User talk:Martin Hogbin/Archive 7

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By the way, on centrifugal force, I left a note for you about that on the Talk:Centrifugal force page. The gist of it is that the content of the page is not just about exposition and due weight for different subtopics. It also is about maintenance of the page. Where readers of an article hold a variety of views, they naturally want to see their views presented. A subset of readers actually think their view is the correct view, and the others are mistaken superstition and should be removed. These factors mean the page will be under assault to reflect these realities, and survival with limited warfare means it has to have content that is battle ready. That is, it has to anticipate the changes a hodge-podge of readers may attempt, and present matters so that readers will come to a wider appreciation of the subject, and feel accordingly that their views are reasonably represented. In particular, reactive centrifugal force may be a minor topic, but WP history shows that many readers think it is the whole story and are willing to fight about it. Brews ohare (talk) 17:27, 15 January 2012 (UTC)

In fact, I feel that this issue of designing a page to be battle-ready is the basic reality underlying WP and the mechanism that ultimately determines its content. For any stability of content, by definition the article must be persuasive enough to keep dissent to a minimum. It is misconception to think that an on-line encyclopedia subject to attack from all comers has the same criteria for content as a Britannica. Brews ohare (talk) 17:39, 15 January 2012 (UTC)

In this context, for example, one may query how the struggle for concise articles pursued with diligence by DickLyon, for example, plays into survivability. On the one hand, a very brief article may be interpreted by a reader as an unambitious presentation, and so not one to seriously dispute. On the other hand, a more extended article may be more useful or clearer but also has more detail that may raise opposition. That means a longer article demands a more careful exposition to survive without continuing dispute. Brews ohare (talk) 17:46, 15 January 2012 (UTC)

I appreciate your aim in trying to make pages more stable but I think it will not work, and at the expense of including more fringe views. Martin Hogbin (talk) 18:00, 15 January 2012 (UTC)

It might be that Centrifugal force is an interesting experiment in this respect. For reasons unknown to me, it is a subject very popular among a large cross-section of readers including some fanatics. In the old form as one article it was subject to unending controversy. Today it is in several pages of different scope. It seems to be less controversial this way. Brews ohare (talk) 18:10, 15 January 2012 (UTC)

To a degree, I can understand this improvement: if an article is clearly an overview, then debate over details is properly directed at a page devoted to that subtopic. If a page is a particular facet, then debate over other facets is properly directed elsewhere. On the page for a specific aspect, there is greater likelihood of a narrow debate, and less likelihood of wandering into generalities that cannot be decided. So the natural tendency of Talk pages to wander off into the desert of digression is curtailed, and a real resolution of opinions becomes more likely. Brews ohare (talk) 18:44, 15 January 2012 (UTC)

Traffic statistics show Centrifugal force has about 2000 hits/month, while Reactive centrifugal force has about 80 and Centrifugal force (rotating reference frame) and Fictitious force about 180. Inertial frame of reference about 350. On the basis that the general overview Centrifugal force is not the right place to argue details, it appears that breaking up the articles may reduce controversy because not many readers are interested enough to pursue the other pages. So they seldom look at the venue where their disputes could be argued. Other related topics are even less looked at: Absolute rotation, Rotating spheres: 20 Bucket argument: 50 . Of far more interest is the Coriolis effect at 3000. Brews ohare (talk) 16:23, 16 January 2012 (UTC)

What is this "struggle for concise articles pursued with diligence by DickLyon" of which you speak? I don't recall pursuing anything like that. As far as I can recall, the only editor that I routinely push back against for unrestrained article bloat is Brews ohare. Dicklyon (talk) 16:18, 17 January 2012 (UTC)

Martin, just a personal note. An enterprise such as Wikipedia, or anything as big as that, should be based on trust. What I mean with that, is that I don't know who I'm talking to, and neither do you; but we must assume on good faith that we both know what we're talking about, or we wouldn't be here wasting our time just for the sake of vandalizing other people's articles. The fact that I introduced my entry in the Talk page should give you a hint that I am willing to build, not to destroy. Should you have any questions regarding my entry there, please do not hesitate to contact me. Best regards, Jordissim (talk) 23:44, 28 May 2012 (UTC).

Martin, from now on I'd appreciate you use the User Talk page to send personal messages. Your last comment is for everyone to see, which is not right. And your first comment was aimed directly to my credibility as a source. I don't like the way you did things, and I don't think anyone would. The second time, and after recieving my message, why didn't you answer back to my User page same as I did? Your reaction was a bit over the top, especially in such a stupid matter, and in such a public place. Please make sure it doesn't happen again. Jordissim (talk) 01:28, 2 June 2012 (UTC)

I intended no criticism of you personally although I should point out that you are not a source but an editor. Reliable sources are externally published books, papers, and articles.

I responded to your comment on the Coriolis force talk page in the same page. I cannot really understand what you are getting at but you are free, like anyone else, to edit the article in whatever way you think will improve it. Why not give that a go to make clear exactly what you are trying to do. If anyone objects to your edits they will let you know.Martin Hogbin (talk) 08:42, 2 June 2012 (UTC)

Hi Martin, There's an ongoing discussion regarding the wording of the lead in the Ian Tomlinson article. SlimVirgin asked one of the editors previously involved in the discussion to if he would like to comment, but she seems to have forgotten to ask you. The discussion is here, if you'd like to share your views. Cheers, 87.113.118.169 (talk) 11:04, 22 June 2012 (UTC)

Hello there. As you expressed interest in hearing updates to my research in the dispute resolution survey that was done a few months ago, I just wanted to let you know that I am hosting an IRC office hours session this coming Saturday, 28th July at 19:00 UTC (approximately 12 hours from now). This will be located in the #wikimedia-office ^connect IRC channel - if you have not participated in an IRC discussion before you can connect to IRC here.

Regards, User:Szhang (WMF) (talk) 07:03, 28 July 2012 (UTC)

I just wanted to thank you for coming in and helping with the Tom Van Flandern article. Please don't feel compelled to get it all done quickly, too many editors have burned themselves out on the TVF article. The fighting adds to the stress as well. I've been editing that article for almost 4 years now, so I have a lot of patience. I see you are a busy editor, so I urge you to take your time and don't let it burn you out and if you can hang in there we can make it a better article. Thanks again. Akuvar (talk) 15:56, 2 August 2012 (UTC)

Glad to be of assistance. I still have some difficulty in working out exactly what the difference is between yourself and Flaubert. You both seem to agree that TVF's contributions to science in certain areas is distinctly fringe so it seems to be just a matter of style that you disagree on. At least you both seem to have stopped throwing accusations and insults an one another. Martin Hogbin (talk) 16:52, 2 August 2012 (UTC)

Thanks for the revert at MHP. I am not up to speed on the topic, but it looked dodgy. On the other hand I couldn't let an edit made by a ban-evading sockpuppet stand. I was glad to see someone sane stepping in to decide what version was best. Cheers! --Guy Macon (talk)

No problem. Who knows what is best. The point is that there was no consensus for such major changes to the article. Martin Hogbin (talk) 20:12, 5 August 2012 (UTC)

(Posted to Martin and Rick's talk page) Please check the RfC below for errors and suggest changes as needed. I did some minor copyediting for clarity, so let me know if you think the old version was better. In particular, look at my solution to the question of the yellow highlighted sections, and double check to see that I started with the correct version. If you wish, I can preload your comments before posting the RfC. -Guy M.

---

{{rfc|sci}} <<-- (this gets uncommented when we go live. -Guy M.)

The aim of this RfC is to resolve a longstanding and ongoing conflict involving multiple editors concerning the relative importance and prominence within the Monty Hall Problem article of the 'simple' and the more complex 'conditional' solutions to the problem. The 'simple' solutions do not consider which specific door the host opens to reveal a goat (see examples here and here). The 'conditional' solutions use conditional probability to solve the problem in the case that the host has opened a specific door to reveal a goat (see example here).

