Wikipedia talk:Requests for mediation/Monty Hall problem/Archive 1

Issues to be mediated
Glkanter has revised his statement of the additional issues to be mediated. This is obvious from the history, but he's deleted a number of specific issues and replaced them with a single, unspecific complaint. I think it would be helpful to restore the original list of specific issues and in general simply add additional issues as needed (or, not that I think this is the case here, indicate specific issues have been resolved rather than delete them). -- Rick Block (talk) 17:25, 23 January 2010 (UTC)

When do we start.?
When will mediation start? Martin Hogbin (talk) 17:47, 26 January 2010 (UTC)

My initial thoughts
How do you all feel about referring to the "unconditional" solution as the naïve solution? While I think there's merit to giving a good deal of equal time to the Morgan paper, I think we should recognize that this "popular" solution is more intuitive and closer to the apples and oranges, supermarket math approach, while the Morgan approach is slightly more convoluted. Having taken Concepts of Mathematics I think the Bayesian solution the most "mathematically correct," but I think we should be able to rationalize the Morgan solution as an intermediate step between Bayesian and common-sense reasoning. After all, Bayes' theorem is really just an expression of conditionality, and Morgan's analysis is an application of the conceptual underpinnings in layman's terms, isn't it? Andrevan@ 06:23, 31 January 2010 (UTC)


 * You may be missing the fundamental point of disagreement which (IMO) is that there are two conflicting ways to interpret the problem. One is that the problem asks "what is the average chance of winning by switching" - this is how vos Savant interprets the question and has the same answer regardless of how the host chooses which door to open in the case the player has initially selected the door hiding the car.  The other is "what is the chance of winning by switching given which door the host opens" - this is how Morgan et al. (and a significant number of others) interpret the question.  The answer to this question depends on how the host chooses which door to open in the case the player has initially selected the car.  This is the topic of the second question at Talk:Monty Hall problem/FAQ.


 * IMO (meaning I'm trying not to speak for others) there's one camp here (including Glkanter, Martin, JeffJor, and possibly others) who think the "Monty Hall problem" is by definition a problem where the question is the average probability of winning by switching regardless of what reliable sources have to say and another camp (including Rick, Nijdam, Kmhkmh, and possibly others) who think the "Monty Hall problem" is a problem where the question is the conditional probability of winning by switching given which door the host opens consistent with what a significant number of reliable sources say. There are reliable sources which ignore this distinction, but those that acknowledge that there is a difference between these two interpretations all say that how the host chooses which door to open in the case the player has initially selected the door hiding the car determines the probability of winning by switching.


 * The question is first whether the article should say anything at all about this controversy (i.e. should the article simply ignore these sources) and then, if the answer is these sources should not be ignored, how much prominence does NPOV require these sources be given. -- Rick Block (talk) 07:42, 31 January 2010 (UTC)


 * I think there is a third way, which is that the two interpretations aren't conflicting but merely represent different levels of abstraction in approaching the problem, much like say instantaneous velocity and average velocity, which are equivalent in a certain case, mean the same thing generally, but possess important distinctions such that they are two different concepts.


 * But I'm confused about a part in the FAQ which I hadn't read previously. "However, in this variant, if the player initially picks door 1 and the host opens door 3 the player knows the car is behind door 2 and has a 100% probability of winning by switching. If the player initially picks door 1 and the host opens door 2, either the player's initial pick was the car (a 1/3 chance) or the car is behind door 3 (also a 1/3 chance). This means, given that the host has opened door 2, the player's chance of winning by switching is 1/2." My understanding of the Morgan solution is that it still results in a 2/3 probability for switching regardless of the door picked. Is this incorrect? Andrevan@ 07:55, 31 January 2010 (UTC)
 * Not quite or rather it depends on the exact wording and to which step "picked" refers. The door the player picks in the first step doesn't matter, but the door the host picks (the revealed goat) may matter. The overall or total probability for switching to win is 2/3. But the conditional probability for switching may change in dependence of the revealed goat door. Whether it does or not depends on how you model the host's behaviour, which is of no consequence for the total probability but crucial for the conditional ones. This essentially what Morgan and others have examined. If you assume, that the host picks a goat door at random (p=0.5 for each door) any time he has a choice between 2, then the conditional probability will be 2/3 (numerically identical to the total probability for switching) regardless which door the host reveals. If you however assume the host favours a particular door, when he can pick between 2 goats, then the conditional probability for switching will deviate from 2/3 but always remain greater or equal than 1/2. The notion of layman versus mathematician is a bit misleading, since both both approaches are mathematical, however one of them (total probability) is arguably much easier for layman to understand. It is is also a trade off in a certain way. The conditional probability might in some be closer to the real observed scenario (player has picked a particular door and the host has open a particular door and the playe makes his decision under these conditions). However this "being closer" comes at a cost, since it requires to model a host behaviour which strictly speaking you have no knowledge about in the first place. The latter therefore is the strong point of the total probability (unconditional solution) it might not be "as close" to the observed scenario, but it requires no knowledge of the host's behaviour, meaning it is free of any additional assumption. So there is trade off between more structural insight ("being closer") and additional assumptions, i.e. you pay for your "increased insight" by being forced to make additional assumptions along the way. Whether it is worth the cost is ultimately a personal opinion and you can certainly argue from the layman's perspective it is not. ---Kmhkmh (talk) 11:05, 31 January 2010 (UTC)


 * But nonetheless, isn't the conditional case subsumed by the general solution? The notion that the conditional solution is the "correct" solution to the problem because it deals with a more specific formulation seems to me, pedantic. Andrevan@ 17:11, 31 January 2010 (UTC)
 * From my perspective both solution are correct and valid answers to the ambiguous MHP and each has of them has their own merits and shortcomings (the described trade off above). I agree that, considering one solution as "more correct" is properly overly pedantic and ultimately not helpful, that's why i used quotes in the explanation above. I'd rather see them as different approaches/perspectives to an ambiguous problem. Imho much of the article's troubled past is caused by people insisting on making their favoured solution appearing as the "more correct one", that is at least my personal reading of endless arguments over ultimately minor issues. What the article however should do is explaining the difference between both solution, i.e. the reader should learn that in literature there are 2 prominent ways to approach or model the problem. However unfortunately the "pedantry"-issue is at the heart of the editing conflict and to resolve the pedantic viewpoints in both camps is the central challenge for this mediation, I'm afraid.--Kmhkmh (talk) 18:15, 31 January 2010 (UTC)


 * In response to Andrevan's question "isn't the conditional case subsumed by the general solution?" - it depends on what you take the question to mean. A specific example might help.  If you run an experiment and simulate 9000 iterations of the problem with the car initially randomly placed and the initial player's pick random, you'd expect each door to be the player's initial pick about 3000 times and if the player picks door 1 only for some portion of these will the host open door 3.  We can ask what is the probability of winning by switching, meaning what fraction of the whole 9000, or what fraction of the (about) 3000 who pick a given door (like door 1), or what fraction of the less than 3000 who pick a given door and see the host open a particular door in response (like player picks door 1 and host opens door 3).   The first two are the same regardless of how the host chooses when the player's initial pick is the car.  If all 9000 switch, about 2/3 will win.  If all 3000 who pick any door switch, about 2/3 will win.  What happens in the fully conditional case depends on how the host picks - but without affecting the first two (!) .  The unconditional solutions say 2/3 who switch will win.  This is correct (regardless of any host preference) if we're talking about all 9000 cases.  If we're talking about the fully conditional case the answer is also 2/3, but only if the host picks randomly if given a choice.  You can say the unconditional solution implicitly assumes this, but another way to look at it is the unconditional solution is simply not addressing the conditional case.


 * And, yes, this at least initially sounds pedantic however it is a point made by numerous reliable sources (see talk:Monty Hall problem), most forcefully by Morgan et al. who call unconditional solutions "false" solutions. An unconditional solution is correct for a problem that is definitely related to the "standard" MHP, but is simplified in one particular way.  The difference is essentially whether the player knows which door the host opens.  There are several equivalent ways to describe the simplified problem:
 * The player is allowed to pick a door, and after picking a door is blindfolded. Now the host says he is opening one of the unpicked doors and does so.  The audience shouts out "it's a goat!".  The player is asked if she wants to switch to the other closed door (still blindfolded).  Only after deciding is the blindfold removed.
 * The player is allowed to pick a door. The host says (before opening a door) that he will open a door showing a goat and asks the player to decide (now, before a door is opened) if the player would like to keep what's behind the initially chosen door or the other door that will remain closed.
 * Rather than doors, the host uses an urn with three marbles, one white (representing the car) and two black (representing goats) as described at talk:Monty Hall problem/Arguments.
 * In each of these variations, the player does not and cannot use any knowledge that may or may not be given by the specific door the host opens. These may still be difficult problems for most people to solve correctly, however none of them are the "full Monty" where the player is asked to decide looking at two closed doors and one open door showing a goat. -- Rick Block (talk) 18:30, 31 January 2010 (UTC)


 * And so it is. To the best of my knowledge, almost no one thinks of the MHP in terms as described above by Rick (unconditional version). It just doesn't fit the description of the MHP. I really have not the slightest clue why some mathematicians, who understand the difference, insist on the unconditional version. The only reason I can think of is they have at some moment given the "simple solution" as the right solution, and now are desperately seeking to defend this. At least this is what MvS is suspected off. Nijdam (talk) 22:11, 31 January 2010 (UTC)
 * No they don't nor did mathematician using the simple solution in their later textbooks have to defend any stakes here. They simply recognize it for its merits, i.e. being simple and providing a sufficient heuristic to the very least. Also in particular in cases like these one needs to distinguish between the actual math (i.e. the mathematical formalism) and its interpretation/application regarding the "real world". There is no issue with the math here but with only with its interpretation/application, which strictly speaking is outside the domain of math and a matter of personal judgement. Apparently there are quite a number of mathematicians (including several involved in the recent discussions), who judge the total probability to be at the very least a useful criterion for the player to decide.--Kmhkmh (talk) 00:32, 1 February 2010 (UTC)
 * What merit is there? It is like the statistician feeling comfortable with one foot in water of 0&deg; C and the other in water of 70&deg; C. Only because the average chance has the same value as both the specific ones, does it seems to be defendable. If in an asymmetric way opening of one door would lead to loosing an amount of money and opening of the other in gaining an amount, no one would use the average probability.Nijdam (talk) 10:50, 15 February 2010 (UTC)

Andrevan, I suggest you read the version of the article as it was following the FARC. I think this is approximately May, 2008. This will give you a sense of the violations of the NPOV that I, and many before and since, have been working very hard to remove from the article. I also suggest you read Martin's user page critique of Morgan's paper. I believe Morgan's paper is not a 'reliable source', and can be ignored/minimized as per Wikipedia policy by a clear consensus of editors. Glkanter (talk) 18:35, 31 January 2010 (UTC)

One other thing. There's nothing 'ambiguous' or 'implicit' in the host choosing randomly when faced with 2 goats. In Selvin's 2nd letter to The American Statistician in 1975 (Morgan is published in the same journal, in 1991), (Selvin introduces and solves the puzzle unconditionally in his 1st letter), Selvin states outright that the host chooses randomly. Morgan appear not to be aware of Selvin's letters. Glkanter (talk) 18:42, 31 January 2010 (UTC)
 * There is no such thing as a consensus among editors that Morgan is unreliable.--Kmhkmh (talk) 19:33, 31 January 2010 (UTC)
 * This is a majority of editors who believe that Morgan should not be given the special status to control the structure of the article or declare other reliable sources wrong. Martin Hogbin (talk) 09:52, 6 February 2010 (UTC)


 * No source has this power. Where there is not a consensus among current research, contradictory sources will be included and treated as neutrally valid - not "true," merely "reliable." Andrevan@ 01:32, 8 February 2010 (UTC)

Reconciling contradictory sources
Let me first state that in terms of considering Morgan a reliable source, according to Wikipedia policy we must consider this paper reliable and a significant minority viewpoint worthy of inclusion in some form. While it may be easily criticized by Wikipedians or other academics, it has not been invalidated, and exists as a peer reviewed academic paper with citations and a considerable length of time since it was published.

Now, that being said, a survey of the historical MHP cannot discuss the entire problem in the context of Morgan. The point of the article is to be an overview about things that others have said about the problem and not an authoritative mathematical description of the solution. This is not about a math problem, but the overall concept that is referred to as the MHP over the course of its entire existence. There exists a considerable body of work on the subject both before and after Morgan that contradicts it; there are also independent Morgan supporters. This easily fits the criteria of a minority viewpoint that must be balanced into the article with nearly equal billing to the "common" solution.

Can we all agree on these generalities before we hash out the wording? Andrevan@ 02:54, 3 February 2010 (UTC)
 * I might somewhat agree on the "equal billing" part and that the article should simply summarize that various reputable literature on the subject (rather than insisting on prominent coverage of one's personally favoured approach), but otherwise i disagree with the whole premise. I'm aware of people pushing for that impression however, but imho it is not true. The bulk of the papers does indeed use support a conditional solution (i.e. the morgan approach) at least as well or even as the preferred treatment (for instance, gillmann, snell/grinstead, Henze (German), Rosenhouse, Behrens, ....). There is even an entire popular science book in German Das Ziegenproblem (Gero von Randow, science journalist), which describes the problem and related aspects and in probability theory and psychology for laymen and he uses the conditonal solution as a center piece. From the publications I've seen, I hardly consider the conditonal approach as minority position in academic publications (it is rather the other way around). What some academic papers implicitly or explicitly disagree in with Morgan, is his tone (towards vos Savant, see rosenhouse for instance) and his notion that conditional approach is the only way to treat the problem. Note even if you consider the conditional approach as the "true" one or closest to reality, you still can consider the total probability as valid "heuristic" argument for switching. In my view no mathematician (and probably even hardly any other academic) disagrees with Morgan's solution, many of them however might disagree with additional conclusion, that the problem can and should be treated that way only.--Kmhkmh (talk) 13:04, 3 February 2010 (UTC)

Something strange is happening here. Morgan, et al are being considered 'reliable', while Selvin, 16 years earlier in the same journal is being ignored. Glkanter (talk) 17:25, 3 February 2010 (UTC)
 * nothing strange is happening here and nobody suggested to ignore Selvin. However as pointed out by Andrevan, the term MHP is not just defined by Selvin's original letters (or any other single publication), but by the (complete) set of reputable literature on it.--Kmhkmh (talk) 17:32, 3 February 2010 (UTC)


 * I agree with everything Andrevan says except the characterization of this source's POV as a "minority viewpoint" and the subtle implication that the existing article discusses "the entire problem in the context of Morgan".  I realize there are editors who think the article is dominated by the Morgan et al. POV, but per an analysis I did of the article recently (see talk:Monty Hall problem) out of 42 paragraphs in the article (at that time) there were 3 references to Morgan et al.  This paper's POV is hardly the context of the entire article.  What I believe is the heart of the issue here is that Morgan et al. (and others) criticize the simple solutions (Morgan et al. call them "false solutions") and it's difficult to include this criticism in the article in what comes across in an NPOV fashion. -- Rick Block (talk) 17:38, 3 February 2010 (UTC)


 * I'm not trying to say that the article currently discusses the problem in the context of Morgan, I actually think the article as it currently exists, or last I checked, is pretty balanced. The point was that even if Morgan obsoletes other approaches, Wikipedia must not present the problem as though unconditional solutions are no longer relevant. Andrevan@ 21:45, 3 February 2010 (UTC)


 * To Glkanter: I think the problem here is that you seem to think that, because Selvin and Morgan contradict each other, only one of the two can be reliable. Actually, both are reliable and worthy of inclusion. Andrevan@ 21:47, 3 February 2010 (UTC)


 * Morgan's paper is full of errors. Martin's user page show this. The whole paper is based on the fallacy that the host might choose between the goat doors unevenly, and that somehow the contestant knows this. Not only did Selvin state this random selection clearly, it's a game show, Monty can't let on to the contestant where the car is. Without changing the problem, Morgan's paper couldn't exist. And Morgan's claim that all simple solutions are false would hopefully not exist either. I don't know if you've read the paper, Andrevan, but almost every sentence is erroneous. They purport to give a 'history' of the problem, mention the 3 Prisoners Problem, but don't mention Selvin. They accuse vos Savant of 'false' deeds in the opening paragraph. They claim their's is the only 'correct resolution', and seemingly call all others, including the combining doors and Selvin's table of possible outcomes 'false'. The paper lacks 'quality'.


 * "Peer review is an important feature of reliable sources that discuss scientific, historical or other academic ideas, but it is not the same as acceptance. It is important that original hypotheses that have gone through peer review do not get presented in Wikipedia as representing scientific consensus or fact. Articles about fringe theories sourced solely from a single primary source (even when it is peer reviewed) may be excluded from Wikipedia on notability grounds. Likewise, exceptional claims in Wikipedia require high-quality reliable sources, and, with clear editorial consensus, unreliable sources for exceptional claims may be rejected due to a lack of quality (see WP:REDFLAG)." http://en.wikipedia.org/wiki/Wikipedia:Fringe_theories Glkanter (talk) 23:39, 3 February 2010 (UTC)


 * Morgan's paper is cited by 62 according to Google Scholar. That's acceptance, and clear evidence that this is not a fringe theory. It is not for us to judge the accuracy of the statements, merely to interpret its status in the literature. Andrevan@ 04:45, 4 February 2010 (UTC)


 * Morgan's paper is in a not very reliable journal and not very highly regarded journal, and it contains serious mathematical errors, as well as being written in a dogmatic and pedantic style and making subtle alterations to history in order to strengthen its (biased) point of view. It is widely cited because it exists. It contains a lot of good stuff too, I don't say it is all wrong. It's just not a particularly good piece of expository mathematical writing, IMHO. OK: it is not for you wikipedia editors to judge its accuracy, but it is my personal duty to do so! Gill110951 (talk) 15:56, 4 February 2010 (UTC)


 * You're missing the point. Reliable sources are not determined by individual editors. We cannot judge a source on perceived errors or the reputation of the journal beyond being serious and academically legitimate. Andrevan@ 20:23, 4 February 2010 (UTC)


 * The point that you are missing is that the Morgan paper has been criticised by other reliable sources as well as many editors here. It may still qualify as a reliable source under some WP criteria but it cannot be elevated to a 'super source' which has the power to control the structure of the article or declare other sources wrong. Martin Hogbin (talk) 23:40, 5 February 2010 (UTC)


 * What you are describing doesn't exist as such. No one source will control the structure of the article or the veracity of other reliable sources. Whether or not Morgan has been criticized by editors or other sources is not relevant to its importance, and the criticism will be reported as well. Andrevan@ 01:30, 8 February 2010 (UTC)

JeffJor's Comments
I'm not sure what the format is supposed to be here, so I'll just add some comments to the above.

Andrevan: "How do you all feel about referring to the "unconditional" solution as the naïve solution?" It isn't naïve, and calling it that implies it is wrong somehow. And just to make one point clear (although it does seem to be already), the issue isn’t which solution is correct for the MHP, it is which interpretation of the MHP is correct. Then, how do the two interpretations help (or hinder) the explanation of why the contestant should switch.

Rick Block: "One [way of interpreting] is that the problem asks 'what is the average chance of winning by switching'". No, it really isn't. That is how the second camp you described shoehorns the solution the first camp gets into their interpretation of the problem. And this is what I mean when I say you use your own POV when you write anything about the MHP. You insist that the arguments used by those on the other "side" must be applied using your interpretation, not theirs.

