their original belief. Missing the unequal chances of those two doors, and in not considering that (1/3+2/3) / 2 gives a chance of 50%, similar to "the little green woman" example (Marc C. Steinbach, 2000).
 * But this is not what Steinbach says. In fact he writes:


 * From the words of the host and the sight of the goat alone the contestant cannot know, whether any rule holds - and all the more not which rule. [...] There only remains coin flipping: so the contestant - independently of the behaviour of the host - gets the right door with probability 1/2.


 * (In the first problem formulation in Germany in 1991 in the newspaper DIE ZEIT Gero von Randow added to the wording of Marilyn vos Savant the words Now I'll show you something, which the host says just before opening the door with a goat. This version has been widespread in the German language area, always together with the claim of a 2/3 solution. For example, Krauss & Atmaca in 2004 start their paper with this version, and they invite the reader to think about the answer. Krauss & Atmaca consider the solution 2/3 as the single correct one, and finally add a footnote where the reader can find the problem description which really has a 2/3 solution. The reader in this footnote is also informed, that almost all test persons assume these rules implicitly (Krauss, S. & Atmaca, S. (2004). Wie man Schülern Einsicht in schwierige stochastische Probleme vermitteln kann. Eine Fallstudie über das "Drei-Türen-Problem". Unterrichtswissenschaft, 1, 38-57). Steinbach writes, that the words Now I'll show you something from the viewpoint of the contestant are nonsense (unsinnig), if he by the rules of the game anyway expects to get shown a goat.)--Albtal (talk) 11:35, 29 June 2014 (UTC)


 * In the section History we can read:


 * The Parade column and its response received considerable attention in the press, including a front page story in the New York Times in which Monty Hall himself was interviewed. (Tierney 1991) Hall appeared to understand the problem, giving the reporter a demonstration with car keys and explaining how actual game play on Let's Make a Deal differed from the rules of the puzzle.


 * What is not reported about this story in the New York Times is the following:


 * Monty Hall: If the host is required to open a door all the time and offer you a switch, then you should take the switch. But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood.


 * Martin Gardner: The problem is not well-formed, unless it makes clear that the host must always open an empty door and offer the switch. Otherwise, if the host is malevolent, he may open another door only when it's to his advantage to let the player switch, and the probability of being right by switching could be as low as zero. The ambiguity could be eliminated if the host promised ahead of time to open another door and then offer a switch.


 * Persi Diaconis:The strict argument would be that the question cannot be answered without knowing the motivation of the host.--Albtal (talk) 13:59, 29 June 2014 (UTC)


 * Yes, those references make it very clear that the question as stated in Parade (for example) does not have the unambiguous answer of "switch", because it does not specify that the host must always open an empty door and offer the switch. In fact, given the "game show" scenario as described in Parade, there's actually a perfectly rational basis for concluding that the player has no clear reason to either switch or not switch.  This has been shown rigorously by math and confirmed by simulations (by people who are not brain dead).  Of course, whether most readers answer on this rational basis, or by some erroneous reasoning, and whether most readers would give the right answer if they had been presented with a scenario in which the host is constrained to offer a switch, are separate questions.  But the current article has obviously been written with the objective of convincing the reader that the Parade question has the unambiguous answer "switch", giving 2/3 probability of winning, so all the well-sourced (and incontrovertible) explanations of why this is not true are suppressed - on the grounds that they will just "confuse" the reader.IVLeeg (talk) 15:21, 29 June 2014 (UTC)
 * Nothing has been suppressed and this has all been discussed at length before. The question as asked in Parade leaves so much out that it is insoluble without making some assumptions.  We know from reliable sources that most people, on hearing the Parade question, make the same assumptions which are that the host always opens an unchosen door to reveal a goat, and always offers the swap. On that basis it is best to swap.


 * It is no secret that with different assumptions, the answer may be different and there is certainly no conspiracy to hide this fact. Martin Hogbin (talk) 16:03, 29 June 2014 (UTC)


 * The standard assumptions are discussed in the section, 'Origin and full assumptions'. An alternative scenario in which the host does not always have to open a door hiding a goat is discussed in the section, 'The little green woman'. Martin Hogbin (talk) 16:10, 29 June 2014 (UTC)


 * Martin, I disagree strongly with every single thing you've said there, except perhaps for the statement that this has all been discussed before, which I can easily believe. I should probably just leave it at that, but just for fun, let me comment on your assertions one at a time:


 * Nothing has been suppressed... That's not true.  All the references noted above, that present a very different view of the subject, are absent from the article. See, for example, the suppressed quote from Monty Hall, where he points out that it doesn't really make sense to assume what you say everyone assumes.


 * The question as asked in Parade leaves so much out that it is insoluble without making some assumptions... Well, every puzzle question leaves out information, and asks us to answer based on the information that is provided.  That's what makes it a puzzle.  Based on the info in the Parade statement (and as Monty Hall himself was smart enough to realize, as he explained in the suppressed quote) there really is no clear preference to switch or not switch.  I realize you would characterize this by saying "the question as stated has too many ambiguities to have a definite answer", but of course that's just another way of saying that the answer is there is no clear advantage to switching or not switching based on the given information.


 * We know from reliable sources that most people, on hearing the Parade question, make the same assumptions which are that the host always opens an unchosen door to reveal a goat, and always offers the swap. You're on shaky ground here.  First, the whole point of this Talk section is that the "reliable sources" quoted in this article are not actually reliable sources according to Wikipedia guidelines. (See above.) Second, most people are not mathematically sophisticated enough to formulate any word problem mathematically, nor do they understand the significance of things like prior versus posterior information, etc., If you were to explain to those people the rational basis that confirms the correctness of their intuitive answer (as explained by Monty Hall, for example), do you doubt that they would say "Yes, that's exactly what I was thinking"?


 * On that basis it is best to swap. The basis you're talking about is a very unnatural and artificial assumption, which is not implied by the actual Parade problem statement.  So, if a person is answering "on that basis", then that's their first mistake.  It's true that many people will answer incorrectly a question with the unnatural assumption explicitly imposed, but that's just because (again) most people are terrible at solving word problems.


 * It is no secret that with different assumptions, the answer may be different and there is certainly no conspiracy to hide this fact. I think you completely miss the point.  The correct answer to the question posed in Parade, taking into account all the ambiguities, is that there is no clear preference for switching or not switching (as explained by Monty Hall in the quote that has been omitted from the article).  If a different question is posed, stating that the host must reveal a goat and offer a switch, then the correct answer to THIS alternative question is that there is an advantage (2/3) to switching.  Your position, as I understand it, is essentially that most people get the right answer to the Parade question but for the wrong reason, i.e., you believe they make two errors that happen to cancel out.  First, you believe they wrongly assume that the host must always reveal a goat and offer a switch (even though this is not stated in the question, and is in fact a highly unnatural and artificial assumption).  Second, you believe they incorrectly think that, on this basis, there is no advantage to switch.  Putting those two mistakes together, they arrive at the right answer (i.e., no clear advantage to switch).  That is possible, and it's certainly true that most people would give that same answer even if the unnatural assumption was stipulated.  But it's also possible that people instinctively get the right answer to the Parade question for more or less the right reason, even though they lack the mathematical sophistication to be able to formulate or articulate their reasoning.


 * The standard assumptions are discussed in the section, 'Origin and full assumptions'. An alternative scenario in which the host does not always have to open a door hiding a goat is discussed in the section, 'The little green woman'.  No, you're completely off the rails with this comment.  The little green woman scenario has absolutely nothing to do with what we are talking about.  The only issue here is that the Parade question does not specify that the host must always reveal a goat and offer a switch.  No one is talking about little green women coming from outer space in the middle of the game.  That section is a perfect example of the poor quality of the existing article, and the POV that is being pushed in the article.IVLeeg (talk) 18:21, 29 June 2014 (UTC)


 * In the section Recent discussion we can read:


 * Accounts of the Hungarian mathematician Paul Erdős's first encounter of the problem can be found in 'The Man Who Loved Only Numbers' and Vazsonyi 1999; like so many others, Erdős initially got it wrong.


 * But what did he get wrong?


 * We can read it in the given sources. In the German Wikipedia ("Ziegenproblem") there is a summary of this strange story in the section Paul Erdös und das Ziegenproblem. Erdös accepted the solution when Ron Graham (by Paul Hoffmann) told him the reason: The key to the Monty Hall problem is knowing ahead of time that the host is always going to give you the chance to pick another door. That's part of the rules of the game which you have to figure into your thinking.--Albtal (talk) 06:07, 30 June 2014 (UTC)


 * Yes, the story told by Vazsonyi is a crock. I once asked Ron Graham about the incident, and he said Vazsonyi (an aeronautical engineer) did not explain the problem to Erdos correctly.  According to Vazsonyi, the host would actually CRY if the contestant won, so he was motivated to make the contestant lose (and so clearly he would offer a switch only if the contestant's initial guess was correct).  As you said, once the requirement for the host to always show a goat and offer a switch was added to the problem statement, Erdos understood the answer to THAT question easily.  But it wasn't the question he was asked by Vazsonyi (nor was it the question the Parade readers were asked by Vos Savant).  Amusingly, Vazsonyi imagined that he was the only person to have ever heard of Thomas Bayes, whose image just "flashed into his mind" when he heard of the Monty Hall problem.  Here's how Vazsonyi explains why he, a lowly engineer, was smarter than his boyhood friend, the world class mathematician and probability expert and renowned problem solver Paul Erdos:


 * "The goat controversy stirred up a question that baffled me. How could somebody like Erdos not know anything about Bayes' theorem, which I consider to be a bridge between the world of pure math and the real world?  So I did some research.  I checked my books and found little, almost nothing, about Bayes' theorem.  If it was discussed, only an incomprehensible formula was included, that no one could understand, much less apply to anything useful.  No wonder the PhDs were jumping on Vos Savant. They didn't know anything about Bayes, and most of them couldn't care less."


 * So, according to our Reliable Source for this slanderous story, Paul Erdos (and other PhDs in statistics and probability) never heard of Bayes theorem! That's the level of thinking that informs this entire Wikipedia article.IVLeeg (talk) 13:09, 30 June 2014 (UTC)


 * In the section Recent discussion we can read:


 * The Monty Hall problem appears in the film 21 (Bloch 2008).


 * When the teacher there states the problem and a student proposes to switch the teacher says (analogously): But it may be that the host is spoofing you.--Albtal (talk) 06:23, 30 June 2014 (UTC)

The rules of the game
Some people who get the answer wrong try to defend their error by pointing to ambiguities in the game rules which, of course, do exist. However, we have good sources showing that most people assume the 'standard' game rules, as given in this article, and that, even when the intended rules are very clearly spelled out to them, they still get the answer wrong.

Vos Savant said, 'Virtually all of my critics understood the intended scenario. I personally read three thousand letters... Very few raised questions about ambiguity...'.

Krauss and Wang have a section, entitled 'Are there possible effects of incomplete information?' in their paper (which analyses the results from subjects who were asked to answer the problem) and after a detailed analysis of the wrong answers given and the reasons for them given by the subjects, conclude that, 'In short people seem to struggle not with the ambiguity of the standard version's assumptions, but with the mathematical structure of the scenario'.

Despite the quotes given by IVLeeg and Albtal above there are no sources which contradict the fact that most people do understand the problem as intended and that assumption must therefore be the main thrust of this article. Of course, we should, and do, have some discussion of alternative interpretations of the problem and their effects on the answer. Martin Hogbin (talk) 08:08, 30 June 2014 (UTC)


 * Some people who in 1990/1991 read the problem and the claim of the 2/3 solution wrote letters to Marilyn vos Savant and Gero von Randow with the hint that the 2/3 solution only holds if the host is committed to open a not chosen door with a goat. But these letters have not been published. (See MvS and Ein Auto und zwei Ziegen). In the German newspaper DIE ZEIT again in 2004 an article about the Monty Hall Problem ("Ziegenproblem") was published (online). The occasion had been studies of Krauss et al. with pupils. (Others may describe and evaluate these studies.) The author of the article in DIE ZEIT added a further phrase to the formulation of Gero von Randow so that now we had: The host says mysteriously: 'Now I'll show you something!' Krauss is cited there with: The crucial trick surely was that we hadn't analysed the task setting, for this had been done for ten years. Shortly after this article in DIE ZEIT and contradicting letters Gero von Randow himself wrote a further angry article. Neither in the first nor in the second article there is any remark about ambiguities in the formulation of the problem. (How could it be there? The author did not at all think that the host is committed to open a not chosen door with a goat, but that he now acts surprisingly and 'mysteriously'.)


 * Again, like 1991, letters who criticised that the crucial rule was missing, were not published.


 * Gero von Randow himself in his book about the problem describes in a separate section My error that he first saw a 2/3 solution in a problem, which excluded the crucial rule.


 * There is plenty of evidence that the 2/3 solution is claimed without assuming the critical rule. Devlin even supposes a shock for those who learn that it is possible that the host acts just like the host in vos Savant's problem, but that the solution is 1/2.


 * And a final remark seems appropriate: That someone (or 90% of a set of pupils or so) gives a wrong answer to a problem, has nothing to do with the publication of a problem together with a claim and explanations for the solution, and a subsequent "storm of protests" in the form of letters. The problem with the critical rule is very simple, and the explanation of the solution, too. But the "explanation" is very difficult, if the reason for the solution is kept concealed. So the MHP is not a "paradox" but a joke.--Albtal (talk) 12:16, 30 June 2014 (UTC)


 * Hogbin's argument is why the assumptions section needs to be beefier than the section on 15 different ways of deriving the solution GIVEN the assumptions. If the Assumptions section, probably split into its own section, showed 1) What the standard assumptions were (some subset of the 9 above) and 2) That most people when given the ambiguous version of the question infer those and exactly those assumptions, then the whole argument about ambiguity would be essentially moot trivia. Can people start collating what the sources say about standard assumptions somewhere on this talk page so we can collaboratively draft a working section on assumptions and fix this article?SPACKlick (talk) 12:19, 30 June 2014 (UTC)


 * This has all been discussed before but the W/vS version of the question was asked in Parade, a popular, general interest magazine so the question we should be looking to answer is 'What would the average person understand the game rules to be?', not what interpretations of the rules are possible. Under plausible game rules (for example that the host only offers the swap when the player has originally chosen the car) the correct solution could be 'never swap'. In addition we should bear in mind what Seymann says in his 'Comment' on the paper by Morgan et al, 'Without a clear understanding of the precise intent of the questioner, there can be no single correct solution to any problem'. In this case we do know the 'precise intent of the questioner' because vos Savant has told us. I some ways, therefore, we do not need any sources other than vos Savant to show that the question was generally understood by the intended audience to have the meaning that was the precise intent of the questioner.


