Wikipedia talk:Requests for mediation/Monty Hall problem/Archive 6

The problem
How should Whitaker's question be interpreted? Should the problem be considered as a simple probability puzzle with all necessary assumptions to make it simple? Should we take the host to always choose evenly between available doors (the fully symmetrical, or standard problem)?

Solutions
Are the simple solutions to the complete for the standard problem? Is this decision tree derived from Carlton's solution or is it OR? Should the so-called "combining doors" solution be included in an initial "solution" section, or is it more appropriate to include this in an "aids to understanding" section?

Presentation
What is the best order of presentation? Should conditional and unconditional solutions be presented in a single "Solution" section (more or less like this draft, in the show/hide box), or should these solutions be presented in separate, possibly chronological, sections (like in this version)

NPOV
Is the article NPOV, in particular with respect to the POV of the Morgan et al. source which criticizes the so-called "unconditional" solutions as not addressing the problem as stated? Should the article fully address the unconditional solution, with (solution-specific) "sources of confusion" and "aids to understanding" sections before mentioning anything about conditional probability - or would this violate NPOV by tacitly favoring the unconditional approach?

current state
Where are we now?--Kmhkmh (talk) 15:47, 14 November 2010 (UTC)
 * We've spent a lot of effort on a rewrite of the "Conditional solution" section (at /Conditional probability solution), which got hung up on Martin's insistence that the conditional probability NOT be described as pertaining to the situation "after" the host has opened door 3. On the same page, we have then done some work on an alternate approach (suggested by one of the mediators) where rather than an integrated summary we summarize by source - and have worked on a summary of the Carlton source that no one seems to have any specific objections to (although restructuring the article in this way has not been agreed to).  I believe we're waiting on the mediators for guidance on how to proceed, although Martin has (today) suggested what he thinks would be a useful direction. -- Rick Block (talk) 22:19, 14 November 2010 (UTC)
 * Rick: Would you be able to provide a link to Martin's suggestion? I've been away for awhile and want to be sure I know what you are referring to. Sunray (talk) 07:38, 18 November 2010 (UTC)
 * /Conditional probability solution. -- Rick Block (talk) 14:45, 18 November 2010 (UTC)
 * I said "link," but I should have said "diff" (or "diffs"). I need to know exactly what you are referring to. Sunray (talk) 07:00, 19 November 2010 (UTC)
 * Further progress: at a consensus seems to emerge that "the simple solution and the application of logical reasoning show together that always switching is optimal", in other words, the conditional solution is only a small step away from the unconditional solution. And on the MHP page itself, I deleted references to my unpublished work and replaced them by references  (Springer International Encyclopaedia of Statistical Science, November 2011) and  (Statistica Neerlandica vol 65 (1), January 2011). Richard Gill (talk) 09:03, 19 November 2010 (UTC)
 * Richard: I agree that this is further progress. Would you be able to comment on whether we might gather all threads of discussion into one area? Or is that desirable? Sunray (talk) 16:34, 19 November 2010 (UTC)
 * Sunray - Martin's proposal was . -- Rick Block (talk) 14:36, 19 November 2010 (UTC)
 * Thanks, Rick. It seems opportune for me to ask participants how they wish to proceed from here. Sunray (talk) 16:34, 19 November 2010 (UTC)

How do we proceed? I'd like to understand what happened to the mediators the last 3 weeks or so, and if we should expect a recurrence. Glkanter (talk) 16:44, 19 November 2010 (UTC)

Background
Mediators have lives. As you all know, I have been traveling. I knew (and told participants) that I would have sporadic access to the web over the past two months, but I overestimated how much I would be able to participate. Nevertheless, I have been keeping an eye on things. Let me explain a couple of things about that.

One of the features of this mediation is that it is relatively inaccessible to outsiders, including mediators. The arguments are complex and the participants possess a great deal of knowledge of the subject. Those are facts. If I may also venture an opinion: Academic discourse predisposes one to debate and dispute. The best academic forums are collegial, but that is, perhaps, a by-product rather than the main objective. The objective is advancing knowledge in a particular field.

Wikipedia also has the aim of advancing knowledge. However the means are rather different that those of academia. Collaboration is the the basis and lifeblood of all Wikipedia policies. The goals of academic discourse are not always compatible. Research is a important aim of academic life that is not a goal of Wikipedia. Fortunately, the requirement to use verifiable sources is common to both. Some participants and the mediators have recognized this as the way forward. Sunray (talk) 17:41, 19 November 2010 (UTC)

Current status
Much has been achieved in this mediation:
 * 1) Several participants have worked hard at finding common ground.
 * 2) Interaction between participants is civil and generally respectful.
 * 3) Participants have generally held to groundrules. The last two remain somewhat elusive (leanness of expression; consensus).
 * 4) There seem to have been recent developments towards collaboration and consensus.

It seems that there are, potentially, grounds for a successful mediation. I know I am in the optimist category on this, but I think that if the above gains are consolidated and participants are willing to continue working on the latter two groundrules, we could pull it off. Sunray (talk) 17:41, 19 November 2010 (UTC)

Next steps
Please comment on the above. If I've got something wrong, feel free to straighten me out. Also, I would like to hear what you want from me. I will have more time to devote in the next while and if progress continues, will be keenly motivated to assist. Sunray (talk) 17:41, 19 November 2010 (UTC)


 * Despite the wide-ranging discussion there is only really one issue on which there is serious disagreement and that it the validity of the simple solutions. Does anyone disagree with this?


 * If there is general agreement on my statement above that I suggest that it would be useful for the mediators to lead a discussion on that subject. I concur with Glkanter in that it will be necessary for the mediators to take a relatively active role for the mediation to be of any value.  I also suggest some clear ground rules must be in place and enforced to keep the discussion on topic. Martin Hogbin (talk) 17:45, 19 November 2010 (UTC)
 * Martin: You refer to "... my statement above." With the volumes of text in this mediation, I find it difficult to have to refer to a statement above - it means I have to stop reading and search for something and I am not always clear that I've got the right one. Would participants be able to either reproduce the statement they are referring to or provide a diff that links to it? Sunray (talk) 11:58, 21 November 2010 (UTC)
 * I was referring to this:
 * 'Despite the wide-ranging discussion there is only really one issue on which there is serious disagreement and that it the validity of the simple solutions.'
 * Whatever is discussed, this is always the sticking point. I think it would be very helpful for you to lead a discussion onthis subject. Martin Hogbin (talk) 13:06, 21 November 2010 (UTC)
 * Thanks, I will pose that question. Sunray (talk) 02:33, 22 November 2010 (UTC)
 * You're right, and its f-ing ridiculous. The simple solutions come from many reliable sources. There should be no discussion allowed questioning their 'validity', or insisting that they are 'wrong', and that the article *must* reflect that. Take a look Sunray, that's what goes on way too much in this mediation. Just like it did on the talk pages. Glkanter (talk) 18:19, 19 November 2010 (UTC)


 * The topic on which there is serious disagreement is not the validity of the simple solutions, but the validity of the criticism of the simple solutions and how to address this in an NPOV manner in the article. I said this in response to Martin's suggestion for a change of approach - rather than change directions yet again, I'd like to see if we can close on any of the topics we've already discussed, i.e. the Carlton summary, or the rewrite of the Conditional solution section, or even the rewording of the single sentence about Carlton's intuitive explanation that's in the "Simple solutions" section.  Changing topics and/or approach every time we reach an impasse seems extremely unproductive.  What I would like the mediator to do is to pick a topic (any topic, but I'd suggest one of the topics we've already spent considerable time discussing) and then drive a discussion on that single topic to closure - keeping the discussion focused on the one topic with as heavy handed intervention as is required (up to and including simply deleting off topic posts, although putting them in collapsible sections would probably be sufficient).  Then pick another specific topic and do the same thing.  Then another, and another, until we're done.  Rather than dictate topics, the mediator could force us to pick each topic (using a multi-vote sort of approach like we did when we decided we'd address the conditional solution section, see above). -- Rick Block (talk) 04:03, 20 November 2010 (UTC)
 * You are saying that there is only one point of serious disagreement: "the validity of the criticism of the simple solutions and how to address this in an NPOV manner in the article." You also list topics that could be closed quickly with some focussed discussion. I will ask for comments on this. You suggest a process: picking topics by vote and address them one at a time. Sunray (talk) 12:10, 21 November 2010 (UTC)
 * Rick, am not going to disagree with your first statement but I do not really see the distinction between what you say and what I have said. If the simple solutions are valid then the criticism of them is invalid and vice versa. I cannot see any benefit in endlessly discussing things that we agree on or in rewriting all or part of a FA from scratch.  Is there anything other than the validity of the simple solutions and their criticism and the way that that should be presented in this article that you feel there is general disagreement on?  Martin Hogbin (talk) 10:05, 20 November 2010 (UTC)
 * Well the key issue would be to rewrite the simple solution section together and possible the sources of confusion section. Why not going directly for this?--Kmhkmh (talk) 13:04, 20 November 2010 (UTC)


 * I am not against that but I do not see that it helps either. The key thing is to try to agree on the validity of the simple solutions and the criticism of them.  If we cannot do that we should try to agree on a form of words that would be acceptable to all.  The sources really do not help either, the majority give the simple solutions without criticism, some criticise the simple solutions, and some criticise the criticism.  We do need to make such a meal of it in the article, we just need to try to find a form of words that sums up the situation to everybody's satisfaction.  I have a suggestion, see below. Martin Hogbin (talk) 14:14, 20 November 2010 (UTC)


 * The sources do not help? Of course they do.  We agree many give a simple solution.  We agree (but you omit) many give a conditional solution without any comment about simple solutions.  We agree some (a not insignificant number) give conditional solutions and "criticize" simple solutions.  I'm aware of only one of these sources that is in turn criticized - although it's more about the tone than any mathematical point.  I think we don't exactly agree what the sources criticize about the simple solutions - I'd say these sources draw a distinction between the unconditional probability (the probability of always switching, also phrased as deciding to switch before seeing which door the host opens) and the conditional probability (the probability of switching knowing which door the host opens), and say the simple solutions are perfectly valid but address the former rather than the latter.  Some editors here seem to disagree with this point (themselves), seem unwilling to accept that this is what these sources are saying (even when this point has been explained to them dozens if not hundreds of times, including by a professor of statistics) and do not want the article to reflect this point even as a POV let alone as an uncontroverted mathematical fact. -- Rick Block (talk) 18:17, 20 November 2010 (UTC)


 * If only there were any 'uncontroverted mathematical' facts about the MHP but the 'facts' depend on what version of the problem you consider, how you interpret it, what statistical approach you take, and what is regarded as too obvious and to need explaining. We then have to consider how best to present these facts for our readership.  The sources do not tell us how to write an encyclopedia article.  We need to decide how to do that ourselves, all based on what reliable sources say, of course.


 * You response really proves my point that this is the subject that we need mediation on. It is the one thing that we constantly disagree on and one on which we seem to have made no progress.  Unless you like my suggestion below? Martin Hogbin (talk) 18:32, 20 November 2010 (UTC)

Question about Morgan et al
Over on Richard's talk page we were, in a round-about way, discussing/questioning just which reliable sources directly say the simple solutions are flawed in some way, and that the conditional decision tree and Bayes' theorem were correct. Morgan didn't say it, without the host choosing uniformly between goats premise, Morgan has the conditional decision tree, as one of the 6 false solutions they address. Rick, can you shed light on this? Glkanter (talk) 06:09, 20 November 2010 (UTC)
 * The problem they're addressing includes their interpretation of vos Savant's clarifications from her columns (most explicit in the experimental procedure she describes in her third column). In this problem the car is uniformly distributed, the player's choice is random, but the host's choice between two goats is not specified.  Their objection to F6 (which is a conditional solution but not a decision tree) is that it assumes the host chooses uniformly between two goats.  They're not criticizing the method of this solution, but the assumptions.  -- Rick Block (talk) 18:32, 20 November 2010 (UTC)

Thanks, Rick, but you are intertwining two different concepts from the discussions on Richard's page.

For purposes of this discussion of the solution sections, my only point is that Morgan says that without the host choosing uniformly between goats premise, both the simple solutions and the verbal description of the conditional decision tree (and Bayes' theorem) solutions are false.

So, I ask, as Nijdam and others repeatedly insist the article must present, which reliable sources say 'the simple solutions are flawed, and the conditional decision tree and Bayes' theorem are valid'? Because Morgan's paper clearly does *not* say that. Glkanter (talk) 19:18, 20 November 2010 (UTC)


 * You're ignoring Morgan's rejoinder to vos Savant where they say "even if one accepts the restrictions that she places on the reader's question, it is still a conditional probability problem" - the restrictions they're referring to include that the host is "an agent of chance who always opens a losing door and offers the contestant the opportunity to switch" (presumably also meaning the host chooses randomly between two goats).


 * Other sources in some way "criticizing" the simple solutions and expressing a clear preference for conditional solutions:


 * Gillman: "This is an elegant proof, but it does not address the problem posed, in which the host has shown you a goat at #3"
 * Grinstead and Snell: "This very simple analysis, though correct, does not quite solve the problem that Craig posed. Craig asked for the conditional probability that you win if you switch, given that you have chosen door 1 and that Monty has chosen door 3."
 * Lucas, Rosenhouse, and Schepler: "This shows that any proposed solution to the MHP failing to pay close attention to Monty’s selection procedure is incomplete."
 * Rosenthal: "This solution is actually correct, but I consider it "shaky" because it fails for slight variants of the problem."
 * Eisenhauer: "Consequently, what could and should have been a correct and enlightening answer to the problem was made unconvincing and misleading. Subsequent work by Gillman (1992) and Falk (1992) applied the correct Bayesian mathematics to derive the general solution, but several other authors continued to perpetuate the "no news" argument, which at best relies on an unstated assumption (see, for example, Engel and Ventoulias 1991; Gilovich et al. 1995)."
 * Devlin (Monty Hall revsited): [after showing the simple solution produces the wrong answer in a slight variant] "Confused? As sometimes arises in mathematics, when you find yourself in a confusing situation, it may be easier to find the relevant mathematical formula and simply plug in the appropriate values without worrying what it all means. In this case, the formula you need is due to an 18th Century English Presbyterian minister by the name of Thomas Bayes."


 * -- Rick Block (talk) 20:07, 20 November 2010 (UTC)

Thank you Rick. I didn't ignore their rejoinder. Morgan's rejoinder also says that when each unchosen door is equally likely to be opened to reveal a goat, the simple solutions are a valid method of solving the conditional door 1 and door 3 problem.

Your list, then, is an excellent starting point.
 * I would remove Rosenthal,as he says 'it fails for slight variants of the problem.' Well, the article isn't about 'slight variants of the problem'. Devlin goes out for the same reason.
 * Rosenthal uses the "variants" to motivate his student Glkanter to realise that he should worry about the conditional probability, not the unconditional. Richard Gill (talk) 15:18, 21 November 2010 (UTC)
 * I would eliminate Eisenhauer and Lucas as the Selvin and K & W assumptions prominent in the article *do* state the assumptions. Also, Morgan pointed out the conditional solutions are false without the 50/50 host, so these sources are not saying 'simple false, conditional valid'.
 * The simple solutions do not even use all the K & W assumptions. You need all of them in order to derive the conditional probability = 2/3 solution. So apparently Eisenhauer and Lucas are also conditionalists. Richard Gill (talk) 15:18, 21 November 2010 (UTC)
 * As to Gillman, and G&S, their argument is specifically refuted by Morgan's rejoinder where they say:
 * "From this and her previous solutions, one is tempted to conclude that vos Savant does not understand that the conditional problem (of interest to the player) and the unconditional problem (of interest to the host) are not the same, and that 2/3 is the answer to the relevant conditional problem only if p = q = 1/2. Certainly the condition p = q = 1/2 should have been put on via a randomization device at this point. It could also have been mentioned that this means that which of the unchosen doors is shown is irrelevant, which is the basis for solving the unconditional problem as a response to the conditional one. But then, it may be that only an academician, and no one connected with a game show, would ever consider p # q."