If we treat the MHP like it asks for a probability (it doesn’t), your first camp thinks the question is more like "Assuming that any unspecified random occurrences are represented by uniform probability, what is the probability of winning by switching?" (Note that this assumption is supported by "reliable sources" like Seymann and Savant). The second camp thinks that some of those occurrences should be treated parametrically and others uniformly. The choice between those options is never discussed, but is based on whether it affects the answer to the actual question "should you switch?" None of the sources cited by the second camp ever suggest a way to tell what value the parameter should have, when used, except to treat it uniformly. So all they do is get the same answer "yes, you should switch" by a far more complicated, and unnecessary, route. It is unnecessary, because it does nothing to explain the controversy surrounding the MHP; and in fact, it detracts from that explanation (at least my opinion).

Rick Block: "... and another camp who think the 'Monty Hall problem' is a problem where the question is the conditional probability of winning by switching given which door the host opens consistent with what a significant number of reliable sources say." Most of the reliable sources ignore Morgan's interpretation of the problem completely, and represent the majority, so I don’t know what you mean by "significant number." You conclude from that, without evidence and a clear example of your POV, that the majority did not consider it at all. But it is more likely that they didn’t feel it necessary to mention an invalid interpretation (which is my POV). But regardless, we can't choose between the sources that interpret in these ways, unless they discuss why. The only source that does actually says that Morgan's interpretation is wrong. Finally, the originator of this version (Savant) of the MHP, and the source of the controversy that makes the MHP significant enough for Wikipedia, said explicitly that Morgan's interpretation was wrong. You have questioned what this meant, since she never mentioned it specifically (she is always vague on details when it might be considered rude to mention them), but there is only one point where Morgan's interpretation differs from hers. The conditional door numbers. That is all she could mean.

Rick Block: "Those [sources] that acknowledge that there is a difference between these two interpretations all say that how the host chooses which door to open in the case the player has initially selected the door hiding the car determines the probability of winning by switching." No, Rick, they do not. Savant said it was wrong. Krauss and Wang, the ones that discuss the interpretation, say that most people take it for granted and use those door numbers, but in a way that treats them uniformly and so that choice does not affect the answers.

Andrevan: "But nonetheless, isn't the conditional case subsumed by the general solution? The notion that the conditional solution is the 'correct' solution to the problem because it deals with a more specific formulation seems to me, pedantic." The formulation itself isn't pedantic, but how it is used can be. Morgan's thesis isn’t that the contestant can know their parameter value somehow, it is that it is irrelevant to the question "should I switch?" It would be pedantic to insist the contestant needs to divine a value for it, or even use their formulation at all, but Morgan does not do that. They say any result of their formulation irrelevant, because all possibilities mean you should switch. However, a similar question, "what is the probability the car was placed behind Door #1?" is just as important to Morgan's interpretation and completely ignored by Morgan. It is assumed to be uniform by all the other sources who use Morgan's interpretation, with no reason given for the different treatments. This is one reason why some editors want to separate (not ignore) Morgan's interpretation, since it is handled inconsistently. It is, as you say, "and a significant minority viewpoint worthy of inclusion in some form." But it does not contribute to the main reason the MHP is in Wikipedia - the fact that people get the wrong answer regardless of interpretation. It is an advanced topic within the MHP, based on possible alternative interpretations that we have no justification to choose between, but it does not contribute to the main points the article should be making. JeffJor (talk) 21:52, 3 February 2010 (UTC)
 * There seems to be a misunderstanding regarding Wikipedia and its articles. MHP is not in Wikipedia simply because laymen (or experts) get the answer wrong (and might want to look it up). It is in Wikipedia because it is a notable problem (=plenty of publications dealing with it and it has acquired it's own name). Of course you can argue that the notability is in particular due to being unintuitive, which might have triggered most publications on it. However from the WP perspective of notability that doesn't really matter, because whether WP contains an article on a subject or not is primarily due to its notability and not due to the cause for its notability (there can be many and completely different ones).--Kmhkmh (talk) 23:49, 3 February 2010 (UTC)


 * I think Kmhkmh's points are important. Jeff, do you agree that we must address the problem objectively as it exists in all contexts, regardless of how it contributes to correct understanding? This is a hurdle we must get past before we can begin hashing out the specific details. Andrevan@ 04:29, 5 February 2010 (UTC)


 * There seems to me to be a simple solution to this problem, which I have suggested several times. This is to fully address the simple treatment of the problem first then consider the more academic details later on in the article.  For the non-expert general reader this has the advantage that the extremely unintuitive problem is fully covered and explained before there is a confusing discussion of academic complications.  This layout will also not be a problem for the expert, who will quickly recognise that the article does fully address the potentially conditional nature of the problem identified in the paper by Morgan et al.  This format is typical of may good quality mathematics text books which often start with a simple treatment of a subject then proceed to cover some of the complicating issues later on.  Martin Hogbin (talk) 00:17, 6 February 2010 (UTC)


 * Does anyone feel that we cannot address the simple problem first before progressing to the trickier conditional treatment? This doesn't mean the two approaches need be completely in separate sections, but even within a single synthesized introduction, the simple problem leads into the conditional problem. This sounds wholly reasonable to me. Andrevan@ 01:28, 8 February 2010 (UTC)

Richard (Gill110951) Gill's POV
Preamble: I write here without anonimity because I'm a professional mathematician/probabilist/statistician and actually often use the Monty Hall problem in my teaching, both within the university and in "public" education related events. (I'm sorry if my pedagogical skills may lag behind my specialist professional skills).

I think that Monty Hall is a gold-mine, and a folk-culture jewel. My opinion is that it does not belong to anyone, in particular not to any particular brand of mathematician. I do like to take as starting point Marilyn vos Savant's formulation, literally, including all its ambiguities. That's the problem which got famous and which intrigues people. It can be formalised in different ways having different though interestingly related solutions. I know it has a pre-history and a post-history; that's all part of the story, but I think vos Savant's correspondent's question is a good starting point. Part of the problem is to decide what is the problem you want to solve.

[I do agree with the supplementary clarification, that we KNOW that Mr M.H. ALWAYS opens a door different from yours and reveals a goat, and that he can do this because he KNOWS in advance where the car is hidden.]

I take it central that we are NOT told by Craig Whitaker that all doors are equally likely to hide a car and we are NOT told that all doors are identical and we are NOT told that the quizmaster chooses a car randomly (or completely randomly) when he has a choice. These are additional ingredients added by people who have a tool (probability theory, in particular Bayes theorem) and want to show off that it works for this problem.

Moreover, we are NOT initially asked to compute a probability (1) or a conditional probability (2). We are asked "What should Craig Whitaker do in this situation (3)"?

Briefly I now state the easier formalizations (1) and (2), giving brief verbal arguments for the usual/right answers, under minimal/conventional additional assumptions. These logical arguements can be translated "word for symbol" into rigorous, formal mathematics.

Then I state the game theoretic formalization (3) and try to deal with it verbally, as briefly as possible too. I like (3) the best, but it is of course the most challenging!

1) "Unconditional Probability Question": What is the probability that switching will give you the car, ignoring the information you got so far (car chosen by you, door opened by Monty)?

It is completely correct to say that IF your initial choice is correct 1/3 of the time, THEN always switching will give you the car 2/3 of the time. QED.

2)"Conditional Probability Question": What is the probability that switching will give you the car *given* that you chose door 1 and Monty Hall opened door 3?

IF your choice is completely random, and IF the car is hidden completely randomly, and IF Mr. M.H. opens a door completely randomly when he has a choice, THEN every probability you like can be calculated, and everything that you calculate remains the same if you renumber the doors arbitrarily (SYMMETRY). By (1) we already know (unconditional or overall) Prob(switch gives car)=2/3. It's a well known fact ("the law of total probability") that Prob(switch gives car) is the weighted average of Prob(switch gives car GIVEN you chose x, MH opened y); averaged over all pairs x, y of different door numbers. But all those conditional probabilities are all the same, by SYMMETRY. Hence they are all equal to their average which equals 2/3. Therefore, under the supplementary conditions giving us SYMMETRY, (the conditional) Prob(switch gives car GIVEN you chose door 1, MH opened 3) = 2/3. QED.

(I like the SYMMETRY argument, which was drawn to my attention by Boris Tsirelson, a whole lot better than a CALCULATION using BAYES' formula).

Now, my favourite:

3)"Game Theoretic Question": What should you DO, only knowing what you know? (only knowing what Marilyn/Craig told us)

You the player choose a door and later decide whether or not to switch. They, the TV people, hide a car and later their man MH decides which door to open. It's a zero-sum two-party game since you want to get the car, they want to keep it. For each party (you and them), there are at most three choices at each of the two steps, so it's a finite game. So by John von Neumann's landmark minimax theorem (1929), there exist minimax stategies for both parties, and a "value of the game", which is the best probability p of you winning/losing that each party can guarantee themselves. This means: if *you* use *your* minimax strategy, then *you* are guaranteed *at least* probability p of getting the car, whatever *their* strategy; conversely, if *they* use *their* minimax strategy, *they* are guaranteed probability *at most* p of losing the car, whatever *your* strategy.

It's easy to check that *your* strategy: "completely random initial choice and then switch"; and *their* strategy: "completely random hiding, completely random opening"; are such a matching pair. If you use your strategy you'll get the car (at least) 2/3 of the time, whatever stategy they use. If they use their strategy, they'll lose the car at most 2/3 of the time, whatever strategy you take.

Our solution = our advice to the player: we hope you did choose your door completely at random; but assuming you were so smart, we can tell you "switch" is the good thing to do. QED.

Concluding advice to wikipedia editors and to our mediator: start the wikipedia page with Marilyn vos Savant/Craig Whitaker's question, literally, and discuss the different approaches which exist out there in the world for tackling the problem. In particular, these three. Do them in the order given above. Neutral POV! Discuss the pre- and post-history. Discuss the legitimate criticism of various approaches. Discuss further variations, alternative proofs, reasons for misunderstanding, psychology, etc etc.

I'm saying that before solving the Monty_Hall_Problem, one has to solve the Monty_Hall_Problem problem. One wants to convert a string of common language words about an idealised real world situation into an unambiguous question, belonging to some natural domain of knowledge, with a unique nice answer within that domain. The first step is to decide how to formalize Craig's question. (1), (2) and (3) are different existing formulations, all with something for and against them. All are legitimate. Within their context, all are correct. Once the question has been formalized in each of those ways, the answer is clear, it can be given in words, or in mathematics; it is indisputible, though there may be different routes to getting the answer.

Postscript: As a professional mathematician I see my role in this mediation merely as someone who ought to point out any real errors of mathematics or logic. What counts for me in this role is the correctness of the logic, not the "authority" of the writers. Of course I can help look for literature references out there, for those who don't trust a logical argument unless someone in the peer-reviewed literature has stood behind it, and in any case, because of wikipedia's purpose and nature.

On the wikipedia entry on Monty Hall, and on the talk pages, and I have always tried to edit in order to correct what I see as errors of fact or errors of logic. I learn about the problems of communicating between mathematicians and non-mathematicians from what goes on here. Editors who don't like my edits can revert them, anytime. I'ld prefer people would try to get the message at least, and maybe help to find a better formulation, if I'm not being clear enough. I don't have a monopoly on clarity of exposition to non mathematicians of mathematical/logical arguments.

You can read some further documentation of my POV at http://arxiv.org/abs/1002.0651 which is a draft paper submitted to a peer-reviewed (but light-weight) statistics journal and which contains what I learnt here on wikipedia from all you great wikipedia editors. There is totally no original research there, nor in my comments here. The game theoretic approach has been around for ages. Everyone in the economics/optimization/game theory field knows it and finds it too obvious to waste their time writing papers about it. Those guys in economics and optimization and game theory go for heavy-weight articles in heavy-weight journals, they don't have time to play. I mean, they can get Nobel prizes! (Like Nash did for his extension of von Neumann's seminal work, cf A Beautiful Mind; he also appropriately got the John von Neumann Theory Prize for that work, too). There is no Nobel prize in statistics or in mathematics. That's because a famous mathematician was playing around with Nobel's wife rather than concentrating on his core business of becoming a famous rich influential mathematician. And statistics is completely out of the picture being one of the most low-status areas of science, only losers would become statisticians, think all the others, though unfortunately they all need us and none of them understand us.

I already have quite a few improvements in mind for arXiv/1002.0651.pdf, so there'll be revisions, and when/if it is published, the final version will go up there on arXiv too. I'm looking for game theory literature references.

Gill110951 (talk) 13:51, 7 February 2010 (UTC)


 * You say, 'Moroever, what counts for me is the correctness of the logic, not the "authority" of the writers.' Unfortunately, this is not how Wikipedia operates. That "authority" is exactly what makes articles verifiable, and to do otherwise is often original research. Do you understand this point? Andrevan@ 04:27, 5 February 2010 (UTC)


 * Please see this well balanced paper. hydnjo (talk) 05:37, 5 February 2010 (UTC)


 * @Hydno, I suppose that was a joke? But anyway: my well-balanced paper is deliberately provocative and deliberately unbalanced. It is also not meant to be taken too seriously. But the maths is meant to be correct.

Gill110951 (talk) 11:16, 5 February 2010 (UTC)


 * @Andrevan, of course I understand that point, that's what I try to make clear in my comments above. I am a researcher in this field. I've read most of the papers and most of the books and I'm qualified to tell you if some paper contains gross mathematical errors or not. But you are free to ignore me. I'm also a wikipedia editor. I consider myself always free to propose corrections to what I consider errors of fact or errors of logic. I am not going to dictate here what the community decides is the Monty Hall problem, as far as wikipedia is concerned Gill110951 (talk) 11:16, 5 February 2010 (UTC)

Drop the math!
I think it is tough to make progress here because we keep delving into probability theory. It is important to understand that Wikipedia entries are dictated by secondary sources and not by mathematics. Can we all agree to stop talking about mathematical truths and instead address the sources, and what they say? Andrevan@ 04:31, 5 February 2010 (UTC)
 * Fine with me. Note that Talk:Monty Hall problem/Arguments was set up some time ago for math discussions. -- Rick Block (talk) 05:34, 5 February 2010 (UTC)
 * yes--Kmhkmh (talk) 10:06, 5 February 2010 (UTC)


 * The secondary sources are the ones which ****-up the issues with mathematics. The original Monty Hall problem is not a problem of mathematics. The Monty_Hall_Problem problem [sic] is to decide what mathematics, in any, can be used to solve it.Gill110951 (talk) 11:10, 5 February 2010 (UTC)
 * This is not that easy. It is correct that the MHP is at the intersection between "real world problem" and mathematics as such and it is of problem of interpreting math. However mathematics is not just used in the sense of "pure" math. In fact many areas as applied math, engineering math, statistics do specifically deal with the interpretation of math in the real world (deciding what math to use to solve a real world problem) and from that perspective it is of course math. It's a math problem published by mathematician in math journal (Selvin), you can't get anymore math than that (neglecting internal quibblings about the real nature of math between mathematicians).--Kmhkmh (talk) 15:30, 5 February 2010 (UTC)
 * @Kmhkmh, if you take as starting point Selvin's problem we might call it a maths problem, but some might not like to take Selvin's problem as the start. Others would point out that Selvin's formulation was ambiguous and his intention can only be determined implicitly, by studying his solution. Selvin is a biostatistician, not a mathematician. Gill110951 (talk) 15:48, 5 February 2010 (UTC)
 * So wikipedia says that 2=3, if this is published by some obscure mathematicians in the so-called peer-reviewed literature and even gets 60 citations according to Google Scholar? Maybe those 60 citations are pointing out that those guys were wrong. This reminds me of discussions with lawyers. "We are not interested in the mathematical truth here, we are not interested in logic, we are interested in the legal facts and what the law says". Lot's of people are sitting in Dutch jails, because of crimes they couldn't possibly have committed, because of this point of view. The fact that the judge's logic is wrong is no reason to reopen a case, since the judge has the last word. Gill110951 (talk) 11:10, 5 February 2010 (UTC)
 * Actually, yes, that's how Wikipedia works. The comparison to the justice system is misleading, though. Of course if the 60 citations say the guys were wrong, we report that as such. I'm not saying we can't interpret the context of the sources. But original research is not allowed, and mathematical fact checking is not necessary. Andrevan@ 15:11, 5 February 2010 (UTC)
 * The WP concepts of relying on "reputable sources" and "authority" doesn't mean authors should give up on common sense (in doubt you can drop misleading or obviously false sources, even if they formally qualify as "reputable". However the scenario you've put forth above is extremely unlikely (and has nothing to do with MHP) and it merely illustrates that the WP concept is not 100% error proof (which is true for almost anything). Moreover I don't see any alternative, because if you argue a mathematician could ignore the sources and simply write, what he "knows" to be "true", then you'll get exactly, what you want to the avoid, because such obscure mathematicians as the one in your example above will start write WP articles directly. If we are lucky they might get blocked or corrected by other (hopefully less obscure) mathematicians, but then we've moved the discussion/evaluation/research of the math community into WP and this is precisely what we have to avoid. WP is supposed to report on results of the work/research/knowledge creation of the math community, but it is not platform or facilitator for that process itself. That process belongs into universities, journals, blogs, usenet, arxiv.org and we report/summarize the results only.--Kmhkmh (talk) 15:30, 5 February 2010 (UTC)
 * Sorry guys, I know, I know. Just blowing off steam. BTW, I do not claim any originality whatsoever in my point of view, including the mathematical results. There is a whole world out there of people who studied MH using game theory. There is a whole world of people who think the short "unconditional" problem and solution is all that is needed. There is a whole world out there of people who think that the tricky question about conditional probabilities is the problem we are all supposed to be talking about. I tried above merely to summarize the "hard content" (the indisputible maths) of those three points of view, and add my own point of view as to what MH is really all about. Not OR, just own POV. And I put it here just to be on record Gill110951 (talk) 15:48, 5 February 2010 (UTC)

Nijdam's POV
In my opinion it is important, how WP presents what in general is considered the MHP. My experience is that most people think of the MHP as something like the K&W version. At least one assumes the player is given the choice of switching, after the door with the goat has been opened by the host. Only because a lot of discussion about the simple solution has come up, some defend the unconditional version, I will not speculate here why. For me it is unthinkable that the unconditioal version (i.e. the audience, unfamiliar with the initial choice of the player, is asked if she should switch, before her choice is made) is presented as being a more common interpretation than the conditional (i.e. the player is asked if she will switch after the host has opnend the goat door). In further sections any other interpretation and analysis may be presented, even Gill's game theoretic approach, but the article has to start, may be after a historic introduction, with the K&W version, or something similar. And if it is necessary to mention a reliable source that really defends the simple solution, or equivalently the combined doors solution, as a solution to the K&W version, I'm glad this may be directly followed by Morgan's remark this is not correct! It is precisely because of this mistake I started the discussion here, as I noticed a lot of web sites, from teachers, students, etc., following this wrong way of reasoning, based on the WP article. Then I must say, anyone normally thinks of the solution in terms of probability. So the main explanation must be in terms of probability. Of course it is not needed to mention the term "conditional probability", but we may speak instead of "before" and "after", but the distinction has clearly to be made. Nijdam (talk) 23:14, 5 February 2010 (UTC)

I forgot to mention that there is no such as an unconditional and a conditional solution. There are unconditional and conditional problems, differing substantially. So who in this stage of the discussion still speaks of an unconditional solution, definitely doesn't know what he is talking about.Nijdam (talk) 23:29, 5 February 2010 (UTC)

And IMO it is idiotic and misleading to present the situation of the conditional version, leaving the reader with two unopened doors, from which the equal chance error stems, and then give the solution of the unconditional version, as a solution to the presented problem. It is exactly this point Morgan is referring to. Nijdam (talk) 23:40, 5 February 2010 (UTC)


 * @ Nijdam, then I am a misleading idiot. But you think that we two agree what is the problem. We don't. My unconditional solution is to a different problem than yours. I call my problem the Monty Hall problem, you don't.