 * I am Being Bold and adding a larger note about assumptions to the lede, as I think it needs to be there to make the article reflect the literature. SPACKlick (talk) 14:52, 2 July 2014 (UTC)


 * If we need more sources I think it would be best to consider specific unstated assumptions about the rules:

The host must open a door
I do not think anyone has ever argued about this one; the rest of the game would make little sense if the host did not open any door.
 * That said, I agree it needs to be stated as an explicit assumption because if the host has a choice in opening the door, the fact that they do gives information and therefore could vary the success rate of switching between 1 and 0. However, I'm struggling to find sources that explicitly discuss this assumption. SPACKlick (talk) 10:03, 1 July 2014 (UTC)
 * Http://www.softouch.on.ca/kb/data/Scan-130122-0004.pdf here Monty hall and Professor Perci Diaconis discuss the need for certain assumptions and the precise effect of ignoring them. SPACKlick (talk) 10:37, 1 July 2014 (UTC)


 * "After the 20 trials at the dining room table, the problem also captured Mr. Hall's imagination. He picked up a copy of Ms. vos Savant's original column, read it carefully, saw a loophole and then suggested more trials. On the first, the contestant picked Door 1. "That's too bad," Mr. Hall said, opening Door 1. "You've won a goat." "But you didn't open another door yet or give me a chance to switch." "Where does it say I have to let you switch every time? I'm the master of the show. Here, try it again.""
 * "On the second trial, the contestant again picked Door 1. Mr. Hall opened Door 3, revealing a goat. The contestant was about to switch to Door 2 when Mr. Hall pulled out a roll of bills. "You're sure you want Door No. 2?" he asked. "Before I show you what's behind that door, I will give you $3,000 in cash not to switch to it.""
 * "Was Mr. Hall cheating? Not according to the rules of the show, because he did have the option of not offering the switch, and he usually did not offer it. And although Mr. Hall might have been violating the spirit of Ms. vos Savant's problem, he was not violating its letter. Dr. Diaconis and Mr. Gardner both noticed the same loophole when they compared Ms. vos Savant's wording of the problem with the versions they had analyzed in their articles."
 * "Still, because of the ambiguity in the wording, it is impossible to solve the problem as stated through mathematical reasoning. "The strict argument," Dr. Diaconis said, "would be that the question cannot be answered without knowing the motivation of the host." Which means, of course, that the only person who can answer this version of the Monty Hall Problem is Monty Hall himself. Here is what should be the last word on the subject: "If the host is required to open a door all the time and offer you a switch, then you should take the switch," he said. "But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood. "My only advice is, if you can get me to offer you $5,000 not to open the door, take the money and go home.""


 * Comments by Monty Hall about this subject are irrelevant. Vos Savant does not mention Monty Hall at all; that name comes from Selvin.  Monty states how we might run a hypothetical version of his show.  Martin Hogbin (talk) 18:31, 2 July 2014 (UTC)

The host cannot open the door originally chosen by the player
I think it is only Albtal here who has ever questioned this rule.


 * Interesting to read that I am "questioning" a rule. But thank you for mentioning here Albtal and not Gardner, Georgii, Monty Hall etc.--Albtal (talk) 16:44, 30 June 2014 (UTC)
 * Can you provide quotes or links to support what you say. Martin Hogbin (talk) 17:36, 30 June 2014 (UTC)
 * I might add that the possibility that the host might open the door chosen by the player is plain crazy. You pick a door and the host opens it to reveal a car and asks if to want to swap.  Seems like a bad idea to me.  If he reveals a goat there is not a lot to lose by swapping.  Martin Hogbin (talk) 17:48, 30 June 2014 (UTC)


 * Please ask Gardner, Georgii, Monty Hall etc. to provide quotes.--Albtal (talk) 23:22, 30 June 2014 (UTC)


 * I would suggest that other people have discussed the possibility that the host opens the players chosen door, I definitely remember reading something about "if the host can open the players chosen door then that choice was a red herring and added nothing to the problem of choosing between two doors". That said, can't find the source to hand. SPACKlick (talk) 10:05, 1 July 2014 (UTC)


 * [This Paper] has this section which refers to the choice of unchosen door


 * THE HOST'S PROTOCOL AND THE GENERAL SOLUTION
 * The statement of the problem provided above specifies almost nothing about the host's protocol. Most important, it does not specify whether the host considers the position of the prize in deciding whether to open a particular door or whether he is required to open a door under all circumstances. Instead, the statement of the problem specifies one instance of his behavior that is consistent with a large number of possible protocols. The answer that the contestant has a 2/3 chance of winning if she switches follows if we assume that the host's protocol requires that he reveal an incorrect, unchosen door regardless of whether the contestant's initial choice was correct, and that, if two doors are available, the host opens them with equal probability. In addition, it assumes that the prize is behind each door with equal probability. None of these assumptions is specified in the problem as it is commonly stated.


 * which I think sums up the standard assumptions pretty nicely, although it doesn't justify treating them as assumed SPACKlick (talk) 11:22, 1 July 2014 (UTC)
 * It follows up by saying "However, this solution is not necessarily implied by the narrative as it is commonly stated. The problem statement specifies only that the host has opened door 3 to reveal a goat, implying no more than p1p13 + p2p23 > 0." SPACKlick (talk) 11:25, 1 July 2014 (UTC)
 * I cannot find anywhere in that source where it mentions the possibility that the host will open the door originally chosen by the player. Martin Hogbin (talk) 19:04, 2 July 2014 (UTC)

The host can never reveal the car
The most discussed rule, since it changes the probability of winning by switching from 1/2 to 2/3. Mentioned by vos Savant
 * [This paper]


 * "First, she stated more explicitly than previously that it is crucial to assume that Monty always opens a losing door. Relaxing that assumption in any way changes the problem completely."
 * "If we now assume that you always choose door one and that Monty only opens goat-concealing doors, we see than in two of the three scenarios you win by switching."
 * "And there was a subtle shift from the correspondent's initial question, in which the host always opens door three, to the listing of the scenarios given by vos Savant, in which it was assumed only that the host always opens a goat-concealing door."
 * "Of the letters from the general public, 92% are against my answer; and of letters from universities, 65% are against my answer." Nevertheless, vos Savant does not back down, and for good reason, as, given a certain assumption, her answer is correct. Her methods of proof, however, are not."
 * "The problem has all it can handle getting itself stated with su�cient clarity to be mathematically tractable. Embedding it in a skit only makes it harder to parse, and typically, as in the two examples above, leads to important assumptions not being spelled out. So knock it o�ff!"


 * It also refers to Alan Bohl, Matthew Liberatore, Robert Nydick, \A Tale of Two Goats...And a Car, or the Importance of Assumptions in Problem Solutions," The Journal of Recreational Mathematics, Vol. 27, No. 1, 1995, pp. 1-9. Which I haven't seen but sounds like an important reference for this section SPACKlick (talk) 11:04, 1 July 2014 (UTC)

The host must always offer the option to switch
Adams raised this question. It can change the probability of winning by switching from 0 to 1. Martin Hogbin (talk) 14:40, 30 June 2014 (UTC)

Quote: "Some people who get the answer wrong try to defend their error by pointing to ambiguities in the game rules which, of course, do exist." That's quite true. People such as Vos Savant who erroneously say the answer is "switch" try to defend their error by claiming that the problem statement is ambiguous enough so that it's possible to imagine that the host is required to always reveal a goat and offer a switch, even though this is not stipulated in the question (and is not an apriori reasonable requirement). However, this doesn't really exonerate them, because it is also possible (and more reasonable) to surmise from the stated question that the host offers a switch only if the contestant initially guesses correctly, and hence he should never switch in this circumstance. Therefore, the unambiguous answer to the question as stated (i.e., with no requirement for the host to offer a switch) is that there is no clear advantage to switch or not switch.

Quote: "The W/vS version of the question was asked in Parade, a popular, general interest magazine so the question we should be looking to answer is 'What would the average person understand the game rules to be?', not what interpretations of the rules are possible." Well, we can consider whatever questions we like, but clearly one of the most important questions is "What is the correct answer to the question that was actually asked?" The correct answer is that there is no clear advantage to switch or not switch. We can then go on to ask other questions, such as "What question did Vos Savant intend to ask?", and "What is the correct answer to the question that Vos Savant intended to ask?" The answers to these questions are that Vos Savant intended to specify that the host must always reveal a goat and offer a switch, and in this case you should obviously switch (2/3 advantage).

Then we could ask (as Martin suggests) "What would the average person understand the game rules to be, when presented with the Parade question?" But this is a very difficult question to answer credibly, not least because the average person doesn't grasp the distinction between prior and posterior information, and isn't mathematically sophisticated enough to complain when someone tells them that the only reasonable assumption is that the host must always show a goat and offer a switch. And yet, instinctively, the average person is influenced by the feeling (without being able to articulate the reason in mathematical terms) that this is not the only reasonable assumption, and in fact that it isn't a reasonable assumption at all - certainly not how any real game show would be played. Street smarts: If your opponent is trying to entice you to switch, it's probably not a good idea for you to switch. And indeed, in the absence of any requirement for the host to always offer the switch, this is a perfectly valid reason for not switching. The fact that the average person isn't able to explain this reasoning, and lets himself be misled into focusing on a completely different question (that differs from the original question in a way that he doesn't really understand) just means that most people don't have much mathematical sophistication. They intuit more than they understand.

We also have the unknown factor of the "10000 letters", 1000 of them being supposedly from PhDs (in what subjects?), not average persons. So we could ask "Does the average PhD mathematician think the host is required to always reveal a goat, even if the question does not stipulate any such requirement?" Now, Vos Savant doesn't say what their PhDs were in, nor do we have any independent knowledge of these letters, and if they really were PhDs. (Has any actual PhD in mathematics actually been identified as one of the original respondents to Vos Savant?) Obviously since Vos Savant was embroiled in a polemical dispute, her summary of the contents of these 10000 letters (1000 from "PhDs") subject to question. But even Vos Savant concedes that SOME of the letters complained about the lack of specification of the "always offer a switch" rule.IVLeeg (talk) 18:19, 30 June 2014 (UTC)
 * Do you have any sources supporting your opinion? Martin Hogbin (talk) 19:05, 30 June 2014 (UTC)


 * Marilyn vos Savant, The Power of Logical Thinking, 1996:


 * p. 14: And a very few percentage of readers feel convinced that the furor is resulting from people not realizing that the host is opening a loosing door on purpose. (But they haven't read my mail! The great majority of people understand the conditions perfectly.)


 * p. 15: Very few raised questions about ambiguity, and the letters actually published in the column were not among these few. [...] When I read the original question as it was sent by my reader, I felt it didn't emphasize enough that the host always opens a door with a goat behind it, so I added that to the answer to make sure everyone understood.


 * Unfortunaltely the MHP was going around the world without these "answers" of MvS in her subsequent columns, stimulating other publicists to decorate MHP with supplements going just in the "other direction" than the crucial but missing rule would do (see above: Gero von Randow, DIE ZEIT, Krauss et al. - and Steinbach ("nonsense")).--Albtal (talk) 10:18, 1 July 2014 (UTC)


 * Can I suggest that we move this discussion to the /Arguments page and leave this section for listing sources. Martin Hogbin (talk) 19:16, 30 June 2014 (UTC)


 * 1984--Albtal (talk) 23:52, 30 June 2014 (UTC)


 * My sources are the same as your sources, except that I would discard some of your sources as not meeting the Wikipedia criteria for Reliable Sources, as discussed in the section I started above, called "Reliable Sourcing - Working Papers?". Aside from those sources that should be deleted per Wikipedia editorial policy, the issue here really isn't about differing sources, it's about crafting a rational and balanced account of the subject from a bunch of mostly primary and sometimes highly polemical, self-serving, and inter-disciplinary sources.  To understand how this affects the article, notice that up above you offered the opinion that "The question we should be looking to answer is 'What would the average person understand the game rules to be?'", and I offered the counter-opinion that we should be looking to answer the question "'What is the correct answer to the question that was actually asked?'"  Neither you nor I have provided sources for these opinions, but the whole structure of the article is based on what we think is the main question to be answered by the article.  The lack of a mature secondary literature on this subject leaves this open to editorial judgment.


 * Every source agrees that, from the Parade problem statement, we cannot infer a clear advantage to switch or not switch. In my editorial judgment, this should be the first point made in the article.  Then we can consider variations on this problem, such as by adding the stipulation that the host must reveal a goat and offer a switch.  We can then, if we wish, go on to review the dubious speculations about what is going on in the minds of the people (some "average" and some with PhDs in unspecified disciplines) and chickens when confronted with different variations of this question.  So this would be one way to structure the article.  I know that this would make your head explode, because you have a very different editorial judgment about what this article is about, what questions it should be answering, and so on.  The point is, these are not differences over sourcing (aside from the unreliable ones that should be dumped, as noted above), they are differences in editorial judgment, which are unavoidable due to the immature state of the literature on this subject.IVLeeg (talk) 20:33, 30 June 2014 (UTC)


 * IVLeeg, not every source agrees that. Some don't because all sources assume certain base facts not explicitly stated. For example, that the location of the car is concealed from the player isn't explicitly stated. The article should deal with the question that most of the sources deal with, which seems to be the problem including the standard assumptions listed above. This is because, as shown in literature, most people assume, when given this kind of probability question, that facts given fall to standard repetitive patterns. So the host opening a door and offering a switch, in this case, is interpreted as a thing that happens, not a thing that happened this time. The real question this article should answer isn't "the one Vos Savant intended" or "The one that is strictly interpreted by a direct reading of what Vos Savant wrote and only what she wrote" but "The Question Vos Savant Asked" which is affected by how the question is interpreted by the reader. SPACKlick (talk) 10:15, 1 July 2014 (UTC)


 * This Paper has this paragraph on switching SPACKlick (talk) 11:27, 1 July 2014 (UTC)
 * The assumption that the host follows a protocol in which he must offer the contestant the chance to switch is also critical. If the host were to choose to open a door and provide the opportunity for the contestant to switch when, and only when, the contestant's original choice was correct, any contestant who chose to switch when given the opportunity would lose every time. Such a strategy for the host amounts to setting p10=0, and p20=p30=1, while maintaining pii=0, i=1, 2, 3


 * Do we really need a source for that? It must surely count as a routine calculation. If the host only offers the switch when the player has originally chosen the car switching always looses; if the host only offers the switch when the player has originally chosen a goat switching always wins.  Have a source if you like but that must surely count as a routine calculation.