 * Let's Make a Deal: The Player's Dilemma]: Rejoinder


 * Author(s): J. P. Morgan, N. R. Chaganty, R. C. Dahiya, M. J. Doviak


 * Source: The American Statistician, Vol. 45, No. 4 (Nov., 1991), p. 289


 * Published by: American Statistical Association


 * No. Morgan et al. make clear that one can use symmetry as a bridge to get the conditional solution from the unconditional. They nowhere give up their criticism of vos Savant that she only thought about unconditional probabilities. Richard Gill (talk) 15:18, 21 November 2010 (UTC)


 * From all this, I conclude you have found 2 reliable sources that find fault with one or more of the simple solutions, but not necessarily all of the simple solutions, and then use the conditional solutions. For example, the Combining Doors solution *does* consider doors 1 and 3. And Morgan, one of the ring leaders of the 'simple is false' movement says the simple solutions *are* valid with the Selvin and K & W premises, which the article incorporates. Finally, all of the reliable sources that publish the various simple solutions do not agree with the critics. From this I cannot agree that the POV that 'simple solutions are flawed, conditional solutions are valid' requires inclusion in any solution section. It's a minority POV, and there is no reason to give it Undue weight in the article. On many occasions, I have suggested a 'Controversies' section later in the article where these criticisms could be noted, but not judged. Glkanter (talk) 20:37, 20 November 2010 (UTC)


 * Morgan's statement "It could also have been mentioned that this means that which of the unchosen doors is shown is irrelevant, which is the basis for solving the unconditional problem as a response to the conditional one" in no way contradicts their assertion that the problem is inherently conditional and it certainly does not "refute" what Gillman, G&S, and Lucas et al are saying. And, dismissing Rosenthal, Devlin, and Eisenhauer because they talk about variants is ludicrous as well.  They show the simple solutions are deficient by applying them to variants where they produce the wrong answer (where the conditional and unconditional answers are different).  If, as you have recently been claiming, you're interested in sticking with what the sources say, you shouldn't be throwing sources out because you disagree with them.  This is the essence of POV-driven editing. That the unconditional probability is mathematically different from the conditional probability, and that the simple solutions address the former but not the latter and hence produce the right answer only where these are the same, is not a "minority POV" but an uncontroverted mathematical fact.


 * If we follow Will's suggestion and reorganize at least the "Solution" section by source, I think this argument becomes moot. The simple solutions don't say whether they're intending to address the conditional or unconditional probability, so in our summaries of these sources we wouldn't either.  Most sources presenting conditional solutions say nothing (at least not directly) about simple solutions, so in our summary of one of these sources we wouldn't either.  Where we put what Gillman, or G&S, or any of these sources have to say about simple solutions is surely a topic we'll eventually discuss.  Instead of arguing about it now, I'd rather wait until Sunray lets us know how to proceed. -- Rick Block (talk) 23:16, 20 November 2010 (UTC)


 * Yes, I can wait for Sunray. I'm not dismissing any reliable sources with my comments above. I'm simply saying that only 2 reliable sources support the POV of many editors here that 'simple solutions are flawed, conditional solutions are valid' must be prominent in the article, especially the solution sections, and that for the 2 that may actually say that, Morgan (!) clearly refutes their conclusion. Thanks very much for your input. Glkanter (talk)


 * Shame on me. I neglected to say that I was using the Selvin and K & W premises as stated in the Wikipedia article as my source for which premises are stated or not. Glkanter (talk) 23:36, 20 November 2010 (UTC)


 * Further, the points made by the reliable sources about unstated premises and ambiguities in the Whitaker/vos Savant version are addressed twice in the article before the solutions. These issues are 'normalized' by the K & W statement of the problem that appears before the solutions, making moot the issue of the host choosing between goats equally, and making the 'variants' clearly a different problem than the MHP being solved. In short, those issues being raised in the solution sections add no value to the reader, serving only to undercut the simple solutions, and instead, confuse the reader. *Those* are the NPOV violations. Glkanter (talk) 00:06, 21 November 2010 (UTC)


 * I'm sorry Glkanter, but I agree with Rick that all those sources which he listed think that you ought to compute a conditional probability. Including Morgan et al. The remarks in the Morgan et al. rejoinder only admit that it is possible to easily deduce that the conditional probability is 2/3 once you know the unconditional probability is 2/3, by the use of symmetry and the law of total probability. This means tha, in their opinion, any simple solution can be used to make the first step, but a few more words are needed to justify the final step. Richard Gill (talk) 15:18, 21 November 2010 (UTC)

The POV in question is specifically 'Simple is flawed, conditional is valid', not which technique they 'prefer'. And how much prominence that POV deserves. I stand by my statement that only 2 of those sources make the flawed/valid conclusion for the K & W formulation. You've said this weekend numerous times that for the symmetrical problem, the simple solutions solve the door 1 and door 3 problem. How can say those 5 sources are calling it wrong, when they are saying it needs to be [what's the word I want?] 'used judiciously? Otherwise, I think you talk in circles. Glkanter (talk) 16:03, 21 November 2010 (UTC)


 * Richard is a professor. He's trying to get you to understand (this is not POV, but mathematics) that simple solutions address the unconditional probability, which is the probability of winning by always switching, and this is not the same mathematical concept as the conditional probability given the player picks door 1 and the host opens door 3, which is the probability of winning by switching in this specific case.  You are apparently hearing this as "simple is flawed, conditional is valid".  Although some of these sources are definitely saying this, it is not what Richard is saying. He's saying the two types of solutions address different mathematical concepts.  He's not picking a "right one" and a "wrong one" - but saying they are indeed different. -- Rick Block (talk) 17:04, 21 November 2010 (UTC)


 * Please don't lecture me about another editor's intent, especially when your comments are based on a flawed reading/understanding of my comment regarding the uses of the reliable sources. Glkanter (talk) 18:40, 21 November 2010 (UTC)


 * Rick was correctly reporting my views and also my intention. I have by the way completely abandoned promoting my own POV in these discussions some time ago, and right now an only interested in increasing mutual understanding so we can at last move on. Richard Gill (talk) 07:40, 22 November 2010 (UTC)

OK, Rick got you right. Please re-read my comment, both of you, and tell me the significance of that fact relative to my comment. Richard, I hope your revised approach contributes to bringing us to closure. Glkanter (talk) 10:27, 22 November 2010 (UTC)

Another suggestion
We leave the article exactly as it is but we add to the simple solution section some wording which in a positive way expresses the fact that the application of the simple solutions to the conditional problem is based on the symmetry with respect to door number that is inherent in the question. By 'positive way' I mean more like 'makes use of an obvious symmetry' rather than 'is incomplete because...' but I am sure we should be able to find better words between us. Martin Hogbin (talk) 14:14, 20 November 2010 (UTC)


 * You mean like the text I first proposed probably about a year ago as part of a consolidated NPOV "Solution" section (e.g. here) - like this:


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


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


 * That is not what I meant, or what I said. Martin Hogbin (talk) 20:44, 20 November 2010 (UTC)


 * As I recollect, you didn't like the "Assuming each specific case ..." wording. Perhaps you'd prefer


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


 * My suggestion is we stick to what the sources actually say. -- Rick Block (talk) 19:06, 20 November 2010 (UTC)


 * Fine let us stick with what most of the sources say, which is a simple solution with no disclaimer. Martin Hogbin (talk) 20:44, 20 November 2010 (UTC)

Sunray, any chance of some mediation here? The sticking point is always the same. Martin Hogbin (talk) 11:32, 21 November 2010 (UTC)
 * Sure. Bear in mind, though, that I don't have any magic to offer. the heavy lifting will have to be done by participants. I will start a new section below. Sunray (talk) 04:46, 22 November 2010 (UTC)


 * Rather than increase the cross traffic with a new section, maybe you could mediate by attempting to filter or otherwise bring into line the editor 'contributions' that are not supported by reliable sources? There's plenty there just from today. Glkanter (talk) 04:51, 22 November 2010 (UTC)

Glkanter's Response to Sunray's Posting
Sunray, you are mistaken on nearly every count. We are not close to any consensus. There is nothing complex about this topic or this mediation. Some editors may make it seem that way, but it is not complex. There are sources, and the sources do not all express the same POV. How to properly represent these differing POVs in the article is the whole argument. The current article is full of NPOV and OR violations. I explained why I thought we have been at this endlessly, but you admonished me for expressing what I believe is obvious. The tag team mediator approach does nothing but halt any momentum. Its happened more than once. As far as I can tell, the mediators seem focused on civility, and not much else. Basic application/interpretation of Wikipedia policies from the mediators would help a great deal. Glkanter (talk) 17:53, 19 November 2010 (UTC)

Getting this back to a single talk page would help, too. But you know what the most important thing is? For the mediator(s) to actually read the postings in, or close to real time. I have the feeling that even when you guys are participating, you're basically waiting to be called in for some reason. I don't know this, but that's how it seems. That doesn't work. The editors who disregard Wikipedia's policies are very evident to those who read all the postings. Calling these civilly-behaving but policy-ignoring editors out, as an opposing editor, does me or the mediation no good whatsoever. First on the list, if we were forced to stick to reliable sources in our discussions, this mediation would be a lot less unwieldy. Glkanter (talk) 18:10, 19 November 2010 (UTC)


 * Mistaken on nearly every point? Well, at least I'm consistent :)


 * As to criticisms of mediators: Fuhgeddaboudit. I'm back and will continue to be your mediator. If I need help of other mediators, I will ask for it. For now you are stuck with me. As to reliable sources: that is policy. Sunray (talk) 15:17, 20 November 2010 (UTC)


 * I stated a number of relevant facts, in response to your query. You're welcome. Reliable sources *are* policy. That several editors in this mediation continue to ignore that policy is also a fact. * (Personal comments removed) ... following the rules is not the point of the mediation. Ending the 2.5 year stalemate is the point, and I'm showing you where the problems are. With the goal of improving what to now has been a goddam waste of everybody's time. Glkanter (talk) 16:06, 20 November 2010 (UTC)

I hope you acted on my suggestion, and in the best interests of this mediation read some of the postings I referred you to. Glkanter (talk) 06:38, 21 November 2010 (UTC)


 * I suggest that we consolidate any gains participants have made and all work at meeting groundrules and policy.


 * Glkanter is sharp and fast and uses rapid intuitive reasoning. Because he's fast and impatient he doesn't seem to see the difference between completing a simple solution to MHP by symmetry in order to derive the conditional solution, and the simple solution itself. For him the answer is "2/3" and "switch", and he wants a short fast route to getting "2/3" and "switch". He seems not interested in the question "2/3 of what?" The simple solutions all tell you (A): "2/3 of all the times you play, switching will give you the car". The authors of those solutions would all agree with that, BTW. The conditional solution, however, tells you (B): "2/3 of the time you chose door 1 and the host opened door 3, switching will give you the car". The "conditionalists" all think this is an important difference and think that (B) is what you really need to know. Most "simplists" probably did not even think about the question.


 * Now it's a Reliably Sourced Fact that by symmetry it can't make any difference which door you chose and which door was opened, so once we have got (A) we can deduce (B) as well, not just for the specific door numbers 1-3, but also for all the five other pairs 1-2, 2-1, 2-3, 3-1, 3-2 simultaneously! This means that the simplist and the conditionalist solutions are not that far apart at all, and there is no need to make a big deal out of the difference. Let's just report the simple solutions first, at face value, and then go on to the conditionalist solutions and their critique of the simplist solutions.


 * As long as you do not mean by "the" simple solutions the simple solutions mentioned by a lot of sources, not concerning the condition, but rigorously mention the unconditional probability of 1/3 and without any further argument state the problem to be solved. A simple solution, that mentions the situation the player is in, and argues that the original (unconditional) 1/3 is not influenced and hence ... etc. is okay. Nijdam (talk) 16:43, 21 November 2010 (UTC)


 * Martin put it really well the other day on my talk page. I rewrite in my own words.


 * Everyone's (except my mother's, but she was one of Alan Turing's computers) immediate intuitive response to MHP is 50/50.


 * After some discussion and thought most people become convinced by one of the simple solutions that the right answer is 2/3:1/3. Most readers are not specialists in probability and statistics and are not even aware of the difference between conditional and unconditional probabilities. A few will think briefly about the fact that the simple argument for switching doesn't address explicitly the case "player chose door 1 and host opened door 3", but probably quickly forget about this again, feeling intuitively there's no reason to address each of the six cases separately.


 * Shame on all those people and the people presenting this simple solutions. Nijdam (talk) 16:42, 21 November 2010 (UTC)
 * There is no reason for shame. As it turns out their intuition is correct. Martin Hogbin (talk) 16:58, 21 November 2010 (UTC)
 * However people accustomed to using probability calculus will do more than feel uncomfortable. They learnt already from the 50-50 versus 2/3:1/3 solutions, that it is not safe to trust to intuition in probability puzzles. They will sit down and do the calculations on the back of an envelope, and find the conditional probabilities are all 2/3 as well. A smaller number of really smart mathematicians will not have wasted time and an envelope, but realise immediately that *all* the 6 conditional probabilities must be equal to one another by symmetry and hence all equal to the unconditional 2/3: one can replace a calculation using Bayes' theorem of probability trees with a few words of plain English.


 * These smart mathematicians will immediately see the simple solutions are plain wrong, as these solutions calculate the wrong probability and seem to be happy with the numerical outcome. Nijdam (talk) 17:16, 21 November 2010 (UTC)


 * Nijdam is at the opposite end of the spectrum from Glkanter. He believes that the simple solutions are wrong because they address the wrong problem, and that this should be said so right from the start. But that is imposing his POV on the whole article.


 * I think the solution is simply to present the simple solutions first, taking care that it is always made clear what they actually give you. 2/3 and switch, yes, but 2/3 of what. That is a matter of fact, not of criticism. Then go on to the conditional solutions making clear what they actually give you. 2/3 and switch, but now 2/3 of something different. Since the easiest correct conditional solution is obtained by building on a simple solution by invoking symmetry, we need not make such heavy weather in discussing the conditional solutions as in the present article. Richard Gill (talk) 15:58, 21 November 2010 (UTC)


 * Why not present the correct conditional (posterior, or whatever we will name it) first. Nijdam (talk) 16:42, 21 November 2010 (UTC)


 * Because 90% of our readers will lose interest and look for help elsewhere. Martin Hogbin (talk) 16:58, 21 November 2010 (UTC)


 * Here are some more reasons.


 * Recall that not all reliable sources agree with your semantic interpretation of Marilyn vos Savant's words, let alone the mathematical formulation which follows from taking a subjectivist view of probability. Not all reliable sources interpret probability in a normative way (you *must* do this or that). The reliable sources which take the conditionalist view do a very poor job of convincing anyone why you *must* follow the indications of the conditional probability given everything known, and many do a very poor job of explaining why their assumptions are the *right* assumptions to make.


 * So if you want to sell wikipedia readers the idea of solving MHP with conditional probability, you had better bring it to them gently. First explain the "simple solutions" clearly and fairly. This includes making it clear what those approaches assume and what they deliver. This is not a question of criticism, this is simply a question of presenting honest consumer information. Then move on to the conditional solutions. Make it clear what those approaches assume and what they deliver.


 * Another reason why this approach won't succeed is because you'll never convince a significant majority of your fellow wikipedia editors to do it that way. If you think of the editors' opinions as spread out in a continuum there are two extremists: one extremist conditionalist, one extremist simplist. If the main body of the editors come to a consensus and just two editors are left with opposing and extreme viewpoints, the job will probably continue without their input. Everyone else will just get more and more annoyed that the extremists are blocking progress and not contributing to the discussion. Richard Gill (talk) 09:01, 22 November 2010 (UTC)


 * Your suggested approach is very similar to the compromise I've been seeking for nearly two years. I think one of the issues here is that certain editors are hearing making it clear what the simple solutions address as POV-laden "criticism" and therefore do not want this clarified.  Although I think it's far better to clarify what they're talking about at the point the simple solutions are presented I'm even willing to defer this clarification to where the conditional solutions are presented, but even this is apparently not acceptable to certain editors.  This situation is what I think led Will to suggest we forgo any semblance of an editorial narrative and simply summarize what a variety of sources literally say.  Since we seem to fundamentally disagree what the coherent narrative should be, letting the sources speak for themselves and letting the reader form his or her own narrative may be the only feasible approach. -- Rick Block (talk) 16:36, 21 November 2010 (UTC)

As I read your response Rick, it implies to me that criticisms of other sources' solutions would not be in either solution section. Is this correct? Glkanter (talk) 16:42, 21 November 2010 (UTC)


 * Where we're summarizing sources, or clarifying? And do you consider saying that the simple solutions address the unconditional probability (that is, the probability of "always switching") as opposed to the conditional probability (that is, the probability of switching in a specific case such as player picks door 1 and host opens door 3) to be "criticism"?  This is the clarification that Richard is talking about. -- Rick Block (talk) 17:11, 21 November 2010 (UTC)

The simple sources do not make any such distinction, do they? Including any such qualifiers with the simple solutions is NOT consistent with "forgo any semblance of an editorial narrative and simply summarize what a variety of sources literally say", is it? Glkanter (talk) 17:21, 21 November 2010 (UTC)


 * The simple sources give various simple reasons why "always switching" gives you the car 2/3 of the time. Saying that out loud does not seem to me to be doing more than summarizing what such sources do. It does not imply criticism for not doing something else. It simply makes explicit what the source apparently considers a solution. Richard Gill (talk) 21:03, 21 November 2010 (UTC)

No, the simple sources are not saying 'Always switch'. The simple solutions to the Selvin/K & W MHP are saying, 'In this instance when you have opened a door to reveal a goat, I shall switch. That it is door 3 rather than door 2 is of no import to me, thank you'. The simple solutions to the Selvin/K & W MHP derive the results of a single instance of play, just like the formal conditional solutions. The simple solutions are no more 'average' or 'overall' than the formal conditional solutions. They are no more 'before' seeing a door open than are the formal conditional solutions. Glkanter (talk) 01:48, 22 November 2010 (UTC)


 * The argument which they give is an argument for always switching - they show always switching is better than always staying. Of course the consequence they draw is that you should switch in this particular instance, but they don't show that the probability you'll win in this particular instance is 2/3. They only calculate the overall probability that the switcher wins. This has nothing to do with "before" or "after".


