Wikipedia talk:Requests for mediation/Monty Hall problem/Archive 4

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?

Discussion
Sunray, I am not sure there is a best place to start, why not just chose where you think is best. Martin Hogbin (talk) 22:12, 25 August 2010 (UTC)


 * A list of raw issues is probably not terribly helpful, except as a rough starting point or framework for the case. A better approach, I suggest, might be to, underneath each issue, detail the different versions of the article as supported by each faction in the dispute: for instance:


 * Just a suggestion, for whatever it's worth. Glad to see healthy levels of activity on here, though I do wonder, after a quick read, if all the parties are participating, or only some? AGK   15:10, 26 August 2010 (UTC)


 * Sunray was trying to find out which issue to start with. If one of you picks an issue to start with I would be happy to attempt to show what the options are.  Martin Hogbin (talk) 15:59, 26 August 2010 (UTC)
 * Start with: 1) correctly understand the meaning of W's question (sources), 2) conception and evident implications of the (non biased) host's role, 3) appreciate the correct simple solution. Gerhardvalentin (talk) 16:43, 26 August 2010 (UTC)


 * I think Martin's list is a good thing to hold on to. It is not prioritized, and it does not have to be prioritized. I think it would be good if Sunray chose just any item (=issue) from this list he likes, and we have a focussed discussion on that issue, well moderated by our mediator. We can all learn from that. After say a week's discussion, whether or not it comes to any resolution, we reconsider the complete list and Sunray makes a proposal what to do next. Gill110951 (talk) 03:03, 27 August 2010 (UTC)
 * I'm impressed with this work and the collaborative tone. Thanks to Martin for getting the ball rolling. Just goes to show what can happen when the mediators are not present :) This leads me to make a comment about successful mediation. The most effective mediations, IMO, are ones in which the mediators facilitate the discussion, (without playing a directive role) and the participants do the work. The advantage of this, particularly if the participation is balanced, is that there is likely to be a high degree of ownership for the resolution. In wiki mediation this can lead to a stable group of editors who collaborate on a particular article.
 * So the mediators would prefer not to pick an issue for you, but I will make a suggestion: Pick something simple to start. If you pick something that will produce consensus relatively easily, you can then build on that success. Sunray (talk) 07:10, 27 August 2010 (UTC)
 * It is really difficult because many of the issues the issues are interlinked. I think there might be some chance of getting somewhere with, 'How should Whitaker's question be interpreted?'.  I am sure that there will be no single definitive answer but I think we might be able to agree on many things. Martin Hogbin (talk) 08:10, 27 August 2010 (UTC)
 * I am going to start a section below on that question. I hope nobody objects. Martin Hogbin (talk) 08:15, 27 August 2010 (UTC)


 * I suggested above we're not going to get much of anywhere until we talk about sources, by which I meant before we can agree about any issue we're going to need to agree about what sources we're talking about and what those sources say. I would suggest we work on a page like Talk:Monty Hall problem/Sources, although probably a subpage of this page (e.g. /Sources) rather than of the general talk page of the article - possibly even a subpage per source. Our task is not to agree about "the Truth" - but to agree about what sources say about the MHP.  Ultimately we'll need to get to the finer points of WP:NPOV like weight, but for now it seems like it should be fairly easy to create a representative list of sources and paraphrase what each one says.  Once we have this list, I think it will be easier to judge any particular claim against what sources say (as opposed to what we individually think).   Basically, I'm agreeing with Glkanter here.  -- Rick Block (talk) 14:16, 27 August 2010 (UTC)

I'm delighted we agree, Rick. But I don't wholly concur with your action plan. My proposed question regarding the placement and frequency of criticisms of the various simple solutions doesn't necessarily require a survey of the sources. Unless there is a demonstrated reason for it, the criticisms of the simple solutions do not belong any where near, or preceding the simple solutions. We're not interested in judging what any source says. We're interested in presenting what every reliable source says. In a NPOV and non-UNDUE fashion. Glkanter (talk) 14:28, 27 August 2010 (UTC)


 * The answer depends entirely on what weight we give to the criticisms. If this is a POV published by reliable sources with roughly equal prominence to the simple solutions, presenting the simple solutions first without mentioning this criticism would (IMO) violate the WP:STRUCTURE aspects of NPOV.  Even if this is only a minority POV, we need to think about whether presenting simple solutions first, with no mention of any criticism, would make the article endorse the POV that these solutions are correct and complete.  Mentioning that there are sources that criticize the simple solutions is not the same as the article taking the POV of the sources doing the criticizing.  I think the bottom line is that this has to be a discussion about sources. -- Rick Block (talk) 19:25, 27 August 2010 (UTC)


 * Well, I think we are discussing the same issue: 'What is an appropriate placement and prevalence of criticisms of the various simple solutions in the MHP article that does not favor a POV or give a POV UNDUE representation'. Making that editorial decision, and especially: 'Which approach doesn't violate NPOV', is the help/decision I was hoping we would get in this mediation. Glkanter (talk) 01:37, 28 August 2010 (UTC)


 * A very closely related question is: 'Does presenting simple solutions (without criticisms) before the conditional solution, even in the same section, constitute a NPOV (or other) violation?' Glkanter (talk) 01:52, 28 August 2010 (UTC)

Do I take it that you would both rather talk about sources? Should we take one of the questions about sources as our first mediation topic? Martin Hogbin (talk) 08:13, 28 August 2010 (UTC)


 * It sure looks like Rick and I feel that way, anyways. Yes. Glkanter (talk) 12:16, 28 August 2010 (UTC)


 * To avoid simply repeating the way these discussions have gone in the past (see      ), I strongly suggest (bordering on insist) the mediators actively lead (or at least frame) this discussion.  I think framing this discussion in a neutral fashion will be crucial to making any progress. -- Rick Block (talk) 16:17, 28 August 2010 (UTC)


 * Instead of strongly suggesting (or borderline insisting), you could just ask us :) You make a fair point, though. I will make a proposal, below. Sunray (talk) 06:42, 29 August 2010 (UTC)

How should Whitaker's question be interpreted?
Let me start with three background statements that I think we can all agree on. Martin Hogbin (talk) 08:15, 27 August 2010 (UTC) Just for clarification, by 'Whitaker's question', I mean vos Savant's restatement of the question in Parade. 12:55, 27 August 2010 (UTC)

Whitaker's question is the most well known and notable problem statement
Possibly contentious are:

Whitaker's question asks us to consider the problem from the state of knowledge of a contestant
More contentious are:

Glkanter's response
Martin, I appreciate the leadership you're showing here, but I would suggest a different approach (of course!). I think we might do better with a simple yes/no type of question. For example, 'Must the simple solutions be accompanied by, or preceded by, criticisms?' Accompanied by, or followed by, 'How many times and how prominent should the criticisms of the simple solutions be?' Just my 2 cents. Glkanter (talk) 11:27, 27 August 2010 (UTC)
 * It was suggested that we start with something where there might be a degree of agreement. With your question, the answers would be simple enough, but some will say yes and others no.  Where do we go from there?  I agree that your question is essentially the fundamental disagreement here but it might be better to nibble at the edges a bit to start with. Martin Hogbin (talk) 12:06, 27 August 2010 (UTC)


 * Right, I had forgotten the part about starting with an easy one. That made me laugh when I first read it. If something was easy, it wouldn't be part of this mediation.


 * Why beat around the bush? Let's find out if this mediation adds any value. I'm beginning to wonder in what way it will lead to any consensus at all. It seems like, once again, the same editors are going to be hashing over the same arguments, and from the discussions on recent user talk pages, nobody is any less entrenched than before. Other than nobody editing the article, which I think is a short-lived contrivance, I haven't seen much come from this mediation, I'm afraid. Glkanter (talk) 12:18, 27 August 2010 (UTC)
 * There is no general agreement on how to interpret Whitaker's question and the way that this is done affects the arguments relating to your question. Martin Hogbin (talk) 12:55, 27 August 2010 (UTC)

That's interesting. Who says anybody but the reliable sources should interpret Whitaker's question as published by vos Savant? The sources all claim to be answering the same question. They all get the same result.

My question is nothing but a Wikipedia policy question. NPOV, UNDUE, that sort of stuff. What else matters? What else are we supposed to be doing as editors of a Wikipedia article? Glkanter (talk) 13:09, 27 August 2010 (UTC)


 * The sources all get the same result, "switch", and often add to this "2/3". The sources disagree with the route for getting either of these answers, and some sources don't give the answer "2/3" but the answer "at least half, and on average 2/3". Moreover the sources disagree strongly on what additional assumptions "automatically" need to be added. The sources disagree strongly, at least implicitly, on what "2/3" means, roughly speaking according to whether they have a frequentist or a Bayesian notion of probability. Other sources are clearly confused about this. A number of the sources don't say "switch" but say "choose your initial door completely at random, and switch". Many sources make more assumptions than are necessary in order to deduce their answer. A number of sources make evident logical mistakes in their derivation of their answer and other sources have pointed this out. The sources disagree about whether the door numbers mentioned by Whitaker are merely there for illustration of if they are a crucial part of the question, while others find them a priori irrelevant, even if a crucial part of the question.


 * It therefore is clear to me at least, that the sources are not agreed on what constitutes a correct route to get a correct answer. It is clear however that most sources think that the route to the answer which they think is correct is more important than the answer itself. In other words, according to many sources, to solve the Monty Hall Problem is much more than merely giving "the right answer". They think that solving the Monty Hall Problem is about reformulating Whitaker's question in a way which allows a definite mathematical answer. They disagree strongly in the reformulation and the answer. Gill110951 (talk) 13:39, 29 August 2010 (UTC)

Every source claims they are solving Whitaker's question as presented in vos Savant's column. That's good enough for me. The rest is all pointless disputes that Selvin already addressed, and criticisms of the simple solutions. Both of these should be presented later in the article, without our 'interpretations' or 'decisions' as to accuracy. Glkanter (talk) 14:58, 29 August 2010 (UTC)

Kmhkmh's response
I'm not quite sure what we are trying to clarify here. As far as our article ìs concerned, it doesn't really matter how we personally might read whitaker, but how reputable sources do. I.e. it seems pointless to me argue what Whitaker really meant according to us (which we've done for over a year anyhow), but we need to report how our sources read Whitaker. There might be a need to debate however how a particular reads whitaker, since that might be not always completely obvious.--Kmhkmh (talk) 09:52, 29 August 2010 (UTC)
 * Just determining how the involved individual editors read whitaker's question?
 * Or do you want to associate various interpretations which various source?


 * I beg to differ here, I think a lot of the debate of the last year has been precisely about showing that other sources do read Whitaker in different ways. Moreover the debate has focussed on the issue of whether one of those readings is so near universal that it takes priority over all others. This is complicated by the fact that editors who believe there is one near universal reading, disagree with what that reading is.