 * @ Richard, You have to read better. You may be an idiot, I really wouldn't know, but you're not misleading. Your last conclusion is right. Your and mine MHP are not the same. And it seemes that you agree with me that the simple solution is not the solution to what I call the conditional problem. There lies my main concern, and I hope you'll also concerned about this. Nijdam (talk) 17:36, 7 February 2010 (UTC)


 * I think the problem is not a priori a maths problem, let alone a priori a probability problem. I read the words written by Marilyn vos Savant (quoting her correspondent Craig Whitaker) on Parade. And I start to scratch my head. Craig's question was what should you do, not what do you think is the conditional probability under such and such additional information. I think you should switch. And I have written elsewhere why I think you should switch. I think furthermore that a conditional probability which you cannot even evaluate because you do not know the TV show's strategy in making their various choices is not "the answer" to the question originally posed. Gill110951 (talk) 06:32, 7 February 2010 (UTC)


 * A complete article may be devoted to the discussion about what the MHP was, is and will be, and how people have looked at it. I'm quite convinced that in the present the popular perception is it is more or less the K&W version and a probability problem. Nijdam (talk) 17:36, 7 February 2010 (UTC)


 * This is a valid point. The article indeed must cover the problem as a historical culture point, and in that sense is much more generally defined beyond the Morgan interpretation. That is why I call Morgan a "minority viewpoint" because most people are laymen and so interpret the problem simply. Andrevan@ 01:24, 8 February 2010 (UTC)

Martin Hogbin's opinion
The Monty Hall problem is essentially and notably a simple problem which most people get wrong. That is how it should, first and foremost, be represented in the article, as a simple, well-defined, non-conditional problem. This treatment should not be described as wrong or incomplete at this point. The simple solution section include clear and convincing solutions for the general reader and a further section giving aids to understanding the simple case. By non-conditional, I mean either an unconditional formulation or a treatment in which the possibility of a condition is postponed until a later section.

For those that believe that the non-conditional treatment is too easy I guess we must have a section on the somewhat academic case described by Morgan et al in which the host is considered to choose an unchosen door to always reveal a goat non-randomly. In this section we might state that some sources consider the non-conditional solution incomplete. This could be followed by other esoteric variants for those interested in such things.

In fact, the changes that I want are fairly minor, consisting principally of putting the current 'Aids to understanding' section, which makes no mention of the possible conditional nature of the problem, immediately under the 'Popular solution' section.Martin Hogbin (talk) 00:05, 6 February 2010 (UTC)


 * Martin, these were your words(?): It is just the same old thing again. Nobody is interested in the average probability of winning by switching. The only question to be answered is the probability after the player has chosen a door (say door 1) and the host has opened another door to reveal a goat (say door 3). If the host chooses randomly which legal door to open then this probability is always exactly 2/3 and, by reason of symmetry, the simple table at the top of this section is a mathematically valid solution. The, so called, condition, that the host has opened a specific door is irrelevant because it can be shown that the probability of interest is independent of the door opened in the symmetrical case. Although you in the last sentences try to deny it, you here state the conditional nature of the MHP, whether you like it (and understand it) or not. Nijdam (talk) 00:40, 6 February 2010 (UTC)
 * What do you mean by 'conditional'? Are you asserting that in the symmetrical case we need to consider which door the host opens? Martin Hogbin (talk) 09:50, 6 February 2010 (UTC)
 * Though I'm in the "conditional camp", I would agree to Martin's suggestion above in particular if this really helps to settle the argument for good rather than being just a pit stop for ongoing neverending arguments by the same actors. This might also an opportunity for the mediator to weigh in here. WP guidelines and sources more or less require to describe the unconditional solution, the conditional solution and the criticism of the unconditional solution. However there is no requirement for a particular order or structure.--Kmhkmh (talk) 15:15, 7 February 2010 (UTC)
 * Obviously I can only speak for myself, but my aim is to make sure that the basic problem section is clear and convincing for the general reader, with conditional issues coming later. See my comments in the 'I love it too' section on the article talk page for my ambitions for the page.  Martin Hogbin (talk) 16:54, 7 February 2010 (UTC)
 * @ Kmhkmh: "WP guidelines and sources more or less require to describe the unconditional solution, the conditional solution and the criticism of the unconditional solution". Yes. But: It is not the *solutions* that are critized, but the specific version of the problem which they solve. Moreover, the conditional *problem formulation* may be criticized too. Gill110951 (talk) 21:58, 7 February 2010 (UTC)
 * I have no objection of a criticism of the conditional solution (providing it is properly sourced).--Kmhkmh (talk) 21:45, 9 February 2010 (UTC)

Where is the mediation?
Andrevan@, if there is to be any chance of successful mediation here, you must make an effort to understand both sides of the argument. So far you have just given us your opinion on the matter. As a new and previously uninvolved editor you are, of course, free to state your opinion here and join in the discussion in any way you like but as a mediator it is your job to try to bring the two sides together, whatever your personal stance on the subject.

The real problem here is that we have reached a stalemate, with each side failing (it would seem) to fully understand the arguments of the other. Each side sits back, completely baffled by the inability of the other side to understand their perfectly clear (in their opinion) arguments. It is to the credit of all concerned that the discussion has not descended into personal attacks, edit warring, and other unwikipedian behaviour.

What would be useful would be for someone to get to understand what is being argued by both sides of the dispute and try to help us to find ways to make clear to the other side exactly what the point being made is. At the moment, both sides repeat the same arguments over and over again in the vain hope that the other side will truly understand the point that they are making. I hope you can help us break this deadlock. Martin Hogbin (talk) 10:01, 7 February 2010 (UTC)
 * Hear, hear! Gill110951 (talk) 13:59, 7 February 2010 (UTC)
 * Before we try to reconcile both sides, we need to make sure both sides are making valid arguments consistent with policy and precedent. That means we need to get past the idea that one's own mathematical opinions are in any way relevant. Andrevan@ 19:13, 7 February 2010 (UTC)
 * I can only repeat what I first said, which is that if there is to be any chance of successful mediation here, you must make an effort to understand both sides of the argument. I have no idea what makes you think that one side is making invalid arguments but, if you start from that viewpoint, mediation will clearly be impossible. Martin Hogbin (talk) 20:21, 7 February 2010 (UTC)
 * He didn't say only one side was making arguments inconsistent with policy and precedent - he also didn't exactly say either side was. The point is the same point I've been trying to make for ages, which is that whatever it is that you or I or even Gill thinks is irrelevant in the context of editing.  Wikipedia simply doesn't care what its editors think.  Wikipedia only cares what reliable sources say.  Very bluntly, if you're not willing to accept this, you really shouldn't be editing here. -- Rick Block (talk) 20:42, 7 February 2010 (UTC)
 * This is nuts. Of course my or anybody elses mathematical *opinions* (opinions about mathematics?) are strictly speaking irrelevant here ... but then what are we discussing here? How to reconcile our different opinions on how the article should be structured! I'm not aware that we are disagreed on any issues of fact. On the other hand, mathematical or logical truths which are evidently true whether or not anyone is around to believe them, cannot be denied. They might be considered to be irrelevant. Gill110951 (talk) 21:50, 7 February 2010 (UTC)
 * Maybe then Andrevan is suggesting that both sides are making arguments that are inconsistent with WP policy. This seems to me to be an odd place for a mediator to start.  It would seem best to me to assume not only good faith on the part of both sides but also that both sides are intending to act in a manner consistent with WP policies and objectives.  Until the arguments of both sides are fully understood it is not possible to even start considering these issues. Martin Hogbin (talk) 21:10, 7 February 2010 (UTC)


 * This is the first time that anyone in this debate suggests that anyone here is pushing own research or arguing against wikipedia policies of "reliable sources". For instance: very reliable sources tell us that Morgan et al. are not a very reliable source. Anyway, that's irrelevant. Morgan et al. present *a* MHP which is adopted by many others and obviously needs coverage in the article.

How much play in the Wikipedia article will the new book that is critical of Morgan get? So, let's get back to the 3 change proposals from December, 2009 where there was a consensus that the article needed some changing. Glkanter (talk) 20:31, 7 February 2010 (UTC)
 * The new book by Rosenhouse pays little attention to Morgan et al. It's a good book (!) and/though the author has a personal point of view and starts off with a personal and idosyncratic version of the Monty Hall Problem and an idiosyncratic proof. (Technically speaking: he asks for the conditional probability that switching gives the car given your choice and given a door has been opened, but not given which door). But he is not writing an encyclopaedia article! Gill110951 (talk) 21:40, 7 February 2010 (UTC)
 * Looks like a useful source to me. I am waiting to get a copy. Martin Hogbin (talk) 21:46, 7 February 2010 (UTC)
 * Amazon got it to me very fast!Gill110951 (talk) 21:53, 7 February 2010 (UTC)

Btw: A partial early version (60 pages) of Rosenhouse's book is available here: --Kmhkmh (talk) 23:13, 7 February 2010 (UTC)


 * We can progress to specific issues once everyone agrees that no sources are being thrown out based on our own analyses. Martin, you are quick to criticize the progress of the mediation, but I notice you are one of the ones calling for Morgan to be de-emphasized or thrown out because of issues you have with his treatment of the mathematics. This is now how Wikipedia source screening works. What is clear is that, since 1991, many have adopted Morgan's interpretation of the problem, and this impact must be covered objectively. Andrevan@ 01:19, 8 February 2010 (UTC)
 * I am not sure what you mean by 'progress to specific issues'; I have not suggested that we must do this. I have said that in order to mediate you must make an effort to understand both sides of the argument.  I hope that you will be able to do this so that mediation can commence.  Martin Hogbin (talk) 21:51, 8 February 2010 (UTC)
 * Is there something you think I don't understand? Have I mis-stated your positions? Andrevan@ 01:15, 9 February 2010 (UTC)
 * I think that would be something of an understatement. You ask if you have mis-stated my position.  As far as I can see you have not stated my position at all, neither have you asked for it. In fact you do not seem to have asked for anyone's opinion.  I can only ask you to re-read my opening post in this section, which is something of a simplification of the problem.  Although there are two broad camps of opinion here, each editor has their own view on the detailed way that they would like to improve the article.


 * Just as a matter of administration, is this the right place to be discussing the article? Might it be an idea to re-start the mediation process on an article talk page, with a view to bringing the two sides closer together rather than arguing about the mediation process. Martin Hogbin (talk) 10:35, 9 February 2010 (UTC)


 * I don't think you really understand how formal mediation works. It's formal in that the mediator dictates the course of the discussion and tries to be impartial, ultimately bringing the parties together to build a consensus. But there is no requirement that we limit our discussion here nor that all arguments are equally treated or represented, nor must everyone state their opinion, although it seems like everyone has done so. If you are unhappy with the way things are going you may suggest anything you wish, or give up on the process and progress to Arbitration, which frankly I think is grossly overestimating the level of disagreement here. All I've been doing so far is try to establish a common ground from which to advance to specific textual issues; I don't feel that I must defend my methods to you. Andrevan@ 20:17, 9 February 2010 (UTC)
 * A number of editors asked for and agreed to mediation because they have differing opinions. I do not understand how you can mediate without finding out the opinions of those editors.  I think several editors, including myself, will give up on the process if their views are not considered important.


 * I have never suggested that our discussion should in any way be limited, and my comment about where it takes place was just a straight question. If you want us to talk here that is fine with me.  Martin Hogbin (talk) 23:49, 9 February 2010 (UTC)


 * I have reviewed considerable amounts of debate. More than I needed to, to be honest. I don't think you could claim that I don't understand your opinions. Andrevan@ 04:45, 10 February 2010 (UTC)
 * Well @ Andrevan, you are not doing too badly, but to give moral support to @ Martin, I think you don't quite realise that some of the issues which people spend their time debating on here are rather subtle and I felt several times you were missing some of the subtleties. Moreover Martin has very carefully and thoroughly investigated Morgan et al (which most people in mathematics had never heard of, by the way) and has very carefully and objectively pointed out some defects of their maths anf of their representation of history, though they also made some nice original contributions too. And: next to their paper in the same issue of that journal was published a short note highly critical of their approach. We're talking about how to report on wikipedia how people have chosen in different ways to convert a popular problem about a quiz-show into something amenable to rational analysis. Rational analysis often means maths, their are disagreements here about that, and you say we should cut the crap (leave out the maths, and anyway, whether or not a certain mathematical statement is true or false is irrelevant. Even though there is (for truth) a simple mathematical proof which any intelligent person can follow, or (for falsity) a counter-example. There are two steps to be covered. The process of mathematization (that is called making a mathematical model). The solution inside of the mathematics of a well-posed mathematical problem (that is called doing mathematics).Gill110951 (talk) 15:51, 10 February 2010 (UTC)

The problem is not the solution: it is the problem. The solution is to admit this
Erase this section if you like but: one must distinguish between a verbal problem which you might pose at a party or in a problem section of a popular magazine; a precise mathematical specification thereof which people might study in books, pose as problem in a maths exam, or play with in a mathematical paper; and a solution of specific mathematical problem. Discussions here go round in circles because we do not distinguish which of the three levels we are talking about and because we make false assumptions that other people have the same informal problem, or the same formal problem, as ourselves. I think we might achieve agreement on the desired structure of the article if we would agree that there are three levels and that it is a matter of taste how each level is filled in. In wikipedia we should fill in each level with a menu of dishes which are "out there in the wild", ie are the subject of reliable sources. Other reliable sources supply criticism of other sources choices. So that should be documented too. Isn't it all rather easy, from that point of view? Room for everyone! Gill110951 (talk) 22:07, 7 February 2010 (UTC)
 * That sounds like a good way to approach it. Does everyone else agree with that? Andrevan@ 01:16, 8 February 2010 (UTC)
 * As long as we don't try to bury the more mathematcial analysis of the "popular" problem, I'm fine. I'll add that I don't think the basic issue is one of problem definition, but rather that proponents here of the "simple" interpretation do not like the fact that mathematicians have criticized this interpretation.  This seems to me to be a straightforward NPOV issue.  -- Rick Block (talk) 02:16, 8 February 2010 (UTC)


 * I made this point before, and tried to discover what the discussiants consider the MHP. So what is considered the MHP is one issue, but not all. As Rick says there is also the issue of the erroneous POV of some that the simple explanation is a solution to the full K&W or similar version. It is this last issue I fight, and I want the article to be clear about this. Nijdam (talk) 20:35, 8 February 2010 (UTC)
 * I essentially agree with Rick & Nijdam here. Looking at the problem from not mathematical perspective is fine (as long as it is properly sourced), also structuring the article towars the different aspects. The big no-no from my side only comes if people are implicitly or explicitly, subtle or blunt pushing to remove the conditional approach (and morgan's crticism) from the article (basically what rick calls the NPOV issue above).--Kmhkmh (talk) 21:52, 9 February 2010 (UTC)
 * The conditional approach is the right approach to the conditional problem. Morgan et al screw up their mathematics (in one of their extensions) and falsify history (in stating the problem so as to make their opponents sound ridiculous). There are correct and clean and fast solutions to the conditional problem, the symmetry argument for instance. But if you prefer to use decision trees or tables of probabilities and Bayes's formula, you are free to do that too. Also, there is a correct and good mathematical analysis of the popular problem. ie one has the option, and many highly reputable mathematicians even took this, to take the unconditional problem to be the good mathematization of vos Savant's question, and to give a correct mathematical solution of this mathematical problem. Since the solution is so easy it is not much use as an exam question in Probability 101. Reputable mathematicians who like the unconditional version go on to extend it to the quantum domain (eg, yours truly), or embed it in a game theoretic point of view. (Where game theorists and economists and computer scientists had already seen it).Gill110951 (talk) 15:37, 10 February 2010 (UTC)
 * I will not object to also pose the "unconditional" MHP, as long as it is crystal clear what it means. IMO most people understand the simple solution as a solution to the conditional problem. And not only is it understood this way, it is often considered to be so. Let us make clear what the "unconditional" MHP is. I actually am only interested in the much more simplier (!) conditional form. Nijdam (talk) 22:35, 10 February 2010 (UTC)

Colincbn's opinion
Hi Guys, I've been waiting a bit to see what the other editors who have spent more time on this had to say before I put my two cents in. In addition most of what I'm about to say I have said before on the MHP article's talk page but I will repeat myself here for the sake of expediency.

I really don't know much about probability and mathematics, I found this article while reading about paradoxes and this one is great in that I got to try it out in the real world (I used a deck of cards and my sister in-law as the contestant). As I understand it the "Monty Hall problem" as is generally understood and taught in most probability 101 courses states that the host chooses randomly, so most people will approach this article with that understanding and any other discussion about host behavior should come after covering that version, possibly in a "Variants" section or under "Other host behaviors" or something along those lines.

I have never read the Morgan paper nor any other paper on the MHP (or on any mathematics for that matter, I'm a biology and history guy), but I do read a lot about WP policies and whatnot. It seems to me that if (and I'm taking statements made by others in good faith for this) the Morgan paper is not dealt with, covered, or discussed by the majority of sources writing about the MHP it falls under the guidelines for (and please believe me that I'm making no judgement on Morgan or his paper at all when I say this) "fringe" theories and the like (I don't like that term for it but those guidelines seem appropriate for covering an opinion that is not held/discussed by the majority of sources; It should be noted that this has nothing to do with whether or not Morgan is right or not, just with the level of coverage of his version of the MHP. As an example: the idea the dinosaurs were warm blooded would have fallen under the same "fringe theory" guidelines when the ideas were first put forth until enough new information in reliable sources became available and the majority scientific viewpoint changed [Edit:after looking at the sources cited here, which all cite Morgan et al., it seems to me most of them are citing Morgan's treatment of the psychology of the MHP and are actually still describing the unconditional version of the problem mathematically]). The fact that Morgan et al. has been discussed and covered to some extant would seem to me to indicate that while it may not be the majority opinion on the MHP, it certainly deserves to be mentioned even warranting its own section and possibly its own page. There could be a section of the MHP article dealing with Morgan's paper with a "main article" link to a separate article that goes into Morgan's conditional problem in detail. I feel this will give the casual reader all the info he/she is looking for on the MHP with an easy way to delve into the mathematics of conditional probability more deeply if they want. It should be noted that while in most of the WP editing world a subject having its own article is considered a 'good' thing for fans/proponents of that subject, in this discussion Morgan's paper getting its own article has been considered "banishing" or "burying" which is not my intention in any way.