One group of editors considers that the 'simple' solutions are perfectly correct and easier to understand and that the more complex, 'conditional' solutions are an unimportant academic extension to the problem.

The other group believes that the 'simple' solutions are essentially incomplete or do not answer the question as posed and that the 'conditional' solutions are necessary to solve the problem. Both sides claim sources support their views.

That argument is unlikely to ever be resolved but two proposals have been made to resolve the dispute. Both proposals aim to give equal prominence and weight to the two types of solution.

One of the points of contention is whether either of the proposals below violates any Wikipedia policies and guidelines (in particular WP:NPOV, WP:NOR, WP:V, WP:WEIGHT, WP:EP, MOS:JARGON, WP:MOSINTRO, WP:MTAA and WP:OPINION). See the individual editor's comments below for arguments on both sides of this issue.

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

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

Considering all Wikipedia policies and guidelines, do should tye Montey hallProblem page be edited according to Proposal 1, Proposal 2, or neither?

Note: Because prior attempts to resolve this conflict have resulted in long discussions with many endless back and forth comments, please place any responses to other editor's comments in your own "Comments from user X" section and limit your comments to no more than 500 words. If you wish to have a threaded discussion, feel free to start a new section on this talk page but outside of this RfC.

User 1's comments go here.

User 2's comments go here.

User 1's comments go here.

User 2's comments go here.

Please create a new section or two if you use up the last one.

That looks good to me. If I understand it, you are suggesting that all respondents, including long-term editors such as Rick and myself, restrict their responses to 500 words.

Possibly you might toughen up, 'If you wish to have a threaded discussion, feel free to start a new section on this talk page but outside of this RfC', by suggesting that anything over the 500 words should be in the respondents own user space, otherwise I can see the discussion spilling out to new sections on the MHP talk page 'outside the RfC' if you see what I mean. You might also perhaps state that we do not want new proposals unless and until we get a 'neither' response to this one. Martin Hogbin (talk) 12:49, 2 September 2012 (UTC)

I fully sympathize with Martin's concerns about argumentum verbosium, argumentum ad nauseam, and wandering off topic. If you want to limit side discussions, perhaps you could place the RfC on its own sub-page. I don't think it is appropriate for those requesting comment to declare a moratorium on all other discussion of the article. Also, within the RfC itself, I don't think it is appropriate to prohibit comments from mentioning alternatives when critiquing these proposals. It is an important component of cogent and constructive discussion, and this should not, IMO, be "just a vote." ~ Ningauble (talk) 17:27, 2 September 2012 (UTC)

After making some suggestions I am happy to leave all the details at the discretion of Guy so that we can get on with it. Martin Hogbin (talk) 17:46, 2 September 2012 (UTC)

I am going to place it in the article talk page (more visibility), encourage rather than discourage side discussions outside of the RfC (no real cost, could be enlightening), and if someone wants to use his 500 words on a counterproposal, that is a Good Thing -- it might very well lead to another RfC based on the counterproposal. --Guy Macon (talk) 01:50, 3 September 2012 (UTC)

Welcome to the first edition of The Olive Branch. This will be a place to semi-regularly update editors active in dispute resolution (DR) about some of the most important issues, advances, and challenges in the area. You were delivered this update because you are active in DR, but if you would prefer not to receive any future mailing, just add your name to this page.

In this issue:

• Background: A brief overview of the DR ecosystem.
• Research: The most recent DR data
• Survey results: Highlights from Steven Zhang's April 2012 survey
• Activity analysis: Where DR happened, broken down by the top DR forums
• DR Noticeboard comparison: How the newest DR forum has progressed between May and August
• Discussion update: Checking up on the Wikiquette Assistance close debate
• Proposal: It's time to close the Geopolitical, ethnic, and religious conflicts noticeboard. Agree or disagree?

--The Olive Branch 19:16, 4 September 2012 (UTC)

Because of your previous participation at Monty Hall problem, I am inviting you to comment on the following RfC:

Talk:Monty Hall problem#Conditional or Simple solutions for the Monty Hall problem?

--Guy Macon (talk) 22:22, 6 September 2012 (UTC)

@Martin: I hope this will open your eyes. The combining doors solution is one of the worst explanations, as the 2/3 probability for the combined doors is no more than the sum of both the 1/3 for each separate door. How will any form of reasoning explain simultaneously the 2/3 - 0 distribution for the remaining and opened door? You see, without some notion of probability before and probability after, you're stuck with the fact that like each door also the door opened by the host has probability 1/3 on the car, and yet shows a goat. Nijdam (talk) 08:29, 16 September 2012 (UTC)

Nijdam, I do not need my eyes opening because I do see the problem. Before the host has revealed a goat, each door has a probability of 1/3 of hiding the car. After the host has revealed the goat the probability that door 1 hides the car remains 1/3 but the probability that door 2 hides the car has changed to 2/3.

The host's actions give us no information that could make the posterior probability probability that the car is behind door 1 different (in value) from the prior probability because we know that the host can, with certainty, reveal a goat and that when he has a choice he chooses randomly.

The host actions obviously do change the probability that the car is behind door 3. We know that it is zero. We can therefore determine the (posterior) probability that the car is behind door 2.

Are we agreed so far? Martin Hogbin (talk) 10:36, 16 September 2012 (UTC)

Well, in so far that remains 1/3 for door 1 is not obvious and need some explanation. And then, it is quite clear that opening door 3 with a goat changes its probability on the car from 1/3 to 0. However for door 2 the probability was 1/3, but we only know now is will change to the complementary change for door 1. I.e either we reason in some way door 1 has again probability 1/3, or we calculate this, or we calculate the probability for door 2 directly. Are we agreed so fear? Nijdam (talk) 11:06, 16 September 2012 (UTC)

Yes, we start with the fact that the posterior probability of door hiding the car is 1/3 and the obvious fact that the posterior probability of door 3 hiding the car is 0 and from this we show that the probability that door 2 hides the car is 2/3, another step which is natural and obvious, we know the car is behind one of the doors..

Our disagreement is over the required degree of explanation required to assert that the posterior probability of door hiding the car is 1/3. I have a suggestion to solve that but first let me show the standard 'combining doors' solution. Do you assert that this solution is wrong?

1 Initially door 1 hides the car with probability 1/3

2 Initially door 2 hides the car with probability 1/3

3 Initially door 3 hides the car with probability 1/3

After the host has revealed a goat behind door 3:

4 Door 1 hides the car with probability 1/3

5 Door 3 hides the car with probability 0

6 The car must be behind a door.

7 Therefore door 2 hides the car with probability 2/3 Martin Hogbin (talk)

Well, step 4 tends to be considered as just the same as step 1. Many people, not only the ignorant layman, just don't differentiate between them. That's why I like to speak in step 4, 5 and 6 of new probability. It looks quite simple to compute the 2/3 chance for door 2, but the calculation of the new probability 1/3 for door 1 is in fact equivalent to the calculation of the 2/3 new probability for door 1. And in presenting it without mentioning, you're hiding the essential part away, and many people will not be aware of it. Nijdam (talk) 15:26, 16 September 2012 (UTC)

OK, so we agree on what we disagree about, step 4 which you say has insufficient explanation. But I should add that no step is actually incorrect.

So, here is my suggestion, we say something like, 'The host's action in revealing a goat does not change the probability that the car is behind door 1'. That is probably not sufficient for you but it does go some way to explaining an important step. But I think we would agree that it is not particularly easy to explain exactly why the host's action does not change the probability that the car is behind door 1 and in any case a more detailed explanation would detract from the simplicity of the solution.