 * It's very like religion (in particular, Christianity). There are lots of churches which believe they are the Only True Church, they fight one another almost more viciously than they fight people outside of any church. Then there are people outside any single established official church, who have the point of view that all the churches are really saying the same thing (as far as they are saying anything good at all). These people think that The True Church is the combined good essence of all particular "churches". There even exist people who take an even higher point of view concerning, for instance, all world religions. Gill110951 (talk) 13:48, 29 August 2010 (UTC)

response of Gerhardvalentin
As known, this is a conflict between "different aspects", different points of view. And so it matters what reliable sources "really say". We should be careful to correctly interpret those statements of sources, and what they (antinomy) really say.

Up to now, obviously misinterpreting some source, sometimes just "the only one aspect" has been "read into" the statements of sources, although the source actually and effectively just confirms "the other aspect" as well. We need to deal very carefully with the sources, to avoid that the sense of source statements is not reproduced unilaterally, obviously misinterpreting the source. Gerhardvalentin (talk) 11:33, 29 August 2010 (UTC)
 * I agree that interpretation of sources will be important to resolving this. I wasn't sure which discussion you were commenting on (or whether you were making a stand-alone observation). In any case, this seems to me to belong with the section below (proposal on sources). Sunray (talk) 06:08, 30 August 2010 (UTC)

Proposal re Sources
Rick Block and Glkanter are proposing that participants discuss sources. Rick has further requested the mediators to frame the discussion. From what I've seen so far, participants have no difficulty discussing issues, but getting agreement is another matter. I would thus propose that we take a concrete example from the article and discuss the sources in it. Two prerequisites: 1) Participants agree to work on sources, and, 2) participants agree to take a section from the article as a case in point. Assuming that there is consensus on these two points, would participants be able to select a passage of text to work on? Sunray (talk) 06:57, 29 August 2010 (UTC)
 * Agree - And, I think a non-trivial but relatively easy one would be the single paragraph in the "Simple solutions" section referenced to Carlton which has been the subject of recent edit warring. -- Rick Block (talk) 16:02, 29 August 2010 (UTC)

There must be something in the water. I agree, and I like Rick's suggestion. To be honest, I'm not sure exactly what all the other suggestions are really promoting. Glkanter (talk) 16:07, 29 August 2010 (UTC)

I am happy to work on any section of the article. Martin Hogbin (talk) 00:03, 30 August 2010 (UTC)

Fine by me too. The single paragraph in which Carlton turns up is presently a one sentence paragraph: "An even simpler solution is to reason that switching loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3 (Carlton 2005)", which was published on internet as. The paper appeared in a journal for statistics education and promotes the use of probability trees, in opposition to an earlier paper (Rossman and Short 1995) who prefer use of two-way tables. The author recommends the Java applet. Gill110951 (talk) 11:04, 30 August 2010 (UTC)
 * I looked it up in Carlton's paper, and (surprise) the sentence isn't there. It is widely copied from Wikipedia. Which by the way shows the influence of Wikipedia. This is what Carlton says:

"Before presenting a formal solution to the Monty Hall Problem to my students, I find that it helps to give an intuitive explanation for the 1/3 - 2/3 solution. Imagine you plan to play Let’s Make a Deal and employ the “switching strategy.” As long as you initially pick a goat prize, you can’t lose: Monty Hall must reveal the location of the other goat, and you switch to the remaining door - the car. In fact, the only way you can lose is if you guessed the car’s location correctly in the first place and then switched away. Hence, whether the strategy works just depends on whether you initially picked a goat (2 chances out of 3) or the car (1 chance out of 3)"


 * Carlton's formal solution is the one using conditional probabilities. Nijdam (talk) 13:54, 30 August 2010 (UTC)


 * The sentence is a paraphrase of the paragraph quoted above, providing a source for this simple solution which Carlton calls an "intuitive explanation" (not a solution). In full context, Carlton is clearly a conditionalist - e.g. from the intro: The misapplication of conditional probability, the haphazard use of "equally likely" outcomes, and the non-use of Bayes' Rule can lead to all manner of incorrect answers.  There may be an argument here that with this reference, this sentence should not be in a "Solution" section but rather an "Aids to understanding" section.  Carlton's solution (presented earlier in the paper) is a full-out conditional solution.  -- Rick Block (talk) 15:20, 30 August 2010 (UTC)

I have no interest in bible-study-like interpretations of each source's perceived 'intent'. I look forward to discussing whether paraphrasing is better than a brief quote, whether the tree is OR or not, and which sources criticize this solution by name. Glkanter (talk) 17:06, 30 August 2010 (UTC)


 * Carlton does present a tree in section 3 of his paper, which he even calls "the right solution". But it's essentially identical to this one and not at all like this one.  Since he does include a tree in the paper which he calls the "right solution", attributing a different tree to this paper as the "solution" would seem to fail WP:V (let alone WP:OR). -- Rick Block (talk) 18:41, 30 August 2010 (UTC)

Rather than bible-study interpretations, I would suggest the following:
 * 1) Move the text you want to discuss to a new section, below.
 * 2) Collaboratively edit the section
 * 3) Provide a subsection for "Discussion." If participants agree on a particular block of text and sources, just confirm your agreement.
 * 4) When you come to text that does not seem adequately supported by sources, or is differently interpreted, discuss it and try to reach consensus. Likewise for paraphrasing. If consensus is not reached after a reasonable amount of time, move on, noting that the text needs to be returned to later.

The key to all this is collaborative editing. The sooner you can begin to do that, the better, IMO. Different views can sometimes lead to superior articles. There is nothing wrong with saying "on the one hand... on the other..." (always supported by sources, of course). In other words, you don't have to agree, just to collaborate. Sunray (talk) 02:12, 31 August 2010 (UTC)


 * Sorry, I'm not seeing how the approach described above will lead to anything much different than our current stalemate. Glkanter (talk) 02:23, 31 August 2010 (UTC)
 * There is one subtle, but crucial, difference: In identifying parts of the article you agree on, you begin a process of agreement. Then you can temporarily "park" points of disagreement. These can be dealt with later, after you have developed the tools for working together. The mediators can help you acquire those tools (skills), but you have to want to go there. Would you be able to signify that you have read this and let me know if you are in agreement? Sunray (talk) 15:02, 31 August 2010 (UTC)


 * Rather than prolong this exercise by arguing semantics (or 'intent', ugh) on the 15th section of the article (for the umpteenth time), we've chosen to discuss perhaps the most volatile item first. Which is good. It's brief, it's simple, it brings out all the Wikipedia issues in dispute. Are you guys gonna be part of a solution? Or will we just end up with more edit warring? Glkanter (talk) 15:20, 31 August 2010 (UTC)
 * I don't know if it's good. It may be. But flying at a core issue like that can be risky. We mediators try to find ways to take smaller steps first. Sunray (talk) 15:57, 31 August 2010 (UTC)
 * We've shown on the talk pages (for years now) that we can't agree on minor issues, either. Actually, we don't know what a 'minor issue' looks like. We're looking for (and have been waiting since December for) your assistance in ending the logjam, not prolonging it. I don't understand what 'risky' means in this context. What is it that is 'at risk' in this mediation? Glkanter (talk) 16:13, 31 August 2010 (UTC)
 * Let's find out. Right now we are all being civil and gentlemanly and/or ladylike (if there are ladies present) so maybe we will get a bit further this time. We have a neutral and respected observer (Sunray). So not everything is exactly the same. Gill110951 (talk) 08:09, 31 August 2010 (UTC)

The test passage
"An even simpler solution is to reason that switching loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3 (Carlton 2005)."

I propose we replace the above paraphrasing that is currently in the article with the actual statement from Carlton, and add this decision tree that I derived from his solution:
 * "As long as you initially pick a goat prize, you can't lose: Monty Hall must reveal the location of the other goat, and you switch to the remaining door - the car. In fact, the only way you can lose is if you guessed the car's location correctly in the first place and then switched away. Hence, whether the strategy works just depends on whether you initially picked a goat (2 chances out of 3) or the car (1 chance out of 3)." (Carlton 2005).



Discussion
I suggest that first Glkanter edits the passage in the way he likes, and that we then discuss whether or not his edits conform to wikipedia policies. And anyone who likes to do bible studies (I do) can in parallel also discuss the content, interpretation, and The Truth. Gill110951 (talk) 08:05, 31 August 2010 (UTC)

Glkanter's Reasoning
As the simple solutions will be criticized at some point in the article, it seems like 'fair play' would suggest that the reader should get to evaluate & judge for himself the solution the source actually presented, not some Wikipedia editor's paraphrasing. In this case, that's Carlton. The direct quote is barely any longer than the paraphrasing, so space & length isn't a mitigating factor.

With the decision tree, the reader who does better with formal-style proofs will also be served by Carlton's simple solution. The 2 rows of 1/3 & 2/3 and each column are derived from Carlton's solution, Whitaker's question as presented by vos Savant, or simple math. This conditional (well, if 100% is considered conditional, but that's kindly what makes the paradox, isn't it?) decision tree is based on outcomes (selects car or goat), rather than doors, eliminating the entire aspect of 'the equal goat door constraint' altogether. How simple. How elegant. How simply elegant! Glkanter (talk) 10:51, 31 August 2010 (UTC)


 * This quote from Carlton is in a section he specifically refers to as presenting an "intuitive explanation", not a solution. His (conditional) solution is presented earlier in the paper, in section 3, and includes a decision tree that looks like this one, not like the one shown above.  Taking this quote out of context and presenting it as a "solution" referenced to this paper with an original decision tree that does not match the tree Carlton shows in his paper is at best OR - it is distinctly NOT what this paper is saying is "the solution".  I'd suggest either moving the quoted text to the "Aids to understanding' section, or restoring the paraphrase and finding another source (one that actually presents this as a solution).  In either case the tree does not come from Carlton, so should be at least renamed.  If we're going to keep it at all we should have a separate discussion about it since it has other problems I haven't gone into here.  -- Rick Block (talk) 14:15, 31 August 2010 (UTC)


 * Are we getting the hang of it, Sunray? Glkanter (talk) 14:19, 31 August 2010 (UTC)


 * Could we please hear from some others about the substance of the issues I've raised here? I suggest at a minimum Richard, Martin, Kmhkmh, and Nijdam. -- Rick Block (talk) 14:43, 31 August 2010 (UTC)


 * I agree with Rick that Carlton gave a decision tree as graphical aid for doing the calculations of the conditional solution of the so-called standard version of MHP. I don't see much point in giving a decision tree for doing the calculation of the solution of the unconditional problem, but of course one can do that, and it can be pedagogically useful to do so. In that case I would delete the names of the doors from the decision tree since they are being ignored, deliberately, because they are understood to be irrelevant to the problem. Since they are irrelevant, from that point of view it doesn't hurt to add them, but it does confuse those who understand the door numbers to be highly relevant. In my opinion the relevance of the door numbers is a major difference of opinion of the reliable sources and it also corresponds to a major difference of opinion concerning the meaning of probability, and to a major difference concerning the semantic understanding of Whitaker/vos Savant's words. Gill110951 (talk) 09:56, 1 September 2010 (UTC)