And please remember my opinion has absolutely nothing to do with Morgan's views on the MHP or his math as I have never read his paper nor would I understand it if I did. Colincbn (talk) 03:08, 9 February 2010 (UTC)
 * Colincbn, I agree with your proposal. Morgan, historical background & auxiliary conditional maths for enthusiasts plentiful and sufficiently elsewhere in detail, separating wheat where it belongs, concentrated and in detail. Could be a good idea, helpful in this difficult mediation process. See also |"unconditional". -- Gerhardvalentin (talk) 20:03, 9 February 2010 (UTC)
 * Elsewhere meaning what exactly?--Kmhkmh (talk) 21:37, 9 February 2010 (UTC)


 * There appear to be independent sources that support Morgan's conditional problem as well as those critical of it. Fringe theories are generally much wackier. That being said, I think the main thrust of your point is a good one. Does everyone feel OK with the idea of presenting the the unconditional problem first before talking about the conditional probability? I don't think a separate article is necessary, but certainly a separate section seems reasonable. Andrevan@ 20:28, 9 February 2010 (UTC)
 * The notion that Morgan (or better his solution the conditional approach) is not used or discussed in other publications is wrong and not in line with the facts. Such claims were unfortunately repeatedly wrongly stated throughout the discussion and seem to perpetuate. I'd like to point out that many/most papers dealing with the problem contain a conditional approach (in particular many of the currently used sources). The 2 books on the problem (randow, rosenhouse) feature the conditional solution prominently as well. So please stick to facts. If we start to argue under the notion that the conditional approach is "fringe" the mediation is going nowhere (not mention in such case we'd through the basic wikipedia principles overboard).--Kmhkmh (talk) 21:35, 9 February 2010 (UTC)
 * Colin, can you respond to this? Andrevan@ 22:13, 9 February 2010 (UTC)


 * Well like I said I was taking things said by others in good faith as my basis for assuming that the conditional version is not the standard approach to the MHP in mathematics papers. That being said in the small amount of time I spent looking at the MHP related papers listed under the Google citations page linked above it seems to me that in almost all of the psychology papers the unconditional, or simple, version is used. I think the main disagreements here are related to whether one feels the MHP is a 'pure math' or an 'applied math' problem. The truth seems to be that it is actually both depending on how you approach it. That being said having the article use the 'simple' unconditional version first then lead into the psychology of the problem would cover how most people who have heard of the MHP came to it (by most I am including those who come to it as a problem in human psychology as well as high school and first year statistics/probability students). Then leading into the discussion on conditional probability and Morgan etc. would follow the path of learning that most accurately represents the way people come to this problem in the "real world"; ie: first one hears about this wacky game show math problem that most people intuitively get wrong, then you learn the 'simple' math behind why they get it wrong, after that those who have a bent towards math look into the formulas more closely and realize that how the initial problem is stated makes a big difference to the solution, as well as the implications of game theory and the like.
 * By structuring the article this way I don't feel that Morgan will get 'buried' or 'banished' or anything like that. I feel that the in-depth mathematical analysis of the problem will be put in the most logical place considering how most people who have never heard of the problem, or have only heard of it in passing and want to learn more, will come to it.


 * In addition let me say that I don't like the word "fringe" either, I only meant that if most people writing about the MHP (and I am including laymen and psychologists when I say "most", I understand that mathematicians will most likely include both the conditional and unconditional versions when they write about the MHP and give their own viewpoints on both) use the simple unconditional version than the guidelines used for fringe/minority viewpoints seem appropriate for covering this issue. That does not in any way mean that Morgan should be discounted or anything like that. It simply means covering his viewpoint should be done in a way that accurately represents the weight it's given in the sources. In this case while the 'majority' of people who write and talk about the MHP seem to use the simple version, Morgan's views and the conditional version are obviously important and are discussed in mathematics circles extensively. So while I feel we should start the article with the simple version and the psychology of why it is counter-intuitive, I still feel that for a thorough dissertation on the MHP Morgan and the conditional version must be included.


 * Also I apologize for the delay in my responses, I live in Japan and am usually logged into WP when most editors are sleeping. Colincbn (talk) 04:46, 10 February 2010 (UTC)
 * This a point that is really bugging me, so to be crystal clear and to avoid that this misinformation is perpetuating through this mediation s well. It is true that there is indeed a large number of publications (in particular books and papers in psychology, puzzle and brain teaser books) that only use an unconditional solution. However in many of these cases the MHP is mainly used for illustrative purposes and is just a "side issue". In such a context it is perfectly reasonable to spend a minimal amount of resources on its treatment and requiring as less background knowledge from readers as possible. But this is not the context we have in WP, since we have the resources (time, space,editors) and a rather wide heterogenous range if readers. Also having said the above, if we now look at mathematical/statistical publication on MHP or non mathematical publication that deal with the problem in greater detail, then in most of them the conditional solution is covered (and often the primary approach to the problem). This is in particular the case for the 2 main psychological papers used in the article (Krauss/Wang and Muser/Granberg), for many of the math paper and books currently cited in the article or throughout the discussion (Morgan at al, Gillmann, Behrens, Rosenthal, Grinstead/Snell, Rosenhouse, Eisenhauer, Selvin (2nd letter), Henze). As mentioned earlier already the 2 popular science books solely devoted to MHP (Randow, Rosenhouse) feature the conditional solution prominently. Almost any college course (as well as advanced highschool classes) for introductionary probability theory or statistics covers basic conditional probabilities and Bayes' theorem. This means, if such a course covers the MHP at all, it is rather natural (and to my experience quite common) that they cover it to the very least with Bayes as well. This is also reflected in recent textbooks that mention the MHP problem, in particular Grinstead/Snell (used in the article) and Henze (a rather popular german probability primer, mentioned in earlier discussions). Other examples that you can check yourself via Google Books are for instance:
 * Elementary probability, p.75, Topics in contemporary probability and its applications, p. 359, Probability and random processes, p.23, Applied probability and statistics, p. 25, First Course in probality, p. 154, Stochastics: introduction to probability and statistics, p. 55, A Probability and Statistics Companion p. 46, probability and statistical inference'' p.90.
 * I hope this clears the issue up for good and we can move on towards an agreement on how to best structure the article.--Kmhkmh (talk) 13:33, 10 February 2010 (UTC)
 * Why is Monty Hall in Statistics 101 (or Probability 101)? Because *if* you pose the conditional version of the problem along with the assumptions of uniform initial hiding of the car and uniform opening of a door, you have a nice exercise in conditional probability for Probability 101. Selvin, Morgan et al, my good old friend Norbert Henze, Grinstead and Snell, ... are more or less reliable sources for Probability 101. Morgan et al make an awful error in one of their proofs. Selvin asks an ill-posed question and wants his students to guess what he would traditionally write down as the "obvious, logical" assumptions, though they are not. The other guys are a bit more reliable and careful in their mathematics. There is a tradition among teachers of Probability 101 that Monty Hall is a conditional probability problem. There is a tradition in game theory that Monty Hall is a game theoretic problem. There is a tradition in psychology that says Monty Hall is a psychology problem. Actually the psychology of why people usually give the wrong answer is the subject of psychology, one can say that the game theoretic approach tells us how one should train one's good psychology. There is a beautiful book about predicting the future by a guy who advises the White House and has a company advising big companies and who is a Princeton professor in the Economics Faculty, who says that you can predict the future by figuring out the various party's prior distributions and their cost functions (ie their aims) from statistical analysis of their behaviour in the past (assuming their behaviour follows traditional game theory); now you are able to predict how they will react in the future to moves by their opponents, since you know what it is they are actually (subconsciously) trying to optimize. This approach also works well in analysing people's behaviour in personal conflict situations. People actually behave logically as if they had followed Game Theory 101. However, they use rather strange prior probabilities and have surprising costfunctions. It is useful to get these out onto the table. BTW the psychology of giving the wrong answer in the MH problem is that one wants to save brain computer processing time, by discarding some of the information, so as to have an easier job making a conclusion. This is why people jump to conclusions and why they are much more confident about their conclusions than they have a right to be. It has been an evolutionary useful strategy. If analysing a problem, and using all information properly, costs money too, the optimal decision solution may well be "the wrong one". Gill110951 (talk) 15:10, 10 February 2010 (UTC)
 * I'm not sure what this supposed to tell me. I don't argue that the problem has to be posed as a conditional problem. I'm just pointing out that the conditional approach or posing it as a conditional problem is common treatment of the problem in probability 101. Apparently this needs to pointed out, because people (still) put forward the notion that the conditional approach or posing would be rare, almost fringe, not covered in probability 101 and mostly only appear in error riddled publications by a very few mathematicians. With that notion i have an issue not with unconditional or non mathematical approaches to the problem. As long people don't drop this false notion, there is nothing to mediate from my perspective. Before we mediate anything we need to get the facts straight.--Kmhkmh (talk) 15:46, 10 February 2010 (UTC)


 * @Colincbn, I have been repeatedly suggesting here to take Marily vos Savant's literally quoted correspondent Craig Whitaker's problem as "The Monty Hall Problem", or at least as starting point for the article, despite the Selvin etc prehistory in Stats 101.


 * Marilyn vos Savant launched MHP into the *real* world; from then on it was open for everyone to put in their two cents worth. And her problem is still the problem which people tell one another in pubs and at parties and which they quote in papers. Only probability professors think that the conditional probability problem is "the" problem; some of them are able to formulate and solve it properly, others screw it up. BTW Rosenhouse (who works in algebra, not in probability) poses yet another mathematical problem as far as a probabilist is concerned; his conditional probability is a different conditional probability -- a different condition (though not coincidentally, it's the the same number: Symmetry!!!) as Selvin's. [I keep thinking of the fantastic Tom Stoppard play "Rozenkrans and Guildenstern are dead"].


 * Then, I propose we realise that if you start with vos Savant/Whittaker there now arises a "Monty_Hall_Problem problem", which is to decide how to tackle the problem. You could ask, for instance, how to get the problem into applied mathematics language so that after that, you could go to a pure mathematician to have your *internal* algebra/logica/calculus checked. (But you might not want to mathematise it, you might want to see it as pure psychology). In the case you go for a mathematization, you might want to reformulate it, or to sharpen it's formulation, so that it is amenable to your pet kind of mathematics (eg: Probability 101 if that is what you are teaching, or Game Theory 101 if that's the course you're giving). So here we get a branching of MHP formulations/reformulations/refinements with all kinds of solutions, correct solutions and incorrect solutions relative to the terms of the problem which have been provisionally (implicitly or explicitly) put down.Gill110951 (talk) 15:24, 10 February 2010 (UTC)


 * It seems to me that I am not fundamentally disagreeing with Kmhkmh, Gill, or any of the major points brought up by either 'side' of this debate. I am perfectly willing to accept that in mathematics books and papers both versions are covered, meaning that the conditional approach is not discounted among experts, and is even possibly the preferred version (although that seems to be a point in contention I cannot personally claim either side to be right), and therefore should not be left out or 'minimized', I just feel that the psychology and puzzle books etc. add to the weight of sources using the unconditional, or 'popular', approach and therefore it cannot be minimized or discounted either. Admittedly when I first came to this discussion I assumed that "The MHP" implied that the host chooses randomly (given that he has a choice) and so any other treatment was a 'variant', however there are a number of editors and referenced sources, including Morgan, that feel this is not the case. I am more than willing to concede that point as I am by no means an expert on this subject.


 * So to bring this into a content/style debate:
 * Because a large number of laymen, psychologists, and puzzle books use the MHP to show how people break up problems in order to be easier understood, I feel that the explanation of the 'Popular solution' should then lead directly into 'Sources of confusion' and then 'Aids to understanding'. This is mainly because those two sections mostly cover the MHP from the popular standpoint, and I feel the article flows more logically that way. After that leading into the 'Probabilistic solution', including the detractors and controversy if there are reliable sources that document it (possibly adding the points raised by Gill above about the "Monty_Hall_Problem problem" if there are sources that can be referenced), followed by the 'Bayesian' section and ending with 'Variants' is, in my opinion the best way to organize the article. And I feel I cannot overstate that I am in absolutely no way trying to 'bury' the conditional approach. It is not a matter of the accuracy of the math for me as I really cannot speak to that considering I am not a mathematician. In fact I think the 'content' of the article as it stands right now is actually quite good and I am mainly just proposing a re-organization for the sake of ease of reading. A simple Cut/Paste would satisfy my main points. Colincbn (talk) 04:01, 11 February 2010 (UTC)
 * I have no objection against separate sections and having the unconditional solution first. In fact I'd like to restate a suggestion that I've made earlier in the discussion and which probably also similar to what Gill110951 has in mind. Create separate sections such as popular solution (as a puzzle problem without math), mathematical analysis (unconditional & conditional,explaing the difference, Morgan's criticism, some of the variants, game theory), psychology (sums up the content and aspect of the psycology papers), history (from gardner, selvin up to now), etc.--Kmhkmh (talk) 04:18, 11 February 2010 (UTC)


 * May I ask how you feel about Martin's suggested organization listed below? Colincbn (talk) 12:28, 11 February 2010 (UTC)
 * I could live with Martin's suggestion for the structure, though i personally prefer a different structuring. However i may not be able to live with exact content Martin's wants to assign to those sections (for details see my answer below).--Kmhkmh (talk) 21:02, 11 February 2010 (UTC)

Glkanter's opinion
The MHP begins with Selvin. There is no reason to ignore or minimize his contribution. In addition to the obvious logical argument that the host of a game show must act randomly (uniformly, symmetrically) when faced with 2 goats, Selvin wrote in his second letter to The American Statistician:
 * "The basis to my solution is that Monty Hall knows which box contains the keys and when he can open either of two boxes without exposing the keys, he chooses between them at random."

Any discussion of host bias, non-random, etc. is clearly taking the MHP into a different direction than intended. Which may be fine for a text book on probability that finds this problem a 'fun' place to start a discussion on conditional probability problems, but it's not the MHP. It only perpetuates one of Morgan's errors. Glkanter (talk) 12:34, 10 February 2010 (UTC)
 * The MHP (or for that matter any term in WP) is not defined by the original intent of its first author (alone), but by what the subsequential reception and treatment in reputable sources made of it. For instance communism is not (solely) defined by Marx (and his intentions), nor is Calculus defined by Newton. The conditional approach by Morgan is no "error", but a standard mathematical treatment of the problem (see other article sources and the textbooks above listed at Google above).--Kmhkmh (talk) 13:52, 10 February 2010 (UTC)
 * Morgan and conditional maths do refer to another scope that could be named "Mathematical treatment of various versions similar to MHP using Bayes", where Morgan et al and Bayes' conditional probabilities could largely and in detail be presented for interested groups of students in mathematics, but such lemma distinctively differs from the virtual paradox of the MHP, and it never should be confusingly mingle-mangled with this famous and interesting paradox. So put it in a separate short section, naming that impressive historical panorama, with a link to a detailed own main article. See also "Simple solution - combined doors". Regards, -- Gerhardvalentin (talk) 14:40, 10 February 2010 (UTC)
 * As long as the article is not too long there is no need for such step. Commonly you treat all aspects of term in one article until it becomes too long and then split in separate articles and overview article (still mentioning all approaches but linking to main articles for the detailed treatment. The conditional solution or approach is really linked to the problem from the beginning and intentionally ignoring that is simply POV pushing. Clearly separating both approaches (and other aspects) into separate section within one article however is fine. I have no objections to that in fact i envision that as one likely outcome of the mediation. However that is only going to work if people stop pushing for the article to describe their favoured approach as the "real MHP" (this goes for people in both camps). Looking at the mediation so far, I still get the feeling that this sometimes reads as a continuation of the spat between vos Savant and Norgan, too much ego and too little information (ala "WP readers needs to see through my eyes and my eyes only no matter what reputable sources might argue").--Kmhkmh (talk) 15:26, 10 February 2010 (UTC)


 * Moreover, @ Kmhkmh, the "obvious logical argument" that the host must act *uniformly* randomly when faced with a choice is actually wrong. Selvin makes this assumption in his solution, but he does not make it in his statement of the problem. He needed the assumption in order to do the calculations leading to his solution. That does not make that argument "obvious" and "logical" and correct argument. Thus from the point of view of mathematics, Selvin's problem is ill-posed and his solution is debatable. The whole MH story is a comedy of errors. Nothing wrong with that. It is nobody's specific property. It is part of living culture with a living, evolving history. Wikipedia should give a neutral point of view. Exactly as on communism or on calculus (as defined by Leibniz, perhaps?).Gill110951 (talk) 14:53, 10 February 2010 (UTC)
 * I'm not sure why you address me here. Glkanter was talking about the "obviously logical argument" not me. I don't disagree with you on that and regarding your last line that was exactly my point. However it is not the point that glkanter and gerhardvalentin are making, they pushing for unconditional (or even non mathematical) only article, which is precisely what I object to here.--Kmhkmh (talk) 15:01, 10 February 2010 (UTC)
 * Not pushing for "non mathematical", but for clearly unscrambling two different aspects that should not inextricably be mixed: Pushing for a clear distinction between the MHP and its various possible meanderings. To distinguish and keeping them clearly apart. Regards, -- Gerhardvalentin (talk) 15:14, 10 February 2010 (UTC)
 * If you only want separate sections within in one article, please state that clearly, because at other times you did not do so (at least it was not apparent to other readers like me).--Kmhkmh (talk) 15:28, 10 February 2010 (UTC)
 * I just say what I read above: I feel OK with the idea of presenting the unconditional problem first before talking about the conditional probability, and I don't know whether a separate article is necessary, "but certainly a separate section seems reasonable." -- Gerhardvalentin (talk) 15:53, 10 February 2010 (UTC)

I don't understand the thought process that allows, when discussing a story problem, to eliminate any explicit premises. Selvin made it clear in his second letter in 1975. Glkanter (talk) 15:25, 10 February 2010 (UTC)
 * True, but not in his first letter. People often only refer to or reference his first letter, not to his second letter. But sorry, there is a problem that people understand "Selvin" to mean different things: the person, his first letter, and the collection of all his letters. Gill110951 (talk) 15:58, 10 February 2010 (UTC)


 * Unfortunately, Whitaker didn't mention this premise, vos Savant wasn't aware of it, and Morgan wasn't aware of it. Did the premise cease to exist because of 3 uninformed 'people', one of whom, at least, who should have known better? I've always presumed papers were written between Selvin and vos Savant. Did these papers also eliminate the 'random 2 goat choice' premise? Glkanter (talk) 16:20, 10 February 2010 (UTC)


 * And, so, for Whitaker's question is it an appropriate assumption or not? Is vos Savant's solution, which as Selvin says is based on this assumption, fully correct if she never mentions this assumption?  Is there anything in the least confusing about saying "Here's a simple solution ...[attributed to some source]  This solution is based on the assumption that ... [attributed to some source - it's tricky to use Selvin for this unless the simple solution is Selvin's]  A solution that explicitly uses this assumption is ... [attributed to some source] The reason this assumption is required is because if you don't make this assumption the situation is like this ... [attributed to Morgan et al. or any of the other similar sources]".  If you want to defer the bit about not making the assumption until a variant section, that's OK with me, but the other bits ("the solution is based on the assumption that ...", and "a solution explicitly using this assumption is ...") seem important to include immediately following the simple solution.


 * Like I've said before, Nalebuff's paper was published (in 1987) between Selvin (1975) and vos Savant (1990). His problem statement (quoted in this thread) doesn't make this assumption clear (not entirely sure about his solution, I don't have his paper handy at the moment but I don't think he makes it clear in his solution either).  Phillip Martin's bridge column included a version as well (in 1989) which doesn't make this assumption clear (in the problem statement or the solution).  As I've also said before I haven't been able to find anything that was published in the interval between Selvin and Nalebuff, although Nalebuff says "This puzzle is one of those famous probability problems, in which, even after hearing the answer, many people still do not believe it is true" - clearly implying it was famous (at least within academia) by that point and, if not through publications, then how?  In Phillip Martin's "afterthoughts" (here) he says he heard of it from a friend in the 1970s who had run across it in a college probability course. Nalebuff doesn't say where he got it from. There was a mention of it in Mathematical Notes (a newsletter) from Washington State University shortly before vos Savant's first column (I don't have this source, the specific reference is in Barbeau's survey - which I also don't have handy at the moment). I don't know where Whitaker heard of the problem, but it would be interesting to compare Whitaker's version to the one in Mathematical Notes from WSU.