 * I agree with your statement '"The Question Vos Savant Asked" which is affected by how the question is interpreted by the reader', but point out again that we do know exactly how that question was interpreted by the readers of 'Parade' because vos Savant tells us exactly that. If it had been a question in a statistics exam it could be argued that it should have been interpreted differently. Also I agree that it is true that the normal understanding when a story of this kind is told is that the actions (where there is not an obvious degree of choice involved) are to be considered as a formula for repetititions.


 * The reason that this problem (with the standard rules) is so infamous is that, even when the problem is clearly and unambiguously stated most people still get the answer wrong. Not only that but, even when the correct answer is explained to them, they still do not believe it. It is probably the hardest simple probability puzzle known, even if all the assumptions are made perfectly clear.  With non-standard rules just we have a rather uninteresting problem with an obvious answer.  There are also interesting and more complicated technical details that should be, and are, discussed but mainly this article should be about a simple problem that fools almost everyone. Martin Hogbin (talk) 16:27, 1 July 2014 (UTC)


 * I think you might benefit from learning about some other probability puzzles, such as the Boy or Girl paradox. Pay careful attention to how just tiny differences in the wording of the question lead to different conditionals and different answers.  This should help you see that (1) the Monty Hall problem is not nearly as unique as you suppose, and (2) many puzzles of this type rely on a STRICT interpretation of the exact stated restrictions or lack thereof.  Also, you greatly overstate the number of people who have difficulty with the simple host-restricted puzzle.  We know at least 35% of the educated letter writers to Marilyn had no trouble solving that trivial little problem, and we've debunked the Erdos anecdote here, explaining that he had no trouble seeing the answer once it was stated that the host must always show a goat and offer a switch.IVLeeg (talk) 18:35, 1 July 2014 (UTC)


 * I know perfectly well that, 'just tiny differences in the wording of the question lead to different conditionals and different answers', but that is not what is in dispute. What you seem to be claiming is that most people who get an answer of something other than 2/3 do so because they interpret the question in a different way from that which vS intended.  I have given you two sources that clearly state that this is not the case.  You have just quoted sources which say in general that one must be careful when interpreting probability questions.  That fact is also not in dispute.  However, you have still provided no sources clearly stating that most people misunderstand what the intended question is for the MHP.


 * The article does state the assumptions on which the answer of 2/3 is based. If you think that point is not made clearly enough I am sure no one would have a problem with making it clearer Martin Hogbin (talk) 22:32, 1 July 2014 (UTC)


 * You say "I know perfectly well that, 'just tiny differences in the wording of the question lead to different conditionals and different answers', but that is not what is in dispute." I disagree.  This is precisely what's in dispute.  The problem as stated in Parade has an answer, which is that no clear advantage to switch or not switch can be inferred from that question.  A slight change in the wording, to impose the peculiar and unnatural stipulation (completely unrealistic for any real game show) that the host is not an active element in the game but is simply an automaton that always reveals a goat and offers a switch, leads to a question with a different answer (switch with 2/3 advantage).  When dealing with puzzle questions like this, it simply makes no sense to claim that the correct answer to Question B is to be regarded as the correct answer to Question A, because many people when asked Question A allegedly believe they've been asked Question B.


 * I disagree with your interpretation, the problem as stated in parade states nothing about the hosts behaviour therefore in order to work out if switching is advantageous you need to consider the average across all host protocols. Any host protocol can be inverted, however some inversions reflect about 50/50 ranging from 0-1 (The host offers the switch only when and every time you pick the car <-> The host offers the choice only when and every time you pick the goat) whereas other host protocols reflect about 2/3 ranging from 0.5 to 1. The average across all protocols, unsourced and by personal estimation, would be slightly higher than 1/2 and therefore switching would be advantageous. That said, you are again sticking with a naive reading. The article ALREADY states that there are some assumptions, necessary for the interpretation of the question, which were intended but not stated. The article outlines these (in a section which isn't brilliant but can be fixed) SPACKlick (talk) 11:18, 2 July 2014 (UTC) Etide to add: In fact I'm going to run a simulation where 1) Whether a switch is offered is random 2) Whether the host can open the contestants door is random 3) Whether the host can open the car door is random in order to account for all possible host protocols, so I can see the final figure for switching v sticking given player picked 1, host opened 3 it was goat. SPACKlick (talk) 13:38, 2 July 2014 (UTC)


 * Having run the simulation for 1,000,000 [All percentages in square brackets are approximate] trials where the contestant picked door 1, The switch was offered in 500,073[50%] In Which the host.
 * 209,048 [42%] trials opened door 2, 83,273 [17%] opened door 1 and 207,606 [42%] opened door 3 Of which the result was
 * 34,822 [17%] the Car was revealed and 172,834 [83%] the goat was revealed of which
 * 75,449 [43.66%] Sticking Won, 97,335 [56.33%] Switching Won. Which gives a win by switch, Given pick 1 and reveal goat behind 3, of 56.333%. Meaning even with a naive reading where nothing can be assumed about the host's strategy, Switching still beats sticking by over 1/8.
 * While I was typing this the 10,000,000 trial run finished. The percentages were the same except the final one which was [56.11%] to [43.89%] in favour of switching. Switching wins in the naive literalistic reading and in the interpretation of the question intended and generally inferred. Switching wins, so no more 50/50 nonsense. SPACKlick (talk) 14:30, 2 July 2014 (UTC)


 * SPACKlick, you say "the problem as stated in parade states nothing about the hosts behaviour therefore in order to work out if switching is advantageous you need to consider the average across all host protocols". Well, you're on the right track (excellent), but you actually need to consider the suitably WEIGHTED average of all the possible host protocols (i.e., weighted according to their probabilities, not just the simple average), and you haven't even begun to consider all possible protocols, let alone quantified the suitable weights.  (In fact, you haven't even mentioned the most likely and reasonable protocol for an actual game show host... think about it.)  Also, even though your commendable effort to simulate the situation is hopelessly flawed (because you haven't even considered the most likely protocols, let alone all possible protocols, nor assigned them their appropriate weights), you HAVE succeeded in demolishing your own position, because you've concluded that the correct answer to the question that was asked in Parade (with no requirement that the host MUST show a goat and switch) is NOT the "standard" answer (2/3 advantage to switch), you say it is 0.56 advantage to switch.  So, according to your analysis, Marilyn did not give the correct answer to the question that her reader actually asked.  She gave the correct answer to a different question.  We're in agreement about this.  Now you just need to think a little more about Monty's actual motivations, to discover the most likely protocol (which you haven't even considered yet).  You see, the question without the stipulation that Monty must always reveal a goat and offer a switch is actually FAR more interesting than the trivial little grade-school probability puzzle that results from the question with that stipulation.IVLeeg (talk) 15:59, 2 July 2014 (UTC)


 * I’m getting rather tired of you asserting what other people have said when you’ve made it quite clear you couldn’t infer your way out of a paper bag IVLeef. Nobody is on the right track if they take a naïve reading of a casually displayed problem. Simple as that, one may reasonably make standard assumptions for that problem as you attempt to do later, by giving any prior probability to the host protocols. “The most likely protocol for a game show host” assumes certain desires on behalf of the game show host (and by the way, assuming the game show host doesn’t want to give the prize away is a mistake of fact). I haven’t shown anything about “THE CORRECT ANSWER TO THE QUESTION ASKED IN PARADE” I’ve answered the naïve interpretation of that question, which WASN’T the question being asked. Marilyn gave, as most of us have given, the correct answer to the question asked. The issue you are having is that you cannot comprehend that the assumptions detailed, that the host always opens a non picked, non car containing door and offers the switch are part of the question. They are implicit not explicit but they are there. Your interpretation is the same as answering the question “If I toss a fair coin 3 times what are the odds I get exactly 2 heads?” with “approximately 0 because you didn’t specify the coin had a head side”. The assumptions are there, in the wording of the question and the context of the asker and audience.


 * You say "I’ve answered the naïve interpretation of that question, which WASN’T the question being asked." Why do you think the literal interpretation is naïve?  In any real game show the host would not always reveal a goat and offer a switch, it depends on his mood (as Monty put it), so why is it naïve to think that Monty exercise some choice?  Wouldn't it be more naïve to think he doesn't exercise any choice?


 * You say "The assumptions are there, in the wording of the question and the context of the asker and audience." Could you show me where, in the wording of the question, or in the context of the asker (Craig Whittaker) and audience, we can find the requirement that the host must always reveal a goat and offer a switch?


 * Bear in mind that Marilyn herself admitted that she didn't think the question made this sufficiently clear, and every other reputable source (from Martin Gardner on down) acknowledges that the Parade statement does not impose this restriction, and we also have Monty Hall telling us that such a restriction would certainly not apply in a real game show. So I think you face a difficult task to successfully argue against all these people, and make the case that, in fact, the restriction "is there".


 * The example you gave actually disproves your own position, because the assumption that a coin has no head side is unreasonable and should not be made, just as the assumption that Monty always reveals a goat and offers a switch is unreasonable and should not be made. Some things can reasonably be taken for granted, and some cannot.  It's perfectly fine to take for granted that the contestant wants the prize and not the goat. But it's not perfectly fine to assume that Monty must always reveal a goat and offer a switch.IVLeeg (talk) 17:33, 2 July 2014 (UTC)


 * As for calling the question with the standard assumptions trivial, it STILL fools most people most of the time. You are missing the entire point of the history of this problem in the English speaking world and have STILL yet to say what you feel should be changed in the article. If you aren’t talking about the page, please take it to the arguments page. This page is for discussing the article which your rants are disrupting. SPACKlick (talk) 16:20, 2 July 2014 (UTC)


 * You say: "What you seem to be claiming is that most people who get an answer of something other than 2/3 do so because they interpret the question in a different way from that which vS intended."  It's undoubtedly true that SOME of the people who give the right answer to the Parade question (i.e., that we can't infer any benefit to switching) do so because they are interpreting the question correctly, rather than as Vos Savant interpreted it, but that isn't what I'm talking about here.  The point I'm making here is simply that the question asks if we should switch, and I claim that the correct answer to the question that was asked is that we cannot infer an advantage to switch or not switch.  There are many possible thought processes that might lead someone to this conclusion, some valid and some not valid.  But in the absence of a requirement for the host to always show a goat and offer a switch, this is the correct answer, as you yourself have said.


 * You say: "You have still provided no sources clearly stating that most people misunderstand what the intended question is for the MHP.  My point is not that "most people misunderstand what the intended question is", my point is that the correct answer to the question asked in Parade is that we cannot infer an advantage to switch or not switch.  You yourself have agreed with this, and every reputable source says this, including Martin Gardner, Marilyn Vos Savant, Martin Hogbin, and everyone else on the planet.  There's an abundance of reputable sources for this central and indisputable fact - and yet it is absent from the article.  That should be fixed.IVLeeg (talk) 01:23, 2 July 2014 (UTC)


 * SPACKlick, you say "not every source agrees that". Can you cite any source that doesn't agree? As far as I know, every source agrees that the requirement for the host to reveal a goat and offer a switch is essential. They also agree that the Parade problem statement did not include this requirement.  In fact, you yourself just posted a quote from Marilyn Vos Savant in which she explicitly agrees with that, and even admits that the original problem statement in her article dis-satisfied HER because it didn't specify this.  I would be interested if you could cite a reliable source that disagrees with any of this.


 * You go on to say "Some don't [agree] because all sources assume certain base facts not explicitly stated. For example, that the location of the car is concealed from the player isn't explicitly stated."  Sure, there are many tacitly assumed conditions, such as the assumption that the player prefers the prize rather than a goat.  But all of these are perfectly natural and rational assumptions that can legitimately be taken for granted.  In contrast, the assumption that the host is always required to reveal a goat and offer a switch is completely unnatural, artificial, counter-factual.  It would not make any sense for a game show host to be subject to such a requirement, as Monty Hall himself pointed out.  The only reasonable assumption, in the absence of a stated requirement on the host's behavior, is that the host has some choice in his actions, and is acting reasonably.  What you call the "standard assumption" about the host's behavior is not even a reasonable (let alone necessary) assumption, so it is completely different from all the reasonable tacit assumptions that we all agree can be taken for granted.


 * Then you say "The article should deal with the question that most of the sources deal with". Sure, the article should include a discussion of the cute and entirely trivial little probability problem that arises when we add the artificial stipulation that the host must always reveal a goat and offer a switch.  However, it is important to point out, as Marilyn Vos Savant herself admitted, that this is not the question she posed in her original article.  We need to deal with the question that was actually asked, and the objection that was raised by (according to her own account) at least 35% of the educated letter writers.  (My guess is it was more than 35%, because that's just the number that complained about the ambiguity explicity, whereas many people probably just gave the answer based on a reasonable assumption that the host is free to decide his actions.)


 * You say "As shown in literature, most people assume, when given this kind of probability question, that facts given fall to standard repetitive patterns." I dispute that the reputable literature shows this.  I've already challenged the reliability of several of the flakey primary sources cited in this article.  Also, most people aren't even mathematically sophisticated enough to distinguish between subtly different conditionals, and they can be bamboozled into accepting whatever subtle premises are smuggled into the discussion by anyone with a clipboard and a white smock.  Anyone who is capable of distinguishing between the question Vos Savant asked and the question she intended to ask already knows the answers to both questions.