 * You apparently find it obvious that the specific door numbers are not important and that the conditional probability that the switcher wins is also 2/3, but the sources which give the simple solutions do not make this explicit. On the other hand, most sources of the conditionalist solution think that you have to do long calculations to get the conditional probability, so apparently all these professors of mathematics don't think this is obvious at all. Only (AFAIK) Morgan et al. (response to Seymann) and Gill (2010, 2011), who learnt it from wikipedia editor Boris Tsirelson, say explicitly that you can logically deduce that the conditional probabilities are all equal to one another and hence to the unconditional probability and hence that conditioning is superfluous - taking explicit account of the specific door numbers in this particular instance won't lead you to change your decision.


 * Carlton doesn't find it obvious. Rosenthal doesn't find it obvious. Selvin presents all kinds of solutions in all kinds of ways without commenting on the difference in their meaning and without mentioning that he could have got the conditional probability from the unconditional by symmetry. It is nice that we have editors here who are smarter than Carlton, Rosenthal and Selvin combined. They ought to be publishing their research results in the peer reviewed literature so that in a few years time the wikipedia article can be a whole lot shorter and easier. Richard Gill (talk) 10:10, 22 November 2010 (UTC)

Richard' response
It seems that the core dispute concerns "the validity of the criticism of the simple solutions and how to address this in an NPOV manner in the article." I would propose that that is done as follows. First the article should cover the simple solutions, taking care that it is always made clear what they actually give you. 2/3 and switch, yes, but 2/3 of what. Secondly, report the criticism of the simple solutions and give some conditional solutions. Focus especially on the conditional solution which can be summarized as "simple solution plus symmetry". Richard Gill (talk) 16:09, 21 November 2010 (UTC)


 * '2/3 of what'? Due to the obvious initial 3 door symmetry inherent in the Whitaker/vos Savant problem statement about the contestant on a game show, and/or the expressed symmetry in the premises of the Selvin and K & W problem statements, the same as the conditional solutions, door 1 and door 3.
 * One would expect then, that Morgan's comments and mathematical justification in their rejoinder that the conditional solutions are not necessary when the simple solutions are derived from p = q = 1/2 would necessarily immediately precede the introduction of the more complex conditional decision tree solution. Is this correct, Richard? Glkanter (talk) 17:06, 21 November 2010 (UTC)


 * Morgan et al don't say that conditional solutions are not necessary, they say that you do not need to go through long computations to find the conditional probability. And yes, I think the section on the conditional solutions ought to start with the simplest way to get it in the standard case, namely simple plus symmetry. Richard Gill (talk) 20:57, 21 November 2010 (UTC)


 * For those who know only subjectivist probability, I agree that due to the *total* symmetry of ignorance inherent in the Whitaker/vos Savant problem statement about the contestant on a game show, it follows that the unconditional probability is the same as the conditional probability. (We are not just completely ignorant about the car-hiding mechanism, but also about the door-opening-behaviour of the host. And by completely ignorant I mean that the totality of our prior beliefs are unchanged on arbitrarily renumbering the doors.)


 * Whether this independence on door numbers by symmetry is *obvious* or not is a personal opinion. It is Glkanter's personal opinion but it seems not to have been the personal opinion of Carlton, Rosenthal, or Selvin, to mention a few reliable sources who do present conditional and unconditional solutions side by side. Morgan et al mentioned it in their response to Seymann back in 1990, but no other "reliable source" else seems to have explicitly noticed it till twenty years later: Gill (2010, 2011). Which resulted from the present wikipedia debate. Please correct me if I'm wrong.


 * Whether the sources which only give simple solutions are even aware that there is an issue is not obvious at all. How do we know that all those psychologists and animal behaviouralists and popular writers are as smart as Glkanter? Richard Gill (talk) 10:23, 22 November 2010 (UTC)


 * Since we can't know, perhaps we should follow Wikipedia policy and not editorialize on the subject. For 2 years, Glkanter has only claimed that the simple solutions are valid, despite the overwhelming POV of the Wikipedia MHP article since he first read it. Nothing more. Here's the link. Find a sourced simple solution in that version's Solution section, if you can. Here's a clue: there ain't one. Look at the bs that starts that section. That image that's supposed to represent the simple solutions is in the current article supporting the conditional solutions. And it's still unsourced, of course. It wasn't until 4 months after I joined the discussion that I was informed that the pertinent point was that there needs to be reliable sources who say the simple solutions are valid. Which, of course, there are many, but for some reason they were not included. So, go ahead and make sport of me and my input. Many Wikipedia editors have. Too many to count. The article follows Wikipedia policy more closely because of my contributions, and the reader is better served. Hopefully my contributions to this mediation will result in an article that will meet those 2 requirements to an even higher degree. Glkanter (talk) 11:00, 22 November 2010 (UTC)


 * By the way, the version of the article I linked to above is after my first (and probably *only* unreverted) edit to the article:
 * I removed the very first sentence of the Solution section:
 * "The overall probability of winning by switching is determined by the location of the car."


 * Here's the diff
 * I'll leave you decide who I'm smarter than, as it seems important to you. But here's another hint: That statement was included in the FA review. Glkanter (talk) 11:23, 22 November 2010 (UTC)

Begin new discussion
Several participants have indicated that they want to focus on one topic at a time. Further, there is a need for moderation. I suggest that we go back to a modified talking stick approach (perhaps call it a "talking circle,") with the following process: How does that sound? Please indicate your agreement below. Sunray (talk) 05:21, 22 November 2010 (UTC)
 * Participants sign-on to discuss a particular topic. Note: Not all participants need to be discussants. Some may take the role of witness and remain silent for a time.
 * Each discussant makes one post at a time (as succinctly as possible) until all who are participating have made a comment within twenty-four hours, before continuing the round (discussants may just say "I agree with (name of participant)" or "I pass."
 * The moderator comments daily.
 * If the discussants come to a consensus on any matter, that will be taken to the wider group of participants and there will be a four-day period in which to indicate whether they agree or disagree.

Agree/disagree with proposed process
Indicate whether you agree or not and sign below.
 * Agree. -- Rick Block (talk) 05:31, 22 November 2010 (UTC)


 * Repeatedly, prior to this proposed broad sweep approach to focus and otherwise manage the discussions, I have advocated for the counseling of individual editors as to Wikipedia policy vis a vis reliable sources, OR, and NPOV with a similar goal. I'm not aware that this was done, so the 'talking stick' approach is not my preference at this time. Is whatever we are currently doing getting us anywhere? No. So I will abide by the mediator's decision.Glkanter (talk) 06:24, 22 November 2010 (UTC)
 * To my knowledge, this approach has not been used anywhere else online. It is used in mediation in the real world. In fact is is one of the most ancient ways for resolving conflict. As far as assisting participants with policy that would be my role. I will attempt to post frequently enough to be able to contribute. Sunray (talk) 17:23, 22 November 2010 (UTC)
 * Well, I appreciate that info. I'm not really interested in taking part in an experiment after all this time, when traditional active mediation has not been tried yet. I will abide by the mediator's decision. Glkanter (talk) 17:29, 22 November 2010 (UTC)
 * On wiki mediation is very new and usually needs to adapt techniques from meatspace. In the most successful mediations I've been involved with, participants adopted their own process, drawing tools from other media.Sunray (talk) 20:44, 22 November 2010 (UTC)
 * Agree Richard Gill (talk) 08:33, 22 November 2010 (UTC)
 * Agree. [comments moved below] Martin Hogbin (talk) 09:40, 22 November 2010 (UTC)


 * Agree--Kmhkmh (talk) 10:48, 22 November 2010 (UTC)
 * Agree Nijdam (talk) 11:50, 22 November 2010 (UTC)
 * Agree. glopk (talk) 14:59, 22 November 2010 (UTC)
 * Agree Gerhardvalentin (talk) 15:29, 22 November 2010 (UTC)

Comments on the process
I think it would be useful for the mediator to make the exact process clear and to enforce the rules of the process until we make progress or give up. This would include the rules on commenting and starting (or not) new threads on this page. I would also ask the mediators to be reasonably prompt in dealing with responses to prevent the discussion spiraling off into other topics as it has done previously. Martin Hogbin (talk) 09:40, 22 November 2010 (UTC)
 * No doubt that the mediator has to be more present. I've committed to signing in daily. There are times when I will need to present more often. Sunray (talk) 20:41, 22 November 2010 (UTC)
 * Rather than rely solely on the mediator to control this, I'd suggest anyone should feel free to deal with rule violations by putting off topic discussion in a collapsible box and/or refactoring off-topic discussion to its own thread. Although we might like the mediators to monitor this page 24x7 and to be "reasonably prompt in dealing with responses" I think it's unreasonable to expect this.


 * Sunray - what do you think about this? -- Rick Block (talk) 17:06, 22 November 2010 (UTC)
 * I like the idea of participants facilitating the discussion by organizing topics. I do agree that side discussion has made resolution of issues more difficult. I think it is important to make decisions by consensus, though. So instead of someone deciding on his or her own to use collapsible boxes, I would rather see a brief proposal to participants: "Hey, I think there this/these side-topic(s) is/are taking us of course. If no one objects, I will put them in a collapsible box..." After 24 hours and assuming no objection, the discussion would be collapsed. If there was argument on the point, the mediator could comment. Sunray (talk) 20:38, 22 November 2010 (UTC)


 * The point is that discussions here (for whatever reason) seem to be able to generate more volume in 24 hours than most reasonable people are willing to read. I'm fine with trying the 24 hour notice approach.  If problems arise perhaps we can reconsider. -- Rick Block (talk) 23:35, 22 November 2010 (UTC)
 * Yes, I agree that the volume problem is one of the biggest challenges we face. However, if we can all remain conscious of this, it will help. Sunray (talk) 22:58, 23 November 2010 (UTC)

Proposed topic
Several participants have suggested that there be a moderated discussion of Martin's proposal, as follows:
 * try to agree on the validity of the simple solutions and the criticism of them.

Agree/disagree with proposed topic
Indicate whether you agree or not and sign below.
 * Disagree. See comments below. -- Rick Block (talk) 05:32, 22 November 2010 (UTC)


 * The topic for about 2.5 years has been in regard to the current Wikipedia MHP article's POV of 'Simple solutions are flawed, only the Conditional solutions are correct'. Why change the discussion now? I will abide by the mediator's decision. Glkanter (talk) 06:29, 22 November 2010 (UTC)


 * I can agree to the proposed topic. Richard Gill (talk) 09:09, 22 November 2010 (UTC)


 * Agree. As Glkanter says, this has been the root of all argument here for over two years. Martin Hogbin (talk) 09:43, 22 November 2010 (UTC)


 * The simple solutions (as in the sources) are not valid, so how can we agree on their validity? Let us first try to agree on the validity of the criticism. Nijdam (talk) 11:54, 22 November 2010 (UTC)


 * Agree, as per and, i.e. considering the question to be answered and the point of knowledge you evidently can,  resp. cannot  base on when answering the question. Gerhardvalentin (talk) 15:36, 22 November 2010 (UTC)

Comments on the topic selection
The topic is phrased as a matter of Truth, rather than interpretation of a specific source or (better) anything directly related to any specific content of the article. I would be happy to discuss this topic in the context of a specific source (e.g. does Morgan et al. characterize "simple solutions" as false) or in the context of a specific change to the article (e.g. is the wording in section "whatever" that says "blah blah blah" an NPOV presentation of what the cited source or sources say). IMO, the question we need to answer is not whether the simple solutions or the "criticism" of them are valid, but whether these are a) published in reliable sources, and b) represented in the article fairly and without bias (i.e. in an NPOV manner). Whether anyone here thinks they're valid or not is completely irrelevant. -- Rick Block (talk) 05:50, 22 November 2010 (UTC)

And, as I've said before, I'd vastly prefer if we closed on one of the items we've already spent considerable time discussing rather than change directions again. -- Rick Block (talk) 05:51, 22 November 2010 (UTC)


 * I agree with Rick that the wording used in naming the topic is unfortunate. After all, the editors' discussion of the validity of simple solutions simply mirrors the reliable sources' disagreement on exactly the same issue. Our own *opinions* as to whether a particular solution or kind of solution is *valid* or not, is pretty irrelevant according to wikipedia policy. Especially since the word *valid* is about as ambiguous as vos Savant's question. But we should be able to agree on a kind of minimal guaranteed content of each solution. Richard Gill (talk) 09:41, 22 November 2010 (UTC)
 * Then we perhaps we should change the wording. I believe the wording that Rick first pointed me to was: "the validity of the criticism of the simple solutions and how to address this in an NPOV manner in the article." When I first read that, it seemed like a valid discussion topic. I am assuming that whatever we do we will be coming to consensus on reliable sources that support the article text.
 * As far as closing on items already discussed goes, that makes good sense to me. Rick, would your be willing to assemble a list of such topics, below. Then we could select a topic from the list, including the above topic (appropriately reworded). Sunray (talk) 17:42, 22 November 2010 (UTC)
 * If I may, the 'validity' of the article's POV of "simple is flawed, only conditional is valid" is not being questioned. There are reliable sources who say it. The more meaningful topic is, "How much prominence/weight should this POV of those sources receive given the K & W problem statement in the article removes any ambiguity about the '50/50 host' criticisms and the 'variant' criticisms." Of course, one *should* really be asking, "why does the Wikipedia MHP article have *any* POV at all?" Glkanter (talk) 17:56, 22 November 2010 (UTC)
 * Topics we have extensively discussed already but not quite closed on (in reverse chronological order):
 * Summary of the Carlton source, written in a style appropriate for inclusion in the article if the article were to be restructured as a summary of sources per Will's suggestion (proposed summary here, with subsequent discussion spiraling out of control)
 * Rewrite of the "Conditional probability solution" section (draft here, with ensuing discussion spiraling out of control in the subsequent sections of the page)
 * Revision of the single sentence in the "Simple solutions" section referenced to the Carlton source (proposal to reword one sentence in here, with ensuing discussion spiraling out of control)
 * -- Rick Block (talk) 18:18, 22 November 2010 (UTC)
 * I agree we need to close discussion on these. We have agreement to discuss the following topic: "the validity of the criticism of the simple solutions and how to address this in an NPOV manner in the article" (which I've re-worded to reflect the discussion). Then I propose we poll on the other topics begin discussing them. I will start the discussion below. In the meantime, Rick would you agree to keeping the text in order and collapsing or archiving completed discussions? Sunray (talk) 17:25, 23 November 2010 (UTC)
 * I'm not sure exactly what you mean by this. If we're talking about only one topic what ordering would be necessary?  I can certainly collapse or archive, but I would suggest you say when to do this.  -- Rick Block (talk) 17:48, 23 November 2010 (UTC)
 * Fair enough. Right now we need to archive almost everything but the "Subpages" and "Groundrules" sections down to the "Begin new discussion" section. We should keep the new process and agreements, then collapse discussions that have been completed. The list of outstanding topics should also remain. Sunray (talk) 22:58, 23 November 2010 (UTC)

Richard's comments on the validity topic
I hope we agree that the simple solutions do give (relatively) easy to follow arguments why, under reasonable assumptions given the context, the switcher will win 2/3 of the games while the stayer will only win 1/3 of the games (A). This does not deny the fact that some editors and apparently many sources seem to think that that is a completely adequate final answer to vos Savant's question.