 * Further remarks: I am sure reliable sources can be found who agree with Glkanter that what Carlton calls an intuitive explanation is in fact a complete and a rigorous solution. In mathematically modeling a real world problem, one has to decide at what level one moves from intuitive real world discussion to mathematization. It is quite legitimate, especially if one uses probability in the Bayesian sense as a measure of belief, to use the verbal description of the problem to decide that one is certain that the door numbers are irrelevant to the problem. Then one can switch to the mathematization of a much simpler problem. Other people prefer to keep the door numbers in, but because they too are using Bayesian probability, they know they know nothing about the host's method of choice of door to open when he has a choice. So this probability is just taken to be 50-50 without any real world motivation, just like the initial probability distribution over the doors of where the car will be found is taken to be 33.33...-33.33...-33.33.... without any real world motivation. The Bayesian approach puts them in the totally symmetric case, the so-called standard problem. They then do computations to prove what they actually knew already, that the door numbers are irrelevant. Carlton himself does not commit himself explicitly to a probability interpretation (subjectivist or frequentist). In his paper he does discuss what we would do if we had seen the show many times before and "knew" that the car is hidden behind Door 1 20% of the time. "Of course" we will initially choose Door 1 and then switch. This gives us an 80% unconditional chance of getting the car, as Carlton well knows. He doesn't seem interested in what are then the conditional chances of getting the car, when the host thereafter opens Door 2, and when he thereafter opens Door 3. To be consistent he should use the host's known track-record and use estimates of the host's possibly asymmetric probabilities. On the other hand he seems not interested in this question. It seems to me that Carlton does the conditional probability exercise because he knows that MHP is often featured in introductory statistics texts and because he is writing in a journal for statistics education and because he wants to talk about problems of conditional probability. Some educators think you cannot even introduce conditional probability in an introductory course. He thinks you can and should. Gill110951 (talk) 10:45, 1 September 2010 (UTC)


 * Perhaps the proposed decision tree with the door #s accomplishes Boris' earlier suggestion of (paraphrasing) 'a peaceful co-existence of the solutions'?
 * I think your proposed decision tree without the door #s puts the two solutions (to the standard problem) on equal footings. I think that Carlton's intuitive explanation becomes a complete solution once you add the preliminary remark that whether by symmetry or by absence of information to the contrary or by the itention of vos Savant, the door numbers may be ignored. By complete, I mean that every sentence has a translation into mathematical formalism, and the resulting sequence of formal mathematical steps form a complete proof that the player must switch. Recall: the question is "should you switch"? We are all agreed here, and more importantly our sources agree too, that the answer is "yes". Reliable sources disagree on what is a valid justification of that answer. Gill110951 (talk) 11:40, 2 September 2010 (UTC)
 * Would you say the tree is OR, or not? Glkanter (talk) 10:52, 1 September 2010 (UTC)
 * I would say that if the editors are agreed that the tree is part of a valid way to solve a valid version of MHP, then the question of whether or not it is OR becomes pretty irrelevant. Gill110951 (talk) 11:40, 2 September 2010 (UTC)

FYI, Sunray and Will Beback, this exact issue is a large part of what got me flagged for edit warring (more than once), and was likely the main contributing factor in the page being protected twice, due to many of us edit warring. Glkanter (talk) 14:58, 31 August 2010 (UTC)

Carlton's solution
If we were to use this one source for the "Solution" section, I think what the text would say would be approximately as follows.

Carlton (2005) presents a solution based on conditional probability using a decision tree showing all probabilities assuming the player initially selects Door 1 (as shown to the right). The probability the host opens Door 3 given the car is behind Door 1, Door 2, and Door 3 is 1/2, 1, and 0, respectively. The total probability the host opens Door 3 is therefore 1/3 x 1/2 + 1/3 x 1 + 1/3 x 0 = 1/2. By Bayes' Rule the probability the car is behind Door 2 given the host opens Door 3 is


 * P(car behind Door 2|host opens Door 3) = P(car behind Door 2)P(host opens Door 3|car behind Door 2) / P(host opens Door 3)

hence


 * P(car behind Door 2|host opens Door 3) = 1/3 x 1 / (1/2) = 2/3.

In other words, a player who sees the host open Door 3 and switches has a 2/3 chance of winning the car. An intuitive explanation of this result is that switching loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3.

I would be quite happy if this is all the "Solution" section said. The rest is just aids to understanding. -- Rick Block (talk) 15:39, 31 August 2010 (UTC)


 * I do completely agree with Rick. Carlton gave a solution in calculating the conditional probability. The "unconditional" mentioning of the 1/3 - 2/3 chances is called intuitive explanation. Nijdam (talk) 16:30, 31 August 2010 (UTC)


 * I think the proposal above is simply a new way of justifying the conditional solution POV as being the lone correct and complete solution. Not reflective of the reliable sources at all. Not in the spirit of this mediation and the recent instructions at all. And not in the reader's best interest. Or per Wikipedia policy. Glkanter (talk) 15:50, 31 August 2010 (UTC)


 * As far as I can tell, you are not following my suggested process. The test passage does not seem to be a section of the article. Perhaps I didn't make that clear: I was suggesting that you take a section of the article. Also, I think that participants should agree to work on this together. Take it slow. I may need to contact some of the others, as they seem to have fallen away.
 * You may want to chose a section of the article that is large enough to allow people to focus on different aspects. Whether you want to tackle one of the more difficult sections right away is up to you. Sunray (talk) 16:45, 31 August 2010 (UTC)


 * I have restored the original text from the article, making it evident that we are, indeed, following the mediators' guidance in that regard. Nearly all the editors (showing agreement, or consensus) are addressing the part of the article that we expect will indicate where this mediation is heading. It's incredibly brief, yet encompasses (in a broad sense) nearly all of the issues. Why divert us from the first thing the editors have agreed on? Glkanter (talk) 17:13, 31 August 2010 (UTC)


 * The point of the text above (the "solution according to Carlton") is to show how the solution section would look taking Carlton as the only source. We're talking about what sources actually say here, not about what we might wish they say, and here we're specifically talking about what Carlton-2005 says.  He DOES NOT say


 * the solution is simple
 * the "solution" is you have a 2/3 chance of picking a goat so therefore you have a 2/3 chance of winning the car by switching
 * the solution is based on a decision tree other than the one shown just above (in this section)


 * He DOES say


 * the solution is based on conditional probability using Bayes' Rule (using the decision tree shown just above)
 * an intuitive explanation is that switching loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3


 * What I'm saying is using Carlton as a source for the simple solution, or quoting his intuitive explanation out of context and presenting it as his "solution", or using Carlton as a source for the other decision tree is not in the least justified. If we want to present the original text from this section as a solution we'll need a different source. -- Rick Block (talk) 17:56, 31 August 2010 (UTC)

Are you saying I am misquoting Carlton in the solution I proposed? I don't believe I am. Why would Carlton's be the only solution? I don't follow that at all. Glkanter (talk) 18:03, 31 August 2010 (UTC)


 * No, I'm saying this quote is being taken out of context and being used to make it seem like the source is saying something entirely different than what it actually says. I'm not suggesting we only use Carlton as the source for the solution section - I'm simply suggesting what it would look like if that's what we did (to illustrate the point about taking his words out of context).  And, regarding the other decision tree, he presents a decision tree in his paper.  If we're going to reference something as "Carlton's decision tree" it should definitely be Carlton's. -- Rick Block (talk) 19:32, 31 August 2010 (UTC)

So we agree that it is a FACT that Carlton is quoted accurately. Do we agree that it is your JUDGEMENT that I am perverting his 'intent'? Glkanter (talk) 20:20, 31 August 2010 (UTC)
 * Yes we agree it is a fact Carlton is quoted accurately. However, it is also a fact that the quote is NOT from the section of the paper where Carlton is presenting his solution, the diagram bears little relation to Carlton's diagram, and the two sentences immediately before this quote are "Before presenting a formal solution to the Monty Hall Problem to my students, I find that it helps to give an intuitive explanation for the 1/3 - 2/3 solution. Imagine you plan to play Let's Make a Deal and employ the 'switching strategy.'"  It is indeed my judgment that these two sentences constitute critical context (they establish that what follows is not meant to be taken as a solution but rather as an "intuitive explanation") and that this means the quote is being taken out of context and is being inappropriately presented as Carlton's solution when it is not actually Carlton's solution.  It might help if some others weighed in on this. -- Rick Block (talk) 21:19, 31 August 2010 (UTC)


 * How about we just call it an 'explanation' and include it? Martin Hogbin (talk) 21:59, 31 August 2010 (UTC)


 * Include it where? Call it an "explanation" how?  If you're suggesting changing the heading of the section from "Simple solutions" to "Intuitive explanations", I'd be OK (but, my guess is that's not what you're suggesting).  If you read the preceding sentence carefully, what he's doing is changing the problem (to the unconditional problem, where the decision to switch or not is made BEFORE the host opens a door) "Imagine you plan to play ... and employ the 'switching strategy'." (emphasis added).  This is exactly the same approach Grinstead and Snell use.  They (Grinstead and Snell) say "We begin by describing a simpler, related question" (which is the probability of winning by using a pre-selected "stay" or "switch" strategy - i.e. deciding BEFORE the host opens a door).  My contention is referencing this to Carlton as if it is his solution to the MHP is wrong.  This is NOT Carlton's solution, implying otherwise is misrepresenting what the source says.  I'll admit I'm being very picky about this, but I think the only way we're actually going to resolve our issues here is by paying very close attention to what the sources we're using really say.  -- Rick Block (talk) 23:29, 31 August 2010 (UTC)


 * This is interesting. Grinstead and Snell discuss mathematical solutions to two different mathematical problems. Both problems are derived from Whitaker - vos Savant's question. The first mathematical problem is to evaluate the relative merits of the two strategies "always switch" and "always stay" in terms of overall success chance. The second mathematical problem is to compute the conditional probability that switching will give you the car given the number on the door first chosen and the number on the door opened by the host. Both of these mathematical problems can be considered to be mathematizations of the Whitaker - vos Savant question. You must see Grinstead and Snell in context. They are mathematicians describing the mathematical solutions to two different mathematical problems. Carlton nowhere states dogmatically that MHP has to be solved with conditional probability. Grinstead and Snell do take the dogmatic position that only one of those mathematical problems is the right mathematization of Whitaker - vos Savant's question. It amazes me that they can read Whitaker's mind and know that he needs to know a conditional probability. It also amazes me that they can know for sure that the three doors are equally likely and that there is no host bias. Still, their dogmatism and lack of contact with the real world is typical of many pure mathematicians teaching elementary probability. Like many wikipedia editors, they want there to be one unique right interpretation and solution to the MHP. The only mathematical gap between the two solutions is the question whether any overall player strategy could do better than having an overall success chance of 2/3. Obviously not. Hence no need to ask the conditional question. But if you want to do elementary exercises with Bayes' rule, or with von Neuman's minimax theorem, the mathematician can prove that there is nothing better than 2/3. Most wikipedia readers will have absolutely no need to see how you can rigorously prove this fact. Gill110951 (talk) 14:04, 3 September 2010 (UTC)