 * Although all of this is relevant for the "History" section of the article, like Kmhkmh says for the main body of the article (the "Solution" section, in particular) what the original historical sources say is much less important than what the preponderance of sources have said subsequently. -- Rick Block (talk) 21:21, 10 February 2010 (UTC)


 * One possibility is to do away with the "Solution" section entirely. Wikipedia is not a math reference work and so "Solution" is really not in the scope of our coverage. Rather, one might make the entire article into a "History" section and incorporate the various problems and solutions into that framework, with chronological structure. Andrevan@ 00:55, 11 February 2010 (UTC)

I would like Glkanter and anyone else who disagrees to respond specifically to Kmhkmh's statement: "The MHP (or for that matter any term in WP) is not defined by the original intent of its first author (alone), but by what the subsequential reception and treatment in reputable sources made of it. For instance communism is not (solely) defined by Marx (and his intentions), nor is Calculus defined by Newton. The conditional approach by Morgan is no "error", but a standard mathematical treatment of the problem (see other article sources and the textbooks above listed at Google above)." Andrevan@ 00:56, 11 February 2010 (UTC)


 * I don't agree. Premises to a story problem exist for a reason, and together comprise the story problem. Changing premises changes the problem. Glkanter (talk) 02:11, 11 February 2010 (UTC)


 * The Wikipedia article isn't about a "story problem." It's about anything and everything that comprises "Monty Hall problem" as that exists in the minds of the public and specifically as the secondary sources describe it. The differing interpretations are similar enough to be in the same article, but if there was also a TV show called Monty Hall problem, we would cover that as well, although most likely in a different article since it's drastically different and would make one article pretty long. The Monty Hall problem is not just a word problem but the whole concept we describe as the Monty Hall problem, just as the original formulation or coinage of a word is not the only way we describe the word in Wikipedia. Andrevan@ 03:38, 11 February 2010 (UTC)

I came to get an answer
OK, I know that this is an encyclopedia so and when I answered incorrectly (50 -50) during a bar discussion (spittin' - lying - etc.) it was suggested that I look it up on Wikipedia. Holy crap! About 5% of the article was enough to convince me of the error of my stupid logic. Thanks for that anyway - the rest of the article made me feel even stupider. OK, so I made this up but I do believe that it represents a the majority view of most lookers (OR and all that aside). hydnjo (talk) 02:14, 11 February 2010 (UTC)


 * You just 'think' you understand your error. Many editors here would tell you otherwise. But I'm not one of them. Glkanter (talk) 02:51, 11 February 2010 (UTC)


 * No, I know my error. It was my misunderstanding (a long time ago) of the fact that my original chance of winning could be improved (by a factor of two) if I behaved rationally and switched when given the opportunity to do so. I didn't think so at first but after reading the rationale in the first few paragraphs (which is why this article exists) I understood perfectly. I have contributed much on this talk page ages ago to help others understand the basic underlying concept of this undeniable conclusion and have helped others  mount a defense of this being a FA (which, I believe it merits).


 * The main question to be resolved by this article's mediation is obvious. This entire dispute has to do do with "other" considerations. These should of course be addressed but this amount of rhetoric is ridiculous. Once the main obfuscation has been addressed then the article (IMHO) has done its job.


 * So, it seems that this entire dispute has to do with how much article space is to be devoted to the esoteric, conditional, well cited, academic (and sometimes conflicting) treatment for whatever reason but out there and published. both good and and bad.


 * I'm subscribing to Martin's organization:


 * Simple (non-conditional) problem and solution - fully explained with aids to understanding and no confusing mention of conditions.
 * Conditional problem
 * Game theory
 * Variants
 * History


 * Because I can't come up with anything better. hydnjo (talk) 05:48, 11 February 2010 (UTC)


 * addendum: Of course we should be rightfully be accused of a Yahoo style article if we ignored the academic treatment of this complex problem. All I'm saying (along with everyone else I suppose) is how much emphasis (article space) should be allocated to each of the above topics. So long as everyone digs in to their respective positions then Andrevan doesn't have a chance. I don't have any fix for this except for all to back off and come back with a respectful response in the form of an article outline with perhaps a "size" for each perspective - it would be a start. hydnjo (talk) 06:17, 11 February 2010 (UTC)


 * addendum #2. I also realize that this amount of article reorganization will certainly call for a FAR by those who have felt "mistreated" with this mediation. Of course they will, their way or no FAirway. So, Rick and all that have stood behind this article and your leadership will continue to do so and keep this a FA. hydnjo (talk) 06:39, 11 February 2010 (UTC)


 * addendum #3. @Andrevan: No disrespect, I'm just trying to get this off of the dime that it's been stuck on for quite a while now. hydnjo (talk) 06:51, 11 February 2010 (UTC)


 * I think the onus is on those who disagree with this organization to propose an alternate. Andrevan@ 08:12, 11 February 2010 (UTC)

The first thing to note is that this is no so different from the current article, which has:


 * 1 Problem
 * 2 Popular solution
 * 3 Probabilistic solution
 * 4 Sources of confusion
 * 5 Aids to understanding
 * 5.1 Why the probability is not 1/2
 * 5.2 Increasing the number of doors
 * 5.3 Simulation
 * 6 Variants - Slightly Modified Problems
 * 6.1 Other host behaviors
 * 6.2 N doors
 * 6.3 Quantum version
 * 7 History of the problem
 * 8 Bayesian analysis
 * 9 See also
 * 9.1 Similar problems
 * 10 References
 * 11 External links

To add some detail to my proposal, I suggest that we should have:


 * 1 Problem
 * 2 Simple solutions
 * The (non-conditional) problem and solutions - fully explained with no confusing mention of conditions or suggestion that it does not answer the problem 'as asked'. These should be given as solutions to the K&W unambiguous formulation of Whitaker's (and Selvins's) question


 * 3 Aids to understanding - There is no mention of any conditions in this section as it currently stands
 * 3.1 Why the probability is not 1/2
 * 3.2 Increasing the number of doors
 * 3.3 Simulation


 * The sections above are for the average non-specialist reader. The simple (non-conditional) problem is plenty hard enough for most people.  It is the problem that most people (including famous mathematicians) get wrong, and the above sections refer to what vos Savant got all the letters about - the notable Monty Hall problem.


 * Having introduced and explained the non-conditional problem, we go on to introduce the possible issue of conditionality.


 * 4 Conditional problem and solution
 * Including the Morgan paper and why they believe the simple solution does not address the problem 'as asked' and the distinction between conditional and unconditional formulations.


 * 5 Game theory solution - With reliable sources
 * 6 Variants - Slightly Modified Problems
 * 6.1 Other host behaviors
 * 6.2 N doors
 * 6.3 Quantum version
 * 7 Sources of confusion
 * This is more about psychology than mathematics so should stay here.


 * 8 History of the problem
 * 9 Bayesian analysis
 * 10 See also
 * 10.1 Similar problems
 * 11 References
 * 12 External links

By 'non-conditional', I do not mean unconditional but that the issue of which door the host might open is not considered. We do not need to state the problem in an unconditional formulation at the start. This should be done in the 'Conditional' section for comparison purposes.

No doubt there will be some continued discussion of details but I think this format could be acceptable to most editors here. It follows the normal format of many mathematics text books or papers of giving a simplified version first followed by a discussion of the details. It also separates the simple and notable problem, which most readers will be interested in, from the more academic complications, which many readers will not be interested in.

I do think most academics will understand and accept this approach as we mention and fully discuss the academic extensions later in the article. If mentioned at the start they simply obfuscate the main problem. Martin Hogbin (talk) 10:26, 11 February 2010 (UTC)
 * I think this is essentially the same thing that myself and others have also suggested (although I might put 'Bayesian analysis' above variants), as such I would fully support this method of organization. And at the risk of being repetitive let me say that this is not because I think that the conditional approach should be 'secondary' or anything like that, for me it is a simple matter of creating a flow from simple to complex mathematics, and from the popular understanding of the MHP to a more specialized understanding. Colincbn (talk) 12:25, 11 February 2010 (UTC)


 * I'm fine with the separate section but i'm not fine with the content you prescribed to them. I t seems to me that there is some confusion about the unconditional problrm is supposed to be. The distinction between ambiguous and non ambiguous versions of the problem is not the same as the distinction between unconditional and conditional versions. This is exactly where the article needs to be precise, because otherwise we end up telling fuzzyy nonsense to the readers.
 * "We do not need to state the problem in an unconditional formulation at the start." <-- this is exactly what we cannot do. If we have a mere "puzzle or non mathematical" - section you could do that. But as as soon as you attempt any mathematical solution/explanation you need to state unconditional or conditional versions of the problem and be rather clear about that, otherwise you just have a faulty analysis.
 * I don't mind having a "layman" or "puzzler" solution first without mentioning conditions, however this cannot come across a unconditional mathematical approach. For the unconditional math treatment, we need a precise unconditional version of the problem. That's why i rather have a "popular section" (comes first) and a "mathematical analysis section" (comes later). The latter then contains precise formulations for an unconditional and conditional versions, solves both and explains the difference between them.
 * --Kmhkmh (talk) 12:40, 11 February 2010 (UTC)


 * I'm not sure we all mean the same thing by "conditional" and "unconditional", or Martin's highly unconventional "non-conditional". I'd rather talk about whether the problem statement is symmetric or not.  What I think we all agree is that the symmetric problem is the one that should be discussed first.  For the symmetric problem there are two different sorts of solutions.  An unconditional solution (like vos Savant's) and a conditional solution (like Morgan's).  In my view, the discussion of the symmetric problem is not complete without including a conditional solution.  Morgan et al. and others are NOT talking about a different "conditional" problem, or some "academic" interpretation of the popular problem.  They're talking about the popular solutions to the popular problem, and criticizing (most of) these solutions because they fail to mention the assumption that the host chooses between two goats randomly.  In Martin's proposed outline, he's saying we effectively have a complete article before we get to anything related to this criticism, effectively making the article take the POV that this criticism is of "academic interest" only.  In my opinion, this is a gross violation of NPOV.


 * My suggestion is to start with the symmetric problem, present both the simple (unconditional) solutions and a conditional solution (explaining the difference) as equally valid approaches in an NPOV manner, present sources of confusion and aids to understanding, and then the variants (including the game theoretic approach). I think the only question is where to put the direct criticism of the unconditional approaches.  It's currently in the "sources of confusion" section.  This seems like a reasonable place to me.  Since the criticism is one-way (I've never found a source that says approaching the problem conditionally is incorrect or incomplete!), putting this criticism in the "solution" section is arguably POV.  -- Rick Block (talk) 19:52, 11 February 2010 (UTC)
 * Well there is still a problem that people talk about different things. My notion of the unconditional problem (not a possibly incomplete unconditional solution) is simply computing the total probability for switching (which iirc most of the cited unconditional approaches) do. That's perfectly alright for a puzzle approach (as a heuristic substitute for the conditional probability) or as mathematical approach if you restate the problem in terms of asking for the total probability. So far no symmetry condition is needed. Symmetry comes only into the equation if you attempt to compute the conditional probability or if you want to prove that the total probability and the conditional probability are the same. However right now I'm not aware, which source (if any) addresses that explicitly. Do we he a source doing a computation without conditional probabilities explicitly citing a symmetry argument?
 * Another thing is, that if you are unwilling to accept the separation of "unconditional" and "conditional" (even if it is not your preferred approach), then there isn't much to mediate unless somehow expect the mediator to miraculously convince the others that your position is the best one. Given the history of the discussion this mediation won't come for free. The cost of an agreement requires that all involved are willing to accept that the article will deviate from their personal notion of the "perfect article" as long as its content is correct and within WP guidelines.--Kmhkmh (talk) 20:59, 11 February 2010 (UTC)
 * What is the(?) unconditional problem? Certainly not the K&W version. Nijdam (talk) 21:25, 11 February 2010 (UTC)
 * "compute the total probability that switching wins". Basically what Devlin, Behrens or Henze are doing in their takes on the problem with conditional probabilities. And yes that is not K&W problem - however you can use that total probability as a heuristic argument for a "puzzler solution".--Kmhkmh (talk) 21:39, 11 February 2010 (UTC)
 * @Kmhkmh: I know, but I'm not sure whether any of the participants here also have a clear view of what the unconditional problem is. It is time we define somewhere what we are talking about.Nijdam (talk) 11:50, 13 February 2010 (UTC)

What I was proposing for the simple solutions section is much like we have now. There are plenty of sources that just do not consider which door the host opens when he has a choice. These sources do not state the problem is either conditional or unconditional and they do not say that their solutions are in any way invalid or incomplete. They just give their solutions That is exactly what we should do in the simple solutions section.

I think that we should also state the common assumption in this section, the host always offers the swap and always opens an unchosen door to reveal a goat, that cars a randomly placed initially, and the host chooses randomly when he has a choice. This is how most people see the problem. Martin Hogbin (talk) 22:33, 11 February 2010 (UTC)


 * Martin - what I'm saying is present the simple solutions without comment on whether they are correct, but also present a conditional solution and (necessarily) explain what the difference is. I agree the sources presenting simple solutions do not state whether the problem is conditional or not.  But what this means is we don't know whether they consider the question to be asking if it is (in general) better to switch or stay (i.e. the problem is unconditional) or whether it is better to switch or stay given the player has picked a door and is looking at (and can clearly identify) the door the host has opened.  We do know (from K&W) that most people understand the problem to be conditional (they try to solve the case where the player has picked door 1 and the host has opened door 3) and that this conditional mental model interferes with the unconditional solution (one point of the K&W paper is that people do better with unconditional variants, particularly from the host's perspective - but these unconditional versions are variants).  We also know (from Barbeau, if not others) that the "standard analysis" is to include the conditions that the car is initially randomly placed and that the host chooses randomly between two goats.  If these are the generally accepted constraints on the host then the problem is symmetric and the unconditional and conditional answers must be the same.  However, avoiding addressing this topic "because it's just too hard to explain" in the initial section of the article on the symmetric version of the problem is simply POV.  I agree it's tricky to address this in an NPOV way, but I think we must.


 * I've drafted at least two different ways to do this. I thought progress on one was being made for a while.  Comments are still being made on the more recent one in this thread on the talk page.  Unless someone is actually going to create a draft with specific content (not just vague outlines) I don't think there's any good way to tell exactly what is being proposed.  I've drafted what I'm proposing.  -- Rick Block (talk) 01:27, 12 February 2010 (UTC)


 * I agree with your first sentence. The disagreement seems to be the order of proceedings.  My suggestion is:

Proposed structure

 * Start with the Whitaker question.
 * Give the standard interpretation of this statement in our own words. That is, standard game rules, host chooses legal door randomly. This interpretation is sourced by vos Savant (who later stated that she had assumed this) and Selvin at least.  Leave out the K&W formulation here.
 * Give the simple solutions based on the many sources which give one. Do not indicate that these solutions do not address the problem as asked or that they only answer some special version (unconditional formulation or average probability) of the problem. The sources do not say this.
 * Have our current 'Aids to understanding section', more or less as it is with more sources if we can find them. This section as it currently stands makes no mention of conditions.
 * Discuss and explain the conditional/unconditional issue fully, based on what all sources have to say on the subject. Martin Hogbin (talk) 09:55, 12 February 2010 (UTC)


 * This is structurally POV. By putting the criticism of the simple solutions an entire section away from where these solutions are presented, the article is taking the POV that the simple solutions are correct and complete.  This structure also fails to present a conditional solution to the popular problem as an equally valid approach - given the "expert POV" that a conditional solution is actually better, not presenting a conditional solution on an equal footing is horribly POV. -- Rick Block (talk) 15:24, 12 February 2010 (UTC)


 * Of course it is not. I am presenting the simple solutions in the manner in which they are presented by the sources that support them. The 'Aids to understanding' section says nothing at all about the conditional problem, it refers only to the simple solutions and clearly belongs with them.


 * Consider how the article will work with different readers. The non-expert readers will not want to be troubled with the complications of conditionality before they have understood the basic problem.  The hardest thing for anyone who has not seen the problem before to do is to understand the basic problem.  You seem to have lost sight of this fact.  Thousands of readers, who had obviously failed to understand the basic problem, wrote to vos Savant, including expert mathematicians.  We must explain the basic problem first.


 * The expert reader, on reading the article, may briefly wonder if we are aware of, and have properly addressed, the issues relating to conditional probability. They will, however, be quickly able to skim through the basic problem and aids to understanding sections to see that we have indeed fully addressed these issues.  No doubt the order of presentation, starting with  'easy' then 'harder' will seem logical to them.


 * Please tell me what is the "expert POV" that tells us that a conditional solution is actually better. This is nothing but your own POV.  Martin Hogbin (talk) 22:41, 12 February 2010 (UTC)
 * If you do it as suggested, you somewhat misrepresent the sources. If the sources do suggest a solution based on computing the total probability for switching, the article should state that clearly.--Kmhkmh (talk) 23:54, 12 February 2010 (UTC)
 * None of the sources which support the simple solution mentions anything about 'computing the total probability for switching'. It therefore cannot be called misrepresentation of the sources not to state what they do not state themselves. Martin Hogbin (talk) 11:05, 13 February 2010 (UTC)


 * Of course it is. If you present the simple solutions "in the manner in which they are presented by the sources that support them" but NOT the criticism of these solutions by the sources that criticize them (even if you're simply deferring this criticism to a later section of the article) you're making the article take a POV.  Deny it all you want, but my distinct impression is that this is exactly your intent.  You think that these solutions are correct.  You don't want any criticism of these solutions to be contextually nearby.  You expect the non-expert reader to stop reading the article before encountering this criticism.


 * I am not sure whether you want to discuss what the sources actually say or our opinions on the matter. The many reliable sources that give simple solutions do not state that their solutions are in any way deficient, limited, or incorrect.  If it is your desire to discuss my personal opinion on the subject, I will be pleased to do so.  Non-expert readers who read and fully understand the simple solutions are, of course, perfectly at liberty to continue reading the article to find out about the criticism of those solutions.  Those who have not understood the basic solution will be wasting their time in attempting to understand the criticism.


 * What you seem to have forgotten is that the unconditional solutions are simply NOT convincing. K&W tells you why.  People interpret the "basic" problem conditionally and once this conditional mental model is created cannot switch to the mental model required for the unconditional solution.  Although I admit I have no evidence for this (just like you have no evidence that it's remotely possible to create a convincing unconditional solution), I'm suggesting that it might actually better serve non-experts by giving them an understandable conditional solution.  Pay attention to Boris's point that the unconditional and conditional can coexist more peacefully.  We can present unconditional and conditional solutions in a single Solution section.  They address the same problem, from slightly different angles.  One might be more understandable for some people, the other might be more understandable for other people.  Insisting that the "simple" problem (by which you mean the symmetric problem, right?) be addressed only by unconditional solutions is obviously an anti-conditional POV (which is not even a published POV - it's just yours!).
 * You start by giving your own personal POV on which solutions are convincing. My reading of K&W is that door numbers, which give rise to the 'condition', are nothing but a distraction and that most people give better answers to the problem when not presented with the distracting and potentially condition-creating door numbers.  No source at all suggests that the conditional solution is simpler or easier to understand.  I have not suggested that the, 'the "simple" problem...be addressed only by unconditional solutions', I have suggested that we give simple solutions first, then we make sure the reader understands the simple solutions, then we consider all aspects of the issue of conditional probability and other complications relating to the subject in full detail.  This is the normal order for the exposition of any subject.


 * What I mean by the expert POV that a conditional solution is better is the POV expressed by Morgan et al., Gillman, Grinstead and Snell, Falk, Rosenthal, Rosenhouse, etc. that an unconditional solution does not quite address the problem, but a conditional solution does. "Better" meaning "actually addressing the problem". -- Rick Block (talk) 00:19, 13 February 2010 (UTC)


 * What you are saying is that some sources state that an unconditional solution does not quite address the problem. This is quite right, and we should state what these sources say, but only one source goes as so far as to claim that all solutions other than its own are 'false'.  On the other hand many equally (or perhaps more) reliable sources give simple solutions without any suggestion that their solution does not address the problem as asked.