I hope we agree that the conditional solutions give a bit more delicate arguments why, under stronger but still fairly reasonable assumptions given the context, the switcher will win 2/3 of those games in which the player happened to choose Door 1 and the host happened to open Door 3 (and the same for all the other 5 combinations), (B). This does not deny the fact that some editors and also quite a few sources have the opinion that *this* is what you need to know to give an adequate final answer to vos Savant's question, and that if you haven't noticed that (B) and (A) are different statements, you have only found the right numerical answer (2/3) and the right action ("switch") by quick but sloppy thinking.


 * Agreement isn't the issue, which are the sources that say:
 * "...conditional solutions give a bit more delicate arguments why, under stronger but still fairly reasonable assumptions given the context, the switcher will win 2/3 of those games in which the player happened to choose Door 1 and the host happened to open Door 3 (and the same for all the other 5 combinations)..." Glkanter (talk) 13:04, 30 November 2010 (UTC)


 * Who are the 'quite a few sources' that say:
 * "*this* is what you need to know to give an adequate final answer to vos Savant's question, and that if you haven't noticed that (B) and (A) are different statements, you have only found the right numerical answer (2/3) and the right action ("switch") by quick but sloppy thinking."?
 * Not 'some editors', but the reliable sources. Glkanter (talk) 12:52, 30 November 2010 (UTC)

We have to leave it to the reader to ponder on this difference, if they are so inclined. Respect for the reader implies that we present both points of view factually and neutrally. Since we are agreed that an appeal to symmetry makes it easy to deduce (B) from (A), so easy that some editors think it goes without saying, there is no need whatsoever to be confrontational about this. Unfortunately, plenty of reliable sources (Carlton, Selvin, Rosenthal) apparently didn't feel that it goes without saying, and even the infamous Morgan et al. only remarked on it in an answer to a discussion of their original paper. How can we know whether the reliable sources which only give a simple solution are even aware that there might be an issue? Richard Gill (talk) 09:41, 22 November 2010 (UTC)


 * You have put your finger on a key problem for this mediation. Participants have written volumes about it, without reaching consensus. If there are no sources for the simple solution that show any awareness of the conditional probability solution, we must deal with that. It could be a linking paragraph at the end of the simple solution section or the beginning of the conditional probably section. Is that worth pursuing? Sunray (talk) 17:51, 22 November 2010 (UTC)


 * More importantly, Sunray, which are the reliable sources that criticize the simple solutions for not specifically mentioning 'symmetry is the link to the conditional problem' as part of their solution? It wasn't until I recently stumbled upon Morgan's rejoinder stating that the 'link' exists (Richard then found other mathematics sources) that this 'link' was even acknowledged by the opposition editors. I'm not aware of the sources that make this criticism. Glkanter (talk) 12:42, 30 November 2010 (UTC)


 * Yes Sunray, I think that is the way to go. Richard Gill (talk) 07:03, 23 November 2010 (UTC)


 * Richard's comment, in my opinion, can be easily mis-read. Selvin, the originator of the problem offers 2 simple solutions and a mathematical conditional formula. He makes no criticisms of any of them, and praises Monty Hall himself for the ultimate in simple solutions. vos Savant argues with Morgan at length, and still stands by her simple solution. Carlton offers both, criticizing neither. Rosenthal, I believe mentions both, and calls the simple solutions, 'correct, but shaky', because it doesn't work when certain premises are changed. Morgan states explicitly that with the (K & W) premises, the simple solution *does indeed* solve the conditional problem. I think certain editors are actually critical of the various sources 'presentation' of the simple solution, no longer arguing that the simple solution is flawed. But they haven't acknowledged this yet, nor has the article been corrected for this. Glkanter (talk) 18:07, 22 November 2010 (UTC)


 * Sorry Glkanter, I disagree. Richard Gill (talk) 13:07, 24 November 2010 (UTC)


 * Glkanter seems not to distinguish between three distinct things:


 * the answer "2/3, switch"


 * the interpretation of 2/3 (2/3 of what?)


 * the reasoning used to get the answer.


 * Of course he is in the good company of the legion of non-professional-mathematicians who wrote "reliable sources" on MHP (journalists, psychologists, animal behaviour researchers, ...).


 * I stand by my reading of Carlton, Rosenthal and Selvin. As a professional within the same scientific community as those authors (mathematical statistics) I am pretty sure that my reading of their works is correct, but of course it is difficult for me to prove that, especially since the distinctions which we professionals have been trained to make, are not even recognised by many amateurs.


 * Carlton says that he uses the simple solution as a pedagogical device to make his students realise that 50-50 must be wrong. He shows how to use Bayes' theorem and decision trees to find the conditional probabiity. Apparently he thinks we need to find the conditional probability.


 * Carlton, in a paper on '...The Misuse of Conditional Probability' does not describe any shortcomings of the simple solution he, himself gives. This statement:
 * "Apparently he thinks we need to find the conditional probability."
 * is totally unsupported conjecture on your part. Glkanter (talk) 11:22, 30 November 2010 (UTC)


 * Rosenthal argues that this kind of problem must be solved by careful consideration of the conditional probability since otherwise you might get the wrong answer. In the Monty Hall case he apparently sees it as coincidence that the simple solution gives the right numerical answer and hence the right decision. In his "Monty crawl" the simple argument is correct (yielding an unconditional probability of 2/3) yet the conditional probability is 1/2 hence - in the case the player chose 1 and the host opened 3 - the player need not switch.


 * I just re-read Rosenthal. He never mentions 'conditional probability', and does not solve the puzzle with a conditional decision tree. He's advocating his 'Proportional Probability' technique. He calls his simple solution 'shaky' because it can't solve the problem when premises are changed, creating 'variants' that are not the MHP. Later, he clarifies a 'trick' he used:
 * The original Monty Hall problem implicitly makes an additional assumption: if the host has a choice of which door to open (i.e., if your original selection was correct), then he is equally likely to open either non-selected door. This assumption, callously ignored by the Shaky Solution, is in fact crucial to the conclusion (as the Monty Crawl problem illustrates).
 * He is in fact saying the simple solutions (his Shaky solution, anyways) are fine with the 50/50 premise. He does NOT say it solves the wrong problem. He does NOT call it a coincidence. And his Proportional Probability solution only returns the 2/3 & 1/3 value with the 50/50 host bias, same as his simple solution. He says the 'shaky' solution does not apply to 'Monty Crawl (Random Host)', which has another changed premise and therefor is not the MHP, not that it yields an incorrect 2/3 & 1/3. Glkanter (talk) 11:05, 30 November 2010 (UTC)


 * Selvin presents many different solutions without comment, including Monty Hall's own one-liner in which it is the player who chooses a door uniformly at random while the location of the car is already fixed. Selvin is focussed on providing any argument for switching. Both unconditional and conditional probabilities give you a good reason to switch. The point is that the conditional probability, if you can know it, gives you a better reason for switching.


 * Selvin highly praises Monty Hall's simple solution, after giving a simple solution of his own. Selvin does not say 'The point is that the conditional probability, if you can know it, gives you a better reason for switching.' Nor am I aware of any reliable source describing this 'better reason for switching' you write about. Glkanter (talk) 11:22, 30 November 2010 (UTC)


 * None of these authors wrote that one can deduce the conditional probability from the unconditional by an appeal to symmetry. It would have made their computations and decision trees superfluous, and would have been a valuable contribution to their students' education. It would have changed the face of MHP studies. Apparently it is not so obvious to many professionals, while amateurs are not sensitive to the logical distinctions. AFAIK, the symmetry argument has only been given in print by Morgan et al, in their response to Seymann's discussion of their paper, and by Gill (2010, 2011) in response to the discussion here. Richard Gill (talk) 07:00, 23 November 2010 (UTC)


 * Perhaps not, but each of these reliable sources gives a simple solution to the problem they have described where the contestant chooses door 1 and the host reveal door 3. Without criticism. Glkanter (talk) 11:22, 30 November 2010 (UTC)

Richard, that is the clearest example of 'Emperor's New Clothes' justification I have ever seen. The unwashed are too ignorant to know why they know. Selvin, Carlton and Rosenhouse (thal?), being educated, are expected to tell the unwashed why what they know is why they know it. Failing that, Selvin, Carlton and Rosenhouse (thal?) are no better than the unwashed. Remember that sentence about 'You...people make me...' that you deleted from your talk page the other day? This is why. Glkanter (talk) 19:07, 23 November 2010 (UTC)


 * Yes, I object to guests to my talk page using language like that. Richard Gill (talk) 13:07, 24 November 2010 (UTC)

A Short Play, Starring Science!
Glkanter: Oh, now I get it! It's 2/3 because the contestant's choice is still 1/3! That is so cool!! I love this paradox!!!

They: NO, THAT'S NOT RIGHT!! You only 'think' you 'get it'. Dr. Selvin, a subject matter expert will now commence to educating you.

Glkanter: No, really, I get it. That's not nec...

They:  Dr. Selvin?

Dr. Selvin: Yeah, I dunno. I think he gets it. That's kinda why I wrote my letter the way I did. I don't really see any need...

They: SILENCE!! Be gone with you, 'Dr.' Selvin, if you really *are* a 'Dr.'!! Do not darken our door posts with your ignorant blasphemies and heresy again!

And take your fellow charlatans and heretics with you! Be gone, Dr. vos Savant, Dr. Carlton, Dr. Devlin, Dr. Williams, Dr. Adams, Dr. Stibel and all the rest of you so-called 'Drs.' who have published papers similar to theirs!! Your credibility ceases to exist in our village!

[User:Glkanter|Glkanter]] (talk) 20:11, 23 November 2010 (UTC)

Prof Kill (hastens to defend Glkanter from the arrogant and pedantic conditionalist hordes): I think I understand Glkanter's reasoning. I believe that he is using probability in the subjectivist sense, it describes his information about the world, rather than in the frequentist sense, as an objective property of the physical world (which would be revealed by observing many independent repetitions). He also intuitively realizes that the obvious symmetry of the problem (the door numbering is totally arbitrary) means that the conditional (subjectivist) probability that switching will give the car, given the numbers of the doors chosen by the player and opened by the host, is independent of the specific numbers involved, hence equal to the unconditional subjectivist probability 2/3 (which is what the simple solution tells us). Strange that this is only mentioned in print by those scoundrels Morgan et al, in a kind of footnote to their pretty unreadable and arrogantly phrased paper. If only Selvin or Carlton of Rosenthal had noticed the symmetry argument, they wouldn't have had to bother us with the intricacies of Bayes' theorem. And the present mediation would be completely unnecessarily since the only matter of dispute is a "matter of taste": should one consider the symmetry argument so obvious that there is no need at all to mention it, or should one write it out (as I just did) explicitly. Richard Gill (talk) 12:58, 24 November 2010 (UTC)

Glkanter: how dare you label me! How dare you claim to be able to read my mind! How dare you use long words which only arrogant mathematics professors know about! You ****ing people make me ****ing sick! This was not posted by Glkanter on this mediation page. Glkanter (talk) 13:47, 24 November 2010 (UTC) No it was posted by me. Richard Gill (talk) 14:23, 24 November 2010 (UTC), Prof. Kill


 * As an editor involved in this mediation, my only point is that 'simple is flawed, conditional is the only valid solution' is not a prevalent POV in the literature, it is unambiguously contradicted by the 3 most prominent sources, and it is not a statement of mathematical certainty. This whole POV business does nothing more than clutter the current article (especially the conditional solution section), and confuses readers, without adding any value to understanding why its 2/3 & 1/3 rather than 50/50, while violating countless Wikipedia policies. And Richard, stop saying, 'If only Selvin or Carlton of Rosenthal had noticed the symmetry argument'. They probably did, and felt it was too minor to make an issue of. Either way, none of us *know* why they wrote their papers the way they did. We shouldn't act like we do. Any of us. Glkanter (talk) 13:47, 24 November 2010 (UTC)


 * My only dispute with the previous is with the statement "it is unambiguously contradicted by the 3 most prominent sources". I would like to see a reliable source which supports this reading of those papers. I'll ask Jeff Rosenthal, with whom I have been corresponding, what he thinks. Richard Gill (talk) 14:40, 24 November 2010 (UTC)

While I look forward to Dr. Rosenthal's confirmation, I don't see why you cannot agree with my statement.
 * Selvin's first letter offers only a simple solution to his box B, box A problem. In response to a letter from Monty Hall, Selvin concludes his second letter with:
 * Monty Hall wrote and expressed that he was not "a student of statistics problems" but "the big hole in your argument is that once the first box is seen to be emprty, the contestant cannot exchange his box." He continues to say, "Oh and incidentally, after one [box] is seen to be empty, his chances are no longer 50/50 but remain what they were in the first place, one out of three. It just seems to the contestant that one box having been eliminated, he stands a better chance. Not so." I could not have said it better myself.


 * Here's Monty's full letter
 * May 12, 1975


 * Mr. Steve Selvin Asst. Professor of Biostatistics University of California, Berkeley


 * Dear Steve:


 * Thank you for sending me the problem from "The American Statistician."


 * Although I am not a student of a statistics problems, I do know that these figures can always be used to one's advantage, if I wished to manipulate same. The big hole in your argument of problems is that once the first box is seen to be empty, the contestant cannot exchange his box. So the problems still remain the same, don't they. . . one out of three. Oh, and incidentally, after one is seen to be empty, his chances are no longer 50/50 but remain what they were in the first place, one out of three. It just seems to the contestant that one box having been eliminated, he stands a better chance. Not so. It was always two to one against him. And if you ever get on my show, the rules hold fast for you -- no trading boxes after the selection.


 * Next time let's play on my home grounds. I graduated in chemistry and zoology. You want to know your chances of surviving with our polluted air and water?


 * Sincerely, Monty


 * http://www.seaofstars.net/math/montyhall/montyhall.htm


 * Vos Savant argued with Morgan that her simple solution was valid until she was blue in the face.


 * Morgan says it explicitly in their rejoinder to vos Savant.

What is it that you don't agree with in "it is unambiguously contradicted by the 3 most prominent sources"? Glkanter (talk) 17:49, 24 November 2010 (UTC)


 * The POV of Morgan et al. is that simple is flawed. They think you must compute the conditional probability. They admit that this can be done by an appeal to symmetry and the unconditional probability determined by the simple solution. Vos Savant's POV is that simple is fine. She is totally uninterested in the conditional probability. Selvin doesn't care a damn, any argument supporting switching is fine by him. His POV is that any argument at all which tells you to switch is OK. First he gives a simple solution, later he gives a conditional, but he also, later still, strongly approves of Monty Hall's solution, who sees the player's choice as random and the location of the car as fixed, and is only interested in the unconditional probability. Monty Hall hereby espouses, avant la lettre, the game theoretic approach. Monty Hall does not buy the K & W assumptions at all.


 * Economists support the game theoretic approach. No K & W assumptions for them. K & W only state that many people seem to make the K & W assumptions, K & W don't say that you have to. I think that their empirical observation supports the observation that I have earlier made: the man in the street mostly uses probability in the subjectivist sense. Since you don't know anything, for you on this single occasion it is 50/50 what the host will do (if you happened to have picked the door hiding the car). Of course, the man in the street is usually sure that there is no point at all in switching, so I don't think his opinion is necessarily a good guide. Psychology actually supports the game theoretic approach. You shouldn't pick your door according to your favourite number since this will make you reluctant to switch: losing after switching will damage you psychologically much more than losing after staying. The game theorist and the psychologist wisely chooses his own door completely at random, does not even take notice of the number of the door opened by the host, and switches regardless. They are totally uninterested in any conditional probability, whether constructed by some rationalization literally out of nothing, or constructed out of theoretization about quiz showmen's behaviour. Animal behaviourists observe that it is better not to think at all: behave like pigeons who start off by switching or not at random, learn from your experience, and hereby rapidly learn to switch, mostly. Humans on the other hand get trapped by their rationalizations and thereby never even learn their mistakes.


 * Pigeons send an important message to all wikipedia MHP editors who think there is one and only one good solution to MHP. Richard Gill (talk) 21:38, 24 November 2010 (UTC)

Gerhard's comments on the validity topic
Considering the question to be answered and the point of knowledge you evidently can, ''resp. cannot'' base on when answering the question As no-one can ever "know" about such bias, it is an arrogant and presumptuous claim to sell "closer results" as a correct answer to the MHP question. Because any such "solution" will forever concern quite another show. But who cares?
 * For frequentists: The "game show" might have been already discontinued after, say, 3 episodes. Who has better evidence? In 100 million shows, if the host has no bias at all, you certainly will thousands of times be sure to "have detected" some host's bias, even if there is none. You never heard of roulette games in a casino yet?
 * And for mathematicians: Without "conditioning on door numbers" anyone can plainly see that (before AND after!) - even if the host should be extremely biased - Pws will at least be 1/2 in 2/3 of games (but never less) and, simultaneously, 1/1 (!) in 1/3 of games, i.e. on average 2/3.