I have been troubled for a long time by the ability of certain editors to discern that what a quoted source's statement 'really' means is different than the quoted statement itself. This is one perfect example. Rosenhouse, or whoever writes about the 'shaky' solution turns out opposite according to these editors as well. And Morgan saying, 'It's 2/3, period' doesn't mean 'it's 2/3 period', either. How does one compromise or find 'common ground' with such bellicose arguments? Glkanter (talk) 23:39, 31 August 2010 (UTC)
 * Sunray - if you could comment here I'd appreciate it. -- Rick Block (talk) 00:11, 1 September 2010 (UTC)
 * I note Martin's comment: "How about we just call it an 'explanation' and include it?" This was my own first reaction. However, as this text and graphic is not in the article right now, I find myself wondering whether it is needed. Isn't this perspective already presented in the article? Is there some particular reason for presenting this? I would like to hear from more than one editor on this. Sunray (talk) 00:40, 1 September 2010 (UTC)
 * Also, from a purely editorial perspective, I must add that it is usually not a feature of good writing to quote text without putting it into some context. To begin a section with a quote does not meet generally accepted standards for a good article. Of course, we are mainly considering sources now, but I do think that the sources have to be put into the context of the needs of the article as a whole. Sunray (talk) 00:48, 1 September 2010 (UTC)

Please note, a paraphrased version of Carlton's simple solution text is in the current article. I will wait for other editors to weigh in before further explaining the value of Carlton's simple solution. Glkanter (talk) 00:48, 1 September 2010 (UTC)
 * Right, I see that (missed it the first time because I spelled Carlton's name wrong in my search). The fact that it is already in the article begs the question: Why are we discussing a different version here? Sunray (talk) 03:08, 1 September 2010 (UTC)

Because it's a disputed item. (Be sure you're starting with the section called 'The test passage', not the section called 'Carlton's solution'.) And it's wonderfully brief. It has led to edit warring, blocks and protection.

What's in the current MHP article is a paraphrasing, not the actual text. The decision tree is something I derived from Carlton's solution, Whitaker's question, and simple math. Some editors say it's OR. Some editors claim it's not even a 'solution', that it's taken out of context, and it doesn't belong in the Solution section. These editors seem able to discern that the source's intent is different than the source's written words.

How this first mediation item is guided by you guys will be a strong indicator of how valuable this mediation will be, so a meaningful dispute is a good place to start. There are no 'easy ones' in this mediation, anyways. I've previously addressed these just-mentioned disputed points very clearly in the discussions so far. The strenuous objections to this solution by other editors is just as clear. Glkanter (talk) 03:28, 1 September 2010 (UTC)
 * Glkanter says: "How this first mediation item is guided by you guys will be a strong indicator of how valuable this mediation will be..." While that seems axiomatic, it actually misses a critical condition: The participants do the work.' The mediators only facilitate. Moreover, sometimes we won't be here (usually due to off-wiki stuff). At times we will miss crucial moments. Thus you (participants) must ultimately carry the ball. I think that there are resources within this group to do that. In the best mediation, the participants will say: "We did it ourselves." [To paraphrase a well-known Asian philosopher]. Sunray (talk) 16:37, 1 September 2010 (UTC)


 * Well, I did use the word 'guided'. What I'm trying to avoid, or at least shorten the cycle of is this:
 * "The definition of insanity is doing the same thing over and over again and expecting different results" - Albert Einstein
 * If it's the same editors making the same arguments, with the mediators cautioning, "At times we will miss crucial moments. Thus you (participants) must ultimately carry the ball.", then I believe we will be doing the same thing over and over again on a new talk page. Glkanter (talk) 16:48, 1 September 2010 (UTC)

Gerhardvalentins proposal
Switching will win in 2/3:

As long as the player initially picks a goat, which happens with probability 2/3, he can’t lose: Monty Hall must reveal the location of the other goat, and the player switches to the remaining closed door - the car. (Carlton 2005)

An even simpler solution is to reason that switching loses if and only if the player initially (only 1/3) picks the car, so switching will lose in 1/3 only (regardless of which goat the host just might have shown), while switching must win with probability 2/3. (Carlton 2005)

In fact, the only way you can lose is if you guessed the car’s location correctly in the first place (1/3) and then switched away. Hence, whether the strategy works just depends on whether you initially picked a goat (2 chances out of 3 switching will win) or the car (1 chance out of 3 switching will lose). (Carlton 2005)


 * no tree at this place. And Rick, please consider: With his well-arranged words Carlton isn't contradictory here to any other kind of approach he takes to solve the paradox, he's just helping to understand the problem. --Gerhardvalentin (talk) 22:36, 31 August 2010 (UTC)

Nijdam's comments about the test passage
I refer to Glopk's comment below. I fully agree and have nothing to add. Nijdam (talk) 18:13, 1 September 2010 (UTC)
 * Nijdam, we should carefully and precisely pay regard on what we are talking about. And it should mind us that the source cited is a textbook on conditional probability theory, and that it just isn't a science book on the MHP. But that this [source] is addressing conditional probability theory first. That's what it's meant to be, and having a look we can see that's what it really is. Foremost addressing the issue of conditional probability theory. Regards, --Gerhardvalentin (talk) 12:59, 2 September 2010 (UTC)
 * Am sorry to bring this up, but am I the only one having a harder and harder time understanding what Gerhardvalentin writes lately? I think he may be adding some valuable contributions, but I just can't be sure because they read like Flemish passed back and forth into English twice through Google Translate (or similar web service).
 * May I hereby make a plea that all non-native English speakers contributing to this page (myself included) to try a bit harder for readability, in addition to concision? glopk (talk) 15:45, 2 September 2010 (UTC)

Kmhkmh's comments about the test passage
Maybe it's just me, but imho the original short sentence is easier to understand and more clear than the longer text passage including the picture.--Kmhkmh (talk) 01:53, 3 September 2010 (UTC)
 * It's not just you! Gill110951 (talk) 18:55, 15 September 2010 (UTC)

Glopk's comments about the test passage
After re-reading the Carlton '05 paper, I believe that:
 * The proposed change in the "test passage" section is an out-of-context quote that perverts the meaning of the source's text: Carlton's own words make clear that he regards as a solution of the problem the one he develops in Section 3 of his article, and that he considers the quoted text (in Section 5) a mere intuitive explanation to help his students instead. By placing the latter in the WP article within a section entitled Simple solutions, and without further qualifications, the proposed change ignores Carlton's own distinction between solution (i.e. a rigorous sequence of logical steps proceeding from givens to answer, presented with the aid of a decision tree, and completed with an algebraic expansion using Bayes' rule) and intuitive explanation (a mere illustration of the intuition underlying the solution, with no pretense of rigor).
 * Even the current text in the article is not an accurate rendering of the source's intent - it would be if the word solution were replaced with explanation.
 * The proposed figure of a decision tree is both OR and a quasi-misattribution (weaseled away using the "As per" device). The Carlton paper shows a different decision tree. The proposed one is quite unlike the cartoonish illustrations currently present in the WP article, as the latter are not attributed AND make no pretense to rigor. Further - to ward off the obvious rejoinder - the proposed picture is quite unlike a formula or sequence of equations with the source's symbols replaced by equivalent ones (e.g. changing the name of variables, or using equivalent operators): WP conventions have long allowed the latter substitutions in formulae, in the interest of clarity and/or uniformity of presentation. However, I have uniformly encountered a much more cautious approach to introducing un-sourced figures.glopk (talk) 15:29, 1 September 2010 (UTC)
 * I disagree, Glopk, that the proposed tree is quite unlike a formula or sequence of equations. The proposed tree can be converted into a sequence of formulas, just as Carlton's original tree. It is a simpler tree and a simpler sequence of equations. The difference is that Glkanter knows or reasons a priori that the door numbers are irrelevant, whereas Carlton shows that a posteriori. It's a matter of taste for the mathematician, whether you get rid of the door numbers in advance or only as an afterthought. Carlton characterized his short solution as "intuitive explanation". If we add to his intuitive explanation the words that knowing the door numbers makes no difference by symmetry (by ignorance) then this short solution is complete, easily formalizable, completely mathematically rigorous. Carlton gives the long calculation with the long tree because he is assuming that you met this question in a statistics class in which you were learning about Bayes rule. He is simply not interested in the fact that you can completely solve the MHP without use of Bayes rule. His paper is incomplete, as a paper about MHP. Gill110951 (talk) 11:53, 2 September 2010 (UTC)
 * This is all somewhat interesting, but not at all relevant. WP editors are not allowed to convert into a sequence of formulae by themselves, nor invent a simpler tree and a simpler sequence of equations out of their reading of a source. It does not matter whether they know or reasons a priori that the door numbers are irrelevant, unless the source states that. Most importantly, editors are not allowed to add to his intuitive explanation the words that knowing the door numbers makes no difference by symmetry (by ignorance). On the other hand, editors must consider the sources as a whole, insofar as their content is relevant. Picking and choosing quotes out of context to support a favourite POV is a disservice to the readers. Please note that we are not engaged in writing the umpteenth OR article on the MHP for our own benefit. The issue at hand is whether a quote from the Carlton '05 article, un-paraphrased and without further context, has place in the Simple Solutions section of the WP article on the MHP. I'll really appreciate if you please stay focused on that, and be concise. Thanks. glopk (talk) 00:01, 3 September 2010 (UTC)
 * We all know what wikipedia editors are allowed to do and not to do. While we write, we had better be aware of what intelligent readers can do by themselves. What we write ought to be simultaneously wikipedia-correct as well as common-sense real-world correct. It's therefore highly relevant to keep an eye on the Truth, on simple logic and common knowledge; we aren't mere scribes. Editors are not scribes, we have to select and order as well as duplicate. Regarding what we are really discussing here, yes, I am pointing out that a little bit of literature search should easily fix this little problem. There are hundreds of sources besides Carlton who take Carlton's "intuitive explanation" simultaneously as the final rigorous answer to the MHP. I don't understand what you have against Glkanter's tree apart from the fact that one might misread his attribution. He does not attribute the tree to Carlton. Carlton presents the simple solution but does not say it is wrong. He presents the long solution because his paper is about doing computations with Bayes formula. Taken in context, Carlton is no authority at all for or against any particular solution. This seems to me to be what the discussion is about and I'm sorry if I can't explain this more concisely. Gill110951 (talk) 06:03, 3 September 2010 (UTC)
 * Let me help summarize concisely what you are saying above:
 * You too are against the change proposed by Glkanter which we are discussing, i.e. inserting the quote above from Carlton's 05 paper. Yes?
 * No. I am not against inserting the quote. I am not against the original text, nor the proposed new text by Glkanter. I think this is much ado about nothing. I am in favour that we editors work in a collaborative spirit, with a constructive approach. I don't take a dogmatic position. I think that all widely current and sensible approaches to MHP should be included in the article, and we are not to put one approach on a pedestal as being the only true deep mathematically correct approach, and all the rest as intuitive crutches for poor dumbos who can't do math or can't understand the difference between a conditional and an unconditional chance. Gill110951 (talk) 12:25, 3 September 2010 (UTC)
 * This is not helpful. The editor proposing the change has repeatedly and forcefully stated that nothing other than the insertion of the direct quote he proposes is satisfactory: no context, no paraphrase. This categorical position has been the cause of edit warring when several other editors have disagreed. Both these assertions are factual (diffs available), and one cannot collaboratively edit away such a position (which is dogmatic). glopk (talk) 14:01, 3 September 2010 (UTC)
 * No, it's not helpful. Anyone who is wedded to the idea that one particular solution is The One True Solution is not going to be a good editor of a controversial topic. Gill110951 (talk) 14:29, 3 September 2010 (UTC)
 * You do not understand my and other editors' criticism of the proposed picture as being WP:OR, even though you attribute it to GLkanter yourself: your honor, I rest my case :-) glopk (talk) 07:05, 3 September 2010 (UTC)
 * I do understand the criticism. I don't like arguing by reaching for the rule-book. It leads to polarization and discord. Of course Glkanter's picture shouldn't be attributed to Carlton. But it is not attributed to Carlton in his draft. Read it carefully. I do think the picture is a) superfluous (overkill) and b) inaccurate (door labels should be removed). Gill110951 (talk) 12:25, 3 September 2010 (UTC)
 * I have read it carefully, and remarked the use of the WP:WEASEL device "As per" to non-attribute it to Carlon. Regardless: it isn't a picture of Carlton's and it isn't a trivial transformation of a picture of Carlton's, hence it is WP:OR. And I agree that it is superfluos. So you too are against inserting it in the article - as it is? (yes/no) glopk (talk) 14:01, 3 September 2010 (UTC)
 * Yes, I am against inserting it, both as it is, and as it ought to have been. I think Glkanter produced it first to add support to his argument that the simple solution is as mathematically rigorous (can be made as mathematically rigorous) as the conditional solution. However, I am not so dogmatic as to say that every time a wikipedia editor comes up with a simple graphic device to illustrate a particular point, that his illustration should be banned on the grounds of being OR. Please let's keep some sanity here. Wikipedia editors do do (collaborative) own research all the time, at the meta-level: not within the topic, but on understanding and organising the material on the topic. We aren't just copying pre-existing encyclopeadia's. Gill110951 (talk) 14:34, 3 September 2010 (UTC)