 * You say "The real question this article should answer isn't .. "The one that is strictly interpreted by a direct reading of what Vos Savant wrote and only what she wrote" but "The Question Vos Savant Asked" which is affected by how the question is interpreted by the reader." The two things you named there are one and the same.  Your explanation for why they are different doesn't make sense, because a big part of answering a puzzle question is interpreting it correctly.  According to your reasoning, there is no such thing as a wrong interpretation, but that can't be right.  If someone reads a question and interprets it incorrectly (e.g., by adding some unnatural and artificial restriction that is not implied by the problem statement), then they are making a mistake. It doesn't matter how many people make this mistake, and it doesn't matter if the added restriction was actually intended by the person who mis-formulated their question, it's still a mistake.  The question Vos Savant asked and the question she intended to ask are two different questions, as she herself acknowledged (when she said the question as posed by her reader dis-satisfied her, because it didn't specify that the host must reveal a goat and offer a switch) and as over 35% of her educated respondents pointed out to her.IVLeeg (talk) 18:07, 1 July 2014 (UTC)


 * With your last paragraph I must vehemently disagree. A poorly phrased question can ask something a strict, minimalist literal interpretation of the wording doesn't ask. I will search for more sourcing to find numbers of people who assume the host always offers what is offered in the written question but the question Vos Savant was, i would say clearly, asking is about a host who behaves this way not a host who behaved this way in one instance and a lot of the literature certainly makes the assumption that this is how the question is to be interpreted. I would say if the intent of the author and the inference of the reader aligns in the majority of instances then that interpretation is "the qestion asked" far more than a literalist interpretation of the words used is. I would be interested in your suggestion for how to structure the page based on your objections and the sources that support them...SPACKlick (talk) 22:03, 1 July 2014 (UTC)


 * You say the question Vos Savant asked was quite clear, but remember that Vos Savant didn't ask the question, it was asked by a reader named Craig F. Whitaker, and you yourself just posted a quote from Vos Savant saying "When I read the original question as it was sent by my reader, I felt it didn't emphasize enough that the host always opens a door with a goat behind it...". In view of this, how can you possibly say the question that was posed was quite clear, when Vos Savant herself admitted it was not?  She tells us that she merely assumed, without justification, a restriction on the host that was not part of the question she was asked.  She also claims that she specified this extra restriction in her answer, "to make sure everyone understood", but unfortunately that's untrue.  She did not clearly specify the added restriction in her answer.  But regardless of that, the point is that the question she was asked did not include that restriction, and she knew it (and you know it).IVLeeg (talk) 00:29, 2 July 2014 (UTC)


 * I still maintain the question asked was clear. It wasn't explicitly and formally worded but the intended meaning was conveyed to most readers. Just because part of the question wasn't EMPHASISED enough doesn't mean that it wasn't conveyed, even implicitly. She most certainly does not tell us that "she assumed without justification" the restriction on the host. The question DOES include the restriction, however it does so implicitly. To analyse


 * Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. All explixit so far. Three doors, one door has car, two have goats, you get to pick one.


 * You pick a door, say No. 1 You have a free choice of doors


 * and the host, who knows what's behind the doors, opens another door So the host, who uses knowledge of what's behind the doors opens the door. This door (another door) cannot be your door by implication (in what sense have you picked a door if the host is unconstrained by that choice and why would you say another door if the host was not constrained?). Here we have the first implication that the host's behaviour is constrained otherwise there would be no mention of his knowledge.


 * Say No. 3, which has a goat. Here the implication is that whichever door you picked (Say 1) and he picked (Say 3) the door opened has a goat behind it. And now the restriction has been implied. Not stated but implied.


 * He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? And so we have the question, "what tactic wins the car most often?". I still say the question clearly includes the restrictions it simply doesn't explicitly state them. SPACKlick (talk) 11:18, 2 July 2014 (UTC)


 * Having read around a lot of non reliable sites looking for anecdata of how people interpret the problem I have found many people reasoning many ways about it but they never seem to misunderstand the intended set up. They regularly say things like "I contend that as one door is ALWAYS eliminated, 3 doors should not be accounted for in calculating probability", showing they are aware a losing door must always be eliminated and a choice must always be offered. The reasoning for the 50/50 answer is inevitably "You have a choice between 2 doors, 1 with the car, that's 1 in 2 or 50%". Rarely if ever does the idea that Monty could have an alterior motive come up from people who don't already accept the 2/3 answer.SPACKlick (talk) 11:18, 2 July 2014 (UTC)


 * How can you say this? I just went looking on the web for examples of people's reasoning, and the very first example I found was Anya, who said "You never know what motivates Monty to reveal a door in the actual game. I would definitely stick with my first choice."  She is clearly indicating that she does not assume Monty must always reveal a goat, and this is what convinces her that it's best to stick with her first choice.  Anya is answering the question as written, which does not include the requirement for the host to always reveal a goat and offer a switch.
 * I'd also like to comment on your repeated use of the doublespeak phrase "misunderstand the intended setup". Do you realize what a weird thing to say that is?  When someone understands the stated question correctly, you describe it as "misunderstanding the intended setup".  (I suppose this is the kind of thing that prompts that editor here to keep posting links to "1984".)  To be reasonable, you should revise your sentence to say "Having read around a lot of sites looking for anecdotal evidence of how people interpret the problem, I have found many people reasoning many ways about it, but they never seem to base their reasoning on a literal and correct interpretation of the question."  That would be a rational statement - albeit false (since I've already provided you with a counter-example)IVLeeg (talk) 16:16, 2 July 2014 (UTC)
 * I am convinced that most arguments about the rules stem from people who got the answer wrong (1/2) trying to justify their incorrect answer by pointing out possible ambiguities in the question. In fact as vos Savant, Krauss and Wang and SPACKlick (when he publishes his work) show that people very rarely misunderstand the intended question. Martin Hogbin (talk) 19:16, 2 July 2014 (UTC)


 * Let's not refer to unpublished things (SPACKlick's work), and let's agree that Vos Savant is just an entertainment columnist, not writing in a peer reviewed journal, and not a recognized expert on, well, anything, defending her column and livelihood, and merely reports that in a selected sampling of letters (how selected?) mailed to her she did not discern that very many of them based their answers on a literal reading of Whittaker's question (despite the fact that she herself admitted that a literal reading of Whittaker's question leads to the answer these people gave). But her criteria for deciding whether the letter-writers were influenced by the possibility that Monty had a choice of actions was invalid.  She thinks that if someone says the probability for switching is 1/2, this proves they are not influenced by the possibility that Monty has a choice of actions - but that's clearly false.  It is entirely possible (and indeed plausible) that Monty acts in such a way that the probability for switching (when offered the opportunity) is 1/2.  The only actual peer reviewed published source you've cited is Krauss and Wang, but again, they do not "show" anything, they say "we argue that", using much the same invalid reasoning as Vos Savant.  I've even posted a nice case study, in which someone says BOTH that the odds for switching are 1/2 AND that you can never tell what motivates Monty to reveal a door so you shouldn't switch.  This completely invalidates the reasoning by which your sole reference argues.  Against this I've cited two published reputable sources that argue the opposite (and my sources have the advantage that they are not brain dead).
 * Hilariously, Krauss and Wang argue that the missing restriction on the host's actions does not affect people's responses, but then later they describe applying four cognitive strategies for improving people's responses, which basically consists of explicitly specifying the restriction on the host's actions, and then they announce proudly that almost everyone answers 2/3 switch when presented with this form of the question! So they disprove their own claim.  How idiocy like that get's published (even in psychology journals), I will never know.IVLeeg (talk) 21:55, 2 July 2014 (UTC)


 * I can say it IVLeeg because I went beyond the first example YOU found and coralled hundreds of examples of people disagreeing with the 2/3 answer. Many of these people state, in their reasoning, that Monty Hall will always open a losing door, some of them caveat "If it was a real game show then..." but they understand that maths problems don't work like real life, and therefore most things are unrealistically constrained. There does seem to be a difference between different language speakers (my personal suspicion as yet untested, is that this is to do with the persistent present used in the question not translating well). Now I'd like to remind you that YOUR interpretation of the question isn't necessarily the CORRECT interpretation, no matter how many times you say it is. The literal interpretation is naive, in that it doesn't take into account the context being a maths puzzle. It doesn't interpret the additional information given. It simply translates the words, computationally, to predicates. It's not incorrect, but it's wrong, it's not what's being asked. Even ANYA appears to understandd the intended set up, because her "host motivation" caveat is itself caveated with " If Monty Hall, the game show host of Let's Make a Deal [was the host]" and " in the actual game. ". It is clear that the comment relates to a real world game and not, explicitly not, the maths puzzle being pondered.
 * Just look at the responses around the web, "The explanation of the Monty Hall effect above is flawed in that Monty's actions are not random when he opens the "losing" door(s). Thus the contestant's odds "reset" (so to speak) every time Monty opens a door he KNOWS is a loser." "i made a simple c++ program in which first the prize winning door was chosen, then the door containing the goat was chosen and then contestants answer and his decision to switch or not was chosen by the computer," The only times you see comments about a devious or strategic host who doesn't follow the pattern every time are when the problem is discussed by people who ALREADY understand the answer.
 * You also still seem to be suggesting that a real world game show host would NEVER work the way the MHP's host does, to which I say, tosh. The opening of a losing door builds suspense, extends the show. The host would never open the car door, that would be anticlimactic as all hell but he might open another door, and it might not depend on what you picked (If it did depend on what you picked it wouldn't be always when you have car or always when you have goat because the audience at home would quickly spot that). The host in the MHP works like the host in many shows, prolonging suspense and ultimatley making it easier for a guest to get a prize without is seeming trivial.SPACKlick (talk) 20:15, 2 July 2014 (UTC)

What the cites sources above say
The sources quoted above mainly point out, perfectly correctly, that the assumptions do matter. Yes, we all know that and that fact is stated quite clearly in the article. There are still no sources which clearly and ambiguously (in the way that vos Savant, Krauss and Wang tell us the reverse) that any significant fraction of people misunderstand the intended question. Martin Hogbin (talk) 19:16, 2 July 2014 (UTC)

Recent Discussion section needs a prune
Ok, so The problem was first written about in 1975 and became famous, mostly, in 1990. That's 24 years ago. The first piece of Recent Discussion is 1998, which is 16 years ago, 2/3 as long ago as the problem became famous, that's not recent (nor is it being a puzzle in a weekly puzzle publication really discussion)SPACKlick (talk) 10:44, 7 July 2014 (UTC)

The debatable Erdos incident is from pre 1999. Not recent, not really discussionSPACKlick (talk) 10:44, 7 July 2014 (UTC)

Mr Mee is from 2000 and is just an appearance. We're starting to get an Appearances in pop culture section which frankly never worksSPACKlick (talk) 10:44, 7 July 2014 (UTC)

Curious incident of the dog in the night time, haven't read this source in a long time, don't know if it discusses it or just presents it. At least this was published closer to now than Savant's columnSPACKlick (talk)

Derren Brown seems to be an actual pop culture discussion of the problem, as does the Penn Jilette one. Chen and Rosenhouse discuss it in depth in their 2008 and 2009 pieces. Myth busters and Man lab both test the problem. So I think those 6 sources showing it is discussed in various venues in the last 6-8 years. I've blanked out the rest because WP:BRD SPACKlick (talk) 10:44, 7 July 2014 (UTC)

Identical paradox in lede and first section
Currently the lede and the first section both quote the Vos Savant edit of Whittaker's question. This seems somewhat redundant. Either an edited version from a later paper correcting some of the missing assumptions should go in the lede or it should be left how it is. The repetition just annoys me somewhat. SPACKlick (talk) 15:27, 8 July 2014 (UTC)

The little green woman, what's it adding
I understand why Vos Savant used the little green woman but The media furor section is disjointed already and should just be a review of the history and events surrounding Vos Savant's relevant parade publications and the response to them. It doesn't need its own section, surely. SPACKlick (talk) 15:36, 8 July 2014 (UTC)

Source for 'Deriving the "standard assumptions"' ?
The current article contains the parenthetical statement "(From the point of view of subjective probability, the standard assumptions can be derived from the problem statement: they follow from our total lack of information about how the car is hidden, how the player initially chooses a door, and how the host chooses a door to open if there's a choice.)" Can someone provide a source for this claim?IVLeeg (talk) 18:40, 23 June 2014 (UTC)
 * I am not sure "derive" is the right word for this. These are the Bayesian uninformative priors using the principle of indifference toward alternatives when no difference between them is known. Gill, Richard (February 2011) describes this at pp. 13–14 ~ Ningauble (talk) 20:25, 23 June 2014 (UTC)


 * "Derive" is definitely not the right word for this. One can't derive assumptions. Also, note that Gill 2011 refers only to some of the trivial and inconsequential ambiguities in the problem statement, which we could agree to forgive by glossing over them with the principle of indifference, but he doesn't refer to the primary ambiguity, i.e., Monty's motives and requirement to always open a door after the initial guess. This ambiguity has a huge effect on the answer, and it certainly can't be resolved in the actual situation by appealing to any principle of indifference.  Even with the underspecified and ambiguous problem statement, the player is not devoid of knowledge about how likely Monty is to offer a second choice.


 * Coincidentally, the wiki editor who inserted the claim that we can derive the assumptions from the problem statement was also named Gill. The Gill 2011 document you mentioned doesn't seem to have been published anywhere (has it?).  I think it would be best to have an independent and published source.  Failing that, we should either delete the claim, or at the very least the article should say "According to Richard Gill, the assumptions can be derived from the problem statement".  That way we won't be misleading readers into thinking it's true.IVLeeg (talk) 22:50, 23 June 2014 (UTC)


 * I think the wording could be improved, but the use of a standard Bayesian uninformative prior could probably qualify as a routine calculation. If this is not agreed there are many sources which discuss this subject in a way in which it would obviously apply to the MHP.  Morgan, in their response to H&N, almost make the point. Martin Hogbin (talk) 08:23, 24 June 2014 (UTC)


 * The main ambiguity in the problem statement is that it doesn't make clear whether the host must always reveal a goat and offer a change of selections, or whether the host has simply decided to do that in this particular instance (perhaps because the host wants the contestant to change his answer for some reason). There's no "routine calculation" or Bayesian uninformative prior that can resolve this fundamental ambiguity in the problem statement.  So, lacking a published reference for the claim that the "standard assumptions" can be derived (or even plausibly inferred) from the problem statement, I think that claim should be removed from the article.IVLeeg (talk) 16:02, 24 June 2014 (UTC)
 * The text does need to make clear what assumptions are being referred to. The standard assumptions about the game rules are just that, standard assumptions. They seem to be the ones made by most people (I think Kraus and Wang and vos Savant confirm this) and most sources. In general they are not very contentious. Martin Hogbin (talk) 18:22, 24 June 2014 (UTC)


 * I agree that assumptions (standard or otherwise) are assumptions, and I'm also saying that assumptions are not derived. The statement in the article claiming that the assumptions can be derived from the problem statement is obviously false and, more importantly, is unsourced, and hence should be removed. (Someone suggested sourcing it to the contributing editor's own pdf file accessible on the web, but an unpublished pdf file does not meet the Wikipedia requirements for a reliable source.)