A little more of honesty and humility would be appropriate instead of evidently wrong "before and after"-perception. Mistaking textbooks for mathematics students as the "answer to the MHP question". Gerhardvalentin (talk) 15:36, 22 November 2010 (UTC) I am happy with presenting a "conditional solution" to the original question, that does not imply more than what is given by the MHP-question "is it to your favor to switch", that is not answering a "900 times" frequentist-question or a "biased-host"-question, assuming an "implied q" deflecting from what is KNOWN. Be aware that other assumptions address other questions. They may be used in textbooks, but they do not address the Whitaker / vos-Savant paradox. Conditional probability is fine but, even without it, everyone can see that Pws in the said "actual game show", that the question is about, will be from 1/2 (and never less) to 1, and without better knowledge exactly 2/3. I am happy with presenting a "conditional solution" to the original question, that does not underhand address "quite other variants", claiming to be "the only correct answer" to the MHP-question. Would be fine if mediation could help to quote what the sources really "are" saying, and not what they are "said to be saying". Gerhardvalentin (talk) 23:27, 22 November 2010 (UTC)
 * How could we operationalize the honesty and humility you refer to? Sunray (talk) 20:25, 22 November 2010 (UTC)
 * Thank you, Sunray, for your question "How to". - Who knows. Read my comment, and read the sources with regard to what "granted suppositions" they are taking for certain, on what assumptions they really are based on, i.e. not just "what they are saying", but still revising what they really "have to say". And you are bound to compare those issues. And read what is written here in this mediation: "Shame on all those people and the people presenting this simple solutions", "These smart mathematicians will immediately see the simple solutions are plain wrong, as these solutions calculate the wrong probability and seem to be happy with the numerical outcome", "the correct conditional solution (posterior, or whatever we will name it)", "The simple solutions (as in the sources) are not valid, so how can we agree on their validity?" and so on,  a.s.o.
 * I take your point. Thanks. Sunray (talk) 17:10, 23 November 2010 (UTC)

Glkanter's Comments On The 'Simple Is Flawed, Only The Conditional Solutions Are Valid' Discussion
Selvin originates the MHP, describes the contestant choosing a box 'B', and the host revealing box 'A. He solves it only with a simple (not conditional) solution in his first letter. Only in his second letter does he add the 50/50 host bias, then uses this as part of a conditional solution to the same problem.

vos Savant edited Whitaker's letter before publishing. She describes the contestant choosing door 1 and the host revealing door 3. She uses only a simple solution to solve this problem.

Morgan writes an entire paper criticizing vos Savant's and (unnamed) other sources use of a simple solution to solve the problem. The 50/50 host bias, which is not needed to calculate her solution is not stated, and they say the simple solutions are not solving the door 1 and door 3 problem. In the same issue that Morgan's paper is published in, vos Savant rejects both of Morgan's arguments. In their rejoinder to vos Savant, again in the same issue, Morgan acknowledges that *had* the 50/50 host bias been included, as both doors would be equally likely to be revealed, the simple solutions do, indeed, solve the door 1 and door 3 problem.

The K & W version of the MHP in the Wikipedia article states that the host bias is 50/50.

So we have Selvin, the originator of the problem comfortably solving the conditional problem with a simple solution.

vos Savant, who popularized the problem, also uses a simple solution to solve the conditional problem.

Morgan, who stirred up the whole controversy in the first place, expressly state in their rejoinder:
 * "From this and her previous solutions, one is tempted to conclude that vos Savant does not understand that the conditional problem (of interest to the player) and the unconditional problem (of interest to the host) are not the same, and that 2/3 is the answer to the relevant conditional problem only if p = q = 1/2. Certainly the condition p = q = 1/2 should have been put on via a randomization device at this point. It could also have been mentioned that this means that which of the unchosen doors is shown is irrelevant, which is the basis for solving the unconditional problem as a response to the conditional one. But then, it may be that only an academician, and no one connected with a game show, would ever consider p # q."

My point is that the 3 most prominent sources in the long history of the MHP all agree that, with the K & W premises that are in the article, simple solutions solve the conditional door 1 and door 3 problem. For the Wikipedia MHP article to promote or favor a POV contrary to this is unwarranted. Glkanter (talk) 07:10, 23 November 2010 (UTC)


 * Selvin does not claim that his solution to the simple unconditional problem solves the more subtle conditional problem. He does not say that the two problems are the same. As a biostatistician he is moreover very much aware of the difference, I can assure you. Morgan et al. agree that the simple solution can be used to solve the conditional problem but they do not say that the two problems are the same. They say "this (no host bias) means that which of the unchosen doors is shown is irrelevant". You have to say that out loud, and I can assure you they know it, in order to bridge the mathematical gap from unconditional to conditional. Kraus and Wang explicitly use Bayes' rule and present the Morgan et al. solution which allows host bias. They write "The standard version provides no information about Monty Hall’s strategy. Is the problem therefore mathematically under- specified and insoluble? The answer is no, because the standard version does not ask for a probability, but for a decision. The general Bayes’s rule for the standard version of the Monty Hall problem in the absence of information about Monty Hall’s strategy is ... Because the strategy-dependent probability p(M3 | C1) varies between 0 and 1, the conditional probability p(C2 | M3) can vary only between 0.5 and 1. Therefore, whatever strategy one assumes Monty Hall to use, the conclusion is that the contestant should switch." Kraus and Wang (psychologists, by the way) use Bayesian formalism throughout their paper. Richard Gill (talk) 08:43, 23 November 2010 (UTC)

Richard, Selvin and vos Savant may have done it wrong, which we're not here to judge, but they solved door 1 and door 3 with simple solutions, which is a valid method of modeling for the symmetrical door 1 and door 3 conditional problem, didn't they? And the Wikipedia article uses the K & W premises, which includes the 50/50 host. So I don't care about 0 and 1, or whatever. And whatever you say about the simple solutions calculating 2/3 goes for the conditional solutions, too. Is my conclusion, repeated and enhanced below, correct?
 * "My point is that the 3 most prominent sources in the long history of the MHP all agree that, with the K & W premises that are in the article, simple solutions solve the conditional door 1 and door 3 problem. For the Wikipedia MHP article to promote or favor a POV contrary to this (as it currently does) is unwarranted."

Or is there a reason to make a Federal case out of a debatable minority point of view which contributes nothing of value relative to actually solving the problem and understanding the paradox? Glkanter (talk) 09:42, 23 November 2010 (UTC)


 * I believe that your conclusion is not correct.


 * I'm confused. What part of my conclusion "is not correct"? Glkanter (talk) 17:50, 24 November 2010 (UTC)


 * Your conclusion was


 * "My point is that the 3 most prominent sources in the long history of the MHP all agree that, with the K & W premises that are in the article, simple solutions solve the conditional door 1 and door 3 problem. For the Wikipedia MHP article to promote or favor a POV contrary to this (as it currently does) is unwarranted."


 * I think it is manifestly false that any of the 3 most prominent sources in the long history of the MHP agree that, with the K & W premises, simple solutions solve the conditional door 1 and door 3 problem.


 * Selvin does not use the simple solution to solve the conditional problem, he uses the simple solution to solve the unconditional problem: his simple solutions tell us that always switching is better than always staying (2/3 vs 1/3, overall) . His conditional solution tells us that 2/3 vs. 1/3 also holds on those occasions when the player chose door 1 and the host opened door 3. Apparently he found both computations good reasons to recommend people to switch but he makes no attempts whatsoever to relate the two results. He apparently did not notice that one could get the conditional probability from the unconditional by an appeal to symmetry. It would have saved him a lot of calculations to do it that way, and it would have saved us all a lot of time and bother, so I doubt he was aware of this elegant short cut. But anyway, his focus was on giving good reasons to switch. Both unconditional probability and conditional probability tell you that switching is better than staying. The conditional probability tells you that switching is better than staying in a stronger sense. And as we know now, the conditional probability can be simply deduced from the unconditional probability plus an appeal to symmetry.


 * Vos Savant apparently was never interested in the conditional problem. The simulation experiment she advocates shows that always switching is better than always staying, she did not ask people to check that switching when the player had chosen door 1 and the host had opened door 3 also won the car in 2/3 of the times. She asked them to check that always switching won the car 2/3 of the times, always staying won it 1/3 of the times. Whitaker did not even mention specific door numbers in his original letter to vos Savant and it looks to me that the words "say, door 1" and "say, door 3" were only added by vos Savant in an attempt to help the reader visualise the set-up. I read somewhere that she even said this explicitly in one of her later writings, but I can't remember where I saw this and I haven't been through her complete works. Her solutions show that she did not intend the reader to do probability calculations within the subset of those occasions when door 1 and door 3 were the two doors chosen and opened in turn.


 * Morgan et al. are at great pains to point out that (in their opinion) we should be interested in the conditional probability, not the unconditional probability. They show that without the K & W premise of no host bias, switching is always at least as good as staying with regards to the conditional probability, which only equals 2/3 in the 50-50 case; in general it can be anything between 1/2 and 1. I find their notation and explanations clumsy. Their result is immediate from the odds form of Bayes rule, but with just a little careful thought the average wikipedia reader and editor could figure it out for themselves. Gerhardvalentin for example takes it as pretty obvious. In their response to Seymann, Morgan et al. point out that in the 50-50 case, by symmetry one does not need to do any difficult computations to find the conditional probability. By symmetry it has to be equal to the unconditional 2/3 found by the simple solution. Basically they say that simple solution unconditional 2/3 plus symmetry implies that the conditional probability is 2/3. Their POV remains unchanged that it is the conditional probability that you want to know.


 * I also disagree that the wikipedia MHP article should promote or favour any POV. There are reliable sources who apparently feel that the MHP is solved by computation of the unconditional probability. There are reliable sources who find it very important to check the conditional probability. Almost all of them do it a long fancy way, almost none of them show that there is an elegant short cut. Richard Gill (talk) 21:10, 24 November 2010 (UTC)


 * Your negative at the start of the first sentence of above paragraph confused me. Maybe you could re-phrase the sentence? Thank you. Glkanter (talk) 01:30, 25 November 2010 (UTC)


 * Which sentence, Glkanter? Which negative? But I'd prefer that you take some time off and try to read carefully, with an open mind, the Statistica Neerlandica (vol 65, January 2011) paper by Richard D. Gill "The Monty Hall Problem is not a Probability Puzzle (it's a challenge in mathematical modelling)". It's posted on Gill's University of Leiden home page at, and on the mathematics ePrint server arXiv.org at . Gill wrote there what Gill wants to say on the matter. He's checking the proofs, just got them from Wiley publishers, so there is still a chance to make the non technical parts of this new source on MHP, which was directly inspired by the wikipedia MHP wars, more readable by the ordinary man in the street. Richard Gill (talk) 08:18, 25 November 2010 (UTC)


 * "I also disagree that the wikipedia MHP article should promote or favour any POV." Glkanter (talk) 08:48, 25 November 2010 (UTC)


 * OK, to say that in a positive way, I agree with you that the wikipedia MHP article should not promote or favour any POV. In particular therefore, it should not promote or favour the POV that the simple solutions are completely adequate. Neither should it promote or favour the POV that the conditional solutions are the only correct solutions. Fortunately the conflict between editors who hold each of these two POV's (I hold neither) is much ado about almost nothing since from the simple solution (an easy way to correctly deduce the unconditional probability) together with symmetry in mathematics and the law of total probability one can immediately conclude that the conditional probability which the conditionalists believe you need to find is equal to the unconditional which is provided by the simple solution. Richard Gill (talk) 09:56, 25 November 2010 (UTC)


 * I believe there is a simple way to reconcile the two main classes of "solutions" under the K&W premise, namely by the remark (due to Morgan et al.) that by the symmetry of these conditions, the conditional probability that switching gives the car given the number of the door chosen by the player and the number of the door opened by the host does not depend on the specific door numbers 1, 3 of vos Savant/Whitaker's question, hence must be equal to the unconditional probability that switching will give the car, 2/3, as given by the simple solutions. In other words, the specific door numbers are irrelevant for deciding whether to switch or stay and it is an utter waste of time to laboriously compute a conditional probability from first principles, except perhaps as some kind of exercise in an introductory statistics class, where of course it has a great deal of merit to solve a problem like this in a number of different ways. See symmetry in mathematics especially the section on randomness, and see the law of total probability. Richard Gill (talk) 13:23, 24 November 2010 (UTC)
 * See also Probability interpretations for discussion of the different notions of probability which are out there. Richard Gill (talk) 14:55, 24 November 2010 (UTC)


 * Maybe it is better to say that Selvin and vos Savant assert that the simple solutions solve the Monty Hall problem, as they saw it. It is not totally clear whether they saw the problem as a conditional one which the simple solutions solve or they considered the problems was unconditional and therefore their simple solutions solved it. Maybe they did not initially notice the difference.  Morgan also seem to have greatly softened their stance on the simple solutions, all but saying that the simple solutions are OK.


 * What more is there to say? The three most important sources agree that the simple solutions solve the MHP.  That is more than enough justification to give them, without reservation, as the first solutions in the article.


 * After that we might consider whether the problem was ever intended to be a conditional one and, for that particular interpretation, what solutions are valid. Martin Hogbin (talk) 22:41, 24 November 2010 (UTC)

My first goal is to end this minority POV of the reliable sources from its domination of the article. I think we've demonstrated more than enough to justify that. Glkanter (talk) 00:30, 25 November 2010 (UTC)


 * It's not a minority POV. And, even so, it doesn't "dominate" the article.  I've been biding my time waiting for Sunray to officially open discussion on whatever topic he's going to open discussion on, with the one post per day rule he's suggested above.  I think it might be good if we all could wait until he does this. -- Rick Block (talk) 00:56, 25 November 2010 (UTC)
 * Is too! Does so! Glkanter (talk) 01:30, 25 November 2010 (UTC)


 * You can't have your cake and eat it, too, Rick.
 * Without the 50/50 host bias premise, the conditional solutions are false as per Morgan.
 * With the 50/50 host bias premise all that nonsense about 'gaining information' goes out the window, as per Morgan.
 * With the 50/50 host bias premise the simple solution is a surrogate for the conditional solutions, as per Morgan.
 * Pick your poison. Both branches of 'simple flawed, conditional valid', the Wikipedia MHP article's current - and your long held - POV, can't be true at the same time. Glkanter (talk) 02:08, 25 November 2010 (UTC)

Discussion topic #1
We have consensus on beginning a new process (see Wikipedia talk:Requests for mediation/Monty Hall problem above). We also have agreement of a majority as to the first topic. Let us begin. Remember, this is a talking circle approach. The key is to speak one at a time and let others comment before speaking again. We have agreed that a 24 hour period is enough to give everyone a chance to speak. Those that prefer to sit out on a particular topic may remain as "witnesses." However, I see no reason why participants shouldn't be able to change roles once during each discussion. Discussants should sign in for each topic. Sunray (talk) 03:20, 25 November 2010 (UTC)


 * Topic #1: the validity of the criticism of the simple solutions and how to address this in an NPOV manner in the article

Discussants: Glkanter, Rick Block, Richard Gill, Martin Hogbin,Nijdam (talk) 14:50, 25 November 2010 (UTC), Gerhardvalentin

Glkanter's Round 1 Comments
First, I'd like to ask Sunray to clarify what he means by 'change roles once during each discussion'? Does 'witness' have a special meaning I'm not familiar with?

It is my opinion that the current Wikipedia MHP article has held a NPOV violating opinion regarding the simple solutions since the article was essentially turned 180 degrees in the spring of 2008. I request each editor address:
 * Whether they concur with my opinion that a POV critical of the simple solutions exists in the article
 * If the answer to the above is 'yes', and the editor is in agreement with this POV, please provide justification for this POV in the form of reliable sources making it clear this is the prevailing viewpoint of the literature on the MHP

Glkanter (talk) 04:14, 25 November 2010 (UTC)


 * The idea was that participants who want to discuss a particular topic so indicate (so we will know when everyone who wants to speak has spoken). However, I'm suggesting that someone can change their status and join the discussion. Witness has no special meaning other than someone who observes.

Rick's round 1 comments
I think we first need to agree that it is not in the least POV to say that the overall probabilistic average of all six possible combinations of player pick and door the host opens is mathematically different from the probabilistic average of the door 1/door 2 and door 1/door 3 combinations and that neither of these probabilistic averages is mathematically (as opposed to numerically) the same as the probability in any one of the 6 possible combinations. Since these are different mathematical concepts, it is possible to construct variations of the MHP where they have the same or different numerical values. Arguing about this strikes me as simply silly.