Now we come to the other important point that puzzles most people when they first see this problem. Why does it matter that the host knows where the car is provided that he does reveal a goat? My point is that this question is easier to answer than the first one. We can say something along the lines that taking a random sample of the two available doors and finding a goat gives us information about what is behind the doors. It is more likely that a goat will be revealed in a random sample if there are two goats. We can then say that in the first case, no such information is revealed. This is still not a full explanation of the facts but I think it goes a long way towards it. I have no objection to covering the subject in more detail later in the article once the reader has got their head around the main facts.

Would you accept this? Martin Hogbin (talk) 16:59, 16 September 2012 (UTC)

Where do we disagree? Indeed is no step, with the right understanding, incorrect. And as you may have seen in my proposals,they show the same idea. iI's fine with me if we follow your suggestions. I do not quite catch your second point. He simply has to know where the car is in order to be sure to reveal a goat. Nothing to explain there, imo. Or do I miss something? Nijdam (talk) 18:58, 16 September 2012 (UTC)

If you are happy to include the 'combining doors' solution with just the comment 'The host's action in revealing a goat does not change the probability that the car is behind door 1' (I think we could cite Adams or Falk here) with no further explanation then we agree.

My second point is twofold. Firstly we must explain to our readers why it matters that the host must reveal a goat rather than just happens to. Many people are puzzled by the fact that if the host chooses a non-chosen door randomly and just happens to reveal a goat then there is no advantage in swapping. It might appear to our readers that the 'combining doors' solution might apply equally to this case but of course it does not.

Secondly, explaining why the 'combining doors' solution does not apply to the random host choice case allows us to add a bit more detail as to exactly why the probability remains 1/3 in the standard case.

My objection to disclaimers for the simple solutions has never been to assert that they are perfectly correct but that disclaimers will put people off following the explanations. Once readers have understood and been convinced that the answer is 2/3 we can discuss the validity of the simple solutions in detail. Martin Hogbin (talk) 19:11, 16 September 2012 (UTC)

No need for disclaimers. Just don't call a simple solution a *solution*. Call it an observation, insight, argument, or explanation. Or add the extra line (perhaps in parentheses) which completes the argument, when an argument can be easily fixed. The whole mess starts because people decided to create a contrast between what they see are simple solutions which are mathematically incomplete, and the one and only true mathematically correct solution via a lengthy computation of conditional probabilities from first principles (thus not even making use of Bayes theorem; instead reproving it in this special case!). Morgan et al were responsible for creating this false image.

Two examples. Simple argument 1. If you always switch you'll win the car 2/3 of the time (assuming the car is equally likely behind any of the three doors) is a perfectly true statement, and extremely helpful to people first meeting MHP. Add to it the observation than 2/3 can't be beaten and there is no longer any reason to consider staying in any situation at all. Simple argument 2 (Devlin fixed). The car is initially behind your door with probability 1/3 and this is not changed by opening a door. Being informed which door is opened can't change it either, by symmetry. Richard Gill (talk) 19:25, 16 September 2012 (UTC)

Adding the extra line to Devlin's solution is exactly what I am proposing. My objection has always been to anything which would put off the reader, such as saying that this solution is incomplete or answers the wrong question. To someone who is struggling to understand the 2/3 answer, a statement that the explanation that you have just given them is wrong is extremely off-putting. Martin Hogbin (talk) 22:45, 16 September 2012 (UTC)

See how easy it is to come to a mutual understanding! I do however prefer not to use the combined doors as an explanation. It is equivalent to the ordinary simple explanation which has less disadvantages. The combined doors way of reasoning has the disadvantage that a reader actually thinks that because together the chance is 2/3 (due to the 1/3 chance for each door) and the opened door has zero chance, THUS the remaining door must have 2/3 probability. We would be imposing a wrong way of reasoning. It's all about the same value for the chosen door, before and after opening a door. So people should have to understand that after a door is opened, the chance for the combined doors is 2/3. But that's the same as understanding that after a door is opened the remaining door has 2/3 chance. So, please, tell me why you're so keen on this explanation. I hope not because of the misleading idea of the combined probabilities. Nijdam (talk) 19:43, 16 September 2012 (UTC)

I think we need as many different explanations as possible, within the rules of WP. Different explanations work for different people. Martin Hogbin (talk) 22:45, 16 September 2012 (UTC)

Martin I've commented to you at Tree Shaping talk about the ref. Martin quote "?oygul removing references to arborsculpture again" What are you referring to, give me diffs. ?oygul (talk) 00:38, 23 September 2012 (UTC)

I'm disturbed by your tone of voice in your edit summary "Well blow me down if it is not ?oygul removing references to arborsculpture again." I've checked through my contributions and I'm still mystified by what you are referring to. Please provide diffs or I expect an apology. ?oygul (talk) 02:50, 24 September 2012 (UTC)

I explained on the talk page. As well as removing a link, you changed 'arborscuptor' to nurseryman and author'. You seem to be intent on removing references to the term 'arborsculpture'. Why is this? Martin Hogbin (talk) 14:19, 24 September 2012 (UTC)

No, you made a statement/question, to which I had already replied. Please read the Tree Shaping talk

What you have not explained is your tone of voice in your edit summary. Please explain or give a diff. ?oygul (talk) 04:31, 26 September 2012 (UTC)

Martin please don't revert my edits. Please discuss them at the talk page as Elonka told us to. ?oygul (talk) 03:18, 24 September 2012 (UTC)

?oygul the standard WP procedure is WP:BRD please have a read of this. In a nutshell, you make an edit, I revert it, then we discuss it and decide whther we want the dit or not. I suggest that we stick to this. Martin Hogbin (talk) 14:16, 24 September 2012 (UTC)

Thank you for this link, now I would appreciate if you would go and do the discussion part of this and not just the reverting. Example you have twice removed the same cited content at Richard Reames and have not once talked on the discussion page to explain why. [1]. ?oygul (talk) 04:40, 26 September 2012 (UTC)

Hi Martin. It seems we have a disagreement over the inclusion of the 7/7 bombings in the main London page. I am glad to see you've already done the right thing by raising it as an issue on the talk page. I will be posting on there if so if you wish to respond to my argument please do so there. Alright then, bye. Enjoy the rest of the weekend. --ThunderingTyphoons! (talk) 22:18, 29 September 2012 (UTC)

Just for amusement, here is my question. Using the K&W formulation and ignoring really bizarre possibilities (who is to ay what they are?) in what fraction of games will a player who initially chooses door 1 and who sees the host open door 3 to reveal a goat win if they switch?

The answer, if you accept Morgan's argument, can be anywhere between 1/2 and 1, even if the host opens a door chosen uniformly at random when he has a choice. Martin Hogbin (talk) 12:14, 16 September 2012 (UTC)

My first reaction is no, so what is the trick you want to show? Nijdam (talk) 15:28, 16 September 2012 (UTC)

You are not going to like it but it depends on how the host chooses his door. Would accept this as a method of door choice? Before the host opens a door, his assistant looks behind them and, if there are two goats he tosses a coin and leaves a marker behind that door to tell the host which door to open. The host then opens the door with the marker. Martin Hogbin (talk) 17:03, 16 September 2012 (UTC)

Sorry, Martin, I'm lost. What is the problem? Nijdam (talk) 19:01, 16 September 2012 (UTC)

Martin: K&W say that the host chooses completely at random when he has a choice. And that the car is hidden completely at random. So the answer to your question is 2/3. (By symmetry the door numbers are irrelevant). Richard Gill (talk) 19:12, 16 September 2012 (UTC)

Suppose the host chooses a door to open (when he has a choice) by the following method. When the player has originally chosen the car, the host always reveals the goat that he does in the Whitaker example (let us follow vos Savant's example here when she numbered the doors to help her explanation and call this goat 'Billy' and the other goat 'Nanny'). We then know that, if Billy is revealed the contestant will win by switching only half the time but, if Nanny is revealed, switching is a sure thing.