Colincbn's comments about the test passage
As a non-mathematician I think Gerhardvalentin's reduction above is fairly easy to understand and covers the main issues people seem to struggle with while considering the MHP. I think that in the first sections of the article this kind of explanation is exactly what is called for and all of the formulas and in-depth explanations should come later. Colincbn (talk) 05:09, 14 September 2010 (UTC)


 * On a related but somewhat separate note; It seams to me, as a mathematically "challenged" person, that one problem here is that we all agree on the answer but there are multiple ways to reach it. I know this is a gross oversimplification, but from my standpoint a lot of this looks like many ways to reach the answer "= 2" from the starting point "1". Now as I said this is not an exact analogy, but to me its like:
 * 1 + 1 = 2
 * vs.
 * 1 X 2 = 2
 * vs.
 * 1 X 4 / 2 = 2
 * vs.
 * 1 X 8 /2 - 2 = 2
 * ect. etc. ad nauseam.


 * Of course the real difference between the above examples and this debate is that there are valid reasons behind the different methods used to reach 2/3 in the MHP. But all of this simply boils down to the single question: "What is the Monty Hall Problem". I think the methods used to reach 2/3 are not as important as what the MHP means, i.e. humans break complex problems into smaller parts in order to deal with them easier and this leads to the mistaken impression that the answer is 1/2. If we start with this as the basis of this article then all of the issues with how you reach the correct 2/3 answer are really only background that is not all that important to the central goal of explaining the MHP. Colincbn (talk) 05:24, 14 September 2010 (UTC)
 * Both Rick and I have both used analogies similar to yours to show that we are each right. The problem with using analogies to prove an argument is that the question soon becomes, 'How good is the analogy?'.


 * I would argue that there is never only one correct method of solution for any mathematical question and therefore, as you say, the methods used to reach the correct answer are not as important as the central goal of clearly and convincingly explaining the MHP. A have always said 'convincingly' because, however valid the explanation given, if the reader is not convinced that it is correct it will not do its job of conveying the central issues to the reader. Martin Hogbin (talk) 09:02, 14 September 2010 (UTC)


 * Oh I think in this case my analogy is particularly poor, and it was not meant to prove anything; just to show what a lot of this debate looks like to someone with no background in maths. But I think we agree on the larger point I was trying to convey. Also it seems to me the only reason there is a Monty Hall Problem is because of the fact that so many people get it wrong when first confronted with it. We don't have articles on every possible math problem. We only have one on the MHP because of the psychology behind it. That is why I have never felt the questions of conditional vs. unconditional were as important as simply describing the formula in a way that clearly shows why the answer is 2/3 to the majority of WP readers. I just don't think leading with a lot of complicated math is the best way to do that. Colincbn (talk) 10:36, 14 September 2010 (UTC)


 * I was not criticising your analogy, just stating that analogies are not a good way or resolving arguments. Rick used a mathematical analogy to show the simple solutions were invalid and I used a different one to show that they were valid.  This does nothing to resolve the issue as we can then argue about whose analogy is the best.


 * With regard to what you say later, I completely agree. The MHP is notable because it is a simple puzzle that most people get wrong.  I believe that a simple solution and explanation is what is primarily required here even if some mathematical licence were required (which it is not). The conditional/unconditional argument is an academic diversion, relevant only to the few who are studying statistics at the appropriate level.  Martin Hogbin (talk) 16:12, 14 September 2010 (UTC)


 * Good discussion. But I would say: there are many different and good routes to the same good answer to any mathematical question. However Whitaker - vos Savant's question is not a mathematical question. To convert it into a mathematical question, to mathematize it, is not a mathematical question. It requires taste, experience, knowledge ... it is an art, not a science. "How to do it" can't be decided divorced from the context of a living and evolving (multi-)culture. Not everyone's taste is the same. Each new generation can give a new answer. That's the other reason why MHP is so fascinating. Gill110951 (talk) 18:46, 15 September 2010 (UTC)

I understand that we're all supposed to be on our best behavior here.
How might one suggest to the mediators that one feels another editor may not be contributing to this mediation in Good Faith? But hides it via various methods of Gamesmanship? Glkanter (talk) 15:58, 31 August 2010 (UTC)
 * If you have a comment about something another editor is doing, you might discuss it in neutral language (observations, facts) and stating your own reactions using "I-messages" as opposed to "you messages" (attaching blame). Another way to approach it would be to use open questions to the individual (that is questions beginning with: "What...?" "How...?"). However, if you are emotionally charged, sometimes it is best to go for a walk instead :) Sunray (talk) 16:08, 31 August 2010 (UTC)

The way I read your various comments and suggestions, this mediation will get results because the mediators are watching the editors have the same discussions and arguments on this page as we've been engaged in on the article's talk pages for years. I think I'm seeing the same techniques used to avoid changing the article that I've been seeing for 2 years. I don't see any benefit in repeating all that. I don't believe I've demonstrated an 'emotionally charged' state. Just valid, carefully worded frustration. :) Glkanter (talk) 16:19, 31 August 2010 (UTC)
 * The meaning, in your first sentence is not clear to me. As to "techniques used to avoid changing the article," I can't tell. As I've said, I don't think you (the group) have yet begun to try my proposal. As to you "demonstrating emotionally charged behaviour," I did not say that you were. My words were: "if you are emotionally charged." I said that because it is common for people to feel an emotional charge when they are in conflict. Sunray (talk) 16:55, 31 August 2010 (UTC)

I don't have 'a comment about something another editor is doing..' I chose to notify you in neutral terms that (based on 2 years of interactions with this editor) I believe an editor is not participating in Good Faith in this mediation. Perhaps you might address that? Glkanter (talk) 16:23, 31 August 2010 (UTC)
 * Sure, just show me examples (diffs). Sunray (talk) 16:47, 31 August 2010 (UTC)


 * Here's the most important one. I point out what I see as the issues here. There are more issues with that editor's post, and the subsequent 'agree with' post from another editor, which I'll be happy to discuss at the right time. Glkanter (talk) 17:00, 31 August 2010 (UTC)
 * I see no violation of WP:AGF in these examples. Sunray (talk) 18:37, 31 August 2010 (UTC)


 * I think a valid point has been raised and it doesn't make a difference whether it was made in good faith or bad faith. You can refer to Carlton for the intuitive (verbal) solution which he presents, and if you want to, you can put a decision tree next to it, and refer to Carlton as being someone who thinks decision tree's are good tools for doing probability problems, but you should be careful not to suggest that the decision tree which you have actually drawn is actually Carlton's decision tree. Gill110951 (talk) 10:57, 1 September 2010 (UTC)

Here's the caption:
 * "Decision Tree Showing Results and Probabilities As Per Carlton's (2005) Simple Solution to Whitaker's Question in vos Savant's Column."

That's the only verbiage I've ever proposed for the article to associate the solution with the tree. And the caption is modifiable within the article, if needed. Glkanter (talk) 11:03, 1 September 2010 (UTC)


 * I just reread your text and the caption to your figure. Since we have no idea whether or not Carlton thought his 'intuitive explanation" could simultaneously be seen as a complete and simple solution, perhaps the caption should be rephrased appropriately. And as I said many times, I think you should delete the door numbers from the figure so that it does correspond one-to-one with this "intuitive explanation" / "simple solution".


 * Many many reliable sources besides Carlton consider Carlton's intuitive explanation to form a complete solution. It is so easy to turn into a formally complete mathematical argument that no-one bothers to do so.