The second thing I think we need to agree is that simply distinguishing which of these mathematical concepts a particular solution directly addresses is not POV criticism but rather an entirely NPOV clarification. The notion that saying what it is that the simple solutions mathematically address constitutes "POV criticism" is, frankly, bizarre. Morgan et al. definitely elevate this to criticism (calling unconditional solutions "false"), but for example Grinstead and Snell saying (in their opinion) the simple solution "does not quite solve the problem Craig [Whitaker] posed" and that he "asked for the conditional probability that you win if you switch, given that you have chosen door 1 and that Monty has chosen door 3" is not criticizing anything (whatsoever).

Is it "valid" to say what a particular solution directly addresses? Of course it is. What is not valid is putting words into the mouths of sources presenting simple solutions claiming that they meant their solution to address the conditional probability when they say nothing about this at all. We don't know whether they think their solutions are addressing the conditional probability due to symmetry, or whether they fail to realize the conditional and unconditional situations are different because they don't say.

IMO, it's perfectly NPOV for the article to say what these simple solutions say, and then to clarify exactly what these solutions address (i.e. one of the probabilistic averages as opposed to any specific case), and then to present a conditional solution that DOES directly address the specific case (for example the case in the Parade problem description where the player picks door 1 and the host opens door 3). The fact that the numerical answer is the same (with the usual assumptions) should certainly be highlighted, but this doesn't in the least mean that the "simple" solutions are addressing the conditional probability and, moreover, there aren't any sources that claim this.

Having read multiple dozens of sources (at least 50), IMO the "weight" of the sources presenting conditional solutions is at least roughly comparable to the weight of sources presenting simple solutions. Although there might be fewer sources by count that present conditional solutions, the ones that do are generally academic peer-reviewed publications or textbooks, which (per Identifying reliable sources) are generally the "most reliable sources". Only some sources explicitly contrast these two types of solutions, and very few outright criticize sources presenting simple solutions. It seems to me it would be POV to favor presentation of simple solutions in preference to conditional solutions. -- Rick Block (talk) 07:37, 25 November 2010 (UTC)

Richard's round 1 comments
I agree with Rick's first two points. In my own words

(1). It does not constitute the adoption of a POV, to distinguish between concepts of


 * (a) overall probability,


 * (b) probability given only the initial choice of the player, and


 * (c) probability given the initial choice of player and door opened by host.

(2) We are not primarily writing about the *opinions* of the sources, but about the *arguments* which they present in order to solve what they see as MHP. That's what wikipedia readers legitimately want to find - arguments, not opinions. Not (primarily) a list of authorities saying A and another list of authorities saying B. They primarily want to see arguments for A, and arguments for B. Then they can make up their own minds.

My conclusion: it's not only legitimate, because of (1), but also an obligation to the reader, because of (2), to present a source's argument, thereby necessarily distinguishing from the context whether the source addresses probability (a), (b) or (c). Otherwise the argument is simply unintelligible. We may do this even if the source does not do it explicitly, and even if the *opinion* of the source does not coincide with the *argument* which the source provides. Our task is to summarize, not to interpret. To summarize arguments alongside of opinions.

This does not constitute OR, it does not constitute violation of NPOV, it does not constitute editor's personal criticism of the source, it merely constitutes intelligent reading and fair summary of the content (and opinion) presented by the source. Richard Gill (talk) 10:19, 25 November 2010 (UTC)

The article should not promote either simple nor conditional arguments. It must fairly report the criticism which the conditionalists have of the simplists. If that criticism can be reported in a way which is intelligible and compelling to the "great unwashed" majority of wikipedia readers, so much the better for the conditionalists. If not, the criticism will be ignored. That's the way it is.

Fortunately, simple plus symmetry gives conditional, there is no need whatever to make a big thing out of this. Anyone who can understand the arguments behind the simple solution should also be able to comprehend the symmetry argument that the conditional probability is the same. Richard Gill (talk) 10:35, 25 November 2010 (UTC)

Martin - round 1
I think we agree on many things which need not be argued about. In his first paragraph, Rick points out that the conditional and unconditional problems and solutions are mathematically different things. I do not think anyone disputes this. Rick also says that we should say which solutions apply to which versions of the problem. The situation is more complex than Rick suggests in that it can be argued (supported by sources) that the simple solutions do apply to the conditional problem. This is an interesting subject that should, in my opinion, be fully covered in the article. The important issue, and one that it seem we do disagree about, is when and how.

As nearly every source points out, the MHP is essentially a simple mathematical puzzle that most people get wrong, specifically that they think the answer is 1/2 not 2/3. This is what the MHP is all about, it is what makes it notable, it is why we have a WP article on it. I agree that there are interesting lessons about conditional probability that can be learned from the MHP but first and foremost we must make sure that the reader properly understands why the answer is 2/3. Most people will probably want to stop there. To introduce the complication of conditional/unconditional formulations and solutions before this task is complete simply makes the article incomprehensible and confusing to many of our readers.

I am not proposing to favour any source, I am proposing that we write an encyclopedia article that is of most use to all our readers, from the general public to statisticians.

The simple solutions should be given first with full explanation and without reservation for two reasons. The two main sources, vos Savant and Selvin give them as solutions to the MHP (however they interpret it) and Morgan no longer seem to dispute this. Secondly, putting the simple explanations first is the logical order for any complex technical article. Just look at any good text book or encyclopedia article. Martin Hogbin (talk) 10:46, 25 November 2010 (UTC)

Nijdam - round 1
As there is no criticism on the (correct) solution based on the conditional probability, the article should mention this solution first, be it in a way understandable for the average reader. Further as a way of understanding, the simple explanations may be mentioned, together with their restrictions. Also the simple explanations may extended, perhaps by just a short remark, to make them correct. Of course the conditional probability may be calculated using symmetry, but it is not the other way around, i.e. that the simple solution leads to the full conditional solution with the symmetry argument. Nijdam (talk) 14:50, 25 November 2010 (UTC)

Gerhardvalentin round 1
Sources seem to contradict now and then, sometimes it just seems so for a facil look, nonobservant that they actually deal with different aspects and priorities. This is particularly true for the "MHP"-subject with its various aspects. Serious sources deal primarily with the psychological fallacy, with our faulty "assessment skills" in reviewing the obvious "50:50"(same odds) versus "2:1"(double chance) paradox. This paradox can be cleared up quickly, and the WP-lemma should try to help and not to confuse. The famous question clearly asks for a decision, and the "simple solution" helps to find the correct decision: You should switch, because switching will "double" your overall chance from 1/3 to 2/3. Without engaging conditional probability anyone by a closer look can see on the basis of reasonable assumptions (even in considering the most extremely biased host who could be giving closest information on the actual location of the car in every single game show) that - under these premises - switching will give you the car with absolute certainty (with a probability of "1") in 1 out of 3 games, whereas simultaneously Pws will be 1/2 (but never less!) for the rest of 2 out of 3 games. Even such extremely biased host can impossibly change the location of the car, neither the overall Pws of 2/3. Nor can this aspect be of any importance or consequence anyway in finding the correct decision asked for in this paradox.

Other sources have built conditional probability theorems to find the "exact Pws". And although such "host's bias" has "to be known to be given" (a matter of sheer impossibility within the MHP) before you seriously can take it into consideration, other sources have included "any" imaginable host's bias "q" in their theorems. Although completely irrelevant for answering the given question "yes or no", they claim to offer an allegedly "more accurate Pws". Some claimed their procedure for calculating the "only correct solution". But, for the decision asked for, they cannot offer any better news.

So you have to see what the sources really "have to say". Not everything is of importance and helpful for new readers to find the correct decision. And the lemma should not contribute to confusion, but to clarify. I see great progress in our efforts to improve and to upgrade the lemma. To make it better legible, esp. for new readers. Conditional probability should be correctly mentioned, yes, but help to avoid absurd statements that never concern the core of the famous paradox. Gerhardvalentin (talk) 21:40, 25 November 2010 (UTC)


 * Seven participants have contributed thus far and we are still within the 24-hour timeframe, so others may wish to comment. However, for the next round, I suggest that we try to identify areas of consensus. Does someone wish to take a stab at that? Bear in mind that consensus is not necessarily unanimity. Sunray (talk) 22:00, 25 November 2010 (UTC)

Glkanter's Round 2 Comments
There are countless reliable sources that have put forth simple solutions to the popular door 1, door 3 statement of the MHP.
 * These are reliable sources
 * They describe the door 1, door 3 MHP (Selvin describes box B, box A)
 * They give simple solutions
 * These reliable sources do not hedge, weasel, criticize or otherwise qualify their own use of these simple solutions

For any Wikipedia editor to construe these reliable sources are *not* using/supporting/promoting/advocating simple solutions to the door 1, door 3 MHP is indefensible, and has already occurred in the Round 1 comments. That conclusion is contradicted by the English language, logic, math, even by common sense. Accordingly, I can only interpret any and every editors' use of that conclusion to support any argument whatsoever as a breach of Good Faith editing. If it continues, I will seek remedies as allowed by Wikipedia.

The Wikipedia article is titled 'The Monty Hall problem'. Not 'The Conditional Probability Monty Hall problem'. Readers come to the article to understand the 1 car and 2 goats MHP, or even the door 1, door 3 MHP. They *do not* come to the article for a lesson on Conditional Probability. Editing the article with the intention that the article must anoint the Condition Probability aspects of the MHP is the very cause of the NPOV violations in the current article. Glkanter (talk) 18:46, 25 November 2010 (UTC)

And now I have stumbled upon the proper names for the separate solutions sections:
 * The One Car, Two Goats problem
 * The Door 1 and Door 3 problem

Tie them together with symmetry, as per Boris...
 * The Peaceful Co-existence Section
 * Explains how with the K & W formulation of the MHP, by the obvious symmetry you get the same result to either problem using either sections' solutions. Morgan says its OK.

Each reader will clearly be led to the solution that relates to how that reader understands the problem. Glkanter (talk) 21:35, 25 November 2010 (UTC)


 * Glkanter: Participants are still contributing to the first round. Would you be able to hold off on this thread for now? Sunray (talk) 21:46, 25 November 2010 (UTC)


 * Of course, Sunray. I'll probably demure until I can be the last contributor in round 3. Hopefully, I'll be joining in to a consensus for the ideas I put forth in round 2, above. Glkanter (talk) 01:45, 26 November 2010 (UTC)

The first requirement for the simple solution criticisms in the Wikipedia MHP article is that they come from reliable sources, and represent a significant minority view. This has never been demonstrated. Brief lists of sources have been given in response to this request, but the lists have included erroneous sources. I don't think it can be shown that the criticisms represent a 'significant minority', and ask that this requirement be addressed. Glkanter (talk) 11:33, 30 November 2010 (UTC)

There has been a long standing argument over what actually constitutes a criticism of the simple solutions. I'd like this mediation to address this as well. Obviously, I think the characterization has been overly broad. There are sources, often from academia, that point out the simple solutions do not work when the host bias is not 50/50, or when the host doesn't always reveal a goat. Which are true statements, but they are criticizing problems that have different premises than he MHP. They are not criticizing the simple solutions when they are applied to the symmetrical problem (50/50) host with a null condition (the host always reveals a goat), both of which are 'standard' premises to the MHP. These sources use these variants of the MHP as a springboard to the need for conditional probability tools, not as criticisms of the simple solutions to the MHP. As the solution sections are for the symmetrical MHP where the host always reveals a goat only, I'd like to go for consensus that these sources that only criticize the variants with the changed premises cannot be referenced in the solution sections, but may be discussed in later sections as appropriate. Glkanter (talk) 15:36, 30 November 2010 (UTC)

Richard's Round 2 Comments
I think we could get *consensus* (though not unanimity) on the following:

The simple solutions will be given first with full explanation and without reservation, in the tradition of vos Savant and the popular literature on MHP.

Each different route adds a bit of new insight.

After all that, we will introduce the POV that MHP ought to be solved by consideration of a conditional probability, not an unconditional probability (Morgan, Carlton, Rosenthal, and numerous other authors writing in probability and statistics text books). Here we will carefully explain the conceptual difference between conditional and unconditional probability. Following Morgan et al. (1991, response to Seymann's comments) we will emphasize that by symmetry the conditional probability we are after is actually the same as the unconditional (under the assumptions that many people seem to find natural). Hence, under the conditionalist POV, a complete and proper solution to MHP consists of writing down these "natural" assumptions explicitly, giving a simple solution, and adding "by symmetry, the conditional probability of winning by switching given that the player chose door 1 and the host opened door 3 is also 2/3". We will also show how the same conditional probability can alternatively be got be direct computation, and by the use of the odds form of Bayes' rule (Rosenthal, Rosenhouse).

Each different route adds a bit of new insight. Richard Gill (talk) 08:21, 26 November 2010 (UTC)


 * How many participants agree with Richard's statement, above? Sunray (talk) 08:01, 27 November 2010 (UTC)
 * I agree. It is pretty much what I gave been suggesting. I take 'with full explanation' to be some or all or the material in the 'Aids to understanding' section.  In other words more explanation on why the answer is 2/3 not 1/2. Martin Hogbin (talk) 11:55, 27 November 2010 (UTC)


 * Agree with Richard's statement. Gerhardvalentin (talk) 16:33, 27 November 2010 (UTC)


 * Agree if and only if "with full explanation" includes explaining what probability the simple solutions address (i.e. the overall probability or the probability given only the initial choice of the player). Per Krauss and Wang, 97% of their test subjects tried to solve the conditional probability given the player has picked door 1 and the host has opened door 3, and "once formed, this assumption prevents the problem solver from gaining access to the intuitive solution".  IMO, Krauss and Wang are explaining one reason why so many people find the simple solutions unbelievable - these solutions don't explicitly address the problem nearly all people try to solve. -- Rick Block (talk) 17:05, 27 November 2010 (UTC)


 * I do not agree. The only part I remotely agree with is:
 * "The simple solutions will be given first with full explanation and without reservation, in the tradition of vos Savant and the popular literature on MHP."
 * Otherwise:
 * 'Each different route' is double talk.
 * That whole paragraph going into detail about conditional solutions is a rehash of other editors' disputed, and certainly not 'consensus' arguments.
 * The repeat of 'Each different route' is once again, double talk.
 * I'm not sure, but maybe Richard's more recent postings indicate he no longer agrees with the above, either. I can never tell. Glkanter (talk) 10:44, 28 November 2010 (UTC)


 * I only agree giving the simple solution first if it is made perfectly clear it does not address the full MHP as it is normally understood. Nijdam (talk) 16:43, 30 November 2010 (UTC)

Martin - round 2
I support Richard's proposal above but fear we will not reach a consensus on it because we have strayed off the subject a little and have not completed our discussion on the validity of the criticism of the simple solutions. Richard suggests, 'simple solutions will be given first with full explanation and without reservation'. I completely agree but others consider that some form of advisory comment is required because the simple solutions are 'wrong'.

The discussion we need to have concerns the facts that:

1 The two main sources plus other primary and secondary/tertiary sources including those in peer-reviewed journals give simple solutions without reservation.