Note that by using this method the host opens the two unchosen doors uniformly and at random because we know that the car and goats are initially placed at random. Martin Hogbin (talk) 22:33, 16 September 2012 (UTC)

Clever. If the goats are distinguishable and the contestant knows about the relationship between the goat's names and the host's strategy, then his conditional probability of finding the car behind door 2 given that the host revealed Billy behind door 3 is different from his conditional probabillty given only he (the contestant) catches a glimpse of a goat (unidentified). This indeed shows that K&W are silently making further assumptions. On the other hand, the new information doesn't change the optimal strategy of the contestant: "always switch". Overall win-chance is 2/3, can't be improved, conditional probabilities not interesting (since never disfavour switching). Pity this is OR! Please publish! Richard Gill (talk) 07:13, 17 September 2012 (UTC)

It would be great fun for me to publish this but I might need your help as I have no academic affiliation. Martin Hogbin (talk) 13:30, 17 September 2012 (UTC)

Interesting, but the player has to tell the goats apart (and know of course of the host's strategy.Nijdam (talk) 09:56, 17 September 2012 (UTC)

Yes, that is quite right, as Richard says above, but it is no different from Morgan's argument. In that case the player must be able to tell the doors apart and must know the host's strategy. The only difference is that in the Morgan case the player can see both relevant doors but in my example the player sees only one goat. This difference though is completely insignificant since the doors were numbered arbitrarily by vos Savant. Martin Hogbin (talk) 10:33, 17 September 2012 (UTC)

Please pardon for one question, because I don't get this point. If in 1/3 the host got two goats to show, then (as per K&W):
1/2 host inevitably opens door 2 (inevitably 1/4 showing Billy and 1/4 showing Nanny), and
1/2 host inevitably opens door 3 (inevitably 1/4 showing Nanny and 1/4 showing Billy).
Do you suppose for these 4 constellations that *secrecy* regarding the car-hiding door couldn't have been regarded, resp. that the rate of probability to hide the car of the first selected door #1 (previously 1/3) – [by what?] – could have "been" changed? Gerhardvalentin (talk) 10:30, 17 September 2012 (UTC)

Yes, of course. I gave this example to show the weakness in Morgan's argument. As Richard and Nijdam have both said, my example requires the player to know the host's strategy, exactly as is the case in the Morgan scenario. In reality this is most unlikely since, as you say, in a real game show the host must not give the player any clue as to the location of the car.

Still some people here insist that, even if the host strategy is unknown (or the host is defined to choose randomly) we must still calculate the conditional probability, even though we know it will make no difference to the result. I am showing that, if you really want to be that pedantic, you should also consider the condition of which goat is shown. It is really just a more concrete example of 'the host sneezes' argument that we have both used to show the pointlessness of the Morgan's approach. Martin Hogbin (talk) 10:41, 17 September 2012 (UTC)

Makes no difference to what result? The result of an analysis of MHP is not a probability. The result of an analysis of MHP is a well-founded advice whether to switch or not, in any particular situation which the player can find themselves in.

I mean that we know that the probability (Bayesian) of winning by switching is not affected by the host sneezing, the words the host says, the host's choice of goat, or the host's choice of legal door, because none of these events convey any information to the reader.

An easy analysis demonstrates the following: if initially all doors are equally likely to hide the car, then someone who always switches wins the car 2/3 of the time and it is not possible to do better. Hence no need to compute a conditional probability at all. Without computing any conditional probability, we know it cannot favour "staying" over "switching". If the player is happy with our assumption (which only pertains to the initial location of the car, not to the host's strategy), then we have a well-founded advice for him. Switch. You cannot do better.

I think a lot of the endless discussions comes from the fact that many editors and many sources think that the problem has to be solved via a probability. And then the fights start: which probability? But we have got to advise the competitor which action to take. Probability is a means to determining a wise action. But not the only means. Richard Gill (talk) 12:47, 17 September 2012 (UTC)

I agree that probability is not the only way to solve the stated problem but just saying that you should switch undervalues the problem. The reader may not realise that switching doubles your chances of winning and they might assume that there is only a very slight advantage in switching due to some technicality. The MHP is not like the infamous TEP where, slight and somewhat theoretical advantages might be claimed in some cases, it is a clear win for the switcher; he doubles his chances of winning. Martin Hogbin (talk) 13:30, 17 September 2012 (UTC)

Yes, you double your overall chance of winning by always switching ... split out over different cases this might be sometimes as low as half but that has to be compensated by higher chances, as high as 1, since overall you get 2/3. As I said: always switching gives you an overall success rate of 2/3, and that is the best you can do. Both statements have very very simple proofs. No need for Bayes, and no need for any discussion of host bias, since in order for these two statements to be true, it is sufficient to assume that each door has a 1/3 chance of hiding the car. Richard Gill (talk) 16:30, 17 September 2012 (UTC)

I am puzzled that you refer to 'overall' chances and continue to mention the Morgan probabilities of 1/2 and 1. With the standard assumptions, the conditional probability (given any legal choice of doors) is always exactly 2/3. Once you allow deviation from the standard rules, the probability of winning by switching can be any number you like. In the Morgan case and my example above this ranges from 1/2 to 1 but, with a little imagination, you can make it any value. For example the vS/Whitaker statement tells us that the host says the word 'pick'. Maybe he only uses this word this when you have chosen the car, so your chances of winning by switching are 0. Maybe you consider this a bizarre and unrealistic scenario but it is no more so than Morgan's suggestion that the host prefers a particular door.

Just saying that you should switch is a cop out (and not even true if you are really awkward). Even Morgan now say that, with the conditions implicit in the question, the answer is 2/3. Martin Hogbin (talk) 16:49, 17 September 2012 (UTC)

I do not make the standard assumptions. Why should I? And saying you should switch is not a cop out: I told you exactly why you should switch. Because it's optimal. An nothing else is optimal. No need to worry about specific wording of Monty.

I'm not interested in the conditional probabilities because I can't know them. 2/3 is the overall chance and that's all I need to care about, since once I have got it as large as it can be, conditional probabilities in special cases are irrelevant.

But I do find precision important, so that'w why I say "overall chance" and not "chance" Richard Gill (talk) 17:20, 17 September 2012 (UTC)

PS choose your door initially completely at random and you'll walk away with the car two times out of three. Guaranteed. There is no better guarantee. Monty Hall could send you signals by his choice of words which you might be able to use to improve that. But he could also use such signals to mislead you and make your success rate fall below 2/3. Note: it is indeed all about 2/3, and that 2/3 is twice 1/3; quite some improvement. But it is not all about conditional chances. Strategic thinking shows us that the conditional probabilities are unimportant. A red herring. Richard Gill (talk) 17:39, 17 September 2012 (UTC)

I think that it is not really the MHP without the standard assumptions. You must make a subset of the standard assumptions or there is no solution at all. For example, the car must be positioned at random or the player must choose at random. What are the assumptions of your game and why are they the 'right' ones? Martin Hogbin (talk) 17:41, 17 September 2012 (UTC)

Read my Statistica Neerlandica paper! MHP is (for me, and I believe for Wikipedia too) vos Savant's question with minimum disambiguation (the addition of the line "Monty Hall knows where the car is hidden so not only can he always open a door revealing a goat, we are told that he always does open a door revealing a goat"). Fact is, different reliable sources give different solutions, relying on different assumptions. The more assumptions you are prepared to make, the stronger your conclusion can be, but the more limited it is (to the situation when those assumptions are all valid). Up to the "consumer" to decide what assumption set he is prepared to buy. Moreover up to the consumer to decide how he wants to understand "probability". Different meanings of probability give different meanings to the assumptions and different meanings to the conclusions. It's up to the mathematician to present the consumer with a menu of assumption and conclusion sets. By the way I'd prefer to call the K&W conditions not "standard" but "conventional". The article has to pay these conditions some attention, but there are plenty of decent solutions which do not require the full set of K&W assumptions. For instance the game theoretic minimax solution for the contestant: "pick your initial door completely at random and then switch". Lots of amateurs come up spontaneously with this solution. It's discussed in the literature. No assumptions at all how the car is hidden and how the host makes his choice. Richard Gill (talk) 11:40, 18 September 2012 (UTC)

As usual, I do not think we disagree about anything significant. At one extreme, the question exactly as asked is insoluble and at the other, the K&W (now with some strengthening) or conventional formulation as you prefer to call it, the answer is exactly 2/3 for all legal door combinations. The conventional interpretation was , I am sure, the intention of Whitaker and, of course, Selvin and, in my opinion it is the only interpretation in the spirit of a simple puzzle. Certainly most sources explicitly or tacitly assume those conditions.