 * The reliable sources who go for a calculation of a conditional probability do so because they have already rephrased the MHP as an exercise for an elementary statistics class in conditional probability. Certainly that is where it comes from (via Selvin). However vos Savant let it loose into the public domain. Gill110951 (talk) 12:04, 2 September 2010 (UTC)

How Much Editorial Interpretation Of Reliable Sources' 'Intent' Is Appropriate?
Well, we may as well address the issue now. The quote I provided from a reliable source addresses the problem as stated in the article. I do not agree with the opinions that the quotation is used out of context in a perverse manner. That other stuff about 'intent', I'm in no position to judge. Neither is any other editor. Carlton's 'intent' seems to be their entire objection to the quotation, although that never came up in prior discussions. Now what? Glkanter (talk) 15:52, 1 September 2010 (UTC)


 * Let's take this one step at a time. Do you agree Carlton's solution is presented in section 3 of his paper (yes or no)? -- Rick Block (talk) 17:16, 1 September 2010 (UTC)

I don't have the source convenient, but I recall the simple solution was preceded by a solution and was followed by the traditional solution. Let's say 'yes'.

Wasn't it you who added Carlton's simple solution to the article, in the 'Aids to understanding' section (yes or no)? Glkanter (talk) 17:53, 1 September 2010 (UTC)


 * Are you saying you're not looking at the source during this discussion??? I thought we were discussing what the source says.  The source is  (there's a link above as well).  Yes, I changed the wording  and added the Carlton reference per this thread on the talk page.  We're talking about this source and the article in more detail now.  I think we agree the solution is presented in this paper in section 3.


 * Next step - is the quote being presented in the article as a solution, attributed to Carlton (yes or no)? And, if you want to skip ahead, the next question is what section of the paper is the quote from, the section presenting the solution or some other section?  And, if it's not absolutely obvious, the last is how can you not think it is a misrepresentation to take a quote from a paper that has a solution but not from the section presenting the solution and to present this quote as a solution in the article (referenced to this source)?  -- Rick Block (talk) 18:59, 1 September 2010 (UTC)

So the solution met your criteria for inclusion in the article in the 'Aids to understanding' section, but it fails to meet your standards for inclusion in the 'Simple solutions' section? How am I to see this as anything other than arbitrary, and favorable to your POV? Glkanter (talk) 21:09, 1 September 2010 (UTC)


 * If I may, it seems to me that Rick Block's editorial decision was entirely consistent with (i.e. neither more nor less arbitrary than) distinguishing in separate sections of the WP article Solutions from Aids to Understanding. If such a separation has any meaning, it is a distinction between the rigorous arguments and the merely illustrative ones. Given that Carlton himself does provide a rigorous solution in Section 3 of his paper, does not call the quoted passage of Section 5 of his article a solution, and given the context preceding the quote (to wit: Before presenting a formal solution to the Monty Hall Problem to my students, I find that it helps to give an intuitive explanation for the 1/3 - 2/3 solution. Imagine you plan to play Let’s Make a Deal and employ the “switching strategy.”), it was entirely legitimate to paraphrase the quoted passage within the Aids to Understanding section of the MHP article.
 * However, IMHO, it is not legitimate to call a solution that which is not, according to the source itself, and in addition slap on it a misattributed OR picture, and finally place it in the Simple Solutions section of the MHP article. glopk (talk) 23:18, 1 September 2010 (UTC)

The two section headings and the bifurcation strike me as arbitrary. Not based on mathematics, or linguistics, or any science, just an editor's (or a succession of editors') best attempt(s) to give the article structure. Carlton's solution plus the decision tree may provide that 'peaceful co-existance' Boris spoke about. Why delegate it to the Aids to understanding section? Glkanter (talk) 23:29, 1 September 2010 (UTC)


 * The basic point here is that since the source itself does not characterize this as the solution and, indeed, characterizes something else as the solution, presenting this as a solution in the article attributed to this source (as if it is what this source presents as the solution) is misleading. And, at this point, we're still talking about the text.  The "decision tree" (assuming you mean this one) is quite another matter that we haven't really even begun to address.  If we could stay focused on one topic at a time I think it would be helpful.  BTW - if "peaceful coexistence" is what you're after then why not start with any of the numerous NPOV "combined solution section" drafts I've suggested (like this one, in the show/hide section, or this one)?  This gets at a topic we're not exactly talking about yet also, which is whether the existing structure of the article is NPOV or whether presenting "simple solutions" first implicitly endorses the POV that these solutions are correct and complete.  If we're going to focus on one topic at a time, this one should be tabled for now (but keep it in mind as a potential way to address our disagreements).  -- Rick Block (talk) 04:30, 2 September 2010 (UTC)

You're splitting hairs over his choice of words. The solution is in his paper. His paper, 'The Misuse of Conditional Probability' is published. He, and the Journal of Statistics Education are reliable. He doesn't say 'this is not a solution'. A 'smoking gun' doesn't exist. You are arguing about a flawed conclusion you have reached, not a 'fact'. The delineation you are making is meaningless. Your argument that the solution is good enough for one section of the article, but not another doesn't withstand the most basic scrutiny. Glkanter (talk) 07:42, 2 September 2010 (UTC)
 * SunRay, Will, would care to comment on these sentences above? "You're splitting hairs". "The delineation you are making is meaningless". "Your argument [...] doesn't withstand".  I am getting mildly annoyed (or borderline offended) at this repeated overlooking of the spirit and the letter of the rules of this mediation, even after one warning. Thank you. glopk (talk) 16:30, 2 September 2010 (UTC)


 * There is a difference between "You're", "You are", and "Your" as compared to "You". I am being critical of "your" arguments, not "you" personally. Glkanter (talk) 12:09, 3 September 2010 (UTC)


 * This is not splitting hairs about choice of words. Carlton is very very explicit. He write, near the very end of his paper, "Before presenting a formal solution to the Monty Hall Problem to my students, I find that it helps to give an intuitive explanation for the 1/3 - 2/3 solution." His formal solution was the calculation using Bayes rule of the conditional probability, and he used the full decision tree with all the standard assumptions to fill in all the necessary probabilities. His intuitive explanation is the short argument for the unconditional probability. His paper is all about Bayes and conditional probability and decision trees in elementary statistics. I think it is fine to give Carlton's "intuitive explanation" and there is nothing wrong with illustrating it with a decision tree provided it is done properly (no door-labels on the tree). But we have to look elsewhere to find a reliable source who is prepared to say that the intuitive explanation can be considered a complete solution. My day-job is writing reliable sources, and some day I will do exactly this job, since I do believe that the direct unconditional approach can be considered a complete solution of Whitaker / vos Savant's problem. That is because a rational subjectivist, who has a priori uniform probability distributions representing his prior beliefs about location of car and host's actions, can take the perfectly logical step of saying that the door numbers are irrelevant *before* proceeding to a probability calculation. "Irrelevant" (regarding beliefs) translates to "statistically independent" (regarding probabilities). Moreover, vos Savant wrote that when she used the words "say, Door 1" and "say, Door 3", she was naming the doors in order to help you visualize the situation. The door numbers weren't even meant to be part of the formal problem. If you like, the player chooses a door, and we name that door "Door 1", after the fact. Then the host opens a door and we name that door "Door 3", after the fact. Gill110951 (talk) 08:29, 2 September 2010 (UTC)
 * Be my guest, but if the game is repeated, it is not necessarily again Door 1 that will be chosen and Door 3 that will be opened. Nijdam (talk) 13:16, 2 September 2010 (UTC)


 * I'm not interested in repeating the game. I am a rational subjectivist who is confronted with Whitaker / vos Savant's question once, and has to answer it once. Alternatively, taking vos Savant's hint that the door numbers are names given posterior to the actions, the next time the game is repeated, I will again rename the door I choose myself as "Door 1" and next rename the door opened by the host as "Door 3". Thus every single repetion I will always choose Door 1 since by definition Door 1 is the door I choose (it can be a different physical door, every time). Every single repetition, by definition Door 3 is the door opened by the host and Door 2 is the remaining door, because the doors are given these names, anew, because of the actions of the player and the host. Gill110951 (talk) 13:51, 2 September 2010 (UTC)
 * Again, be my guest, if that makes you happy. Nijdam (talk) 13:58, 2 September 2010 (UTC)
 * It makes me very happy to understand and appreciate different approaches to MHP, especially since my day-job is publishing OR on MHP, among other things. But my happiness is irrelevant to a wikipedia editor who has to study the reliable sources without personal prejudice. Gill110951 (talk) 15:23, 2 September 2010 (UTC)

Oh, yes it is splitting hairs. Had Carlton not included his simple solution in his paper, I would have no argument. However, it exists. He must think it is accurate and has value, or he wouldn't have included it. To suggest otherwise is contrary to the fact that it is in a paper written by a professional in a professional journal. Glkanter (talk) 08:38, 2 September 2010 (UTC)


 * Carlton (like you, Glkanter, and me) thinks it is accurate and has value as an argument for switching. Whether or not it is "the solution" depends on what is "the problem". Carlton shows how to compute the conditional probability because some statistics text books ask you to calculate it and his paper is about misstakes when operating on conditional probabiliy. He doesn't say he would give the student a 100% pass mark if the student presented his intuitive explanation as the answer to Whitaker's question (should I switch?) in a statistics 101 class. Gill110951 (talk) 08:47, 2 September 2010 (UTC)


 * Is the source assuring '...a 100% pass mark ...' the standard for a solution's inclusion in the Wikipedia Monty Hall problem article's Solution section? Not so much.
 * The argument that (paraphrasing) 'how a solution is prefaced and introduced is critical to whether the statement that follows is of importance/merit' is unsupported. I'm sure there have been many great advances, from people like Fermi, for example, that were not the 'main point' of a paper. But they were of merit, advanced the understanding and were worthy of subsequent citation, nonetheless. Glkanter (talk) 10:41, 2 September 2010 (UTC)

I am not sure that a 'yes/no' discussion is going to get us anywhere. Carlton's simple solution is just one of many published solutions that do not make a distinction between the goat door that the host opens when he has a choice. It might be argued that solutions which do not make that distinction are somewhat less rigourous or complete than those that do and that those which show all the player's possible original door choices are more complete still, but I do not believe that there is a sharp dividing line between 'good' and 'bad' solutions. Carlton's simple explanation/solution has, like it or not, been published in a reliable source and it is therefore open to us to use it in the article of we choose. The question that I believe we should be asking is how well this particular solution helps clarify and explain the resolution of the Monty Hall paradox to our readers. No OR and NPOV are not particularly relevant in this respect, the solution is published in a reliable source, and editors here have to use there own judgment to decide which of the many available sources solutions to make use of in producing this article. The sources confirm the facts, they do not tell us how to write the article. Martin Hogbin (talk) 10:30, 2 September 2010 (UTC)


 * I was trying to get across to Glkanter that he should easily be able to find a reliable source who thinks that Carlton's intuitive explanation is a complete and rigorous solution to MHP, it doesn't have to be Carlton. Carlton's paper is not about MHP. Carlton's paper is about conditional probability in introductory statistics courses. I agree with you Martin that there is no sharp dividing line between "good" and "bad" solutions, and even if there were, that would be irrelevant to wikipedia, since our job is to report what reliable sources say. At the same time we want to try to be a useful introduction to the MHP, not a boring catalogue. I don't see a conflict between our main job and our hope that the article could again be a featured article because it is more than just a catalogue. I felt that Glkanter was weakening his own support of the simple solution by giving us the wrong probability tree (he should remove the labels of the doors) and by giving the impression that Carlton thinks that this solution is equally "good" as the posterior probability calculations. His paper gives no clue to that issue. Gill110951 (talk) 14:03, 2 September 2010 (UTC)


 * If no one is going to offer a different source, then how about if we change


 * "An even simpler solution is to reason that switching loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3 (Carlton 2005)."