2 The simple solution are in fact correct, with the application of an obvious and intuitive symmetry.

3 Criticism or disclaimers will make the simple explanations to one of the world's hardest simple mathematical puzzles even harder to understand and less convincing. Martin Hogbin (talk) 10:56, 26 November 2010 (UTC)


 * How many participants agree with Martin's statement? Sunray (talk) 08:02, 27 November 2010 (UTC)


 * Agree to Martins statements, also. Gerhardvalentin (talk) 16:33, 27 November 2010 (UTC) Glkanter (talk) 23:46, 27 November 2010 (UTC)


 * I disagree with the premise that distinguishing what probability the simple solutions address is confusing or constitutes "criticism" or is calling these solutions "wrong". I also disagree with the premise that a "full explanation" of the simple solutions can be presented without saying what probability these solutions address.  I agree with point #1, would rephrase point #2 as "the simple solutions solve the overall probability or the probability given the player's initial choice of door, and with the application of an obvious and intuitive symmetry easily extend to the probability given both the player's initial choice of door and the door the host opens", and completely disagree with point #3. -- Rick Block (talk) 17:19, 27 November 2010 (UTC)


 * I was puzzled for a while what Martin was saying. After discussing this with him let me say that I now interpret his contribution as saying that discussion is needed about his *topics* or *statements* 1, 2, 3. He used the word "facts" but, in my opinion, "fact 2" is an issue for discussion, "fact 3" is an opinion.
 * Anyway, my *opinions* on Martin's three statements are
 * 1.) is beyond reasonable doubt a true fact.
 * 2.) is an important issue for discussion.This is what I think aboout it. I believe that for those who want to know "the conditional probability" under "the usual assumptions", the simplest route is to glue an appeal to symmetry (see subsection on randomness) with reference to the law of total probability onto one of the simple solutions. More importantly, reliable sources say this too. I believe that this argument is simple enough even for many of the great unwashed to understand. This doesn't obligate anyone to buy it. So in my opinion, the MHP page should present the simple solutions, without embedded "health warnings", immediately followed by an accessible discussion of conditional versus unconditional probability together with the just mentioned fast route to the conditional solution. I think that such a combo is a sensible compromise. We have editors who hold the point of view that the simple solutions are all anyone needs to know. We have editors who hold the point of view that the only correct solution to MHP is through conditional probability. The MHP page has to be a useful resource to readers with different backgrounds and interests, in fact, it has to accomodate the same spectrum as the present editors.
 * 3.) is an opinion about the presentation of the simple solutions, to which I also subscribe. But the reliably sourced criticism of those solutions has to be on the MHP page too. Let the readers judge for themselves.
 * Richard Gill (talk) 08:09, 28 November 2010 (UTC)


 * Richard, I'm very interested in knowing who are the '...editors who hold the point of view that the simple solutions are all anyone needs to know.' As that is a argument even I do not put forth for the article. Thank you. Glkanter (talk) 08:51, 28 November 2010 (UTC)


 * Glkanter, I was thinking of several editors - yourself, Martin, Gerhard probably. I was refering to these editors' personal opinion about the Monty Hall problem. Not to their opinion about how the wikipedia article ought to be written. But I should have written "almost anyone". Richard Gill (talk) 09:32, 1 December 2010 (UTC)


 * No, you just shouldn't have written anything of the sort, in any way, at all. That you later focus on some meaningless semantics point, 'almost all' rather than the (lack of) honesty, (lack of) validity, and (detrimental) consequences of your post to the other editors is instructive. I, for one, will not miss future 'contributions' of this caliber in this mediation. Quite the contrary, actually. Glkanter (talk) 13:04, 1 December 2010 (UTC)


 * Yeah, that's pretty goddam deceptive, and disruptive. Makes it look like I don't understand Wikipedia's 'reliable source' policies. I've asked you repeatedly to keep your comments about what I, Glkanter is, in your opinion thinking, off of Wikipedia. Especially when it is contrary to what I actually post. I have written repeatedly, unambiguously, and recently that I understand that all reliably sourced solutions belong in the article. I've had more than enough of you ignoring my polite requests for you to cease this behavior. Why you think it belongs in this mediation, or anywhere on Wikipedia continues to confound me. KNOCK IT OFF! Kapisch? Glkanter (talk) 09:58, 1 December 2010 (UTC)


 * And your comment creates a False Equivalency with the editors in this mediation who *do* post their personal opinion that the simple solutions are *WRONG*. And that contrary to many reliable sources, that POV should dominate the article. Try this on for size, once more, Richard, 'YOU F-ING MAKE ME F-ING SICK!' Glkanter (talk) 10:02, 1 December 2010 (UTC)


 * Martin's point


 * Is more or less fact


 * Is a misinterpretation of the simple solution and seems a wrong effort to mend it. Symmetry may be used to calculate the conditional probability, but not to mend the simple solution, as the simple solution does not intend to calculate any conditional probability.
 * I do not understand. The simple solution is wrong and hence cannot be understood as a solution. Nijdam (talk) 16:57, 30 November 2010 (UTC)

Rick - round 2
The current topic is "the validity of the criticism of the simple solutions and how to address this in an NPOV manner in the article". Reading the comments of the six (not seven) participants in this discussion so far I think there is a clear consensus that

1) It is perfectly NPOV to describe the difference between (using Richard's phrasing) A) overall probability, B) probability given only the initial choice of the player, C) probability given the initial choice of the player and door opened by the host. [Rick, Richard, Martin, Nijdam, Gerhard ]

2) There are multiple reliable sources that present simple solutions without (themselves) distinguishing which of A, B, or C they claim to be addressing. [Rick, Richard, Martin, Gerhard ]

3) There are multiple reliable sources that point out that the simple solutions address either A or B, and that they (these multiple reliable sources) interpret the problem to be asking about C. [Rick, Richard, Martin, Nijdam, Gerhard]

If we consider the "criticism" of the simple solutions to be merely pointing out which of A, B, or C they address, I think consensus on the points above implies a consensus that the criticism is "valid".

The consensus regarding how to address this in an NPOV manner in the article is less clear. There is a clear consensus that the simple solutions can be presented first [Glkanter, Rick, Richard, Martin, Gerhard], but no clear consensus about how or where to distinguish what probability these solutions address.

If I've omitted or included your name inappropriately in any of the supporting lists above, please correct. -- Rick Block (talk) 20:26, 26 November 2010 (UTC)


 * Two questions: 1) Do other participants agree with Rick's summary? 2) Are there other participants who agree with any of the above statements. Sunray (talk) 07:57, 27 November 2010 (UTC)


 * I do not agree that Rick's summary accurately reflects the opinions of many editors here. Martin Hogbin (talk) 11:56, 27 November 2010 (UTC)


 * I would like point out that we will need Rick's response to this question, as well, based on this unique theory of published reliable sources, above:
 * ..."What is not valid is putting words into the mouths of sources presenting simple solutions claiming that they meant their solution to address the conditional probability when they say nothing about this at all. We don't know whether they think their solutions are addressing the conditional probability due to symmetry, or whether they fail to realize the conditional and unconditional situations are different because they don't say.... -- Rick Block (talk) 07:37, 25 November 2010 (UTC)"
 * My own response will come later. Glkanter (talk) 13:33, 27 November 2010 (UTC)


 * I do not agree with Rick's summary, he did not mention the wordings of the sources he likes to address. A mathematical theorem, based on "S1-H3" (door 1 selected by the guest - door 3 opened by the host), that is not able to show superior arguments for the decision asked for, can never be said to be superior to any other awareness and comprehension securing for any conceivable variant (S1-H2; S1-H3; S2-H1; S2-H3; S3-H1; S3-H2) that Pws (Probability to win by switching) will definitively be constricted to the range of 1/2 (min!) to 1/1 (max!), with an average of 2/3 and, without better "knowledge", to 2/3 anyway.


 * Furthermore, we should reject the nonsensical claim that awareness and comprehension securing for any conceivable variant (S1-H2; S1-H3; S2-H1; S2-H3; S3-H1; S3-H2) allegedly does only address A) the overall Pws, but neither B) the initial choice of the player, nor C) the initial choice of the player and door opened by the host, as to being an inaccurate and faulty claim.


 * And we should reject the nonsensical claim that, in addressing as well A as B and C also, such awareness and comprehension [securing for any conceivable variant S1-H2; S1-H3; S2-H1; S2-H3; S3-H1; S3-H2 that Pws will definitively be constricted to the range of 1/2 (min!) to 1/1 (max!), with an average of 2/3 and, without better "knowledge", to 2/3 anyway] allegedly do not even distinguish A, B and C nor were considering C. We should reject such inaccurate and faulty claim that evidently and clearly ignores the facts. Gerhardvalentin (talk) 16:33, 27 November 2010 (UTC)

Those of you claiming to not agree, please delete your name from the specific points you do not agree with. -- Rick Block (talk)


 * Rick's description of a consensus is a work of fiction. Glkanter (talk) 23:44, 27 November 2010 (UTC)
 * The only place your name is mentioned is supporting presentation of simple solutions first. Does this mean you don't agree with this?  If so, feel free to delete or strike your name above.  -- Rick Block (talk) 01:06, 28 November 2010 (UTC)


 * As what you describe above does not represent even a *majority* of the editors, Rick, calling it a 'consensus' is, by definition, a fiction. More correctly, maybe you could call it 'wishful thinking'. But not a 'consensus'. Glkanter (talk) 08:57, 28 November 2010 (UTC)


 * I agree with Rick's 1) to 3). Furthermore, consensus is not unanimity, there will always be editors and readers who object either to the merest hint of criticism of simple solutions, or who object to anything remotely resembling approval of the simple solutions. Fortunately there is a "simple wikipedia" for the former, while I suggest that the latter start a new page The Conditional Probability Monty Hall Problem. The main standard wikipedia page on MHP has to report both points of view, since both are in numerous reliable sources. We should better discuss how to present them in a harmonious way to people of reasonable intelligence, than to continue a childish fight to impose one or other point of view on all the other editors here. Even if that would succeed, new editors would come along and start it all over again. Richard Gill (talk) 08:23, 28 November 2010 (UTC)


 * On second thoughts, Rick's statement 2) needs rewording, since it implicitly claims that the simple solutions *claim* to address A, B or C. They make no such claim since they do not make a distinction. So my approval is with the probable intention of 2), not with its implied claims. Richard Gill (talk) 08:41, 28 November 2010 (UTC)


 * Of course there is a difference in principle between Rick's A, B, and C in his statement 1, just as there is a difference between 'Is the world flat?' and 'Is the world flat, given today is Tuesday?'. To mention the difference between A,B, and C in some appropriate way is not POV but to insist on mentioning, right at the start, a distinction that is irrelevant to the basic paradox is POV. Martin Hogbin (talk) 09:52, 28 November 2010 (UTC)

@Glkanter - I list 4 out of 6 supporting point 1 (I thought it was 5 as Gerhard certainly seems able to distinguish A, B, and C). I list 3 out of 6 supporting point 2 (and again thought Gerhard supported this - I actually think you and Nijdam do as well although neither of you said anything specifically supporting this in this thread). I list 5 out of 6 supporting point 3. Points 2 and 3 are about what sources say, so I'm somewhat perplexed how we're not unanimous about these.

@Richard - Please read point 2 again. It says there are sources which do not distinguish which of A, B, or C they address. Perhaps we could all agree these sources don't distinguish A, B, or C at all.


 * Yes, "these sources don't distinguish A, B, or C at all" is a clearer statement, and one to which I fully agree.Richard Gill (talk) 02:02, 29 November 2010 (UTC)

@Martin - I'm not saying we agree where the difference between A, B, and C should be mentioned, but (as you say) that mentioning there is a difference is not in and of itself POV. -- Rick Block (talk) 17:59, 28 November 2010 (UTC)


 * That depends, Rick. The unexpected, and unnecessary mention alongside the solutions of this matter will create doubts of some sort to some readers, which is the POV of not those sources, of course, but the POV of a minority of other sources, and certain editors. The exact wording, placement, frequency, other references to the same topic, etc. could, and *do so* in the current article, create an NPOV violation. Glkanter (talk) 02:13, 30 November 2010 (UTC)

Glkanter's suggestion
 * Wikipedia MHP Article Scope

I posted this section on Sunray's talk page yesterday. He suggested it was more appropriate for discussion on this mediation page. I am including it as my response to Rick's "consensus summary":

[Begin paste]

I believe a fundamental roadblock to bringing the mediation to a conclusion is the scope of the article, hence the scope of the arguments in the mediation.

The article is being treated as an academic article on the Science Of Probability. Which is way too broad.

The article need be no more complex that '1 car and 2 goats' and 'door 1 and door 3'.

The overly broad scope causes the mediation discussions to take off into esoteric discussions of Probability. Its all 'The Emperors New Clothes' stuff.

Please consider my suggestion that the article and mediation scopes be narrowed dramatically. Thank you. Glkanter (talk) 20:03, 26 November 2010 (UTC)

[End paste]

1. The great unwashed, in the collective, or the average punter, singular, do not come to the Wikipedia MHP article to learn Probability. The come to understand the 50/50 vs 2/3 & 1/3 paradox of the 1 car 2 goat puzzle they have heard orally. Oftentimes, not thinking it important, the great unwashed will tell the puzzle to another, leaving out the door 1 and door 3 information.

2. There can be no argument that nearly all sources describe the 1 car and 2 goats, and door 1 and door 3 problem as the MHP. (Selvin describes box B and Box A.)

3. There can be no argument that many reliable and prominent sources give a simple answer to the 1 car and 2 goats, and door 1 and door 3 problem.

4. There are reliable MHP sources (Morgan), Wikipedia Probability articles, and Wikipedia subject matter expert editors who say the simple solutions with the K & W symmetry are capable of solving the door 1 and door 3 interpretation of the MHP.

5. There need not be any determination made as to which interpretation of the problem the reliable sources giving a simple solution are addressing.

6. The claim that the simple solutions are flawed for the K & W version come from very, very few reliable sources. If this POV needs to be addressed in the article at all, it can be later in the article, certainly not in the solution sections.

7. The vos Savant Parade Magazine controversy involving the general public and over 1,000 mathematicians is addressed early in the current article. This is appropriate.

8. The other controversies regarding ambiguities and variants, which are rendered moot by the K & W premises, can be discussed, if at all, later in the article, and certainly not in the solution sections.

9. There are plenty of well written Wikipedia articles on probability that an interested reader can be directed to.

10. The Wikipedia article is appropriately titled The Monty Hall problem, *not* The Conditional Probability Monty Hall Problem.

11. The discussion in the article, and on all these talk pages demonstrates how dense, and unnecessary the Conditional Probability explanations are. They should be eliminated, as they are not appropriate for the Wikipedia MHP Article about the 1 car and 2 goats and door 1 and dooe 3 puzzle.

11.A. The decision tree and Bayes Theorem solutions are reliably sourced, and of course, should be included in the solution sections.

12. If you *must* solve the puzzle with the conditional probability solutions, it ain't much of a paradox.

The article should be stripped of the current emphasis on host bias, variant premises & problems, and lengthy and obtuse discussions of conditional probability. Glkanter (talk) 00:26, 28 November 2010 (UTC)


 * Sunray - I thought we were trying to focus on one topic at a time with the intent of bringing one topic to closure before moving on to other topics. Do you really want a discussion of this topic now, before we close on the very first topic we're attempting to stayed focused on until it's closed?  -- Rick Block (talk) 18:18, 28 November 2010 (UTC)

Imho, no other sequence would be logical. Glkanter (talk) 19:51, 28 November 2010 (UTC)


 * My opinion is that discussion can only take us so far. There are many insights that may be gained from the discussion, but we won't likely get consensus unless we actually agree on text (i.e., edit collaboratively). It will take the will of participants to tackle that and a topic that we can make some headway with. I'm o.k. with continuing the current discussion for now. However, when participants are ready, I would like to propose that we begin editing together. Sunray (talk) 19:57, 28 November 2010 (UTC)


 * I don't understand how this mediation has shown that the editors are ready to edit the article without a resumption of the edit warring. I don't see any consensus anywhere. Please help me understand. Glkanter (talk) 22:31, 28 November 2010 (UTC)


 * Regarding this suggestion (to change the scope of the article), rather than argue each point individually I'll just say I do not agree whatsoever. The scope of the article is the Monty Hall problem, which (per WP:FACR 1c) includes a "thorough and representative survey of the relevant literature" - this includes both the popular literature about the problem (as represented by vos Savant) and the academic literature (as represented by, say, Morgan et al.).  Suggesting well sourced discussions of what probability "simple" solutions address should be omitted, because Glkanter or any other editor disagrees with these sources, is blatant POV pushing.  -- Rick Block (talk) 21:24, 28 November 2010 (UTC)


 * Your self-adjudicated conclusion of 'blatant POV pushing' is exactly the technique you used to thwart the will of the consensus of editors a year ago, forcing us into this mediation. The wail of 'NPOV VIOLATION' has no more merit now than it did a year ago.
 * Further, that you are so cavalier and rapid in this accusation, yet turn a blind eye to the repeated blatant NPOV violations espoused by editors in this mediation with whom you share a POV, calls into question the sense of fair play at work here. Glkanter (talk) 04:04, 29 November 2010 (UTC)


 * Conditional probability explanations are not necessarily dense: everyone can understand the difference between "2/3 of all the times", "2/3 of the times the player chose door 1", "2/3 of the times the player chose door 1 and the host opened door 3".


 * They are necessary, (a) since many reliable sources present conditional probability solutions, and (b) since *wrong* intuitive thinking about conditional probability is one reason why most people jump to the answer 50:50.


 * Since simple solution plus symmetry yields the conditional probability, answering the conditional probability question is hardly more difficult than answering the unconditional probability question.