 * In your arguments you seem to be overlooking the fundamental purpose of Wikipedia and the article. The purpose of the article is to explain to a wide range of readers all aspects of the Monty Hall problem, based what on reliable sources say.  If we pile in at the deep end with discussions of conditional probability that are only spoken about in academic circles we are likely to lose the interest and understanding of a good proportion of our readers before they have understood what the paradox is all about.  This serves the interests of nobody.  I am not suggesting that we omit the contribution of any source (whatever my personal opinion of it) or that we fail to properly state any valid point of view, just that we compose the article to give a logical progression from 'simple' to 'complex'.  No source tells us, or indeed can tell us, not to do this.  I could give many analogies but it is easier to suggest that you open most mathematical or technical text books to see that way that they do things.  Martin Hogbin (talk) 11:05, 13 February 2010 (UTC)
 * WP is for wide range of readers, but to provide them with accurate information. This if we present mathematical unconditional treatment of the problem (such as Devlin,Henze or Behrens) we should represent them accurately, i.e. we should clearly state that they compute the total probability for winning byswitching. That's the very least we need to do. We may not have to mentions Morgan's caveat at that time, but we need to give an accurate description of the unconditional solution. There is nothing wrong with going from simple to complex, but there is everything wrong with giving inaccurate or misleading descriptions.--Kmhkmh (talk) 11:27, 13 February 2010 (UTC)
 * When you say, 'we should represent them accurately' you seem to actually mean that we should state your POV accurately. None of those sources states that, 'they compute the total probability for winning byswitching', they present their solutions as complete solutions to the actual problem. Martin Hogbin (talk) 11:55, 13 February 2010 (UTC)

I went and gave the answer
I agree 100% with the proposal to go with Martin's proposal for the organisation and I agree it isn't really different from what is there. Now I have said many times and I'll say it again: if one takes as starting point, for pedagogical and historical reasons, Craig Whitaker's question to Marilyn vos Savant, you solve all the problems of what is the problem and all the things we have to agree on. Namely, we don't have to agree at all. The Monty_Hall_Problem -problem is to turn an ill-posed problem into a more exactly posed problem with a more or less rational solution. The forks are out there in the world. There is a solution to a problem about conditional probabilities. There are many mathematical ways to get that solution. Direct computation (aka Bayes formula), by symmetry and the law of total probability.... And so on and so on... The arguments and positions which everyone fights about here are what the article is to be about. And Martin's outline does that.

Now people who are biased to a particular "solution" of the meta-problem will want to start by quoting eg Selvin or whoever... Well, I think that's a bad idea. I don't dispute the existence and importance of a conditional probability formulation of the problem and historically MHP has its roots there. But putting that in front is I think a bad idea from the point of view that wikipedia is an encyclopedia which should be accessible, uncontroversial, should be based on reliable sources... BTW uncontroversially correct logic or uncontroversially correct elementary mathematics (which is the same thing) ought in my opinion to be considered a reliable source. We also use common language. No one is saying here that English cannot be used on wikipedai because many important concepts cannot be expressed satisfactorily in English. Gill110951 (talk) 08:25, 12 February 2010 (UTC)
 * The problem however is, that the basic math (and its application) seem to be controversial here. So far I don't see the involved mathematicians (as editors) agreeing. Although I prefer a slightly different separation in a popular/puzzling and mathematical analysis, I can live with Martin's structure, but i cannot not live the way he assigns the content and how he intends to use the sources (in particular point 3 in his last posting). We have few mathematical sources just giving an unconditional solution (i.e. simply computing the total probability) without saying anything else. However from a mathematician's perspective it seems to be clear that they are just computing the total probability and are making no claim about the conditional probability. In fact 1 source (Behrens) points out the difference explicitly. I see absolutely no reason that this gets colluded or buried. From my perspective there is no "we don't know" for the unconditional solution, we very well know that they compute the total probability. I would agree that in a popular/puzzler section that doesn't need to mentioned. However wherever we give mathematical treatment (or cite mathematical sources), we need give the reader a precise description of what is done. That means, if we have an unconditional section that comes across as a (simple) mathematical solution (quoting Devlin, Henze or Behrens for instance), we should mention that they compute the total probability for switching. Anything else is an unnecessary obfuscation and imho bullshitting our readers.--Kmhkmh (talk) 11:18, 12 February 2010 (UTC)
 * You seem to be conflating two things, what the sources actually say, which cannot be altered by what we think they ought to have said, and our own opinions on the subject. Whatever you think they meant or ought to have said, there are many reliable sources that give simple solutions, but none of these sources states that they are 'simply computing the total probability'.  They present their solutions as complete and correct solutions to the question that was asked.  I am not necessarily insisting that we should take such an uninformed view and literal view of what our sources say but it is important to understand where this strict reliance of what the sources actually say leads.


 * Perhaps you could make clear exactly what part of my proposal you object to. Essentially I am just talking about moving the current 'Aids to understanding' section to immediately follow the 'Popular solution' section.  I do not propose to add anything to the popular solution section which specifically states ' this is the 'real' or 'correct' solution.  I want to do exactly what the sources do and, as we do now,  just present these solutions to the reader, without excessive claims or unjustified reservations.


 * Against my better judgment and in the interest of compromise, I am happy to follow the simple exposition with a discussion of what I consider to be an academic distraction and a perverse obfuscation of the essential paradox, the conditional nature of the problem, based on what the sources say. Martin Hogbin (talk) 11:38, 13 February 2010 (UTC)
 * That's probably a reading comprehension problem. If you look at the reputable math sources (such as Devlin, Henze, Behrens) they do compute the total probability even if they don't say the word "total probability" explicitly they do so implicitly by the computation. Any mathematician who looks at the computations, knows that they compute a total probability. And frankly no you don't want to do what the sources do, you want an obfuscating description computing something "simple" without clearly stating what it is.
 * To be precise where the current problem with your proposal is:
 * 2 Simple solutions
 * The (non-conditional) problem and solutions - fully explained with no confusing mention of conditions or suggestion that it does not answer the problem 'as asked'. These should be given as solutions to the K&W unambiguous formulation of Whitaker's (and Selvins's) question
 * What is the non conditional problem? It's not K&W nor Selvin! Which sources do you plan to cite here? (If you cite any math source, you need to describe accurately what they compute). How does the given solution remove the ambiguity?
 * To be clear here from my perspective that section doesn't need to mention conditions at this point, but it needs to mention what it actually computes to the very least. That's non negotiable from my perspective and your description so far does not ensure that.--Kmhkmh (talk) 16:14, 13 February 2010 (UTC)

Let's stop arguing for the moment and formulate our MHP
As I tried before, it is time we come to know what has to be considered an acceptable version of the MHP. I invite any of the participants to stop arguing for the moment and just formulate what has to be considered the unconditional and the conditional version of the MHP. This may also come off hand in the Mediation process.


 * The problem is, as Gill and I have said before, that the most notable statement of the problem, Whitaker's question, is ill-defined. Much of the argument here is about how to precisely formulate the problem, based on his question. A common unambiguous interpretation is that given by K&W, but this may well not be what Whitaker actually wanted to know.  He might well be considered to be asking the unconditional version of the problem, along the lines of, 'If a person were to be going on the show described, what would be their best strategy and how much would it gain'.


 * The best interpretation of Whitaker's question was, in my opinion, that of vos Savant, who was the best qualified to interpret and answer a question from one of her readers to her regular column. Her interpretation was that what we now call the standard game rules applied (host always offers the swap and always opens and unchosen door to reveal a goat) and that the car was initially placed randomly and that the host acted as the agent of chance and thus his choice of door was insignificant.  This is also in line with Selvin's original problem.


 * My conclusion is that we here do not have the right, or the means, to determine exactly what the problem is. There is no agreement as to the precise formulation in the sources or amongst editors here.  We have to accept that the problem is not well defined.  Its most notable aspect by far is that it is a simple problem that most people get wrong.  We should address that first with clear, simple, and convincing solutions. In order to do this we should do as vos Savant did and disregard the complications potentially resulting from the host's legal choice of door.  After we have properly dealt with the simple and notable problem, we can then report the various ways the problem has been formulated and solved, bearing in mind that these are only of somewhat academic interest.


 * I am sorry to be so negative about what might appear to be a logical approach to the article, but the facts are as they are and we have to deal with them. We cannot make our life easier by giving our own verdict on how exactly the problem should be formulated.  Martin Hogbin (talk) 13:39, 13 February 2010 (UTC)
 * The issue that I have with your suggestions above has nothing to do with the ambiguity of the problem. It has to do with an accurate description of the solution. If you want to describe the "simple" problem you can do that, but you need to do that accurately and this requires stating that you compute the total probability for switching to answer the problem and not that just compute some probability (which might be understood as an unconditional approach (via symmetry) to compute the conditional probability). That is exactly an obfuscation the article should avoid and this is not a question of "simple first, complex later" versus "all in one" but a question of clear accurate descriptions. It is also not "describing as it is" versus "removing the ambiguity in a POv manner" but it is about clearly stating which solution addresses  which problem. Nijdam's objection above was not against having a correct unconditional solution, but he required giving an accurate description of the problem the unconditional solution solves. This is something that you lack if you simply state the problem as Whitaker or K&W and give the unconditional solution without further comment.--Kmhkmh (talk) 13:51, 13 February 2010 (UTC)
 * That's what it comes down to, no more and no less. All the discussion from Rick's, Kmhkmh's and my side is about this point. And it seems as if others avoid this point. That's why I asked to state one's personal opinion about what one considers (the sources) to be the versions of the MHP. To make clear whether everyone has the same interpretation. Why not state your POV about what you think the sources mean the MHP to be? Why the hesitation? Nijdam (talk) 14:23, 13 February 2010 (UTC)
 * @Kmhkmh I was responding to Nijdam who is requesting us to decide precisely how to formulate the problem. My point was simply that we have no proper basis on which we can formulate the MHP.  The sources do not agree and neither do we.  The fact is that what is by far the most well-known and notable description of the problem is ill-defined and we cannot change that fact.  It is simply not possible to give a mathematically precise solution to an ill-defined problem and thus we should not try to do this.  There are many sources that give what are essentially correct (but arguably not totally rigorous) solutions to Whitaker's question.  These solve the paradox for most people. Those who are interested in the details can read on to find more clearly defined statements of the problem more precisely answered.  Martin Hogbin (talk) 15:17, 13 February 2010 (UTC)
 * @Nijdam I can give my opinion of what the real MHP is if you like. It would be the K&W statement with the added direction that the door opened by the host (being randomly chosen when the player has originally chosen the car) is to be taken as unimportant and thus not a condition of the problem, but that is just my personal opinion.  The sources do not agree on what the precise problem is and neither do editors here.  Martin Hogbin (talk) 15:17, 13 February 2010 (UTC)
 * That's not a correct description, the sources do not agree on what they consider personally as a sufficient answer. However they do very much agree on the difference between the conditional and the total probability and source like Behrens spell that out explicity. Also sources computing the total probability just do that, they don't make any claims that they compute anything other than the total probability, so there is no good reason for the WP article to suggest otherwise or leave room for interpretation on that aspect. And again the issue here is not about giving (an impossible) rigorous answer to an ambiguous problem, but to accurately describe what precise problem the given solution actually solves. Anytime you give a solution, this particular solution includes a way or an assumption to remove the ambiguity and hence allowing the solution to begin with. So if we have a solution, we can and should describe how it resolved the ambiguity. Giving any solution for an ill defined problem always includes as a first step of the solution to make it well defined. Otherwise there is simply no solution at all.--Kmhkmh (talk) 15:25, 13 February 2010 (UTC)
 * We cannot state exactly what problem the simple solutions solve because many sources themselves do not make that clear. We cannot state our own view on the exact problem that they solve because there is no consensus on that here either. There are several rationales for the simple solutions being correct including: that they applied symmetry, that they considered the unconditional problem, that they used information theory to show the posterior probability must be equal to the prior probability.  They do not say which of these principle they use although a couple do give hints.  Nevertheless, these sources are described as correct by many commentators. I think that order to make our simple solutions clear, simple and convincing we can reduce the level of rigor that we apply at the start of the article.  Martin Hogbin (talk) 16:20, 13 February 2010 (UTC)
 * We can exactly state what the sources do. This has nothing to with our view. We see whether a source computes the total probability or not and that's exactly what we write in the article then and yes computing the total probability is "correct" provided you are not claiming it to be something else, which the sources I've seen don't do. Of course we cannot speculate about additional (unstated) rationales behind behind the computation of the total probability and why it might identical to the conditional probability, i.e. if the source doesn't mention symmetry or conditional probabilities neither do we, when citing it. However a source computing a total probability has to described as computing a total probability and nothing else. And this is exactly what you seem to attempt to avoid here, you're given a false/incomplete description of the math source with an unconditional solution, which is a no-go for the article.--Kmhkmh (talk) 18:22, 13 February 2010 (UTC)
 * You seem to be saying that your opinion can determine what a source should say. Tell me, does vos Savant calculate the 'total probability' (as you call it) of winning by switching or the probability given that the player has chosen door 1 and the host has opened door 3, having chosen randomly which legal door to open? Martin Hogbin (talk) 18:56, 13 February 2010 (UTC)
 * No, I'm not saying my opinion determines what sources say, the sources do (however comprehension of them is required). And above I was referring to the math sources giving an unconditional solution and not about vos Savant. However vos Savant's original explanation (treating 2 doors as one/using the complement probability) is computing the total probability as well. Also note there is nothing wrong with computing the total probability and using that as an argument, but it is different to the conditional probability and that difference needs clear and should not be obfuscated. The everlasting discussion here under which circumstances the total probability becomes identical to the conditional probability (not just numerically) is separate issue. This involves the symmetry condition or alternative conditions, however these issues have nothing to do with published unconditional solutions by vos savant, Devlin, Henze, Behrens, etc. .If there is a reputable source that has examined that aspect, we can integrate that, but that is not a good candidate for the simple solution, since it would have to deal with the all that condition stuff, that you do not want to have in the simple solution.--Kmhkmh (talk) 20:06, 13 February 2010 (UTC)
 * Vos savant does not say anything about computing total probability she, fairly obviously, claims to be answering the question asked by Whitaker. She does not use the words conditional or unconditional or overall probability, she just answers the question.  Maybe she took Whitaker to be asking about the unconditional probability, this is not a totally unreasonable assumption in the circumstances.   On the other hand, maybe she was answering the problem of what the probability of winning by switching is given that the player has chosen door 1 and the host has either door 3 or door 2 to always reveal a goat.  Or maybe she is answering the conditional question of what the probability of winning by switching is given that the player has chosen door 1 and the host has door 3 to reveal a goat, knowing that this probability must be equal to the previous one if the host opens a legal door randomly.  Whatever question she is answering, that is the question that we should be answering here, as she must be regarded as the expert in interpreting questions to her column.  She gets the answer correct. Martin Hogbin (talk) 20:39, 13 February 2010 (UTC)


 * vos Savant clearly computes the total probability of winning by switching: "The benefits of switching are readily proven by playing through the six games that exhaust all the possibilities. For the first three games, you choose #1 and "switch" each time, for the second three games, you choose #1 and "stay" each time, and the host always opens a loser. Here are the results."  We all know that this is the same as the probability given that the player has chosen door 1 and the host has opened door 3 (if the host chooses randomly which legal door to open), but she never says anything like this.  Instead, she is examining "all the possibilities" where the player has picked door 1, not just the possibilities where the player picks door 1 and the host opens door 3. -- Rick Block (talk) 19:44, 13 February 2010 (UTC)Martin Hogbin (talk) 20:39, 13 February 2010 (UTC)
 * Which is however a reasonable heuristic to answer the puzzle.--Kmhkmh (talk) 20:07, 13 February 2010 (UTC)
 * @Rick, are you now claiming that the true MHP is only the case where the player chooses door 1 and the host opens door 3? Martin Hogbin (talk) 20:39, 13 February 2010 (UTC)
 * Did I say anything at all about the "true MHP"? You keep saying it's ambiguous what vos Savant's answer means.  What's ambiguous is whether she is taking the question to mean "what is the total probability" or "what is the conditional probability" since she never exactly said.  Her ANSWER is NOT ambiguous.  Her answer is the total probability.  So, either she's taking the question to ask about the total probability, or her answer is not (exactly) addressing the question.  What I'm suggesting for the article is that we give one or more solutions that address the total probability and also one or more solutions that address the conditional probability.  Saying that only the total probability is the solution to the "simple MHP" is POV.  -- Rick Block (talk) 21:54, 13 February 2010 (UTC)
 * See my reply below. You seem to want to give your own POV as to what question vS answers.  She does not mention total probability anywhere.  Martin Hogbin (talk) 23:45, 13 February 2010 (UTC)

Nijdam
General conditions:
 * A car is put randomly (=uniformly) behind one of 3 doors
 * The player chooses one of these doors (not necessarily at random), independent of the position of the car
 * The host opens a door with a goat (randomly, if needed)
 * The player is offered to switch to the remaining closed door

Unconditional MHP
An outsider unfamiliar with the choice of the player and which door has been opened, decides what to do. He calculates for instance the (average) probability of the car being behind the chosen door.

Conditional MHP
The player did choose door 1 and the host opened door 3. The player is actually confronted with two closed doors, number 1 and number 2. She calculates the conditional probability of the car being behind door 1 given the situation. (BTW: how she does the calculation is unimportant. Also may the problem be solved with other combinations of door numbers.) Nijdam (talk) 12:15, 13 February 2010 (UTC)


 * @Nijdam your general conditions are general conditions often adopted by some (maybe many, so what!) interpreters of the MHP.


 * They are not stated or alluded to in Whitakker/vos Savant. As I said many times below, please lets distinguish between the MHP as literally quoted by vos Savant, and the many interpretations/mathematizations of that problem that our around.


 * The wikipedia page must tell the history and tell the different interpretations neutrally. Gill110951 (talk) 11:44, 14 February 2010 (UTC)

Let's stop arguing and talk about what the SOURCES say
We have, of course, been here before. But just for the record, I'd like to repeat some content from http://en.wikipedia.org/wiki/Talk:Monty_Hall_problem/Archive_11, specifically these two threads.

It's well past time to stop with the WP:OR and personal opinions, and instead stick to what reliable sources actually say. Can we all agree on the following?

1) Numerous reliable sources (mostly popular, e.g. vos Savant's Parade columns) don't rigorously say what problem they're solving but simply present a "solution" to "the MHP" (usually phrased in a way that is identical to or very similar to the Parade version) that determines the total (or, if you prefer, average, overall, or unconditional) probability of winning by switching. Since they don't say, we simply don't know whether these sources think the MHP is asking about the total probability or think their unconditional solution is addressing the conditional problem (because of symmetry or any other reason).