You seem to forget our earlier conversations where we agreed long ago that to choose randomly and switch was a good plan.

The thing I do object to is the continual mention of Morgan's probabilities of 1 and 1/2. They only apply given a very specific, and somewhat contrived, set of assumptions. Firstly we need to assume that the conditional probability, given a specific set of doors is required. Then, in addition, we need to assume that the producer places the car randomly but the host does not choose a legal door randomly, an unlikely combination for any real game show.

In my opinion, the only formulation that stands out from all the others is the conventional formulation. People are quite free to solve other formulations but they should state exactly the assumptions that they are choosing to make. You may do that but some others do not. Martin Hogbin (talk) 14:27, 18 September 2012 (UTC)

Everybody should state the assumptions that they make. And people should only state the assumptions that they use. If you state assumptions but do not use them then there is obviously a gap in your understanding and/or in your arguments.

I hate the word "solution". I would like to know, however, if the following three sentences qualify for you as a "simple solution": Suppose all doors are initially equally likely to hide the car, and that Monty is equally likely to open either door to reveal a goat, if he has a choice. By symmetry, specific door numbers are now irrelevant to the problem. The player's initial choice hides the car with probability 1/3, and the other door left closed by the host therefore has probability 2/3 to hide the car. Richard Gill (talk) 12:29, 20 September 2012 (UTC)

I have no special love for the word 'solution', if you want to call them all explanations or whatever that would be fine with me. I would not be so happy with calling the simple ones 'explanations' and the conditional one 'solutions' but I could live with that if it was done unobtrusively so as not to give the reader the impression that the explanations were not 'correct' in some way.

For the purposes of the article, I would call you three sentences simple but not unfettered, even though there is no mention of conditional probability. My reasons are are given below and they are based on the likely thoughts of someone who sees this problem for the first time. It is far too easy to develop a certain arrogance concerning the MHP, we are all so familiar with it that we cannot put ourselves in the position of seeing it for the first time. Intelligent and well-motivated people still get it wrong and are often willing to argue persistently for the wrong answer. Anything which detracts from their understanding of the correct answer is a distinct negative.

I understand what you are trying to do and agree in principle that we should try to make all explanations in the article as mathematically sound as possible but, I would say that initially this should not be at the expense of clarity. We can always explain the details later.

Here is a simple explanation that I would be happy with. I guess you would too. It unobtrusively avoids inaccuracies.

Now back to your suggestion. This is what I do not like about it:

Suppose all doors are initially equally likely to hide the car, and that Monty is equally likely to open either door to reveal a goat, if he has a choice.

I think this sentence is superfluous, but I do not object too strongly. Of course the stated conditions are necessary for the following explanation to be solid but most people naturally assume that all doors are initially equally likely to hide the car and do not even consider the way that the host chooses legal doors to open. As it happens these common assumptions turn out to be the ones required for your solution.

By symmetry, specific door numbers are now irrelevant to the problem.

This is going to be confusing to most general readers. What is meant by 'symmetry' (I know what you mean)? What do you mean by, 'specific door numbers are now irrelevant', are you suggesting that the host might have opened door 1? This is what the general reader is likely to make of this.

The player's initial choice hides the car with probability 1/3, and the other door left closed by the host therefore has probability 2/3 to hide the car

This is indeed a statement of fact but I am not sure that i would call it an explanation. Martin Hogbin (talk) 09:55, 21 September 2012 (UTC)

Pardon for the question how this could be improved:
"Suppose all doors are initially equally likely to hide the car in this one-time problem.
The door selected by the player has 1/3 chance, and the group of unselected doors, although there is at least one goat behind, has 2/3 chance to hide the car. And Monty, observing secrecy regarding the car-hiding door, is equally likely to open either of his two doors if he should have a choice, in intentionally having to reveal a goat. By this *symmetry* it is unimportant which one of his doors the host has opened, specific door numbers is irrelevant. The player's initial choice still hides the car with probability 1/3, and the other door left closed by the host therefore has probability 2/3 to hide the car in this one-time problem."
Better suggestions? Thank you! Gerhardvalentin (talk) 12:58, 21 September 2012 (UTC)

(re: the original claim) Oh come on! When K&W say that the host chooses randomly, of course they mean that he does the equivalent of secretly flipping a fair coin on the spot (a "mental coin flip" if you will). That's how people take it, and that's how it's meant. There is no problem here. If it's really necessary, then instead of saying "he chooses one [uniformly] at random" we could drop the "[uniformly]" from the quote and add a clarification that "random" means e.g. secretly using a fair coin, like in the Three Prisoners problem, which is considered equivalent to the MHP.

The K&W wording is suggestive enough, I think. Your suggested misconception that the readers might have is very far-fetched. The host being forced to open a particular door is not "host choosing randomly" in the ordinary meaning of the words. It's neither a choice nor random to the host, not at that point in the story anyway, even if you could argue that the host has chosen that door before the show by rolling a fair goat and so it was a random choice at that time. The text says chooses, not has chosen in advance. Further, unlike with the coin flip, the goat roll's result is not kept secret.

Of course you can argue that K&W is ambiguous if you wish. It doesn't even stipulate that the host doesn't open all the doors right before you choose whether to switch, or that the host doesn't shoot all switchers. Maybe he does?! After some point, though, you have to acknowledge that all reasonable points of contention/confusion have been dealt with. I think in practice K&W succeeds at that. -- Coffee2theorems (talk) 18:00, 30 September 2012 (UTC)

Random is random and it makes no difference whether the host tosses a coin when he finds he has a choice of goats or takes the next card from a stack showing the result of a series of coin tosses that he did at the start of the year or he uses the random process used by the producer to place the car and goats to choose which door to open. The point is that the player has no way to predict what the host's choice of door will be.

My main intention though was to draw attention to the arbitrary nature of the Morgan analysis in which the host's choice of legal door to open is artificially made a critical condition of the problem. We could just as easily make an issue out of which goat the host revealed. We know that there are two goats and we are told that the Monty reveals one of them. If he, in fact, has a preference for the goat that he actually did reveal then the probability of winning by switching is only 1/2 just as is the case if Monty has a preference for door 3.

The Morgan solution and concentration on the door opened by the host is nothing but a conjuring trick, an arbitrary selection of an event that occurs between the player choosing a door and his making a decision to swap or not.