 * Some sources do have the opinion that MHP *must* be solved with conditional probability, and that information has to be reported. The reader can decide for themselves. Richard Gill (talk) 02:12, 29 November 2010 (UTC)


 * Yes, I'm basically calling for the removal of all 'health warnings' from the solution sections, plus the removal of the 3 repetitive POV paragraphs in the conditional solution section. Beyond that, I think a new look at Undue Weight is needed, and the off-loading of nitty gritty Probability tangents onto the appropriate Wikipedia articles. I've always said the differing opinions of the reliable sources belong in a 'controversies' section of the article, not elimination. So that the readers can decide for themselves. Glkanter (talk) 04:04, 29 November 2010 (UTC)


 * Since whole books have been written about MHP including much technical literature there is nothing against there being advanced technical sections to the article.


 * As long as the sections for a broad readership are easily distinguishable, there is no need to "offload" material which the "great unwashed" don't need to see.


 * Sections on various controversies and specifically a major section on the conditional versus unconditional debate is a great idea.


 * Health warnings are a silly idea but there's nothing against stating somewhere clearly what each solution actually achieves. Richard Gill (talk) 09:12, 29 November 2010 (UTC)

Sunray - I suggest we close this thread as unproductive, and resume our focused "talking stick" approach to the topic we started above. Glkanter's protestations notwithstanding, I believe we DO have substantial consensus regarding the validity of the "criticism" of the simple solutions - where by "criticism" I mean clarifying what probability (A, B, or C) these solutions address. IMO, this thread has done nothing but disrupt progress toward this consensus. -- Rick Block (talk) 05:36, 29 November 2010 (UTC)

I agree that this thread should me moved somewhere else. Richard Gill (talk) 09:20, 29 November 2010 (UTC)


 * Richard, how do you reconcile these 4 statements?:
 * 'Health warnings are a silly idea but there's nothing against stating somewhere clearly what each solution actually achieves. Richard Gill (talk) 09:12, 29 November 2010 (UTC)'
 * 'Brilliant idea to have a "to condition or not to condition" section.' - Richard
 * 'Topic #1 - the validity of the criticism of the simple solutions and how to address this in an NPOV manner in the article' - Sunray
 * 'I agree that this thread should me moved somewhere else.' - Richard
 * It seems to me that having a separate, later section *is* the way forward to having the criticisms in a NPOV manner. Glkanter (talk) 12:13, 29 November 2010 (UTC)

Glkanter: This is not following the process we agreed to above. I propose to collapse it for now. Sunray (talk) 15:59, 29 November 2010 (UTC)


 * I suppose it doesn't. I think it should be the primary focus, though, as it will bring an end to the passive-aggressive, civil, but unproductive style of discourse present in this mediation thus far. I will abide by the mediator's decision. I'd still like to know how a return to editing the article at this time will *not* result in edit warring. Thank you. Glkanter (talk) 16:26, 29 November 2010 (UTC)


 * Btw, I posted a briefer version of this 'Scope' suggestion on an editor's talk page first, for guidance. It was suggested there that I post it more appropriately on this mediation page. Glkanter (talk) 01:08, 30 November 2010 (UTC)


 * I wasn't suggesting a return to article editing at this point. The one way I could see making some progress would be to have an agreed process of editing parts of the article that would be moved here. The discussion would continue to be moderated. Sunray (talk) 17:31, 29 November 2010 (UTC)


 * So a section (or 2) will be copied over here, and with the same talking stick rules, we each make or propose an edit each day? Glkanter (talk) 20:36, 29 November 2010 (UTC)
 * Yes, but I thing we should give the current process a good try. Sunray (talk) 01:55, 30 November 2010 (UTC)

I didn't mean to change your intentions. So we're still working for consensus on the criticisms of the simple solutions? Glkanter (talk) 02:13, 30 November 2010 (UTC)

Martin - round 3
I hope it is OK to start round 3. Round 2 seems to have stalled because we were possibly answering different questions. Two possible versions of the question are:

1 Is it possible to criticise the simple solutions as solutions to the MHP? Here the answer must be 'yes'. If we take the MHP to refer the conditional formulation then I think everyone agrees that the simple solutions may be criticised for not solving slight variants of the problem, for example if the host is known to choose a goat-door non-uniformly.

2 Must we always criticise the simple solutions? I think that we would all accept that there are cases where the simple solutions are satisfactory. For example, in the unconditional formulation.

I have given the above questions to at least set limits on our disagreement which falls in between the two cases above.

For example:

4 Must we criticise the simple solutions to the conditional formulation in which the host is defined to choose a goat-door evenly (K&W). Very few, if any, sources actually do this.

5 Must we criticise the simple solutions to Whitaker's question, where the issue of whether the question was intended to be conditional or not is not specified? Many sources give such solutions to his question. Martin Hogbin (talk) 18:07, 30 November 2010 (UTC)

Nijdam - round 3
To make sure: The normal formulation of the MHP where a door is chosen and one opened, is what is referred to as the conditional formulation. The so called unconditional formulation is where a decision has to be made before the player has chosen a door. Nijdam (talk) 21:26, 30 November 2010 (UTC)


 * Is there anyone who disagrees?? If no, these facts may be a good starting point for the article.Nijdam (talk) 13:59, 2 December 2010 (UTC)

Let's begin with your reliable sources, shall we? The only place I've ever read 'before vs after' is in G & S, and they specifically describe a different, bogus, problem that they say is *not* the MHP. Glkanter (talk) 14:32, 2 December 2010 (UTC)


 * I disagree Nijdam. In the most well known problem statement, by Whitaker, it is far from clear that the questioner intends to specify door numbers, especially as it is known that these were added by Vos savant, who has since regarded this as a mistake. Martin Hogbin (talk) 20:59, 2 December 2010 (UTC)


 * Well, what do you then consider to be the unconditional formulation of the MHP???Nijdam (talk) 22:36, 2 December 2010 (UTC)


 * This is vos Savant's solution in the Wikipedia article. Her solution shows *indifference* as to whether door 2 or door 3 *has been opened*. Clearly not the personal POV Nijdam is advocating. Glkanter (talk) 22:05, 2 December 2010 (UTC)

All okay?
Just checking in to make sure that mediation is progressing well, that the mediator and most of the parties are satisfactorily active, and that no action is needed to steer things back on course. Replies here will be noticed, and my e-mail and talk page are also open. Regards, AGK   19:20, 30 November 2010 (UTC)


 * We've made zero progress to date. You'll get the whole picture in the 2 diffs preceding Martin's diff today.Glkanter (talk) 19:27, 30 November 2010 (UTC)


 * Is that because of the difficult nature of the dispute, or because of inactivity on the part of any party or the mediator? AGK   21:08, 30 November 2010 (UTC)

There is nothing complex about this topic, or this mediation. There is a serious inability by some editors to either base their discussion on sources or to understand the sources as per what they wrote. I brought this up a few months ago vis a vis gamesmanship, etc and was informed that was not an appropriate topic for a mediation. The diffs I recommended that you read are an excellent example of the first problem. Glkanter (talk) 21:45, 30 November 2010 (UTC)


 * Based on the mediators' participation so far, one might easily conclude the only purpose of this mediation is for editors to be civil and otherwise follow the groundrules of this mediation. Encouraging or admonishing wayward editors to focus on reliable sources as they relate to the Wikipedia MHP article does not seem to be a priority. Which, in this section asking for personal comments, is in my opinion the main reason this mediation, as well as the previous protracted talk page discussions never accomplish a single thing of value towards improving the MHP article. The topic of this article is not that complex, although certain editors are able to prevent the removal of NPOV violations from the article through the endless usage of 'The Emperors New Clothes' style arguments and other verbose tactics that are allowed to inexplicably and frustratingly continue. Glkanter (talk) 10:25, 1 December 2010 (UTC)


 * My view of this mediation is basically as expressed in this post on November 19. Briefly stated, my view was, and remains, as follows: 1) this is a difficult mediation, 2) participants have made progress in their ability to discourse civilly, 3) discussion has a tendency to get mired in technical detail, 4) despite greater clarity on common interests, consensus remains elusive.


 * Discussion appears to be getting bogged down once more. I've said that I think that one way forward might be to take portions of the article and begin editing them collaboratively. This was attempted some time ago, but didn't initially yield success, partially because neither mediator was able to devote enough time to moderating then. The circumstances have changed now, both in the participants' willingness and the mediator's (i.e. my) availability. Participants have been experimenting with a "talking stick" approach and have made some gains in their ability to operationalize that method in discussions. However, I think that a more practical task (taking some text and editing it) would work better. I have been waiting for the current discussion to run its course before proposing a new task.  Sunray (talk) 22:52, 30 November 2010 (UTC)
 * Sunray, I really think that is not the right way to approach this dispute. The article is already and FA and needlessly rewriting sections of it will serve no useful purpose for our readers.  There is only one real point of disagreement, which is the validity of the simple solutions and the way in which criticism of them is best presented to our readers.  Almost every other disagreement can be traced back to this one point.


 * Editors will find it very hard to cooperate on most topics until this conflict is resolved simply because they will always have in mind the impact that any edit will have on their position regarding the above dispute.  I would ask you to take a more active role in resolving the core disagreement.  Once that is done, everything else will fall into place more easily.  Martin Hogbin (talk) 10:45, 1 December 2010 (UTC)
 * Martin: I did not say "rewriting the article." I said "taking some text and editing it." By that I mean: Take a block of text, ask individuals to approve it or suggest changes, and discussing all proposals for change. As the article is FA status, I would not expect to see major changes as a result. More likely, we would be able to see specific points of agreement/disagreement that we could then discuss and resolve. However, I am open to other suggestions as to way to solve the disagreements. How would you describe the "core disagreement"? Sunray (talk) 01:01, 2 December 2010 (UTC)
 * I think the same point applies. There is a fundamental disagreement about what is essentially an undecidable point of principle.  Whatever we try to do together will be dominated by this disagreement. We must resolve that in some way before we can work together.
 * Although my personal opinion is closely aligned with Glkanter and Gerhard in that I consider that the conditional problem and solution are little more than an academic diversion from what was clearly intended to be simple mathematical problem with a simple solution, I have tried several times to find some middle ground, often finding myself taking the same stance as Richard in the hope of achieving a resolution. Along with Glkanter, Gerhard, Richard, and others I cannot accept what I see as an extreme (and with scant support by sources) position that the conditional solution is the only complete, proper, and correct solution to the MHP as it is generally understood although I do accept that it has some advantages (and also disadvantages) compared with the simple solutions.
 * I am personally happy with the balance between these two positions achieved in the article as it is now, which is also in broad agreement with Richad's latest suggestion and indeed earlier suggestions of mine.
 * My acceptance of the current state, which does not completely reflect my personal views on the subject, is based on a completely different approach to the subject which I suggest is of use here. This is to ask about who our readers are and what they might want from this article.  This is exactly what Richard's latest suggestion proposes.  It is also the way in which most good text books and encyclopedia article tackle complex subjects.  We provide a simple explanation first for the average, non-technical, reader, which has the greatest chance if showing to that reader why the answer is 2/3 and not 1/2.  Can there be anyone who does not agree that this is what the MHP is all about?  After we have completed that task, without distracting the reader with problems that they will almost certainly never have though of, we can then proceed to a full and comprehensive discussion of the subject of conditional probability and its application to the MHP, based on what the sources actually say.  Martin Hogbin (talk) 10:12, 3 December 2010 (UTC)

I am quitting the mediation. Thanks to the mediators for doing their best. We did achieve a lot of understanding and uncovered a lot of fascinating material on MHP. But where there's no will there's no way.

It seems to me that the only workable compromise is to have 1) a good section on the simple solutions, 2) a good section on the conditional probability solutions, and 3) a good section reporting the fight between the two in a neutral way. The real problem is that we don't have a strong group of neutral editors interested in doing job 3). Any attempts in this direction are shouted down by both sides, claiming that reporting an unsettled controversy will only confuse poor wikipedia readers.

Of course, job 3) is very difficult. I am not aware of any reliable sources giving a really good analysis and neutral overvew of 1) versus 2). Rosenhouse tries but this is not his best chapter. Rosenthal tries but his argument is not very well formulated. Attempts to explain his argument are shouted down as "own research" by editors who state as a matter of principle that they are not going to invest any time in exploring the relevant professional literature.

There is a continuing discussion in the literature about whether simple solutions or conditional solutions are the right way to solve MHP. Many editors here seem to have a strong personal opinion one way or another and seem more interested in trying to win the others to their side so as to be able to comfortably edit an article which is slanted to their side, than in accepting any kind of compromise. Richard Gill (talk) 11:04, 1 December 2010 (UTC)
 * I agree--Kmhkmh (talk) 20:17, 1 December 2010 (UTC)
 * I didn't realize there was consensus among all the editors to defer the criticisms to job #3 above. If I didn't know better, I would *swear* that just yesterday an editor once again made very clear his vehement, but unsupported, insistence that the simple solution section must reflect that 'the simple solutions are wrong'. Maybe that's what he posted, but he was thinking differently, Richard? I don't have that 'mind reading' skill, I'll have to rely on your demonstrated abilities in that area. Otherwise, your criticism above makes no sense.
 * Of course, our responsibility for task #3 can be as simple as reporting what the sources say. Without any editorializing.
 * I'm glad you 'achieved a lot of understanding and uncovered a lot of fascinating material on MHP'. That may explain my frustration with this mediation, as I'm just trying to reach a consensus with the other editors on how to improve the Wikipedia MHP article. Glkanter (talk) 12:38, 1 December 2010 (UTC)


 * I did not say there was a consensus about defering criticisms to job #3. I suggest this scheme #1, #2, #3 as a possible workable *compromise*. I agree that task #3 consists of reporting what the sources say without editorializing. It will be a hard job since it will involve actually understanding both points of view, something which no other editor in the mediation seems capable of doing, since they all "know" that the "other" point of view is wrong. But no single editor seems to have moved one millimeter from their position that "their" approach is the only right approach (whether simple or conditional). No one seems to have any interest in compromise. No one seems interested in learning from the others. Nijdam's contribution which Glkanter mentions is a good example of this intransigeance. Glkanter's own total misreading of what I just wrote and his belligerent response is another splendid example. Richard Gill (talk) 13:20, 1 December 2010 (UTC)
 * I agree as well. That's why I'm currently basically observing only. As long as the main opponents are unable to overcome their differences or are at least willing to pass the decision or overhaul to comptetent and trustworthy 3rd party and refrain from editing themselves, the whole thing is a rather futile effort and wasting a lot of energy that is spent more productively elsewhere. I'd probably support any agreement between martin/glkanter and rick/nijdam as long as it doesn't contain any serious mathematical error, but the chances for them to reach a compromise seem to be as slim as ever.--Kmhkmh (talk) 20:19, 1 December 2010 (UTC)


 * Maybe you could explain how I'm to follow fundamental Wikipedia policies, yet compromise with acquiesce to Nijdam's demands which are not based on an NPOV response to the reliable sources, but rather his personal viewpoint? Glkanter (talk) 20:53, 1 December 2010 (UTC)


 * I will give Richard credit for *finally* stating the offense Nijdam has been perpetrating since he joined this article's discussion. So, the above question is open to anyone, especially Rick, Sunray and Will Beback:
 * How am I to follow fundamental Wikipedia policies, yet compromise with acquiesce to Nijdam's demands which are not based on an NPOV response to the reliable sources, but rather his personal viewpoint? Glkanter (talk) 21:52, 1 December 2010 (UTC)

"The real problem is that we don't have a strong group of neutral editors interested in doing job 3... [User:Gill110951|Richard Gill]] (talk) 11:04, 1 December 2010 (UTC)"

I dunno, I think I reads real good. Glkanter (talk) 13:28, 1 December 2010 (UTC)


 * Sorry to lose you Richard. I think your comments about compromise are a bit unfair.  I have suggested exactly what you propose several times before and supported your suggestion above.  I have also made clear that I can very well see both points of view but that, as suggested by you, discussion of deficiencies in the simple solutions should be deferred until the simple solution has been fully explained.  This is purely in the interests of clarity for the general reader rather than to push any particular POV. Martin Hogbin (talk) 18:21, 1 December 2010 (UTC)
 * Everyone has his own perception of what is going on. Although I prefer to present the full correct conditional solution first, I made a step towards consensus and said I'll accept the simple explanation to be presented first, be it with the criticism. Now the other camp still wants the simple explanations to be presented first, without the criticism, making no step whatsoever in the direction of there opponents. So?? Nijdam (talk) 22:43, 2 December 2010 (UTC)


 * Which criticisms, and which significant minority (or greater) of reliable sources are you referring to, Nijdam? Glkanter (talk) 01:30, 3 December 2010 (UTC)