 * to


 * "An intuitive explanation is to reason that a player planning to switch loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3 (Carlton 2005)."


 * This changes the characterization ("intuitive explanation" rather than "simple solution") and includes the "planning to switch" phrasing - both of which match what the source says. -- Rick Block (talk) 14:11, 2 September 2010 (UTC)

I don't see a need for corroboration, as Gill110951 suggests, or paraphrasing, as Rick Block suggests. Carlton's solution is reliably sourced, and is brief enough to be quoted, including the 1st sentence or two that I did not include. As the simple solutions will be criticized in the article, it strikes me as 'fair play' that the criticized solutions should, as much as reasonable, reflect exactly what the sources say. Carlton's simple solution is very brief, it does not lead off the section, so I see no reason not to use a full quotation. Glkanter (talk) 14:38, 2 September 2010 (UTC)


 * Unless there's a compelling reason to include them, I think direct quotes should generally be avoided. To compare, I think you're suggesting instead of the one short, simple sentence we include:

"Before presenting a formal solution to the Monty Hall Problem to my students, I find that it helps to give an intuitive explanation for the 1/3 - 2/3 solution. Imagine you plan to play Let’s Make a Deal and employ the “switching strategy.” As long as you initially pick a goat prize, you can’t lose: Monty Hall must reveal the location of the other goat, and you switch to the remaining door - the car. In fact, the only way you can lose is if you guessed the car’s location correctly in the first place and then switched away. Hence, whether the strategy works just depends on whether you initially picked a goat (2 chances out of 3) or the car (1 chance out of 3)."


 * IMO, there's absolutely no need to do this. If there's anything about the paraphrasing you don't like, specifically if you think the paraphrase is in any way changing the meaning of this paragraph, please say what it is.  -- Rick Block (talk) 15:24, 2 September 2010 (UTC)

As I stated above, as the simple solutions will be criticized in the article, it strikes me as 'fair play' that the criticized solutions should, as much as reasonable, reflect exactly what the sources say. I don't really believe the extra sentences are required or add value. I only mentioned them because of the 'out of context' red herring. I regret that already. Glkanter (talk) 15:44, 2 September 2010 (UTC)


 * The point several editors are making here is that this source is not presenting a "simple solution" - objecting to taking a quote out of context and presenting it as if it is a solution advanced by this source is not a red herring (and using pejorative terms like this for other editors' points is completely out of line here). The criticisms that will be presented (later) do not pertain to this source or to this "intuitive explanation" precisely because this source is NOT presenting this explanation as a "solution".  If we're quoting, the "extra" sentences are distinctly required - without them the context establishing the source IS NOT talking about a solution and IS talking about a scenario where a player is "planning" (effectively deciding whether to switch BEFORE seeing what door the host opens) is missing.  I believe the paraphrase suggested above provides this missing context and succinctly captures the point of the paragraph.  Again - is there anything in the original that the paraphrase is missing or misconstrues in any way? -- Rick Block (talk) 17:23, 2 September 2010 (UTC)


 * Yes but .. I think it is a red herring (or at least: a POV) to suggest that the situation of a player who must decide at the moment of opening of the door, is different to the situation of a player who may decide in advance. You can't suddenly "know" what you have to know, at the moment you are standing in the lights ... you must have known these things in advance. Hence you had the opportunity to think in advance. If you didn't, well, silly you. Secondly, it is a red herring (or a least: a POV) to suggest that the actual door-labels which the player sees when he should make his decision should play any role at all in his decision. There are very very good arguments why they are totally irrelevant. Glkanter "knows" this intuitively. With his understanding of the meaning of probability, his intuition is perfectly justified and his solution is complete and correct. So I think that Glkanter should look for a different reliable source (surely he can find one) and though Rick may criticise the attribution of this "solution:" to Carlton, he shouldn't criticise the solution itself. So I disagree with both editors, and neither editor agrees with me. I think the disagreement comes because Rick and Glkanter have different notions of probability. Both are totally reasonable, both are rational, both are even reliably sourced. My collection of books and articles on MHP is 160 Km away from me at the moment. I have to postpone giving reliable sources for my assertions till the weekend. Gill110951 (talk) 18:31, 2 September 2010 (UTC)


 * If someone comes up with a different source, then we can talk about what that other source says. So far, we're talking about Carlton, and Carlton does not present his intuitive explanation as a solution - implying it does by taking several sentences out of context and calling them a solution is what I'm objecting to.  I'm not saying there is no reliable source that presents the same argument as a solution - but rather that Carlton does not.  If we want to leave the reference as Carlton, then I think the paraphrase I've suggested is accurate.  If someone wants this argument presented more directly as a solution, another source needs to be found.  I'm happy enough with Carlton, but only if we accurately represent what he says.  -- Rick Block (talk) 18:59, 2 September 2010 (UTC)

I think we've each had our opportunity to state, without interruption, our positions on 'The test passage'. The situation looks to me about the same as our previous impasses on the article's talk pages. I request that the Mediators get involved at this point. Glkanter (talk) 19:08, 2 September 2010 (UTC)


 * A number of editors are unhappy about the reference to Carlton. Carlton's position isn't clear, I think. I shall do some literature search this weekend to find some more reliable sources who see Carlton's intuitive explanation as the full story. Gill110951 (talk) 06:16, 3 September 2010 (UTC)


 * This is not how I read the situation. A plurality (4 out of 8) of the editors engaged in this mediation (Rick Block, Nijdam, Kmhkmh and myself) are clearly against inserting both the proposed quote from Carlton's 05 paper and the "As per Carlton 05" picture. They are also OK with having a paraphrase very similar to the one currently in the article, but moved into the "Aids to understanding" section.  You are taking an even stronger position, that is eliminating references to Carlton altogether - that makes a 5/8 majority against the proposed change. Of the remaining editors, only two (Glkanter and Gerhardvalentin) are in favor, the latter only for the text and not the picture. I'll leave to everyone (mediators included) to decide whether all this constitutes:
 * A consensus against inserting the proposed picture
 * Add Gill110951 (per this diff ) to the list of editors against inserting the picture. Now it is 6/8, and Martin Hogbin has consistently not taken a position on this issue. Now I do believe there is editorial consensus against inserting this "As per Carlton decision tree" picture proposed by Glkanter in the MHP article. May we consider this particular issue solved? If you agree, please state so below:
 * Agree: glopk (talk) 14:50, 3 September 2010 (UTC)
 * Agree: Nijdam (talk) 15:29, 3 September 2010 (UTC)


 * A near consensus against inserting the proposed quote. glopk (talk) 07:21, 3 September 2010 (UTC)

I object. "Consensus" is not achieved by a show of brute force. Mediation is not about playing games to crush your perceived opponents. If there seems to be a clear majority for a certain action it is up to a neutral player to confirm that, or for the holders of the minority view to gracefully concede. We are trying to prepare the ground for constructive collaboration to resolve bigger issues than these couple of sentences and a disputed picture. The entrenched positions are as entrenched as ever which means that editors are not trying to understand opposing views and not prepared to modify their own. Gill110951 (talk) 17:30, 3 September 2010 (UTC)
 * Sunray, Will, care for weighing in on this? I refer to the relevant WP policy section (which I hereby invite Gill to read in its entirety), and remark that:
 * It does not take a neutral player to count up to 6 after reading the explicitly stated views on this particular issue (Glkanter's picture) of the editors engaged in this mediation. We can all read English. But to play safe I asked explicitly above for agreement to close the issue by consensus.
 * Given the article's recent history, and the text of this mediation so far, I believe it may well be impossible to reach unanimity of consensus on this particular issue, so a significant majority will have to sufice, if we are to make any progress onto further issues.
 * Noone here is using any brute force, playing games nor crushing anybody. Let us please adhere to the spirit and letter of the rules of this mediation, by at least watching our language. Thanks. glopk (talk) 18:27, 3 September 2010 (UTC)

Oppose, and I didn't address Glkanters suggestion by now. Difficulty in counting? Within "The test passage", after Glkanter made his suggestion, I have made an alternative suggestion, showing Carlton's perspective of a clear and plain answer to Whitaker's question, not even needing conditional probability theorems nor Bayes. --Gerhardvalentin (talk) 18:35, 3 September 2010 (UTC)
 * Thanks, I can count, but wonder if there are difficulties in reading. The alternative suggestion includes this text at the end: No tree at this place. Does that "No" means "Yes", as in "Yes, include the picture of the decision tree proposed by Glkanter"? glopk (talk) 18:54, 3 September 2010 (UTC)


 * I said "no tree at this place". You can show that later, when proving that conditional probability gives the same result). --Gerhardvalentin (talk) 19:11, 3 September 2010 (UTC)
 * And it is OK to insert later an unsourced picture? glopk (talk) 20:55, 3 September 2010 (UTC)
 * "Sourced". – Glopk, a much more important issue is Carlton's lecturing on Bayesian conditional probability theory, based on plain examples. Please pay attention to the fact that, in applying conditional probability formulations on the MHP – in lecturing conditional probability – he uses to present his students the correct solution first, to show them what it's all about, before teaching the correct formulation of Bayes' theorems. – It was advantageous to realize in the first line what we are talking about, particularly in evaluating and using of sources. -- Gerhardvalentin (talk) 23:00, 3 September 2010 (UTC)
 * Gerhard - are you talking about section 5 of the paper - the paragraph quoted above starting with "Before presenting a formal solution to the Monty Hall Problem to my students, I find that it helps to give an intuitive explanation for the 1/3 - 2/3 solution."?  He's not saying the following sentences constitute a "correct solution", or even an informal solution - but specifically contrasts "formal solution" (clearly the solution presented in section 3) with "intuitive explanation".  I believe the paraphrase I suggested above says exactly what this paragraph of this source says - no more and no less - i.e. "An intuitive explanation is to reason that a player planning to switch loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3 (Carlton 2005)."  Is there anyone here who objects to this wording?  Let me rephrase that, is there anyone here how cannot live with this wording?  If so, why?  -- Rick Block (talk) 01:51, 4 September 2010 (UTC)
 * Rick, Carlton is not saying that he is correct in striving for evidence by presenting the clear solution before his lecture on Bayes, but I guess we may suppose that he is correct. --Gerhardvalentin (talk) 05:51, 4 September 2010 (UTC)
 * You're focusing on the wrong word here. He's not calling this a solution (of any sort).  Do you object to the paraphrase?  If so, why? -- Rick Block (talk) 06:08, 4 September 2010 (UTC)
 * Rick, please pay regard to what Carlton's paper is about: Carlton lectures on Bayes and is lecturing on how to do it correctly. So his "solutions" do address conditional probability theorems, not the MHP. We should avoid to misinterpret Carlton's endeavor. Regards, --Gerhardvalentin (talk) 06:20, 4 September 2010 (UTC)
 * Excuse me, but section 3 of his paper is titled "The Monty Hall Problem". He certainly says he's talking about the MHP.  He quotes the problem description from Parade.  It's one of three example problems that he's clearly saying SHOULD be solved using conditional probability (using carefully designed tree diagrams).  On what basis do you claim that he is not addressing the MHP?  -- Rick Block (talk) 06:33, 4 September 2010 (UTC)
 * You just gave the correct answer yourself. In his paper Carlton is "on Bayes" and on correctly using conditional probability theorems. He does that using the famous MHP as an "example", saying that Marilyn vos Savant's solution 2/3:1/3 was correct, and showing that the famous 50:50 error can easily be refuted by using conditional probability, also. Please note what his paper is about. It's on conditional probability, it's on Bayes. And so he shows how Bayes gives the correct answer. Because his aim, his paper is to show it on Bayes. It's on conditional probability, not on "solving the MHP", nevertheless he showed the correct answer in section 5, again, saying student's do better in Bayes theorems when knowing the correct result in advance. --Gerhardvalentin (talk) 06:56, 4 September 2010 (UTC)