 * Most of the "standard assumptions" are indeed uncontroversial and inconsequential (hardly even deserving of mention), although one of the "standard assumptions" is quite artificial and unrealistic. Claims that "most people make these assumptions" are unfounded, but that's a separate discussion.  All I'm saying here is that none of the assumptions can be derived from the ambiguous problem statement, so the unsourced claim to the contrary should be deleted.IVLeeg (talk) 22:15, 24 June 2014 (UTC)


 * Gill (February 2011) was the first source that came to mind because he devotes a paragraph to explaining the concept. (Other sources mention it, but this in one of the few sources that even attempts to compare treatments of the problem in an even handed way.) The citation in the article states that it was published in Statistica Neerlandica 65 (1): 58–71. Does anyone have reason to believe that the citation is false? (Or that the online pdf is substantially different from the paper as submitted and published?) Author Richard D. Gill's presence on Wikipedia is has been duly noted at the arbitration case for this article, and he appears to have complied with the reminder given to him therein. ~ Ningauble (talk) 00:51, 25 June 2014 (UTC)


 * I see the article includes three references to Gill in 2011, denoted as "2011", "2011a", and "2011b", and the first of these evidently appeared in "Statistica Neerlandica". However, I don't find anything in that paper saying that the assumptions can be derived.  (The word "derive" appears only once, and not in relation to assumptions.)  He does discuss some of the assumptions, but only the trivial and inconsequential ones, and he does not say that even those can be derived from the problem statement.  And he doesn't even discuss the only really significant assumption (about the host's required actions and motivations).  So I don't think Gill 2011 supports the statement in the article.  This brings me back to my point:  If no one can produce a published reference for the (patently absurd) claim that the standard assumptions can be derived from the problem statement, then I think the claim should be deleted from the article.IVLeeg (talk) 03:23, 25 June 2014 (UTC)


 * Hi guys! Well I am not going to edit the page but I hope it's OK I join in the discussion. Of course there is no way one can *rigorously derive* the formulation of an unambiguous mathematical problem (which might or might not have an unambiguous solution) from a short verbal discussion of a quiz show. That's the whole point of my Statistica Neerlandica paper (did you notice its title?). IMHO model building ie building a bridge from real world to mathematics is an art as much as a science. And depending where you stand you may build different bridges. A carpenter everywhere sees nails to be hammered in. A probabilist sees a chance to do some pedagogical conditional probability calculation. A games theorist sees a wonderful illustration of the minimax theorem. I prefer to go to a higher level, a meta-level, and see an opportunity to discuss model building. To show that there is not one unique right answer but many different answer. The different formulations of problems have different assumptions and different conclusions. There are no free lunches. The more you assume in, the more you can get out. The interesting thing (I think) is to lay bare the various options and show the "consumer" a menu of choices. Expensive five star dishes and light three star dishes. They have different prices, right? It's also pretty pointless to derive assumptions by guessing what went on in Marilyn's or someone else's mind. Once the problem is out there in print everyone's mind approaches it in their own unique way. There's no monopoly to how it should be interpreted. One can discuss historically, culturally, how and why and when different interpretations were made. Richard Gill (talk) 11:54, 28 July 2014 (UTC)

Rewriting the paragraph
I agree with IVLeeg and think the paragraph should be amended to make it clearer. This is the text as it is now:

'Most statements of the problem, notably the one in Parade Magazine, do not match the rules of the actual game show (Krauss and Wang, 2003:9), and do not fully specify the host's behavior or that the car's location is randomly selected (Granberg and Brown, 1995:712). Krauss and Wang (2003:10) conjecture that people make the standard assumptions even if they are not explicitly stated. (From the point of view of subjective probability, the standard assumptions can be derived from the problem statement: they follow from our total lack of information about how the car is hidden, how the player initially chooses a door, and how the host chooses a door to open if there's a choice.)'

The first sentence does not really make sense as it is. There was a game "Let's make a deal" but it never had three doors, two goats and a car, and it never offered the contestant the opportunity to swap doors. I suggest that we move 'do not match the rules of the actual game show (Krauss and Wang, 2003:9)' and make it clear that the assumptions mentioned do not refer to the game rules but the car placement and host door choice.

The paragraph now becomes:

'Most statements of the problem, notably the one in Parade Magazine, do not fully specify the host's behavior or that the car's location is randomly selected (Granberg and Brown, 1995:712). Krauss and Wang (2003:10) conjecture that people make these assumptions even if they are not explicitly stated. (From the point of view of subjective probability, these standard assumptions can be derived from the problem statement: they follow from our total lack of information about how the car is hidden, how the player initially chooses a door, and how the host chooses a door to open if there's a choice.)' Martin Hogbin (talk) 07:58, 25 June 2014 (UTC)

Looking through the article, comments about the game rules and the standard distributional assumptions seem to be in several places. It would be a good idea to clearly separate these issues and deal with them each once only. Martin Hogbin (talk) 08:02, 25 June 2014 (UTC)


 * There is a section called Origins and Full Assumptions. Would it be worth discussing the assumptions there? What are the "Full Assumptions"?


 * 1) The car is placed at random.
 * 2) The contestant has no knowledge of the car's location.
 * 3) The host IS aware of the car's location.
 * 4) The host must always open a door.
 * 5) The host must always reveal a goat.
 * 6) The host must always offer the chance to switch doors.
 * 7) The host never opens the contestants chosen door.
 * 8) When the host has a choice of doors they pick at random.
 * 9) The contestant desires the car.


 * Some of these are referred and sourced in that section (3-8), 2 is never in dispute. 1 makes little difference and 9 is pretty obvious. So how best to beef up the full assumptions section? [SPACKlick]


 * I tend to agree that this is the best place to discuss the assumptions and that we should avoid excessive duplication. Although they overlap, I think we should try to separate the generally-agreed rules of the game (2,3,4,5,6,7,9) from the assumed distributions (1,8). In some interpretations 1 is very important, the car might always be placed behind door 1, for example.[Martin Hogbin (I presume)]


 * Martin, I don't agree with your proposed revision of that paragraph, because it removes sourced material and leaves in place unsourced material. My complaint about the existing article is that the parenthetical claim that "the standard assumptions can be derived from the problem statement" is unsourced, so I think it should be deleted, whereas you've proposed to leave it in place.  In contrast, the phrase that you proposed to delete actually is sourced.  So, I suggest the paragraph be revised like this:


 * 'Most statements of the problem, notably the one in Parade Magazine, do not match the rules of the actual game show, and do not fully specify the host's behavior or that the car's location is randomly selected (Granberg and Brown, 1995:712). Krauss and Wang (2003:10) conjecture that most people make a certain specific set of assumptions even if they are not specified in the problem statement.'


 * Here I'm assuming that the existing statements are sourced more or less accurately, and I've taken the liberty of adjusting the wording slightly to make the paraphrases at least marginally intelligible. If I've thereby misrepresented the sources, please feel free to correct.IVLeeg (talk) 23:26, 25 June 2014 (UTC)


 * I don't like referring back to the game show in this section as that will just add further confusion to matters. Would we be better off with a shorter.


 * Most statements of the problem, notably the one in Parade Magazine, do not fully specify the host's behavior or that the car's location is randomly selected (Granberg and Brown, 1995:712). Krauss and Wang (2003:10) conjecture that most people make a certain set of assumptions even if they are not specified in the problem statement.


 * and then going on to detail those assumptions? I agree with Martin that 2-7 & 9 are the basic assumptions. I also think that 1&8 are pretty standard assumptions as I've only ever seen them varied when the mathematician doing the varying was consciously varying the problem. I also still think this paragraph should be in the "full assumptions" section rather than sources of confusion. SPACKlick (talk) 11:26, 26 June 2014 (UTC)


 * Edit to add, the second paragraph of the full assumptions section is, content wise, almost identical to this one. Again we are repeating ourselves. SPACKlick (talk) 11:28, 26 June 2014 (UTC)


 * I agree about removing reference to the game show (Cf. Hall 1975, already cited in the "History" section) because I do not believe this is a significant cause of confusion, and the source in the article does not say that it is. (Rather, the whole point of the cited section entitled "Are There Possible Effects of Incomplete Information?" (pp. 9–10) is to say ambiguous or incomplete rules are not the cause of confusion.) Regarding the following sentence, "conjecture" is really not the right word. They do not conjecture that it might be so, they argue that it is so. (p. 10: "We argue that...") ~ Ningauble (talk) 12:00, 26 June 2014 (UTC)


 * Spacklick says we should not mention the fact that the standard problem statement doesn't refer to the rules of the game show "because it will just add further confusion", whereas Ninguable says we should not mention it because he "does not believe this is a significant cause of confusion". Hmmm...  Well, Krauss and Wang definitely mention this (they say "Besides not mentioning Monty Hall's strategy, the standard version refers neither to the exact rules of the game show nor to the a priori probability distribution of car and goats"), so I don't think we should selectively omit this (neither because we believe it adds confusion, nor because we believe it does not add confusion!).  The article already mentions that Krauss and Wang argue that many people misunderstand the problem even when the important ambiguity is removed from the problem statement, so that's already covered.  The only thing that is not well sourced in this paragraph is the obviously false claim that the important assumption (that Monty must reveal a goat) can be derived from the problem statement.  So, I repeat that the unsourced claim should be deleted, and the sourced material should be left in place.IVLeeg (talk) 03:31, 27 June 2014 (UTC)


 * IVLeeg, I think you have missed the two different types of confusion Ningauble and myself are referring to. I was saying including the reference to the show could be a source of confusion to the reader of the article, Ningauble was saying, as the sources say, that the loose relationship the problem has with the show is not a cause of confusion for the reader/hearer of the puzzle. All that being said, has anyone actually disagreed about removing the reference to deriving the assumptions? I thought we were discussing the best way to do that elsewhere on this page.[note: I do think the standard assumptions can be calculated from the lack of information and symmetry so derived isn't wholly wrong, but it's not an argument worth pursuing] SPACKlick (talk) 08:38, 27 June 2014 (UTC)


 * I recognized that you were saying that mentioning the unspecified show rules could confuse the readers of this article, and Ninguable was saying the unspecified show rules does not confuse the contestant in the puzzle (or the reader of the puzzle), but I contend that those two propositions are inconsistent. And I further contend that what you or Ninguable (or me) think is irrelevant, since the sources do mention the unspecified show rules.  In fact, they report that 13% of respondents answer "switch" when they are told that Monty must always reveal a goat after the first selection.  So it obviously was a source of confusion for at least 13% of the respondents.  Also, your belief that this information (Monty must always reveal a goat) can be calculated or inferred from the lack of information and symmetry is obviously false, and more importantly, not sourced, so I think you should stop pushing that (unless, of course, you can provide a reliable source).  Yes, someone has disagreed (implicitly) about removing the reference to "deriving the assumptions".  Martin made a counter-proposal after I posted my proposal, and the offending claim was put back.  Also, others (including you) have continued to drag your feet by saying "derived isn't wholly wrong", etc.  If everyone is in agreement that the unsourced parenthetical statement should be removed, I will go ahead and remove it.IVLeeg (talk) 12:04, 27 June 2014 (UTC)


 * To your first contention, it is clearly not inconsistent. The majority of readers of this page will not be familiar with the TV show let's make a deal, nor with the specifics of the choices offered on that show. That is my reasoning for believing it is of better form not to mention that extraneous information. As for the unspecified rules confusing the reader of the puzzle, I cannot find a source to show that the 13% is based on any confusion with the rules of the original show, merely that 13% is the proportion who switch when first told the problem with minor variations fromt he original wording (Granberg and Brown 1995). To attribute that 13% to any form of confusion with the original show is to miss the point of the study. I must admit, I was not directly referring to the fact that Monty must always reveal a goat in my above note, but the three examples specified in the paragraph how the car is hidden, how the player initially chooses a door, and how the host chooses a door to open if there's a choice . I think we can clearly see from this we need a stronger section on the standard assumptions, about which there is further discussion on this talk page. But yes, there is no source for deriving the standard assumptions so the claim should be removed until sourced.SPACKlick (talk) 13:01, 27 June 2014 (UTC)


 * Okay, I will go ahead and remove the unsourced parenthetical statement.


 * The point of mentioning that the proposed game does not follow the rules of the TV game show is simply to explain that the Parade problem statement is insufficient to actually imply the claimed solution. If, in the TV show, Monty had played a game as described in the Parade problem, and if he had always been required to reveal a goat after the contestant made their initial choice, then Parade could argued that the problem statement was sufficient (if they had referred to the TV show explicitly), because they could argue that readers should have researched the game show to find out the rules (if they weren't already familiar with it).  But of course the game intended by Parade bears no real resemblance to the TV game show, nor to any reasonable behavior in any known circumstance, so the reader has no way of knowing, either by researching the old game show or by any other means, that Monty must always reveal a goat.  That's why it's worth mentioning that even though the problem statement refers to a game show, it doesn't actually follow any known game show rules.  If it did follow a defined rule, it would eliminate the ambiguity in the problem statement.IVLeeg (talk) 02:35, 28 June 2014 (UTC)


 * Regarding the TV show specifically: IVLeeg, I endorsed removing it not just because I believe it is not a source of confusion, but because, as I specifically pointed out, the cited source does not say it is a source of confusion. Yes, they mention that the rules are different from the TV show, but only as a strawman when arguing that not knowing the rules is not what causes confusion in the Monty Hall problem. The article already describes the TV show in the "History" section, citing Monty Hall himself. Anyone familiar with the TV show must realize this is a completely different scenario: on TV you never get to switch doors. This section is about things that cause confusion, and there are no sources that say anyone is confused about this problem by the rules of the TV show. ~ Ningauble (talk) 16:01, 27 June 2014 (UTC)


 * I completely disagree. Every reliable source acknowledges that the standard problem statement fails to specify (or even imply) that Monty must reveal a goat, and yet the problem has the standard solution only if the reader makes this (quite unnatural and artificial) assumption.  When this rule is made explicit in the problem statement, 13% of respondents correctly say "switch".  So the lack of this specification affects how some people answer the question, and part of this lack of specification is due to the fact that the intended problem does not follow the rules of the TV game show (or any other known set of rules).  When you say that "not knowing the rules is not what causes confusion", I have to smile.  Not knowing the rules certainly OUGHT to cause confusion, since the answer is totally dependent on the rules (i.e., whether Monty is required to always reveal a goat, and if not, what his rule is for deciding whether to reveal a goat).  It is lamentably true that many "reliable sources" on this subject are brain dead, so there's a limit to how much this article can be improved under Wikipedia editorial policies.  I notice that many of the sources in this article are primary sources (including many from clueless psychologists), which have been synthesized into a narrative by a collection of editors, somewhat contrary to the Wikipedia preference for secondary sources.IVLeeg (talk) 02:35, 28 June 2014 (UTC)


 * Yes, you are completely right. See my comments in the English and in the German Wikipedia. In the German Wikipedia ("Ziegenproblem") the fact of the missing critical rule of the game is emphasized, but there is too much unessential and superfluous stuff. In a separate section Übersicht über die Fachliteratur zu „dem“ Ziegenproblem we can also find there a nice proof of your observation that many "reliable sources" on this subject are brain dead. The Wikipedia editorial policies are very strange in this case where "reputable nonsense" is used against clear arguments. But, as I wrote earlier, there is a Wikipedia rule, too, which allows that we may simply ignore those rules because we are treating here a very simple problem. We only have to source the strange history of the problem in publications. The original problem in Parade, combined with the claim of a 2/3 solution, is an unintended joke. And that almost all publicists have been fallen for this joke is responsible for the widespread opinion that the pure fact that the host opens a not "chosen" door with a goat leads to a 2/3 solution; not being aware that this solution requires that the host is committed to do so. And the often emphasised "simulations" may be the main reason for this error: For the "simulants", which often present the result as a surprise for themselves, apply the critical rule implicitly.