 * I thought this was meant to be about what the sources say, yet you start by giving us your POV about what question these sources are answering. Why do you not stick to what question the sources themselves say that they are answering, which is simply the MHP.  Martin Hogbin (talk) 23:43, 13 February 2010 (UTC)

2) Numerous other reliable sources (including peer reviewed academic papers such as Morgan et al. and probability textbooks such as Grinstead and Snell), specifically referring to the Parade version of the problem and/or vos Savant's analysis, distinguish the total probability from the conditional probability and unequivocally say the MHP is a conditional probability problem and that unconditional solutions address a slightly different problem.
 * I'd caution that a bit. Many of those sources do not necessarily claim the MHP must be understood as a conditional problem, but they clearly distinguish between 2 different problems. I.e. the issue is not so much about, which of the 2 is the "real MHP", but about being precise to what problem is actually addressed in their respective solution.--Kmhkmh (talk) 01:00, 14 February 2010 (UTC)
 * OK. I'll grant that there's a 3rd camp which as you say distinguishes but does not express a preference for either interpretation.  You do agree there are (numerous) sources that do express a preference, right? -- Rick Block (talk) 01:23, 14 February 2010 (UTC)
 * Yes--Kmhkmh (talk) 11:57, 14 February 2010 (UTC)
 * Yes, and this should indeed be mentioned in the article, after the basic problem has been properly explained.Martin Hogbin (talk) 23:43, 13 February 2010 (UTC)

Does anyone disagree with either of these points about what SOURCES say? I'm not asking and really don't care whether you agree with what these sources say, but do you agree there are numerous sources that actually say one or the other of these? If it would help, I could enumerate a list of sources in each "camp" - but I suspect we all know which sources they are by now. -- Rick Block (talk) 22:24, 13 February 2010 (UTC)


 * Yes I disagree strongly. In the first case you do not want us to say what question the sources actually say that they are answering but what question you think they are answering.  You seem to want to have things both ways.  When a source agrees with you, you insist that we must stick strictly to what it says without consideration of its integrity, correctness, intention, and relevance. When a source says something that you do not like, you say that we should work out for ourselves what question it is answering rather than look at what the source itself actually says about the subject. Martin Hogbin (talk) 23:43, 13 February 2010 (UTC)


 * Are you disagreeing that there are two ways to interpret the question and that some sources do not clarify which of these two ways they're interpreting the problem? Or that the solution these sources present determines the total probability?  This is not about what question I think they are answering, it is about what the solutions say.  A solution that says "The benefits of switching are readily proven by playing through the six games that exhaust all the possibilities. For the first three games, you choose #1 and "switch" each time, for the second three games, you choose #1 and "stay" each time, and the host always opens a loser. Here are the results." [emphasis added] is clearly talking about the total probability, not the conditional probability given the player has picked door 1 and the host has opened door 3.  Do you disagree with this?


 * And, what do you mean by the "basic problem"? The sources in the 2nd camp say they're talking about the same problem vos Savant is talking about, and that this problem is conditional but her solution is unconditional.  Do you disagree that these sources say this?  It seems to me your POV is that there is some "basic problem" that vos Savant (etc.) address and that this problem is different from the problem these sources are talking about.  But this is directly contradicted by what these sources actually say.  It seems to me you're the one who wants it two ways.  On the one hand you're unwilling to characterize the specific problem vos Savant is talking about because she doesn't explicitly do so herself.  Yet on the other hand, you want to posit the existence of a "basic problem" that is somehow distinct from the problem other sources directly say she (and they) address.  -- Rick Block (talk) 01:23, 14 February 2010 (UTC)


 * Yes, sources disagree. Some sources claim to solve the problem in a simple manner.  They do this without stating any reservation or restriction.  Other sources may indicate that these sources are incomplete or answer a slightly different question.  We need to use our knowledge and understanding of the subject to produce a good encyclopedia article supported by reliable sources.  The suggestion that we must stick only to literally what the sources say cannot work because the sources conflict.  We need to resolve this conflict in a way that produces an article that will be of interest and help to all our readers.Martin Hogbin (talk) 10:45, 14 February 2010 (UTC)


 * Conflicting sources are dealt with all the time. The solution is to say what the sources say without promoting one over the other.  Are you agreeing that there are 2 types of sources that all purport to discuss the same problem:


 * 1) Sources that, as you say, present a "simple" solution (e.g. vos Savant).


 * 2) Sources explicitly referring to the same problem addressed by the first set of sources that distinguish two types of solutions, one consistent with the types of solutions presented by the first set and the other explicitly using conditional probability. These sources are mixed between those that


 * 2a) take the stance that the simple solutions do not, in their opinion, address the MHP "as stated" but rather some slightly different problem (e.g. Morgan et al.)


 * 2b) consider the interpretations to be different but don't express a preference (e.g. Behrends)


 * Regardless of how we present this, do you agree that this is a reasonable categorization of sources? -- Rick Block (talk) 04:29, 15 February 2010 (UTC)
 * I'd essentially agree to that but add a possible 3rd item:
 * 3) sources that only provide a conditional solutions and do distinguish between 2 types and hence do not take an explicit stance against the simple solution (for instance Stirzaker (in list of books further up))
 * --Kmhkmh (talk) 05:48, 15 February 2010 (UTC)

Reputable sources and comparing degrees of reputability
Martin's argumentation above in his last postings (vos Savant as "question answering expert or authority", "it is not a total probability becomes vos savant didn't say explicitly" and possibly Parade magazin as reputable source) raises another issue, where the mediator might be able to help out or comment on. From my perspective I would treat vos Savant and the original publication in Parade or on her website mostly as primary sources or at best a very minor secondary source, which are definitely not on the same level peer reviewed journals. Also the "question answering authority"-notion makes hardly any sense to me. If you want reputable sources for the psychology of a question (for instance regarding the intent of the question or its common perception) then one should resort to psychology literature/publication on the subject and if you want reputable sources regarding the math aspects one should resort to math publications. Vos Savant is rather smart person, who gave a smart answer to a problem posed in her columns. However that does not turn her into an authority on the subject or Parade into a reputable source on math or psychology. As an encyclopedia WP primarily needs to rely on reputable (whenever possible peer reviewed) academic sources even more so on topics or content that is subject to debate.--Kmhkmh (talk) 00:52, 14 February 2010 (UTC)
 * You mean "definitely NOT on the same level as peer reviewed journals", right? -- Rick Block (talk) 01:26, 14 February 2010 (UTC)
 * Yes sorry (fast typing omission) - i corrected it.--Kmhkmh (talk) 01:56, 14 February 2010 (UTC)
 * Indeed, is there some disagreement on this? I think Kmhkmh understands it correctly. Andrevan@ 03:12, 14 February 2010 (UTC)


 * If you mean Selvin's 2 letters equivalently to Morgan, OK. Glkanter (talk) 09:05, 14 February 2010 (UTC)
 * Selvin's letter would be somewhere in between. As math or statistics professor you can consider Selvin himself as an authority on the subject. Moreover his letters like vos savant's column play an important part in the history, his letters coined the problem and vos savant column caused the controversy and made the problem famous/welll known to large audiences. However Selvin's letter are not the same as published peer reviwed journal articles. Also note that there is no contradiction between Morgan and Selvin's arguments. I would suggest that people get over the Morgan versus vos Savant issue, to me much of the discussion still seems like proxy continuation of vos Savant-Morgan quarrel. Both of them made an important contribution to the problem, however strictly speaking neither of them is needed as a source as far as the solutions are concerned.--Kmhkmh (talk) 11:22, 14 February 2010 (UTC)


 * So, Selvin's contribution to the MHP is of less value than Morgan's? Is this also non-negotiable? Glkanter (talk) 16:13, 14 February 2010 (UTC)
 * I think you're confusing contributions with reputables sources (being used/cited by he article). A reputable source on a problem can be (and often is) no significant contribution to the problem itself. For instance a popular textbook on relativity or calculus usually represents no contribution to the theory at all, it does however provide a reputable source that can be used in according wikipedia articles on those subjects. Similarly do most probability primers covering MHP provide reputable sources but no contributions to MHP. As far as Selvin and vos Savant are concerned personally I consider their contribution more important than Morgan (due to their influence on the history of the problem). Their publications however are less reputable sources than Morgan.--Kmhkmh (talk) 16:44, 14 February 2010 (UTC)


 * The only difference between Selvin and Morgan (both of whom are published in The American Statistican) is that Selvin's letters preceded Morgan's vos Savant criticism by 16 years. And Seymann was permitted to write some sort of qualifier immediately following Morgan's paper. This, with Martin's thorough critique of Morgan's paper leads me to a very different conclusion as to both the importance and the reliability of the publishings. Glkanter (talk) 21:24, 14 February 2010 (UTC)


 * You are mistaken. Selvin's letters were letters to the editor, not peer reviewed papers.  Morgan et al.'s contribution was a peer reviewed paper.  The way peer review works for most academic journals is you submit your paper, the editor finds one or more reviewers (independent experts in the field - generally professors), the reviewers (who remain anonymous to the submitter) make comments, and the submitter makes modifications in response to those comments - all before the paper is published.  Subsequent commentary, such as Seymann's, is more like a letter to the editor in response to a particular paper and the authors are typically invited to publish a rejoinder (Morgan et al. did, in response to Seymann's comments and also to vos Savant's comments).  Martin's critique of the Morgan et al. paper has absolutely no standing here. -- Rick Block (talk) 00:21, 15 February 2010 (UTC)


 * Nothing in WP policy tells us that a peer reviewed source can overrule all others. WP policy says, 'Proper sourcing always depends on context; common sense and editorial judgment are an indispensable part of the process'.  In the case of the MHP there is a wide variety of sources  which we need to use properly to create a encyclopedia article.  We should not omit the opinion of any significant source and we should not allow any one source to dominate the article.  Some specific responses to Kmhkmh's points are:


 * There are no peer reviewed sources which study exactly what Whitaker's question may have meant. This is a critical issue, as made clear by the reliable, peer-reviewed, source, Seymann.


 * The (most notable statement of the) question was not asked in an academic context, it was asked in a popular general interest magazine. That is the primary context in which it should be addressed.  The academic context should be discussed later. Martin Hogbin (talk) 11:08, 14 February 2010 (UTC)
 * This is becoming a pointless discussion. From my perspective you currently seem to pursue a "vos savant must be completely right no matter what"-strategy.
 * Of course can peer reviewed sources in doubt overrule others. But that is not even the issue here, nobody suggest to omit sources, the issue is with describing them correctly and as such, that vos Savant computes the total probability. Also completely wrong is your claim that no academic sources or peer reviewed papers deal with "what Whitaker's question may have meant". Aside from the fact that almost any publication on MHP in some way deals with that issue by assuming a particular meaning for solution, there are various psychological papers dealing exactly with that issue (for instance Krauss/Wang), all published after Seymann's statement from 1991. Moreover Seymann's point was that the problem as stated by Whitaker is ambiguous, that is something everybody here agrees on. The issue was, that our article needs to describe exactly how the ambiguity is resolved for a particular solution and that requires telling that vos Savant, Devlin, Henze, Behrens and others compute the total probability. That is something you can see directly from their computation (and actually Behrens states it explicitly as well when he is comparing both approaches).--Kmhkmh (talk) 11:54, 14 February 2010 (UTC)


 * I have no wish to continue the vos Savant/Morgan quarrel, provided that this issue is left open. There is no consensus that Morgan were right and vos Savant wrong.  We have two sources that disagree, let us leave it at that.
 * Kmhkmh, you are the one making all the fuss about the subject. All I want to do is move the 'Aids to understanding' section to its logical place in the article, where it will be most helpful to our readers.  All I am asking is a little licence to defer the criticism of the simple solutions until the really hard part, understanding why the answer is 2/3 not 1/2, is fully explained, an approach that is in the best tradition of mathematical and scientific text books.  I agree that it is possible that some readers may begin to get wrong understanding of the potential conditional nature of the problem, and I agree that we should word the section so as not to say anything that is actually incorrect, but to start adding in restrictions and conditions relating to the simple solution will do nothing to help our readers' understanding.  Through familiarity with the subject you have lost sight of just how hard this seemingly simple probability problem is to understand for people who see it for the first time.  As the article itself says, 'Even when given a completely unambiguous statement of the Monty Hall problem, explanations, simulations, and formal mathematical proofs, many people still meet the correct answer with disbelief'.  It is our duty to do our best to overcome this problem before complicating the subject in a way that is of little interest to most people.  Martin Hogbin (talk) 13:57, 14 February 2010 (UTC)


 * All this stuff here is besides the point. Of course Selvin is a reliable source, of course Morgan et al are a reliable source - according to conventional definitions. My own paper on arXiv (mathematics section) is now being processed by a peer-reviewed journal in statistics and I'm getting all my friends to add to their papers: "On a totally unrelated topic, Gill (2010) has written a crappy/brilliant paper on the Monty Hall Problem". In a year's time my paper will be a more reliable source than Morgan et al. or Selvin.


 * The Monty Hall Problem is a social and cultural phenomenon. Selvin, Morgan and whoever else you care to think of, offer mathematizations of the problem, and solutions of their mathematizations. Other reliable sources dispute their mathematizations, others dispute their solutions. Morgan et al are definitely not that reliable by any objective reasonable definition since they prove a theore which anyone who is qualified to read their paper can see contradicts other results in their own paper. The 70 citations of their article show that citations don't mean that every word contained in the article is correct. In fact most citations in science are part of a social cultural phenomenon and don't even mean that the author has read the article he/she cites. Many citations are made by people who are not able to judge the correctness of the results, even if they did read the paper.


 * Selvin was a biostatistician, so not even a mathematician. His own answer to his own question demonstrates him adding assumptions to the question which had not been stated there, which make him a bad mathematician and a bad teacher of mathematics. Similarly Morgan et al's apparently misquoting other people so as to claim that they solved a wrong problem and therefore are stupid, means they are bad scientists, whatever else they may be; and you don't have to be an expert mathematician in order to verify that.


 * I notice that scientists are nowadays copying material almost verbatim from wikipedia without checking sources or correctness and getting this published by peer reviewed journals whose referees are busy people who also often check things on wikipedia. I am afraid that the days are over that it is enough to count Google Scolar citations in order to determine whether something is a reliable source. It is also necessary to use one's intelligence, commons sense, logical analysis, in order to figure out whether so-called reliably sourced material makes any sense or not. Gill110951 (talk) 12:33, 14 February 2010 (UTC)


 * No offense but judging te teaching qualities of other academics by an individual publication or letter is as careless as alleged "misquotings" by Morgan or the redefining of the problem by Selvin. Moreover the criticism of various flaws/errors in those papers is somewhat beside the point, because they are cited for their correct arguments and not for errors that don't matter in this context. Also there is no disagreement regarding the social and cultural phenomenon of MHP and descriptions of that. However if we describe a solution than we need to be precise about what that solution actually does and how it resolves the ambiguity, i.e. in vos Savant's case that it computes the total probability that switching wins. This has nothing to do with vos Savant or Morgan being right/wrong (and the nauseating "proxy war" regarding that issue). This simply has to do with giving an accurate (and hopefully illuminating) description of what the various solutions actually do. As far as your comment concerning "scientific" use of WP is concerned, I don't quite see what this has to do with the discussion here. Careless citing (of WP or others) is primarily a problem of the scientific community and its quality assurance and not WP. Also nobody suggests, that the content of what WP considers "reputable sources" should be used without common sense and a personal analysis and of course can editors in doubt dismiss or not use "questionable" papers. "Relying on reputable sources" is neither meant to parrot their content mindlessly nor that every formally reputable source on subject needs to be used. It just means that you should select your sources among reputable ones, which pass your common sense and personal analysis as well. So the sources ideally are "the very best of the reputables". The reputability is used to assure that authors don't start to quote any source, because otherwise we'll get things evolution theory ala Ann Coulter or in the math domain crackpot theories of any conceivable kind. In short reputable sources are not meant to be a perfect solution and to replace common sense, they are meant to prevent worse.--Kmhkmh (talk) 13:10, 14 February 2010 (UTC)
 * We need to use our common sense to write an encyclopedia article that will be interesting, informative, and above all, understandable to our readers. This should be supported by reliable sources.  See my comments above, inserted after an edit conflict.  Martin Hogbin (talk) 14:04, 14 February 2010 (UTC)
 * There is no disagreement about that other than the word accurate lacking in your description. However the pickle here is that we do not agree, what "interesting", "informative", "understandable" means in the context of the article. Nor do we seem to agree how to use and assess sources.--Kmhkmh (talk) 15:04, 14 February 2010 (UTC)

What can be negiotated/mediated?
Ok looking at the discussion so far, we are heading for dead end. With each side pursuing things that are completely unacceptable by the other. I think we need to restart this by focussing on issues where there is actual room for compromise. Below I'm listing a few points which are nonnegotiable from my perspective and where there might be room for negotiation.--Kmhkmh (talk) 12:27, 14 February 2010 (UTC)


 * You cannot start a discussion on how to reach a compromise with a list of non-negotiable demands. Martin Hogbin (talk) 14:13, 14 February 2010 (UTC)
 * Yes we can, because otherwise we get nowhere. It is useful to be upfront here and to clearly see what other people consider as acceptable and what not. Otherwise we are leading neverending (pointless) discussion again and the mediation turns just into another chapter of what we've seen for the last year anyhow.--Kmhkmh (talk) 15:00, 14 February 2010 (UTC)

non negotiable

 * NN1: peer reviewed sources take priority over non peer reviewed
 * NN2: reliable academic sources take priority over general sources
 * No, see above. Martin Hogbin (talk) 14:13, 14 February 2010 (UTC)
 * Answer see above.--Kmhkmh (talk) 14:54, 14 February 2010 (UTC)


 * NN3: WP descriptions of solution (simple, unconditional, conditional, whatever) needs to describe what the compute exactly and how they resolve the ambiguity
 * No, at the start of the article it is better to leave the exact problem slightly vague in the interests of clarity. The details can be discussed later.  Martin Hogbin (talk) 14:13, 14 February 2010 (UTC)


 * NN4: The article itself should give a complete overview/summary of the problem and all its aspects
 * Agreed.

General comment here: Please move further discussions of the items to the comment/discussion section which was created for that purpose. This imprives the readability.--Kmhkmh (talk) 14:54, 14 February 2010 (UTC)

possibly negotiable

 * PN1: the order of various sections
 * PN2: a "simple solution"-section may not need to mention Morgan's caveat at that time but it needs to be precise about what it computes and how it addresses the ambiguity (the total probability thing)
 * PN3: There might be a short "puzzler" or "solution by vos Savant" section which quotes vos savant solution without further comment. A detailed treatment of what that solution actually does can follow is separate "psychological", "matematical" or "scientific" analysis sections.
 * PN4: possibly different arragements or even a splitting up of the "sources of confusion"-section
 * PN5: writing the article (essentially same content as before) but without citing Morgan?

discussion/comments
moved down:
 * NN1: peer reviewed sources take priority over non peer reviewed
 * No 'Proper sourcing always depends on context; common sense and editorial judgment are an indispensable part of the process' Martin Hogbin (talk) 14:13, 14 February 2010 (UTC)
 * That editorial judgement is part of the process is a given. However that does not change the guideline above. If editors agree on not using a particular reputable source (due to being difficult to understand, containing too many errors or obsolete knowledge, being ill suited otherwise) and use a less reputable source having a better content, then that's fine. However if editors do not agree than in doubt the reputability (i.e. the peer reviewed) source takes priority. If you attempt to use your personal editorial judgement rather than the editorial consent to overrule he priority of peer reviewed sources in a particular instance, then that's a no-go. This refers in particular to your suggestions above of considering vos Savant as an "question answering/questioner understanding authority (no matter what domain the question belongs to)" and consequently the Parade magazin as a reputable source for probability arguments. If you really insist on that, than from my perspective the mediation is kinda over, since this is an unacceptable position for an encyclopedia (in my view).--Kmhkmh (talk) 14:54, 14 February 2010 (UTC)
 * Many editors, in fact I would say a sizable majority here, do not think that the Morgan paper is a particularly good or relevant source for this article. It needs to be taken in context, it is just making a rather academic point about conditional probability. Martin Hogbin (talk) 15:02, 14 February 2010 (UTC)
 * Morgan is not a great paper. However as far as computing probabilities are concerned it is definitely better source than vos Savant/Parade. Obviously there is no editorial consent regarding Morgan as source for the article in general.--Kmhkmh (talk) 16:01, 14 February 2010 (UTC)
 * This is the problem, you say you do not want to discuss Morgan vs vos Savant but you insist on asserting that Morgan is better. There is so much wrong with the Morgan paper that I and many other editors would say that vos Savant is the better source.  Why not just leave it that we have two sources that disagree?  Martin Hogbin (talk) 16:56, 14 February 2010 (UTC)
 * Yes, Morgan is a better resource for probability issues. Reputability side (peer reviewed statistic journal versus parade) the Morgan paper (despite various short coming) provides mathematical insights and details, that the parade publication does not have. Hence it obviously is better source in that regard. Note this has nothing to do with vos savant or Morgan being right or wrong. It simply means that vos Savant publication does not discuss the ambiguity of the question nor the conditional approach and other math aspects, hence it is obviously not a better source on that content. That aside as far as I'm concerned the article doesn't need to use Morgan at all, but instead use other math papers. Strictly speaking Morgan is not needed for anything. My issue here was with your attempt on turning Parade/vos savant into a reliable source for probability issues. It is neither reputable source in the WP sense nor reliable source to get information on probability related issues.--Kmhkmh (talk) 00:55, 15 February 2010 (UTC)

I think that NN1 and NN2 are useful general guidelines but not universal laws. Various items of a country's constitution, or the Universal Declaration of Human Rights for that matter, can contradict one another when applied to specific problems. I think you are putting these "non-negotiable" items here to force your point of view.