I defy you to give me any good reason why a 'complete' solution must take account of the door number that the host opens but not the goat that he reveals. Martin Hogbin (talk) 19:58, 30 September 2012 (UTC)

If you start tossing around contrived interpretations of K&W to ordinary people, many (but not all) will acknowledge that it's still "host choosing randomly" if the host flipped a coin the day before and memorized the result (say, "heads, left door; tails, right door"), even though the host now knows what the result is and it is no longer unpredictable to him (or random to a strict objectivist, for that matter). They may even acknowledge that any esoteric interpretation of K&W you can come up with that can be shown to be equivalent to a secret coin flip on the spot is also possible (maybe a stretch, but not beyond the breaking point). But when you get to something that is provably different, it's a creative reading on par with assuming a host who shoots switchers (or "but I want a goat!"), i.e. interesting and amusing but clearly not what is meant.

"He chooses randomly" is code for "the choice is uniform and physically independent of everything else". Not just statistically independent, mind you, like "what's behind your door is independent of the host's choice of door" (difficult to understand); but physically independent, like "what happens on one side of the universe is independent of what happens on the other side" (easy to understand). Unlike much of probabilistic code-speak, this is a wording that actually gets through basically ungarbled to everyone! (even the utter pedant who adds "[uniformly and independently of everything else]" is not actually confused, they are just sticklers for "proper use of language") This wording gets everyone on the same page.

I argued about possible confusion about K&W wording above, because that's the context where you pointed me to this discussion (and the name of this section!). The "Morgan/Hogbin issue" you speak of is an entirely different kettle of fish. With the K&W wording, the host's choice among the two remaining goat doors is independent of everything, be it door numbers or goat names. It doesn't matter whether you are Morgan or Hogbin; K&W want to specify things so that you agree! That's the point of it. In symbols, (the statistically relevant part of) the meaning is:

P(host chooses door x | you didn't pick x, you picked the car door, LeftDoorNumber=1, MiddleDoorNumber=2, RightDoorNumber=3, RevealedGoatName=Billy, everything the host or even God knows) = P(host chooses door x | you didn't pick x, you picked the car door) = 1/2.

The choice is totally random no matter what; no ifs, no buts, no maybes. -- Coffee2theorems (talk) 10:43, 1 October 2012 (UTC)

I started by saying that this was, 'Just for amusement' and as I said above I was only drawing attention to the obsession with door numbers that some people seem to have.

Regarding your conditional probability statement above there is a much better way of defining the host's action to reflect the question's original intent and that is to say that nothing he does that changes the probability that the car is behind the originally chosen door. The whole door choice thing is a red herring, an arbitrary complication, conjured up by Morgan et al and which now somehow seems to have become an essential part of the problem. As you say above, and many others here have commented, there are many possible events that might be considered essential conditions of the MHP, the words the host says or the goat he reveals but it clearly was not the intention of either of the main problem setters to consider ways that the host might change the probability that the car is behind the originally chosen door.

By the way, I am still waiting for someone to show me how a solution that has as a condition the door chosen by the host is any more 'complete' or 'answers the question as asked' any better then the simple solutions.

You might also show me how the proposed conditional solutions prove that the probability that the car is behind the originally chosen door remains 1/3. Martin Hogbin (talk) 14:48, 1 October 2012 (UTC)

OK, just for amusement, then :) The following is an unabashedly Bayesian take on the issue in the context of the vos Savant version (not K&W; no "unbiased host" assumption).

The goats are not conditioned on, because the goats are assumed to be indistinguishable. Poor goats! :) Treated like two exactly identical marbles placed exactly at the centers of their respective rooms behind the doors! If the problem had "a red pill and a blue pill" instead of "two goats", revealed pill color probably would be conditioned on, just like spatial door location (or its proxy, door number) is. Even if it were "a pair of shoes", Morgan et al. might have conditioned on it being the left shoe vs. the right shoe (but probably not if it's a pair of socks, even though those also have minute differences!). In the end it's about what people notice, and that affects what they would still consider an instance of "the MHP".

A lot of this silliness is cut out if the vos Savant version is slightly modified like this:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, nothing. You point at a door, but the door is not opened. The host, who knows what's behind the doors and is not allowed to reveal the car, opens another door, showing no car. He then says to you, "Do you want to switch to the remaining door?" Is it to your advantage to switch your choice, given that you know the rules of the game?

In my view, this is equivalent for most useful purposes. This version does away with door numbers and goats, which are both superfluous. In contrast, one cannot do away with the distinguishability of doors without resulting in something that is, in my view, no longer the MHP. This isn't just me, either: many illustrations, tables, simulations and solutions have indistinguishable goats, but few have indistinguishable doors (vos Savant even suggested a simulation without goats, but with spatially distinguishable doors). A simulation with indistinguishable doors would be e.g. one where you have a red marble and two blue marbles in a bag, you blindly draw one but keep your hand closed around it, the host looks in the bag and pulls a blue one out, and then you decide to either take the one in your hand or the one in the bag. I doubt this simulation would convince as many people; it's not "the MHP".

An "objective" Bayesian would argue that because nothing special is told about any of the doors in the problem statement, any objective rational belief would have to be symmetric wrt door identities, so switching wins with probability 2/3. A subjective Bayesian believing that everyone else plays the game optimally (or who agrees with the objective Bayesian about the use of symmetries being one honking great idea) will basically agree. Other subjective Bayesians will have different answers.

In any case, a Bayesian analysis of MHP has the following steps:

1. Build a model of the situation. The obvious one has distinguishable doors.
2. When computing probabilities, condition on all information available to the model's player at the decision point.
3. Use the probabilities to find out the optimal decision.

Conditioning on all available information is axiomatic to the approach. Of course conditioning on "all available information" is only possible within a model of reality, not reality itself, and you are left with the question of what the model should look like (step 1). In my view, it should look like reality as much as practical. In this case, having distinguishable doors in the model is no hardship, and is more clearly the same thing as described in the problem statement than a model with indistinguishable doors would be. -- Coffee2theorems (talk) 01:58, 2 October 2012 (UTC)

That's a very good explanation, Coffee2theorems. I agree with you entirely about the distinction between a problem with distinguishable and indistinguishable doors. You say: conditioning on all available information is axiomatic to the approach. Yes. Except: information that is irrelevant need not be conditioned on. Irrelevance is operationalized by probabilitistic independence. In the objective Bayesian approach, the problem solver can point out that the problem can be separated into two parts. We have the four roles of the doors: chosen by player, opened by host, left closed by host, and hiding the car. The simple solutions tells us the car chosen by the player has 1/3 chance to hide the car and the car left closed by the host has 2/3 chance to hide the car. Symmetry tells us that all possible ways to assign numbers to these doors are equally likely. Hence in probability terms, the numbers on the doors are independent of the roles of the doors. Hence conditioning on the numbers is a waste of time.

The symmetry was pointed out already by Bell in the discussion of Morgan et al. Yours truly has written about it in several publications. One of the German textbooks uses it right from the start by numbering the doors according to function, i.e., ignoring the actual numbering.

Yes, some sources and some wikipedia editors have an obsession with door numbers. Deal with them (shut them up) by giving a short snappy mathematically rigorous argument why the door numbers are irrelevant, as most lay persons know intuitively! And as quite a few reliable sources already pointed out. Richard Gill (talk) 17:08, 8 October 2012 (UTC)

The point is that once you accept distinguishable doors into your model (step (1)), the irrelevance of the door numbers has irrevocably become something to be proved. Based on your comment, it seems that you acknowledge that. Then it becomes an issue of how to do it, and I don't think the conditionalists are opposed to any valid proofs. What they are opposed is to (1) having indistinguishable doors in your model, and/or (2) omitting any necessary proof (at least without comment). These are very different things, and conflating them has often muddied discussions.

You may note that I applied symmetry in my explanation; that's because it is, one way or another, needed in the objective Bayesian approach with the vague vos Savant version. My preferred way is to note that symmetry leads to the uniform priors of K&W (noting that I have a consistent model) and then simply apply Bayes' rule. Which way of proving the same thing is the bee's knees for a mathematician is IMO more than a bit silly thing to argue about for something this trivial.