 * So we have the situation that people, confronted with the original problem, saying that there is no reason to prefer one of the two remaining doors, are right, and those who say that switching has a 2/3 chance are wrong. And the "1/2 fraction" doesn't need to add a rule to the original formulation of the problem; but the "2/3 fraction" should add the critical rule, but they didn't - or later they did it in footnotes, later sections and so on ...


 * There is a German article from 2009 which treats all this. We also can find there two letters from 1991 written by the author concerning these critical points.


 * In the environment here it is very difficult and time-consuming even to bring on simple facts. I wish you success!--Albtal (talk) 12:09, 28 June 2014 (UTC)


 * @IVLeeg: Please do not keep misrepresenting what other contributors have said here. I did not say that not knowing the rules is not what causes confusion. I said that this is what Krauss and Wang say. I said it to clarify that they emphatically do not say the rules of the TV show cause confusion. Many sources point out deficiencies in common statements of the problem, but there are no sources that say anyone is confused by assuming the rules of the TV show apply to the problem at hand. I can agree with the proposition that saying "Suppose you're on a game show" in vos Savant's famous version of the problem (which does not mention Let's Make a Deal or Monty Hall by name), though suggestive in some respects, does little to clarify the rules. Pointing this out does not really add anything to the observation that most statements of the problem do not fully specify the host's behavior. Incidentally, the first published version of the problem, (Selvin, February 1975), is much more true to the TV show (and does mention Let's Make a Deal and Monty Hall by name), but is subject to some of the same ambiguities if one is not familiar with how the TV show works. ~ Ningauble (talk) 13:52, 28 June 2014 (UTC)

Ningauble, as I was typing that last message I originally said "When you say that Krauss and Wang say that not knowing the rules is not what causes confusion...", but that sentence seemed awkward, and it occurred to me that you seemed to be endorsing the statement, so I didn't think you would mind my saying you said it. And I wasn't smiling at who made the statement, I was smiling at the statement. If in fact you do not agree with that silly statement, then I apologize for attributing it to you. On the other hand, if you do agree with that silly statement, then I don't apologize. On the third hand, if you decline to state whether you agree with that silly statement or not, then I decline to reveal whether or not I apologize. (I tell you, Wikipedia is the best entertainment value around...)

But seriously, we seem to be talking past each other. You keep saying no one is confused by assuming the rules of the TV show apply to the problem at hand, but what I'm saying is that people are confused precisely because the rules of the TV show do NOT apply to the problem at hand (nor do any other defined set of rules). In your latest message you allow that Vos Savant's mentioning of a "game show" does little to clarify the rules, but then inexplicably you say "Pointing this out does not really add anything to the observation that most statements of the problem do not fully specify the host's behavior". Why on earth do you say that? She tells the reader they are on a "game show", and you agree that the intended problem doesn't follow the rules of any game show, and yet you say pointing this out doesn't add anything to the observation that her statement of the problem leaves the rules unspecified. It doesn't ADD to the observation, it IS the observation.IVLeeg (talk) 15:08, 28 June 2014 (UTC)


 * What I believe (though the fact I believe it is irrelevant for Wikipedia's purposes) is that the "standard" problem, as fully disambiguated, is very counterintuitive despite its simplicity, and most people get it wrong. I also believe this is much more interesting than trivial observations about not knowing the rules. Furthermore, though I would not have chosen the wording that Krauss and Wang used, I would make a similar statement: ambiguity in the rules does not explain why most people decline to switch – most people decline the offer even if the problem is stated unambiguously at the outset. But seriously, although it is appropriate to observe that many statements of the problem, notably the one in Parade, do not fully specify the host's behavior, it is not necessary to point out individual parts of the statement that do not do so. The observation is about what is missing, not what is there. "Suppose you are on a game show" does not impart confusing information, it is merely window dressing. Consider that if mentioning the game show in the problem statement is uninformative, which seems to be Krauss and Wang's point, then mentioning it in this section is also not informative. I think that mentioning the TV show in this context is excess verbiage in what is already an overly verbose and digressive article. Insofar as this objection may be considered a matter of style, it is not a big deal. Insofar as language in the article may be taken to suggest the analogy with the TV show is a misleading source of confusion, rather than merely uninformative, the article is somewhat misleading and should at least be reworded. Being a matter of substance, this is a somewhat bigger deal. ~ Ningauble (talk) 15:36, 30 June 2014 (UTC)
 * Yes, this is what the propblem is all about. As you say, 'the "standard" problem, as fully disambiguated, is very counterintuitive despite its simplicity, and most people get it wrong'. Martin Hogbin (talk) 02:56, 29 July 2014 (UTC)

Case Study: Average Person's reason for not switching?
Here's a good sample of one, found on the web, where a somewhat dull-witted person explains his and his wife's reasoning with the Monty Hall Problem:


 * Since there are only two doors left for consideration, you now have a fifty-fifty chance of guessing the correct door. After analyzing this far, my wife, Anya, who received a B in Technical Mathematics in St. Petersburg University (in Russia, not Florida), added: "Besides, you never know what motivates Monty to reveal a door in the actual game. I would definitely stick with my first choice."

Here we have someone mixing clearly erroneous reasoning ("fifty-fifty chance since there are only two doors remaining") with an observation that clearly shows she doesn't think Monty necessarily must reveal a door, since she is worried about his motivation, and she thinks it's safest to stick with her first choice, rather than letting Monty entice her into switching.

Later in the web page, after the husband learns the "correct" answer, he belittles his wife for her "error" - but of course it's clear that she was NOT assuming that Monty must reveal a door, and she had a perfectly good reason for not assuming this - it doesn't make sense, and isn't how any real game show is played. On this perfectly rational and correct basis, her answer (that we can't infer a clear advantage to switching) is perfectly correct.

Of course, she also had some incorrect reasoning (although that may have been her husband's reasoning), mixing two different problems, but she had taken Math in college, and was probably more sophisticated than the average person. Now, according to Martin Hogbin, everyone reading the Parade problem assumes that Monty must always reveal a goat, but that is obviously not true. This woman is CLEARLY influenced by the idea that Monty doesn't always have to reveal a door. Does anyone doubt that many people who answer "don't switch" are guided by the same kind of mixed and garbled reasoning, combining some lack of math aptitude with some perfectly valid reasoning about the fact that we don't know Monty is required to reveal a goat, and we don't know his motivation?

Anya was clearly right, along with all the others like her. And yet her nitwit husband was bamboozled into believing she was wrong, because he can't even identify the fact that her reasoning (correctly) allowed Monty to have some choice, whereas the official explanation of the allegedly "correct" answer is based on the unwarranted assumption that the host has no choice. This just shows that it's hopelessly naïve to think ordinary people are even capable of distinguishing between the different conditional bases. They reason intuitively and informally, taking all factors into account.IVLeeg (talk) 06:03, 2 July 2014 (UTC)


 * The non standard rules are not supported by any source and are completely illogical. See my latest two posts on the /Arguments page. Martin Hogbin (talk) 08:54, 2 July 2014 (UTC)

Now It seems that Martin Hogbin and SPACKlick try to find a way to ignore German sources which contradict them. But the web is full of examples where people show that they don't at all assume that the host is committed to open a not chosen door with a goat and to offer a switch. There is a nice example: A woman said: I would switch, for I don't think that the host is spoofing me.

And "decorating" the MvS problem with formulations which go "just in the other direction" does not only happen in the German speaking area.

And why does Devlin guess a shock for those people who accepted the 2/3 solution, when they later have to take into account that the solution is 1/2 when the host opens a door randomly with the possibility to open the door with the car? - If they had based the 2/3 solution on the crucial rule, they would hardly be "shocked" by this fact. (@Martin Hogbin and SPACKlick: Devlin, like Gardner and others, argues here with a counterexample.)

The "paradox" was born by people who claimed the 2/3 solution without assuming this rule. They thought that the simple fact that the host opens another door with a goat leads to this solution. It is worth to read Selvin's letter which seems to be the starting point of this error.--Albtal (talk) 20:52, 2 July 2014 (UTC)


 * Devlin's "shock" is that if you were convinced by poor reasoning (The two groups interpretation) then you'll be stunned to see it doesn't work. It makes no comment on whether there is an assumption, implicit in the question, that the host opens a door that is not your door and not the car. If, as seems to be being claimed, the general interpretation of the question in the English speaking world (and I am now convinced that there are significant differences across languages which could be interesting to note in a later section if we could find sources for it) is that monty doesn't have to act how he does then surely you would be able to find people whose response to the myriad explanations was "But, you're just assuming the door he opens is always a goat" or "You didn't account for the host opening the car door" or "So it depends on that assumption, but the question doesn't say that" but you never get that response except from people who have already come to understand the question asked before they interpret it hyper literally to defend their instinct. And again, what is the change being proposed to the article with this discussion?SPACKlick (talk) 21:50, 2 July 2014 (UTC)


 * The paradox (double chance when switching doors) arises because of the extremely biased role of the host to never show the car, and the article did show the essential difference as follows:


 * Consider a different problem that contradicts the assumptions of the standard paradox: the host, instead of intentionally revealing a goat, reveals one of the remaining two doors at random. If the host opens a door at random and reveals a goat simply by chance, then the odds are reduced from the standard paradox's 2:1 in favour of switching to 1:1 only. The latter odds track the common intuitive wrong answer because half of the potential winning cases are wasted when the host accidentally reveals the car and by that discards such plain winning event. references 


 * This indispensable hint has been deleted by SPACKlick on 09:52, 18 June 2014, and it is necessary to re-insert that evidence. Gerhardvalentin (talk) 22:15, 2 July 2014 (UTC)


 * That version of the puzzle is referred in the other host behaviours section as monty fail but you may be right that having it earlier in the article as well might help improve the clarity. Any ideas where and how to insert it without reducing the quality of the article? Could go at the end of standard assumptions something like, "These assumptions are key to the 2/3 result it has been shown (Rosenthal) that changing one of them, say 2, leads to the probaility being 1/2" but written better then I can manage at 20 past 1 in the morning. SPACKlick (talk) 23:19, 2 July 2014 (UTC)


 * Leonard Mlodinow stated concerning the intended veridical paradox that is based on the extremely biased role of the host to never show the car: "The Monty Hall problem is hard to grasp, because unless you think about it carefully, the role of the host goes unappreciated." (Mlodinow 2008) That extremely biased role of the host should be emphasised very early, at best immediately after the section "Standard assumptions". Gerhardvalentin (talk) 08:11, 3 July 2014 (UTC)


 * I agree, a short paragraph at the end or beginning of the assumptions section making it clear that when differing assumptions are made the problem is very different. Possibly linking to the "Different host behaviours" section. As a second Draft [Full section changes highlighted]

'' The behaviour of the host is the key to the 2/3 solution. Ambiguities in Craig Whitaker's quoted description of game play do not explicitly define the protocol of the host. However Marilyn vos Savant's (1990a) solution printed alongside Whitaker's question implies and both Selvin (1975a) and vos Savant (1991a) explicitly define the role of the host as follows.


 * ''the host must always open a door that was not picked by the contestant (Mueser and Granberg 1999),
 * ''the host must always open a door to reveal a goat and never the car
 * ''the host must always offer the chance to switch between the originally chosen door and the remaining closed door

'' When any of these asumptions is removed the probability can vary as shown in this section below. It is also typically presumed that the car is initially hidden behind a random door and that if the player initially picked the car, then the host's choice of door to open is completely random. (Krauss and Wang, 2003:9) Some authors, independently or inclusively, assume the player's initial choice is completely random as well. Selvin (1975a) ''


 * I don't really like my changes but they're a start...SPACKlick (talk)


 * Splendid, thank you.  I like the awareness of Mlodinow  in spotlighting this "key" that lets the mathematical paradox arise, especially his "conditioning" on the extreme bias of the host's role, in strictly avoiding to ever show the car. A hint to that aspect – the earlier the better – will help to improve the article, helping to grasp the basics. Gerhardvalentin (talk) 14:38, 3 July 2014 (UTC)


 * I like it too. One quibble is that I would change 'Craig Whitaker's quoted description' to 'the Parade version'.  The Parade version was actually written by vos Savant based on an unpublished letter from Whitaker (who played no further role in the problem).  She added 'say door 1' for example.