But lots of people have made decent proposals for a structure of the article, which allows all existing Points of View to be explained in the article, in peaceful co-existence. The only problem is people who think that their solution is the only solution. But if you distinguish between Monty Hall phenomenon (meta level) and Monty Hall problems (solutions to the MHP-problem) no one need complain, all points of view can be accommodated. It's like paganism versus monotheism. The monothesists are always a nuisance but always finally one of the monotheist groups wins by eradicating all the others and all the pagans. Gill110951 (talk) 12:38, 14 February 2010 (UTC)
 * Of course the non-negotiable points represent are my point of view, i.e. what i consider as nonnegotiable in this context. That's why I've explicitly stated that above. Also you miss (my) issue again. I have no objection against various point of views and to accommodate all them in the article, I have an objection with inaccurate (intentionally obfuscating?) descriptions of the various solutions. Again this is not about solutions being right or wrong, but about describing them accurately. I'm not arguing Morgan is right and vos Savant is wrong (on the contrary), I'm arguing for describing both solutions accurately. It's rather bad thing to represent solutions to an ambiguous problem and not explaining how they resolve the ambiguity and what they actually compute (that would be classical case of lousy teaching btw., essentially fuzzy arguments). As far as your comparison with religion is concerned, from my perspective it is rather religion versus science. Personally I have little regard pagans or monotheist and their dogmatic views. I don't care for either vos-Savant-believers or Morgan-believers, but I care for accurate descriptions (the scientific approach if you will).--Kmhkmh (talk) 14:30, 14 February 2010 (UTC)
 * [Mainly copied from above] I have no wish to continue the vos Savant/Morgan quarrel, provided that this issue is left open. There is no consensus that Morgan were right and vos Savant wrong. We have two sources that disagree, let us leave it at that. All I want to do is move the 'Aids to understanding' section to its logical place in the article, where it will be most helpful to our readers. All I am asking is a little licence to defer the criticism of the simple solutions until the really hard part, understanding why the answer is 2/3 not 1/2, is fully explained, an approach that is in the best tradition of mathematical and scientific text books. I agree that it is possible that some readers may begin to get wrong understanding of the potential conditional nature of the problem, and I agree that we should word the section so as not to say anything that is actually incorrect, but to start adding in restrictions and conditions relating to the simple solution will do nothing to help our readers' understanding. Through familiarity with the subject you have lost sight of just how hard this seemingly simple probability problem is to understand for people who see it for the first time. As the article itself says, 'Even when given a completely unambiguous statement of the Monty Hall problem, explanations, simulations, and formal mathematical proofs, many people still meet the correct answer with disbelief'. It is our duty to do our best to overcome this problem before complicating the subject in a way that is of little interest to most people. Martin Hogbin (talk) 15:22, 14 February 2010 (UTC)
 * Again as i said several times before, though the article might cite Morgan's criticism somewhere, it should not describe vos savant solution as false. However it needs to describe what vos Savant's solution actually computes and how she (or others like Devlin, Henze, Behrens) resolves the ambiguity. This means describing that the computes the total probability that switching wins, that can be done without mentioning Morgan or any conditions. However as a whole a reader going through the article should not just learn a somewhat fuzzy solution to an ambiguous problem, but he should learn why the problem is ambiguous, how the ambiguity can be resolved and how that leads to different solutions and computations. So again this is not about Morgan or vos Savant or whoever being right, but about a (thorough) understanding of what they've actually done. Another thing is, that I find it a troublesome attitude towards readers you seem insinuate here, i.e. assuming them being incapable of understanding the precise solution descriptions. Furthermore I'd like to point out that this is an encyclopedic article and not "the complete idiot's guide to MHP".--Kmhkmh (talk) 15:45, 14 February 2010 (UTC)
 * Your last sentence demonstrates your complete misunderstanding of this subject. There is nothing idiotic about finding it hard to understand the basic MHP.  Thousands of people, including many professional mathematicians and statisticians wrote to vos Savant to tell her that the answer was 1/2 and not 2/3.  They were not all complete idiots.  The question of whether the answer is 1/2 or 2/3 is the only issue that is even know about outside a very small band of academics and other interested parties.  It is the notable MHP and the only reason that we have an article on the subject at all.  We must explain this problem first and we must explain it clearly and convincingly without making any incorrect statements.  I believe that glossing over some of the details at the start is essential to get this job done.
 * After we have explained the basic problem then I have no problem at all in explaining exactly what problem various sources address. I have never suggested that the reader should, 'just learn a somewhat fuzzy solution to an ambiguous problem', and I completely agree that the reader should, 'learn why the problem is ambiguous, how the ambiguity can be resolved and how that leads to different solutions and computations', if that is their desire,  but this is only possible once the reader has fully understood the basic paradox. Martin Hogbin (talk) 16:04, 14 February 2010 (UTC)
 * I think it might just demonstrate your unfamiliarity with Complete Idiot's Guides. The comment was referring to the book series and its writing styles/goals, which somewhat differs from that of an encyclopedia. However professional mathematicians and statisticians writing vos savant (in a rather rude language) were idiots, not for their failure to grasp the prolem but for their attitude and lack of manners. Also the notion, that we just have an WP entry because of the Parade controversy, is false. Generally speaking WP has (potentially) entries on any science problem that has acquired its own name and/or is somewhat notable in academic literature (see for instance Category:Mathematical_problems). MHP has that with or without vos savant/parade.--Kmhkmh (talk) 16:21, 14 February 2010 (UTC)
 * Sorry I missed your point, had you put it with capitals I might have got it. If it were not for the Parade question, I doubt that the problem would be generally known about at all.  The problem is too close to the three prisoners problem to be that interesting to statisticians.  My main point was that it is the fact that the answer is 2/3 and not 1/2 that makes this problem notable.


 * I am not against a full clear and accurate description of the problem that covers all the views on the subject and discusses all the issues that you raise but I still think that you and some others completely underestimate the difficulty that most people have with this problem when they first encounter it. A clear solution is not enough, we need to explain and convince, otherwise the rest of the article is pointless.  All I am suggesting is moving 'Aids to understanding'.  Let me say again that it is very interesting to discuss exactly what formulation various sources solve and that this is something we should do here.  Martin Hogbin (talk) 16:56, 14 February 2010 (UTC)


 * I'm pretty sure MvS was mistaken. Her supposed IQ of above 798 doesn't prevent her for blunders. She made a laugh of all the people reacting negatively, but it is quite obvious she gave the simple solution as a solution to the full conditional problem. And then instead of admitting this, tried to find a way out. If I remember well, she came also with the combied doors solution. Anyway, MvS is not to be considered a reliable source.Nijdam (talk) 20:30, 14 February 2010 (UTC)
 * It is quite possible that Vos Savant did intend her solution to be to the 'conditional' problem, but maybe also she was smart enough to realise that, if the host 'acted as the agent of chance', there was no significant condition. Whatever we may think, some sources and some editors here agree with here and some sources and some editors her disagree with her.  She is another reliable source. Martin Hogbin (talk) 20:53, 14 February 2010 (UTC)
 * Vos Savant did compute the total probability for switching to answer Whitaker's question. Whether she was aware of the conditional interpretation or under which circumstances those 2 approaches become identical is idle speculation and irrelevant for WP. Also vos Savant's solution is neither "wrong" nor "incomplete" per se. It would have been only wrong, if she had explicitly suggested that her computation (total probability) is theoretically identical to conditional solution without giving a justification (such as symmetry). However afaik at least in her original column she did not do that, hence to the very least she made a valid heuristic argument using the total probability and being an appropriate answer to the puzzle. Also the last 2 posting again symbolize the vos Savant versus Morgan proxy war that has become the curse of this article. Don't spend your time on "proving" vos Savant or Morgan wrong, but spend it on writing an article that covers all aspect of the problem in a clear, precise and factually correct manner.--Kmhkmh (talk) 23:23, 14 February 2010 (UTC)
 * You claim not to want to discuss vos Savant vs Morgan but you insist that she is wrong. It seems that you want to drop this subject, provided everyone accepts that you are right.  I am perfectly happy to discuss the details of Morgan's and vos Savant's solutions with you (I suggest in the arguments page) if you wish.  On the other hand I am happy to drop the subject with you on the basis that we both accept that there are two reliable (as defined in WP) but conflicting sources.  You cannot have it both ways, either we accept both sources or we analyse them both in detail.  Martin Hogbin (talk) 11:28, 15 February 2010 (UTC)
 * I suggest you carefully reread my postings. I do not not claim that vos savant is wrong, i claimed for probability issues Morgan is (obviously) a better source. This is about comparing the reliability/reputability of sources and the domain expertise as far as the usage of source goes. But it is not about who is right, which is a separate question. That means you should resort to other reputable sources confirming vos Savants approach, i.e. from the WP perspective vos Savant is not right because she says so in the Parade magazine or her website, but she is right because some reputable sources say so or confirm her approach (such as Devlin, Behrends, Henze, Seymann). To give you an unrelated analogon (not involving vos Savant) to understand the point here. Suppose some amateur (or even a professional mathematician) discovers an short elementary proof to Fermat's last theorem and publishes it in some local newspaper, yellow press or general interest magazine. How does WP assess the discovery is accurate and for real? And what sources would it use for the description in WP? It assesses the accuracy by looking at reputable (secondary) sources (in the concerned domain) and how they judge the discovery and exactly these reputable sources will be used to write the article. The original publication in the local newspaper or general interest magazine however is not used as a (main) source for the article, but at best it is used as primary (as opposed to secondary) source for illustrative purposes. That is a core principle of WP. You need to distinguish between 3 different things: vos Savant role in the history of he problem, vos Savant's argument being right or wrong and vos Savant's/Parade's use as a (reputable) source. I was only talking about the latter and not the former two.--Kmhkmh (talk) 13:40, 15 February 2010 (UTC)

I have never claimed that vos Savant was "right" in her arguements about probability or whavever. I have merely said that I'm in favour of starting the article with the vos Savant/Parade story, and in particular with the literally quoted Whitaker Three Door question. Is it unthinkable that we all agree on that? Secondly we would make progress if we could agree that the Whitaker question has been interpreted in many different ways by different authorities. I don't claim any particular way has priority. I am in favour of the wikipedia article taking a factual, reliably sourced, descriptive and neutral point of view. We should represent the various mainstream points of view in an honest way, and represent the criticism of them which also exists out there, in reliable sources, in an honest way. Gill110951 (talk) 11:41, 20 February 2010 (UTC)

Suggest we defer to the mediator
I think this is rapidly devolving into the same sort of unstructured discussion that has been going on for over a year with more threads popping up every day at a furious rate and resolving nothing. I think folks here are eager to proceed and are honestly trying, but for whatever reason we seem to be unable among ourselves to bring any single discussion to a resolution. On this page we already have 5-10 open threads, with nothing resolved. Rather than continue in this same manner, I suggest we let or even insist Andrevan guide this discussion. To this end, I suggest we all (except Andrevan) refrain from creating new threads (and, yes, I realize this is a new thread). I, for one, am loathe to continue participating in this circus again and was expecting something different from mediation. One way to make sure it is different is to act differently. Perhaps we should comment only in our own sections (arbcom style). Perhaps we should adhere to a rate limit (e.g. one comment per day). One thing we can certainly do is stop commenting completely and ask Andrevan how we should proceed.

So, Andrevan, how should we proceed? -- Rick Block (talk) 19:27, 14 February 2010 (UTC)
 * +1--Kmhkmh (talk) 00:56, 15 February 2010 (UTC)
 * As I have said before, I would welcome a mediator, but not an adjudicator or referee. We all need to be helped to see the other side's POV. I am very willing to accept help in this respect, so long as it is even-handed.  Martin Hogbin (talk) 21:02, 14 February 2010 (UTC)

This is a common problem with on-wiki mediation. In the past we have done real-time mediations, either on IRC or via phone/VoIP. I think we have a strong enough basis here to move to a different format. Does everyone feel comfortable with that sort of approach? Andrevan@ 01:53, 15 February 2010 (UTC)
 * irc is fine with me however i think we might have scheduling issues (participants are spread over 2 continents at least faik).--Kmhkmh (talk) 02:19, 15 February 2010 (UTC)


 * There are enough people involved that I think it might be kind of awkward to schedule a real time meeting, even not considering time zone issues (we span at least 7 time zones, possibly more - I'm UTC-7, I'm pretty sure Martin is UTC+0 but Nijdam and Gill are likely UTC+1). -- Rick Block (talk) 03:02, 15 February 2010 (UTC)
 * Nijdam and I are presently UTC+1 when at our homes or offices (=Central European Time, winter)Gill110951 (talk) 15:51, 24 February 2010 (UTC)

I think that Andrevan is either quitting or wishing he didn't sign-up for this one. An IRC or phone "free-for-all" would be like one of those Sunday morning political shows with everyone talking at once and trying to out-loud the others. This forum at least provides a documented evidence trail. A different format wont change anyone's mind and just make things more confounding for the next mediator. hydnjo (talk) 05:47, 15 February 2010 (UTC)
 * I agree. I do not use IRC or a VoIP service and would not accept mediation through those media.  As Hydnjo says, at least we have a record of what has been said by everyone here.  It would seem to me that Andrevan's style of mediation is not working here and that it might be worth asking for another mediator, someone whose style is more aimed at helping  everyone to see the other side's POV. Martin Hogbin (talk) 10:03, 15 February 2010 (UTC)
 * That's not quite true. IRC has become something of standard tool for many WP editors to facility live discussions on WP topics and live discussion can help to clear things in a time fashion. Also such discussions are not without written record, since you can log them (each participant can do that) and if you feel the need post/save the logs on thus mediation page or somewhere else in WP. You can find some more info at IRC--Kmhkmh (talk) 13:55, 15 February 2010 (UTC)
 * I can see no advantage in using IRC. We are all familiar and happy with the current environment.  Martin Hogbin (talk) 14:26, 15 February 2010 (UTC)

In case we won't be able to pursue in irc approach. So far the meditation has gotten us nowhere but I think that more our fault than that of Andrevan. As I said earlier this mediation can only work if all of us accept an article that is not optimal from our perspective but still acceptable for WP. So everybody has to give in, but not just some token or something that follows from WP guidelines anyway. Then we have to take it item by item (rather than discussing all sorts of aspects all over the place) and stay focused on that until we get it resolved.--Kmhkmh (talk) 14:11, 15 February 2010 (UTC)
 * I am not casting any blame. This is an intractable problem which some styles of mediation may suit better than others.  Martin Hogbin (talk) 14:26, 15 February 2010 (UTC)

The problem with the on-wiki mediation is the rate at which it gets out of control. I would argue this is pretty close to a "free for all." In IRC, it is easy to ask a question, and give speaking power to 1 answerer at a time. It is also trivial to save a log of the conversation. As far as scheduling goes, if someone in the USA would agree to meet in the morning or afternoon, then it would be the evening for the UK. If one or two people can't or don't want to, we may be able to appoint representatives for each side or something. Andrevan@ 20:27, 15 February 2010 (UTC)
 * If you want to speak to one person at a time just do that here. Just start a section 'Questions for Rick Block' for example and make it clear that you do not want others to answer in that section.  I am sure everyone will respect your wishes. Martin Hogbin (talk) 23:01, 15 February 2010 (UTC)
 * OK, fine. Give me a day or two to come up with content for everyone's section. Andrevan@ 19:16, 17 February 2010 (UTC)

Wikipedia MH friends, just FYI: I was asked to contribute to a Springer book and internet encyclopaedia project "Lexicon of Statistics", and have contributed the following: http://www.math.leidenuniv.nl/~gill/1002.3878.pdf, http://arxiv.org/abs/1002.3878 Comments are welcome. Gill110951 (talk) 12:39, 20 February 2010 (UTC)

''I have never claimed that vos Savant was "right" in her arguements about probability or whavever. I have merely said that I'm in favour of starting the article with the vos Savant/Parade story, and in particular with the literally quoted Whitaker Three Door question. Is it unthinkable that we all agree on that? Secondly we would make progress if we could agree that the Whitaker question has been interpreted in many different ways by different authorities. I don't claim any particular way has priority. I am in favour of the wikipedia article taking a factual, reliably sourced, descriptive and neutral point of view. We should represent the various mainstream points of view in an honest way, and represent the criticism of them which also exists out there, in reliable sources, in an honest way. Gill110951 (talk) 11:41, 20 February 2010 (UTC)'' Yes, I strongly feel that this will be the right approach to solve the problem-problem-problem, showing the core paradox and to help the article to be what it is supposed to be. To distinguish and to clearly separate the central issue from the chaff, to keep the core issue clearly apart from the paraphernalia and the historical burden. To present everything clearly structured. Concentrating on the true aim of the paradox. And I welcome the progress in the discussion pages, also. Please stay tuned to this article, your help here is important. Thank you ! -- Gerhardvalentin (talk) 14:06, 22 February 2010 (UTC)
 * Thank you, Gill110951, for showing your contribution to that encyclopaedia project "Lexicon of Statistics" and your structure-centred view. On 20 February 11:41 you wrote here:
 * You are aware that "I don't claim any particular way has priority" suggests that there is no such thing as a "core issue" or "true aim"?--Kmhkmh (talk) 16:13, 22 February 2010 (UTC)
 * @Kmhkmh, you are just teasing my fan @Gerhard Valentin, I think! Seriously though, don't we all have the same true aim here, namely to solve the core issue, which is to decide how to structure the wikipedia article? Gill110951 (talk) 15:41, 24 February 2010 (UTC)
 * Gill, a very interesting article, I think the game theory approach should be added to WP, once we get started again. Martin Hogbin (talk) 17:39, 23 February 2010 (UTC)
 * I cannot remember now, but did you consider that the producer might try to improve his odds by the choice of initial car position. As with host's door choice, I think he can do no better than to choose randomly. Martin Hogbin (talk) 15:09, 24 February 2010 (UTC)
 * @Martin: yes I considered the game-show-producer and the host to be one party, who must decide first where to hide the car and secondly which door Monty will open if the player leaves him a choice. The minimax strategy for the game-show-producer-and-host is to hide the car uniformly at random and for the host to open uniformly at random. This is the simplest, but not the only way to game-theoretize the problem. One of my correspondents suggested to turn it into a three-party game (producer, host, player). Monty Hall gets a cookie from the producer if the player switches from the good door to a wrong door. The producer is only interested in keeping the car, and has an infinite stock of cookies. The player wants to get the car. Now it is a three-person non-zero-sum game. I have to take Game Theory 201 to solve that (what is a Nash equilibrium, is there one? how do you find it/them?), so far I only did Game Theory 101 (study materials: wikipedia pages; content: von Neumann minimax theorem) Gill110951 (talk) 15:41, 24 February 2010 (UTC)