More important is which approach(es) is/are the best for presenting to the reader. There's something to be said for directly using the definition conditional probability: As you can see from the illustration in my proposal, every piece of reasoning can be understood and given a common sense justification. It is a complete proof, and understandable to anyone, with nothing that could leave any doubt. It does not take a Kolmogorov-fearing statistician to be convinced by it and to see how belief update is done properly and why (instead of going "new situation, new uniform distribution!").

Why are the two cases not 50:50? People have in their heads some funny Kolmogorov-incompatible idea of how to do belief updates. If we do not give them a generalizable way of reasoning to replace it, how can we expect them to (emotionally) let go of a rule (i.e. to realize it's not perfect) that has served them well? It's simple enough to do: simulate, and look at what happens on average in the exact situation in question. That always works (for some reasonable values of "always" and "works"). Or reason what would happen in that simulation, as in the illustration. People are very good at generalizing from examples, so I think the one illustration is quite enough for that purpose. Leaving it essentially at "your way is bad because it makes Kolmogorov very very sad" is not the best we can do. -- Coffee2theorems (talk) 07:40, 9 October 2012 (UTC)

I am not sure what you mean by 'distinguishable doors'. All doors are, in principle, distinguishable, as are all goats. The question is whether we choose to distinguish between them, and that decision can made only on the basis of whether it matters whether we do or not. In the case of the MHP it is obvious (to most people) that in any reasonable interpretation of the game, we do not need to distinguish between the doors that the host might legally open or the goats that he might reveal. Martin Hogbin (talk) 22:29, 9 October 2012 (UTC)

I take it that you mean a solution that conditions on "door 2 or door 3", instead of "door 3", and are saying that it is obvious to to most people that these are the same thing. In that case, you might like an explanation like e.g.:

If there is to be a definite solution at all, the answer must be the same no matter how the doors are numbered in the problem statement. In particular, the probability of winning by switching has to be the same whether the host opened door 2 or door 3. The door number therefore must be irrelevant (the answer does not depend on it), and we may pretend that the player only knows that one of the doors was opened, but not which.

When the host opens one of his doors (door 2 or door 3) to reveal a goat, no new information is revealed: the player already knew the host would do that. As no new information is revealed, the chances of the car being behind one of the host's doors are still 2/3. The car cannot be behind the opened door, so the chances of it being behind the remaining door must be 2/3.

Is this what you mean? -- Coffee2theorems (talk) 06:07, 10 October 2012 (UTC)

Not really. What you say is fine but why do we also not need to say?

'If there is to be a definite solution at all, the answer must be the same whichever goat the host chooses to reveal. In particular, the probability of winning by switching has to be the same whichever goat the host has revealed. The specific goat revealed therefore must be irrelevant (the answer does not depend on it), and we may pretend that the player only knows that one of the goats was shown, but not which.'

I can see why, superficially one might think that your statement is required but my one above is not, vS chose to give the doors numbers but did not give the goats numbers (or names). Is that your logic for insisting that we explain why door number is irrelevant but do not mention goats? Martin Hogbin (talk) 08:13, 10 October 2012 (UTC)

No, it's not that superficial. As I said, IMO for most useful intents and purposes, the problem can be stated without door numbers. What the player sees is door locations. Left, middle, right door (or so we illustrate). He picks a door, and needs to do it somehow, so some easily seen distinction exists. Maybe it's top, middle and bottom door; it doesn't matter. Easily distinguishable. The doors themselves may be exactly identical. Maybe they are doors on a huge computer screen instead of real doors, and are pixel-by-pixel identical. Doesn't matter. So what about the goats? Well, they could be pixel-by-pixel identical, too. Indistinguishable. Sure, you can say that they are distinguishable by location, like the doors, and fair enough; but that isn't sufficient by its lonesome (you can't make your initial pick by saying "I pick goat A" or something) nor does it buy you anything extra (the revealed goat is exactly where the open door is; revealed goat location = opened door location, same variable). So you end up conditioning on door locations. Or door numbers, which stand for door locations; "door 1" is just an unnecessary synonym for "the leftmost door" and so on. Well, unnecessary for the player anyway. If we neutrally say "door 1" then the analysis more clearly applies whether it's left/middle/right or top/middle/bottom or front/middle/back or whatever.

Of course you could assume that the goats are distinguishable, if you wish. Maybe they have big stickers "A" and "B" on them, and the player knows that in advance. Then "A" and "B" are like the door locations. But most people don't think that way. A goat is a goat is a goat to most people. You'd have to be some kind of goat aficionado to notice gender or whatever. Most people don't cry foul even when the goats are removed entirely (like in vos Savant's pea/shell simulation). Making the doors indistinguishable by location is a more drastic change. Pragmatically, how would you even illustrate that without risking confusion? You'd end up having spatially distinguishable doors in an image along with an explanation like "imagine that you cannot distinguish the doors here" (??) or "imagine you are blind" or something. If you try to remove door identities this way, to many it feels like you're changing the problem, and would have to give some kind of reason why it's still the same thing. And for what purpose? It's a simple enough problem even with distinguishable doors. Also, it's easier to suspect that the door locations could matter somehow, be entangled with the final result somehow, because they are not as easily removed from the problem (the door locations are less superficial to the problem than the easily removed goats, which are there just as a gag anyway). -- Coffee2theorems (talk) 17:10, 10 October 2012 (UTC)

You have given me no fundamental reason why we should distinguish between the possible doors opened by the host rather than possible goats revealed by the host. It is the host who must distinguish between them, not the player. The somewhat perverse argument given by Morgan et al is that the host might have a preference for one of the doors and always open it when possible. From a frequentist perspective it makes no difference whether the player distinguishes between the doors or not. It is no more perverse (in fact, less IMO) to assume that the host might have a favourite goat, which he reveals whenever possible. Martin Hogbin (talk) 21:33, 10 October 2012 (UTC)

The explanation was from the Bayesian POV, which is "subjective"/epistemic. From that POV, what matters is the player's state of knowledge (the known facts) and his beliefs (the probabilities), not the host's. The host may pick his door completely deterministically using the phase of the Moon or his astrological charts or the digits of pi. What matters is that the player doesn't hold such beliefs.

I'm not a frequentist, but I can understand Richard's POV: Assume nothing, choose your initial door by a fair die, switch, and get 2/3 unconditional objective probability of winning. All objective conditional probabilities are completely unknown; the only guaranteed uniformly random quantity is your initial pick. That makes perfect sense. The problem is solved, but you can't say anything about the objective probabilities of the car being behind the two remaining closed doors, except that they are both between 0 and 1 and sum to 1.

I think Richard's is the most sensible "frequentist"/"objectivist" position, and that any analysis getting you definite probabilities for the two remaining closed doors makes far more sense from the Bayesian POV. The only reason you want to compute these probabilities at all is that people think they must be 1:1. That's wrong from any POV. Richard might say that the "1:1" part is right, but the "must" part is wrong. The objective probabilities could be 1:1, but that is unknowable; they could just as well be anything. No "must" about it. A Bayesian, on the other hand, will say that the "must" part is right but the "1:1" part is wrong; it must be "1:2" instead because those are the epistemic probabilities that result from your ignorance of initial car location and host's choice. Why do people think it's "1:1"? Because they use an ignorance argument about the car location twice. People are trying to assign epistemic probabilities, they're just doing it wrong to the point of inconsistency. The objective POV is not particularly useful to fix/explain an epistemic error.