 * I would also want to add, before 'When any of these assumptions is removed...', something like, 'These are in fact the assumtions that most people make when they first read the problem' citing vos Savant and K&W. Martin Hogbin (talk) 15:17, 3 July 2014 (UTC)


 * I'd like a phrase like that as well but I struggle to justify it.K&W's relevant section is
 * If the rule were, instead, that the host has to reveal a goat if the contestant first chooses the car-door and should otherwise do nothing, then p(M3 | C2) = 0, which makes the probability of winning by switching 0 (see Equation 2.1).14 Nickerson (1996) writes: “... without information or an assumption about the host’s behavior, the situation is ambiguous, and the question of whether one should switch is indeterminate.” (p. 420). Most experimental psychologists consequently insert the intended rule “Monty has to open another door and reveal a goat” into the standard version to avoid criticism about ambiguity in the wording.15 But this does not seem to help participants: Although Granberg and Brown (1995) stressed this rule, they observed only 13% switch decisions.
 * Which doesn't exactly say that most people make this assumption, just that this assumption and incorrect versions of it are not the major cause of the systemic failure to get the correct result. I can't find Anything in Any of the Vos Savant references to show this, the closest I found was "And a very small percentage of readers feel convinced that the furor is resulting from people not realizing that the host is opening a losing door on purpose. (But they haven’t read my mail! The great majority of people understand the conditions perfectly.)" although I haven't seen the 2006 article or her book. — Preceding unsigned comment added by SPACKlick (talk • contribs) 12:05, 4 July 2014‎

The host is bound to maintain strict secrecy concerning the actual location of the car

 * I added "The host is bound to maintain strict secrecy concerning the actual location of the car (Henze, 1997)". This central requirement for the intended paradox is well sourced, Norbert Henze is professor in probability theory and stochastics. Imo this is the most important typical requirement to the intended paradox, see inconsistent Morgan et al. 1991, e.g.: In case the host opens his strictly avoided door, then his preferred door actually is very likely to hide the car. Gerhardvalentin (talk) 21:34, 4 July 2014 (UTC)
 * I have reverted this added emphasis. It is redundant with "reveal a goat and never the car" already in the list. ~ Ningauble (talk) 13:16, 5 July 2014 (UTC)


 * No, this argument is unsourced. You suppose that "reveal a goat and never the car" to be redundant with "the host is bound to maintain strict secrecy concerning the actual location of the car". But there is no redundance at all, it's two quite different things, as Henze targets quite another aspect. He says Marilyn vos Savant had been accused of lack of knowledge in stochastics, but he strictly contradicts that criticism in characterizing the MHP as follows: It's actually quite mundane, and really has nothing to do with conditional probabilities. Henze says so in contrast to other sources, e.g. in contrast to Morgan et al. who said the probability that the car is behind door 2 can actually be p=1. So, Henze's "keeping the location of the car secret" has quite another direction and purpose that is never covered by "always a goat and never the car". Please re-insert that characteristic and distinguishing aspect that is specifically important for depicting the MHP. — Preceding unsigned comment added by Gerhardvalentin (talk • contribs) 15:07, 5 July 2014)
 * Might not this comment be better somewhere else. As I see it, it is is more criticism of the Morgan approach in that it says that, by his actions, including his choice of door to open, the host cannot reveal any information which would give a clue as to the likely location of the car.  I think it is an important point, made by a reliable source, but it is not one that affected most people's thinking on the subject until Morgan raised the spectre of host door choice. Martin Hogbin (talk) 17:18, 5 July 2014 (UTC)
 * Quite the contrary, I strictly oppose. More than 20'000 pages about "simple puzzle" versus "can only be solved by conditional probability", including ArbCom, is enough. And enough is enough. Henze's "It's actually quite mundane, and really has nothing to do with conditional probabilities" versus thousands of arguments "after the host has opened door #3 to show a goat, the probability that door #2 contains the car can obviously be p=1". Never trivial, Henze raises a very clear light on this eternally ongoing conflict and it is a real coup. His saying "the host has to maintain secrecy concerning the actual location of the car" is not just incidental. After naming the necessary requirements of the paradox, he then expressly adds: "In all these considerations, of course, it is crucial that the host is bound to maintain strict secrecy concerning the actual location of the car", naming the central importance of this condition as the starting point for the arising of the paradox. Henze explicitly denies that conditional probability is "necessary" to solve the puzzle that actually is "quite mundane, and really has nothing to do with conditional probabilities". For solving the paradox, "BEFORE" versus "AFTER", did he open #2 or #3, is addressing the host's choice of two goats in 1/3 if the contestant initially picked the prize. More than 20'000 pages are enough, so this central prerequisite has to be in the article. It's really not about "never shows the car". It's the central prerequisite for the arising of the clean paradox that "has nothing to do with conditional probability" (Henze). Gerhardvalentin (talk) 19:14, 5 July 2014 (UTC)
 * That was my point. Henze is is pointing out that the Morgan solution is wrong, or at least not necessary. I agree, but how can we demonstrate that the Morgan solution is wrong before we have given the Morgan solution.  No one is going to understand what point is being made by Henze.


 * All that I am suggesting is that we do have the Henze point, as made by you, but later on in the article. Martin Hogbin (talk) 19:46, 5 July 2014 (UTC)


 * You say "No one is going to understand what point is being made by Henze." It's on a famous "paradox", and it is necessary to name the basics of this paradox, just in the beginning. The crucial point of the matter is the role of the host that generates the paradox, and the most important constitutive basis that does distinguish this famous paradox from quite other variants (of "differing results") is that the host is bound to secrecy regarding the actual location of the car, in any way and whatever that means. It is necessary to name this prerequisite just in the beginning, in first place. It is well sourced and may not be neglected. Later on in the article, you can still refer to this very important main argument. More than: 20'000 pages is enough. Gerhardvalentin (talk) 20:59, 5 July 2014 (UTC)


 * The point made by Henze is a subtle one and it surely will not be obvious to a first time reader of this article why 'the host must always reveal a goat and never the car' is not the same as, 'The host is bound to maintain strict secrecy concerning the actual location of the car'. We know that Henze is having a swipe at Morgan but who would possibly know that until they have seen Morgan's argument.  So, are we agreed that the Henze quote should go after the conditional solutions are explained in the article. Martin Hogbin (talk) 21:59, 5 July 2014 (UTC)


 * Seems you forgot about ArbCom. You suggest to maintain turbidness. My answer is no, never. Because the world famous intended paradox forever has suffered from ambiguity, and ambiguity forever had been the pitfall of this intended paradox. This is the very reason that, from the start, unambiguous specification is indispensable. The possible question "why must the host ..." is secondary. But – indispensably – absolute clarity must be the foreground. The "why" is secondary, space enough to give the answer "why". See all the deviant variants of cross-purposes that were presented in abundance and abundance. Evidence enough. And btw: Morgan et al. are not wrong, they did address a special variant. Henze does make that clear distinction. Finally, the article owes clarity to the readers, in distinguishing the intended paradox from other variants. In case you don't dare to, then I will have to be going to revert Ningauble's deletion of sourced contents, his deletion having been based on an inapprehensible comment "already in the list". Again, evidence enough. It's on cross-purposes, and it's on distinguishing the clean paradox. No way out. Gerhardvalentin (talk) 00:14, 6 July 2014 (UTC)


 * I agree that it is important to maintain clarity at the start of the article, so that it is accessible to as wide a range of readers as possible. For that reason we should start with the simple, 'clean' as you call it, paradox, without complicating factors. That is what was agreed at the, very thorough, RfC that followed the ArbCom case.  After the simple paradox has been presented we can go on to discuss the various distracting complications that turn a simple puzzle, that (almost) everyone understands perfectly well but still gets wrong, into an endless discussion between mathematicians, statisticians, psychologists, and anyone else who wants to give an opinion.


 * The fact that the host must keep the location of the car secret is assumed by nearly everyone without much thought. Henze's point is that, if we are to consider the door opened by the host to be important then we must take it that the host is providing information as to the whereabouts of the car, contrary to our initial assumption.  Thus Henze's point is a clever argument against an unnecessary complication, the Morgan solution.  As such it is itself a complication that can wait until later to be discussed.


 * More important, in my opinion, is to state that the standard assumptions are the ones that most people do, in fact, make thus closing another loophole for people who get the answer wrong. I have no way of proving this but I would bet that nearly everyone who thinks up a complicating factor, from Morgan onwards, did so in response to having got the answer wrong themselves.  Having done so, and having been embarrassed their error, they look for ways to justify their wrong answer; what if the host did not always offer the swap, what if the host had a door preference, what if the host could reveal the car etc.  In my view, 'nice try but no cigar'.


 * So, I agree with you. Let is keep it clean and simple at the start and on that basis the Henze statement is not necessary there, although I agree it is a good argument that should be included in the article somewhere. Martin Hogbin (talk) 10:05, 6 July 2014 (UTC)

Yes, it is necessary to start with outlining of the host's role "the host must always ..." – No redundancy, Henze was not joking in expressively adding the top priority to that outlining of the host's role: "In all these considerations, of course, it is crucial that the host is bound to maintain strict secrecy concerning the actual location of the car". No redundancy at all, Henze knows exactly what he says. This top priority belongs where Henze decided it to belong, it belongs to the outlining of the host's role and its restrictions. I'm going to add this top priority to the outlining of the host's role, verbatim according to Henze's judging, and I hope you agree with his judging. Gerhardvalentin (talk) 12:20, 6 July 2014 (UTC)
 * Not somewhere, but where it belongs.
 * I agree with you about the importance of Hentze but am trying to come to some kind of compromise that other editors will accept. If you want to fight it out I will leave you to it. Martin Hogbin (talk) 14:05, 6 July 2014 (UTC)


 * Have to say, I agree with Gerhardtvalentin, It's an important assumption, mentioned in various forms in several sources, that the Host doesn't reveal and the Contestant doesn't know where the car is. I'm not certain Henze's phrasing is the best for the beginning of the article, because it's phrased that way to account for other details. They key implication of it, in the simple form, is the contestants ignorance not the hosts complicity in that ignorance. SPACKlick (talk) 06:33, 7 July 2014 (UTC)

I agree that it should go in the article; it is a very important point. I personally have no objection to its going near the beginning although I am sure that most readers will not understand its real significance. That is one reason that it would be better placed later on. The other reason is that it seems that others do not want it included near the start, if at all, and I am trying to find a compromise. If it turns out that there is actually a consensus to have it where Gerhard put it that is fine with me.

Regarding its meaning, I take it to mean that we should ignore the host's choice of door when the player has originally chosen that car, because there can be no information about where the car is revealed by the hosts choice of door because the host is strictly forbidden from revealing such information in any way. I have not read what Henze says in full because I think it is only available in German. Does anyone have a translation? Martin Hogbin (talk) 08:33, 7 July 2014 (UTC)


 * Exact. Other sources "seem" to base on the host's complicity in that ignorance, in predicating "if the host has the choice between two goats, then he chooses equiprobable" – but not really meaning that the host really did flip a coin, but that we have to base on "... as if the host did flip a coin". Henze draws attention to that important principle in another way in saying that the actual location of the car remained secret. — Preceding unsigned comment added by Gerhardvalentin (talk • contribs) 08:24, 7 July 2014)


 * I may have misundestood Henze's point. If this premise is merely a rewording of the equiprobable decision between two goats, then it can go in and should be phrased this way, not in a roundabout fashion to do with secrecy. Whereas if you are sourcing a need to include the assumption that the contestant holds ignorance then it would need to include secrecy. That said, if you are simply trying to include the equiprobable choice assumption it really doesn't need to go in this early. SPACKlick (talk) 10:03, 7 July 2014 (UTC)

the actual location of the car remains secret
Should it read:
 * 1) the host must always open a door that was not picked by the contestant (Mueser and Granberg 1999),
 * 2) the host must always open a door to reveal a goat and never the car
 * 3) the host must always offer the chance to switch between the originally chosen door and the remaining closed door
 * 4) In all these considerations, of course, it is crucial that the host is bound to maintain strict secrecy concerning the actual location of the car (Henze 1997)

or it could read:
 * 1) the host must always open a door that was not picked by the contestant (Mueser and Granberg 1999),
 * 2) the host must always open a door to reveal a goat and never the car
 * 3) the host must always offer the chance to switch between the originally chosen door and the remaining closed door
 * 4) after the host has shown a goat, the actual location of the car did remain strictly secret (Henze 1997 and others)

What is the best way to show this principle? Or should it read   4. if the host, in opening of a door, did have the choice between two goats to show, then it is assumed that he chose one of them with equal probability

Anyway, this standard assumption has top priority for understanding the clean paradox. This standard assumption should be shown where it belongs. Other proposals? Gerhardvalentin (talk) 08:34, 7 July 2014 (UTC)

To make the wording accurate and consistent, closer to your last option is best


 * 1) ''If there is a choice between two goats to reveal, the host chooses at random with equal probability.

Does that define the assumption you're looking to add? SPACKlick (talk) 10:06, 7 July 2014 (UTC)


 * Thank you yes, that is the way and the standard formulation that most sources use to underline that – in that situation – we "are to assume" that there was no host's door preference, and both goats were equally likely to be revealed. Yes, that's exactly the point. But I do not like that many sources say "the host flipped a coin", as this sounds like a procedural instruction to the host, whereas it only signalizes that "we have to take it as given" that the host chose at random. So I suggest the wording
 * If there is (or was ?) a choice between two goats to reveal, it is assumed that the host chooses (or chose ?) at random with equal probability.
 * I would appreciate if you could enter it as the most used standard assumption ("it is assumed that" is my addition). But I do like Henze's formulation as well, as it is a more general way to express that fundamental principle of the standard paradox and its surroundings. Gerhardvalentin (talk) 11:15, 7 July 2014 (UTC)
 * You are aware the article currently says "It is also typically presumed that...if the player initially picked the car, then the host's choice of goat-hiding door to open is completely random."SPACKlick (talk) 08:10, 8 July 2014 (UTC)
 * I think that it is wrong to single out this particular distributional assumption from the others that are generally made, which are that the car and goats are placed uniformly at random at the start of the game and that the player's original choice is uniform at random. The three assumptions together then lead to an obvious and intuitive symmetry with respect to door identity which means that we can forget door numbers completely.