So it's not really the reliable sources...
It's how a majority vote of currently active editors interprets the intent of the reliable sources.

I wish y'all had told me that 2 years ago. Glkanter (talk) 02:57, 4 September 2010 (UTC)
 * You're right again, Glkanter. It's not about what reliable sources are saying, it's about what sources are said to be saying. Regards, --Gerhardvalentin (talk) 05:36, 4 September 2010 (UTC)


 * I can't see a huge difference (actually, any significant difference whatsoever other than length) between:

"Before presenting a formal solution to the Monty Hall Problem to my students, I find that it helps to give an intuitive explanation for the 1/3 - 2/3 solution. Imagine you plan to play Let’s Make a Deal and employ the “switching strategy.” As long as you initially pick a goat prize, you can’t lose: Monty Hall must reveal the location of the other goat, and you switch to the remaining door - the car. In fact, the only way you can lose is if you guessed the car’s location correctly in the first place and then switched away. Hence, whether the strategy works just depends on whether you initially picked a goat (2 chances out of 3) or the car (1 chance out of 3)."


 * and


 * An intuitive explanation is to reason that a player planning to switch loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3 (Carlton 2005).


 * We've talked about reasons not to quote only the last three sentences of this paragraph (starting with "As long as ..."), which (IMO) the paraphrase adequately addresses. What are the reasons not to use the paraphrase I'm suggesting?  Specifically, does this sentence say something the full quote does not, or not say something the full quote does?  If so, how about if we fix the paraphrase so that it is completely equivalent but simply shorter and more succinct?  -- Rick Block (talk) 05:41, 4 September 2010 (UTC)

I'm referring to the election taking place on these pages to determine: A. If Carlton's simple solution is a 'solution', and: B. If it merits inclusion in the 'Solution' section, or in the 'Aids to understanding' section, or in the article at all. Glkanter (talk) 06:13, 4 September 2010 (UTC)


 * I understand that. However, if we can ALL agree the paraphrase is acceptable the "voting" becomes moot.  Are you OK with the paraphrase, or not?  If not, why not? -- Rick Block (talk) 06:17, 4 September 2010 (UTC)

Can I ask you both if you agree that the main purpose of this section is to explain as clearly and convincingly as possible to our readers why the answer is 2/3 and not 1/2? Martin Hogbin (talk) 09:14, 4 September 2010 (UTC)
 * Yes Martin, of course you can. --Gerhardvalentin (talk) 10:04, 4 September 2010 (UTC)
 * How about Rick and Glkanter? Does anyone disagree? Martin Hogbin (talk) 11:31, 4 September 2010 (UTC)

I agree to a large extent, but I hesitate with the words 'explain' and 'convincingly'. I don't think that's the editors' role. For example, we've all been arguing for a long time, and we don't all agree on 'why' it's 2/3 - 1/3. Further, I would say our responsibility is to present what '... the sources say as clearly as possible.' And all this ' perceived intent' business runs counter to that. Glkanter (talk) 14:57, 4 September 2010 (UTC)


 * Can we table this new topic for a bit? It seems like we may be close to an agreement on Carlton and I'd like for us to stay focused on that if possible.  Can Glkanter, Martin, and Gerhard directly answer this question - are you OK with the paraphrase, or not?  If not, why not?  Thanks. -- Rick Block (talk) 15:52, 4 September 2010 (UTC)

I'm not close to any agreements. The paraphrasing is less important than the claimed unsuitability of Carlton's simple solution as a solution fit for the Solutions section. After that is worked out, then we can discuss paraphrasing a source, and imaging a source. Glkanter (talk) 15:58, 4 September 2010 (UTC)


 * I'm suggesting a minor rewording of the sentence in the "Simple solution" section, and leaving it in in this section, as an alternative to the proposal to include a quote from this source at the same point. And, to scope this down further, I'm not implying we're agreeing either way about an image.  I'm specifically talking about the text. -- Rick Block (talk) 16:20, 4 September 2010 (UTC)


 * It would seem logical to me that, if we can agree on what it is we are trying to achieve, we might better be able to work together to achieve it. (An interesting aside is that to table a topic in British English means to put it up for discussion whereas I have taken it that Rick wants to defer discussion until later.  This difference nearly caused a major problem in WWII)


 * With regards the paraphrase, how about 'One explanation is to reason that a player who switches is certain to win if they have initially chosen a goat, which happens with probability 2/3, so switching must win with probability 2/3 (Carlton 2005)'. Martin Hogbin (talk) 17:21, 4 September 2010 (UTC)
 * This version omits the entire second sentence of the paragraph which establishes the context as "planning to employ the 'switching strategy'". Whether or not we agree on what this means, since I (and I suspect others) think this is contextually important are you OK with including it?  -- Rick Block (talk) 18:49, 4 September 2010 (UTC)
 * There is no need to say "planning to employ the 'switching strategy'". 'A player who switches' means much the same thing. Martin Hogbin (talk) 23:14, 4 September 2010 (UTC)
 * I do not believe it does: saying that a player switches is weaker than saying that he is following a strategy to switch (say, as opposed to deciding to switch on the spur of the moment). In fact, how about we modify the paraphrase staying a little closer to the source's text as follows:
 * An intuitive explanation is to reason that a player whose strategy is to switch loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3 (Carlton 2005).
 * How about this? glopk (talk) 03:20, 5 September 2010 (UTC)
 * I think "planning to switch" captures what the source is saying more clearly, but "a player whose strategy is to switch" would be OK with me as well. @Martin - whether you think "a player who switches" means the same thing or not, given that there are others here who think this distinction is important, and it is what the source says, would you be OK with including this? -- Rick Block (talk) 05:30, 5 September 2010 (UTC)

As I can best determine, the topic of Carlton's simple solution's suitability for the Solution section was being voted on. Well, I have a question about the suitability of that discussion and that vote. That's why I started this new section. The paraphrasing seems to be a priority for only one editor. Glkanter (talk) 19:01, 4 September 2010 (UTC)
 * We've been talking about changing the sentence in the "Simple solution" section that is attributed Carlton to be a quote, and including the image. Glopk was summarizing what he's heard in the discussion so far, i.e. clearly no consensus FOR this change (a plurality against among those who've directly commented) and a fairly clear consensus against including the image (based on comments so far).  I've been attempting to shift the conversation from this sort of win/lose approach to something we might ALL be able to agree on, focusing first on the text.  Based on the discussion about using the quote, I think a change to the text is warranted - but rather than using a quote I'm suggesting a minor rewording to make the sentence closer to what the source says (in the spirit of collaborative editing - per Sunray's advice).  If we can ALL agree on a change to the text we'll have actually accomplished something, hopefully providing a basis for the ability to make further changes.  So, are you absolutely dead-set against the change I'm suggesting, or not?  -- Rick Block (talk) 22:04, 4 September 2010 (UTC)

I am dead set against the argument that: Due to his perceived intent, as adjudicated by a majority vote of active editors, Carlton's simple solution is not appropriate for the Solution section, and maybe not appropriate for inclusion in the article at all. Glkanter (talk) 22:22, 4 September 2010 (UTC)

I explained what I hoped we would accomplish in regards to Carlton's simple solution here. The question you repeatedly ask is not one I will find of interest until the other issues are resolved. Glkanter (talk) 00:09, 5 September 2010 (UTC)
 * I don't believe that this reading is at all correct. I did a count of the editors' opinions, as expressed on this page, strictly on the two issues being proposed by Glkanter, namely (1) the insertion of the unedited quite from Carlton and (2) the insertion of the proposed "decision tree" picture. I did this cout AFTER Glkanter expressed what looked like a call for an end of the discussion by writing I think we've each had our opportunity to state, without interruption, our positions on the test passage, and then calling for the mediators' intervention. Whether Carlton's simple solution is not appropriate for the Solution section, and maybe not appropriate for inclusion in the article at all was absolutely NOT the subject of my count, or indeed of the discussion. I also expressed my personal opinion that, if unanimity proved impossible to achieve on either issue, a significant majority would have to sufice, and I believe that this opinion is entirely consistent with Consensus (see the first Section: What consenus is").
 * Now, constructively, I Agree with Rick Block's proposed rewording, perhaps with the last addition I suggest above. glopk (talk) 03:20, 5 September 2010 (UTC)


 * Here's where the votes were counted. This statement:
 * "They are also OK with having a paraphrase very similar to the one currently in the article, but moved into the "Aids to understanding" section."
 * comes directly from the editors' personal interpretations of, and following a long discussion of (most of it, including the above diff, taking place in a section titled "How Much Editorial Interpretation Of Reliable Sources' 'Intent' Is Appropriate?"), Carlton's 'perceived intent', and therefore Carlton's simple solution's suitability for the Solution section. My reading of the situation is accurate. I will now take the requested break. Glkanter (talk) 10:35, 5 September 2010 (UTC)


 * I completely agree with Glopk's assessment of the situation. Sunray hasn't edited since Wednesday.  I've asked Will to comment as well.  I suggest we simply wait for comments from a mediator at this point. -- Rick Block (talk) 05:30, 5 September 2010 (UTC)


 * Could everyone take a break for a couple of days while us mediators catch up?   Will Beback    talk    05:38, 5 September 2010 (UTC)