 * Henze's comment is more of a general principle,which has further reaching consequences that most people imagine and I therefore think that should go in the article somewhere. I personally would be happy with it here, even though its real significance would not be clear to most readers, but I would also be happy to have it later on when condition probability is mentioned. You will have to argue that one out.  By the way, I have asked Ninguable to explain his removal of the Henze comment.Martin Hogbin (talk) 09:11, 8 July 2014 (UTC)


 * Thank you. Imo the first suggestion above (the actual location of the car remains secret) is best, item #4 is what Henze says:


 * 1) the host must always open a door that was not picked by the contestant (Mueser and Granberg 1999),
 * 2) the host must always open a door to reveal a goat and never the car
 * 3) the host must always offer the chance to switch between the originally chosen door and the remaining closed door
 * 4) In all these considerations, of course, it is crucial that the host is bound to maintain strict secrecy concerning the actual location of the car (Henze 1997)


 * It's all about the unequivocal representation of the abstract paradox. Henze, well knowing about crucial misunderstandings and controversial issues, considers this generalization to be absolutely necessary, and for that very reason the article should pay regard to it. Imo this generalisation should be mentioned as item 4.
 * Hoping Henze's general clarification can and will help to avoid further upcoming doubtings and misunderstandings as to the intendedd abstract paradox. Gerhardvalentin (talk) 10:02, 8 July 2014 (UTC)
 * I'm sorry but that wording is just unhelpful in the context of the assumptions section. I imagine it reads much better in Henze's paper, although I can't access it or a translation. I also believe that it is right that this assumption is separate from the major three because the major three must be true and known to the contestant in order for the paradox to work. This one must either be true, true and known or false and unknown. The only time it makes a difference is if the distribution of probability of the host choosing between two goat doors is not 1:1 and the contestant knows that. Even then unless the distribution is 1:0 then the decision, switch, is unchanged. SPACKlick (talk) 11:40, 8 July 2014 (UTC)
 * And even the distribution is 1:0 the decision, switch, is unchanged, because staying never can be better. But Henze makes a clear distinction between exercises in conditional probability, and the clean paradox. You may not see Henz's motive as a professor in probability and stochastics, but he calls it: "Bei allen diesen Betrachtungen ist natürlich entscheidend, dass der Moderator die Autotür geheimhalten muss ..." (crucial, essential). As the paradox does not allow for 1:0 or other variants besides 1:1. And, not least, it's on the clean paradox, also. Not only on exercises in probability theory. Gerhardvalentin (talk) 12:02, 8 July 2014 (UTC)
 * Sorry, I don't see it as an essential for simple considerations of the paradox or basic understanding of the paradox. It is already referred, as I said above, as a general assumption just not one of the 3 key assumptions. Can you link me to a translation of henze's paper (or the original German and I'll struggle through it). It gets detailed treatment later in the article. Why needs it be here? SPACKlick (talk) 12:08, 8 July 2014 (UTC)
 * Have only the German edition and can send it if required. Henze is professor in Karlsruhe in probability theory and stochastics, and he knows exactly what it needs to distinguish the intended paradox from variants, and his clarification is important enough. He is also on youtube: (http://www.youtube.com/watch?feature=player_detailpage&v=NcCGgdornVs#t=4150] Gerhardvalentin (talk) 12:48, 8 July 2014 (UTC)
 * Ok, having read, to the best of my ability [I'm not good at German], the relevant chapter of the paper, the comment reads quite throwaway. My best translation of the section is


 * This argument becomes even more clear when considers the prospects of success of the strategies of a switcher" and a stayer. The stayer wins the jackpot, if the prize is behind the door initially chosen and the probability of this is 1/3. A switcher, however, wins the car when he first chooses one of the two "goat's doors", probability 2/3, because after the opening of the other goat's door by the host, automatically leaves the changing strategy with the car door. In all these considerations, of course, it is crucial that the moderator must keep the car door a secret and is also required to open a goat's door."
 * It doesn't appear to be about the equiprobable choice (it may go on to that later I haven't translated more than 3 paragraphs of the paper). It seems to be expressly about not opening the car door and being forced to open the car door. SPACKlick (talk) 14:00, 8 July 2014 (UTC)
 * It looks that way to me too. I have never seen the original so maybe I have misunderstood it. Martin Hogbin (talk) 14:13, 8 July 2014 (UTC)

Sources showing most people make the standard assumptions
Krauss and Wang have a whole section on this subject entitled 'Are there possible effects of Incomplete Information'. At the end of that section (page 10) they say, 'In sum, when solving the standard version, in which the required assumptions are not made explicit, people seem to assume the intended scenario anyway'. They go on to quote vos Savant (no doubt we can find her original) who said, 'Virtually all of my critics understood the intended scenario...'.

K&W go on to say, 'In short people struggle not with the ambiguity of the standard versions assumptions, but with the mathematical structure of the scenario'.

This is two solid reliable sources who say very clearly that most people did make the standard assumptions. Martin Hogbin (talk) 13:36, 4 July 2014 (UTC)


 * Can we add this somewhere please. It really is not disputed by reliable sources. Martin Hogbin (talk) 14:14, 8 July 2014 (UTC)


 * I still disagree that the sources show what they claim. I'm not overly objecting to putting it in because it matches my personal experience but I'd like it to be noted that I think there should be a stronger source for that claim. K&W have


 * "After the Monty Hall problem became famous, many questions on possible effects of incomplete information in the standard version arose. Besides not mentioning Monty Hall’s strategy (1), the standard version refers to neither the exact rules of the game show (2) nor to the a priori probability distribution of car and goats (3) (c.f., Nickerson, 1996; von Randow, 1993; Mueser & Granberg, 1999).
 * ...[A section on each or (1)-(3) including] It can be read in Ichikawa (1989, p. 271) that letting participants know Monty Hall’s strategy does not help them find the solution either...
 * ...Most experimental psychologists consequently insert the intended rule “Monty has to open another door and reveal a goat” into the standard version to avoid criticism about ambiguity in the wording.15 But this does not seem to help participants...
 * ...[followed by]We argue that most of the criticisms of the standard version regarding its unstated assumptions are mathematically relevant, but not psychologically relevant, since the intended assumptions will hold anyway....
 * ...Let us give examples of how assumptions, different from the intended ones, would make this “uniformity belief” in the standard version impossible:...
 * ...[a section on various strategies of Monty that don't result in 1/2''...
 * ...In sum, when solving the standard version, in which the required assumptions are not made explicit, people seem to assume the intended scenario anyway. Along the same lines, vos Savant observed...
 * ...In short, people seem to struggle not with the ambiguity of the standard version’s assumptions but with the mathematical structure of the scenario....
 * ...what blocks correct intuitive reasoning is not lack of information, but lack of the right information representation.


 * What I take from this is "The failure to express the assumptions is not what causes people to make mistakes of reasoning" not "Here it is clearly shown that people make the absolutely correct assumptions about the rules of the game". They make that claim, in passing, but fail to support it. That's why I'm dubious here. SPACKlick (talk) 15:07, 8 July 2014 (UTC)


 * The MHP is a *joke*, a *trap*. The whole point is that people instinctively give the wrong answer for different intuitive reasons some connected to bad probability reasoning some to unwarranted psychology thinking. And when you tell it to people there is a *dialogue*. So you explain the "real assumptions" if and when needed. Richard Gill (talk) 11:58, 28 July 2014 (UTC)


 * Maybe we are taliking a little to cross purposes here. By standard assumptions I am talking about the game rules: that the host must always open an unchosen door to reveal a goat and must always offer the swap. As vos Savant said, 'people seem to assume the intended scenario anyway'.  That seems pretty clear to me.


 * What people assume about the host's door opening strategy is another matter. The probably do not think about it at all. Martin Hogbin (talk) 02:49, 29 July 2014 (UTC)


 * Richard, I do not understand in what sense the word 'joke' applies to the MHP. Martin Hogbin (talk) 03:19, 29 July 2014 (UTC)

Vos Savant and the media furor out of place
With the recent changes to the article the media furor section seems out of place. The article goes [Lede, Paradox, Assumptions, media furor, Solutions, Confusion and criticism]. I think it may work better if Furor was moved between solutions and Confusion and criticism, particularly in light of "A second controversy" That said, I think a lot of it can be removed. The article is about the puzzle, not necessarily about the parade column. I'm not confident enough in this to WP:BRD it and would rather discuss first. SPACKlick (talk) 15:31, 8 July 2014 (UTC)
 * It is what changed the problem from being an obscure mathematical puzzle to being famous. It shows just how many people not only get the answer wrong but are ready to berate those who give the correct answer in print. Martin Hogbin (talk) 02:59, 29 July 2014 (UTC)
 * I fully agree it needs to be there, it's why it's famous. However the way the article reads right now is a little odd and it takes nearly 1500 words to get to actually discussing the problem. SPACKlick (talk) 09:23, 29 July 2014 (UTC)
 * I don't think the section is misplaced, but I agree that it is overlong: the "little green woman" and the "second controversy" subsections are misplaced here. A brief treatment of how the Parade column met with utter disbelief, and was picked up in mass media (The New York Times), is appropriate here. The shell game analogy and the little green woman scenario do not belong here, before solutions are presented. (Years ago we had an "aids to understanding" section for this sort of thing.) Material about criticism of solution methods, now shown as "A second controversy", definitely belongs in the section on "Criticism of the simple solutions", after all of the solutions have been presented. ~ Ningauble (talk) 10:14, 30 July 2014 (UTC)
 * I agree about the second controversy not being appropriate in this section but the little green woman, or some discussion of why it matters what the host knows, should be presented somewhere near the beginning, before mention of the 'conditional' solutions. That is one of the most surprising things about the problem to most people, that (what the player knows about) what the host knows, completely changes the odds.  Martin Hogbin (talk) 14:12, 20 September 2014 (UTC)

Lost sight of the purpose of this article
In what was one of the most comprehensive RfCs in the history of WP it was agree that the article should first deal with the fundamental paradoxes of the problem before delving into mathematical pedantry. Since then, the article is article has drifted away from the principle agreed at that time without there being any sign of a significant change of consensus. I think the article should, with the cooperation and support of all editors, be returned to the agreed format. Martin Hogbin (talk) 08:39, 19 September 2014 (UTC)

Proposals
Proposal 1 is for the initial sections including 'Solution' and 'Aids to understanding' to be based exclusively on 'simple' solutions (with no disclaimers that they do not solve the right problem or are incomplete) then to follow that, for those interested, with a section at the same heading level giving a full and scholarly exposition of the 'conditional' solutions.

Proposal 2 is for the article to include in the initial 'Solution' section both one or more 'simple' solutions and an approachable 'conditional' solution (showing the conditional probability the car is behind Door 2 given the player picks Door 1 and the host opens Door 3 is 2/3) with neither presented as "more correct" than the other, and to include in some later section of the article a discussion of the criticism of the 'simple' solutions.

Admin decisions
Eraserhead1: To briefly summarise I find that there is a consensus in favour of proposal 1, and a consensus against proposal 2.

I Jethrobot: Proposal 1 had the most support among participants; it was also best supported by policies and editing guidelines. It is most sensible to present the lead and body of the article in a way that the more accessible approach should be made apparent first. This also is consistent with due weight, as present coverage of the problem in reliable sources is generally based off of vos Savant's approach.

Churn and change: The RFC veers around to the view the first set should be presented before the second. While some arguments from a one-off psychological study were presented to show people had trouble with the solutions, the general feel is we need to go with our understanding of what lay people think like. Adding to that, I see the best order as Krauss & Wang, vos Savant, Carlton, Adams/Devlin and the multi-door point of view, followed by the solutions of the second set. The Krauss & Wang solution, on page 5 of their paper, does not use probability, and is a simple enumeration of all possible scenarios of the problem and the results from switching and not switching. The figures in that paper compactly cover the solution space. One can argue technically probabilities are implied, but that is no objection; the solution needs no background in math.

The RFC discussed objections to Savant's solution, and seemed to settle more on accepting it as good enough for the lay person. I note in these footnotes of the closers' original discussion that the criticism is mostly in journals catering to undergraduates or math teachers, and are written with them in mind. These are not refutations of correctness, they are more objections from a math pedagogical point of view. Lack of math precision and lack of generalizability seem the main objections, and neither matter for the average Wikipedia reader. The solution is accurate for the problem posed, as mentioned in greater detail here. As such, criticism of Savant's solution does not meet our neutral-point-of-view policy, since the context of WP is different from that of a journal intended for math pedagogy. Martin Hogbin (talk) 08:53, 19 September 2014 (UTC)

Nobody cares?
Will I only get a response when I start editing, with every change being edit warred over, or does nobody care now?Martin Hogbin (talk) 13:19, 23 September 2014 (UTC)

I will put the page back into the agreed order then. Martin Hogbin (talk) 09:45, 7 October 2014 (UTC)
 * I will agree that the article is now inappropriately focused not on the subject of the article "The Monty Hall Problem" but on the "Controversy surrounding vos Savant's response to the Monty Hall Problem" and I think 1) would be a better framework to keep the article focused on the subject of the article.-- TRPoD aka The Red Pen of Doom  11:43, 7 October 2014 (UTC)
 * After the most comprehensive RfC I have seen on WP, the order that I have put the article into was the agreed one. Martin Hogbin (talk) 21:14, 15 October 2014 (UTC)
 * Also, do not see how it now focusses on "Controversy surrounding vos Savant's response to the Monty Hall Problem". The 'Vos Savant and the media furor' section focusses mainly on how people get and are convinced of the wrong answer, which is what the problem is all about. Then we have 'Confusion and criticism', which talks mainly about the why people get the wrong answer. Martin Hogbin (talk) 21:24, 15 October 2014 (UTC)

Furor is a little NPOV
Is there a reason we're using the term "Media Furor?" rather than "Media Response"? Furor suggests anger and rage and over excitement. Response seems more neutral. SPACKlick (talk) 15:40, 8 July 2014 (UTC)
 * 'Anger and rage and over excitement' is a pretty good description of the letters vS got. Martin Hogbin (talk) 03:00, 29 July 2014 (UTC)
 * Is the term sourced to anywhere though? I'd rather keep within Wiki Standards and use a neutral term like response than a loaded one like Furor SPACKlick (talk) 09:20, 29 July 2014 (UTC)
 * We do not have to source every word. Have you read the letters that vS got? Martin Hogbin (talk) 08:30, 19 September 2014 (UTC)
 * I have read it, I just think that response is a more neutral term, there were supporters and detractors but it just reads a little...for want of a better word...puffy. Furor is someone who wrote this articles interpretation of the response, I don't think the term is warranted therefore it either needs a source or a more neutral term unless there is broad consensus it's appropriate.
 * I think just 'response' is too weak. There were thousands of letters to vS, many quite blunt and insulting.  The response was much more than would be expected for a simple puzzle in a popular magazine.  It was this response that made the problem so well known.  Martin Hogbin (talk) 14:16, 20 September 2014 (UTC)
 * Letters from outraged individuals is not a "media" furor, nor was the coverage of her getting the swarm of letters anything so widespread as to merit the term. -- TRPoD aka The Red Pen of Doom  12:30, 17 October 2014 (UTC)
 * What name for the section would you propose? Martin Hogbin (talk) 16:12, 17 October 2014 (UTC)