Some people assume uniform objective probabilities for initial car placement and/or host choice. These assumptions are not necessary for solving the problem; Richard does not need them, nor does a Bayesian. Why assume such things? Because people want the probabilities for the two doors. They don't just want to solve the problem, they want to resolve the paradox. But still, why assume such things, when the paradox is Bayesian in nature and a Bayesian does not need these assumptions? Well, pragmatically, it avoids having to discuss philosophies of probability. It makes the Bayesian analysis correct no matter what you think "probability" means or whether you buy Bayesianism or not. That makes the problem accessible/palatable to a wider range of people. If you're a frequentist or if you simply want to present the problem to people who might be confused/bugged by Bayesian probabilities, you end up tacking on these assumptions. Most people don't think that changes the problem, so it's no hardship. These assumptions are just a crutch, if a useful one. -- Coffee2theorems (talk) 12:48, 13 October 2012 (UTC)

Well said, coffee2theorems. Richard Gill (talk) 15:46, 31 October 2012 (UTC)

I am not a frequentist either, in fact I would describe myself as a strong Bayesian, so we can start by agreeing that, from a Bayesian perspective, we know nothing about the producer's original car placement, the player's door choice, the host's door preference, the host's goat preference, or anything else that might make us believe that the answer is not 2/3.

I also tend to agree with you explanation about why people get the answer wrong. They do not see the difference between all the above choices, about which we know absolutely nothing, and the final choice between two door, where we do know more that we might at first think. This is the very thing, in my opinion, that the article should be concentrating on, rather than bogus conditional probability arguments.

I also agree that there are many ways of approaching the problem, some of which might circumvent or render irrelevant some of the above choices. I think I was actually the first here to point out that, under the normal game rules, if the player chooses a door uniformly at random, the answer is 2/3, whatever the host does. (By 'answer' I always mean the probability that the player will win the car if they swap, after having seen a door opened to reveal a goat). There is therefore no need for you or Richard to remind me of this fact. There are also game theoretical approaches, which are also interesting.

The main problem I see is that much of the literature uses a vaguely frequentist approach to the problem, which they get wrong. From a frequentist perspective there are a number of unknown distributions that we must decide upon before we can answer the problem. These include the distribution with which the producer places the car, the distribution for the players initial door choice, the distribution for the host's goat choice, and the (connected) distribution for the host's legal door choice. If we make no assumptions about any of these distributions the problem is simply insoluble. That is a logical but rather uninteresting position. If, on the other hand, we assume all these distributions to be uniform, then the answer is exactly 2/3, always. This is also a logical position, and one that gives a simple answer to the problem, in line with the Bayesian approach.

Once we stray from these two logical option we need to justify our decision. For example, making players' initial door choice uniform is attractive because it neatly gives us a definite answer of 2/3.

The Morgan decision, which forms the basis of the 'conditional' solution diagrams that some want to show, assumes that the initial car placement is uniform, ignores the players initial door choice (and any potential goat preference that the host may have) but envisages that the host's door preference might not be uniform. This is an arbitrary and unhelpful set of assumptions leading to a confusing diagram that explicitly shows the two doors that the host might have opened, but does not show the doors that the player might have chosen or the doors that might have hidden the car. The reason usually given for ignoring these possibilities is that they obviously make no difference, but neither does the host's door choice, unless we make a bizarre and confusing set of assumptions. This is Morgan's conjuring trickMartin Hogbin (talk) 10:22, 14 October 2012 (UTC)

I agree that the Morgan set of assumptions are strange whether from a Bayesian or from a frequentist perspective. Their contribution is a minor mathematical addition to the MHP saga. Do let's forget them. Let's focus on the natural Bayesian assumptions of uniform distributions on everything in sight, and the player's choice of Door 1 being fixed. Then we agree that (A) switching gives the car with probability 2/3, and that (B) switching gives the car with probability 2/3 whether the host happened to open Door 2 or Door 3. Statement B is stronger than Statement A. Statement B is the final conclusion of "conditional solutions". Statement A is the final conclusion of most "simple solutions". But it is also easy to go from from A to B by explicitly mentioning symmetry. So what the hell is the big deal? Sure, the tricky thing is getting the reader to understand A. That's what the newcomer struggles with coming to terms with. "Upgrading" from A to B might well be considered a superfluous optional extra. But on the other hand it is such a triviality that I don't see the point of trying to hide it from the readers. Some people will find it useful. It will keep the formal conditionalists quiet. At last people can start constructively and collaboratively editing the article again. Richard Gill (talk) 15:43, 31 October 2012 (UTC)

I am all in favour of forgetting the Morgan paper, but we need to also forget their legacy which was to ignore the door numbers when it comes to the players original door choice but not to ignore them when it comes to the host's door choice. Either we see a symmetry with respect to door number or we do not. If we do not, we must show all possible initial choices of the player in our diagrams, if we do ignore the door numbers then we should ignore them for both the players and the host's choice.

Why not have statement (C) switching gives the car with probability 2/3 whether the player picks door 1,2, or 3and whether the host happened to open Door 2 or Door 3? This is stronger still than B. That is what I mean by selective pedantry.

Why not (C)? Because people aren't interested in it. Some people are interested in (A), some are interested in (B). The authors of the sources, and the editors of wikipedia, and the readers of wikipedia, seem interested in either (1) "the chance of winning by switching" (initial choice random?), or (2) "the chance of winning by switching given the player chose door 1", or (3) "the chance of winning by switching given the player chose door 1 and the host opened door 3". These three chances might have the same numerical value 2/3 under certain conditions, but they mean three different things. My personal opinion is that it is somewhat a matter of taste which concept you find the most interesting/relevant. What I don't understand is why you seem to refuse to admit that these three things are three different things. It is not being pedantic to distinguish carefully between them. It is a matter of being careful and precise. MHP is a brain-teaser. It's a logical puzzle which we solve by being careful with logic. You want to be careless with logic. Richard Gill (talk) 15:13, 3 November 2012 (UTC)

I am not suggesting that we ignore these symmetries, just that we defer talking about them until later on in the article. Rather to my surprise, and without your support, it looks as though the result of the RfC is that there is a consensus for my proposal. We have the simple stuff first then the discussion of door numbers, goat ID and symmetry. Nothing is hidden from our readers and everyone can work together. Martin Hogbin (talk) 22:17, 31 October 2012 (UTC)

I am delighted with the conclusion of the RfC. It reached the conclusion which I hoped it would reach. After re-thinking my position (I sometimes change my mind after gaining new information) I decided that I could not 100% support either position, though I much prefer position 1 to position 2, so I'm glad it came out on top.

My personal opinion is that we should be trying to have synergy from the two kinds of solutions. I do not see them in opposition to one another. I do not think one kind is correct and the other incorrect. Both kinds have their advantages and disadvantages. Moreover there are simple bridges between them. I dislike both sloppy-thinking populism and pedantic academistry. But it seems that only myself, Boris Tsirelson, and Coffee2theorems have this opinion. Everyone else is either dogmatically in favour of conditional solutions or blind to the issues. Richard Gill (talk) 15:13, 3 November 2012 (UTC)

PS. I am afraid that because the article now will focus first on simple solutions, we will later get a big section arguing that the simple solutions are wrong and we need to carefully compute conditional probabilities from first principles. In other words, we perpetuate the idea that there is an opposition and that academics like one solution and think that the popular solutions are wrong. While if only we had been able to include a simple proof of a conditional solution in the simple part, this would not have occured. The article has not yet been saved from insanity and unimaginative pedantry. Richard Gill (talk) 15:23, 3 November 2012 (UTC)

The point is that I am consistently sloppy (as you call accepting the blindingly obvious without proof). You are sloppy about everything except one, the door number opened by the host. How do we know that the door number chosen by the player is unimportant (except for the fact that it is obvious)? How do we know that the ID of the goat revealed is unimportant (except for the fact that it is obvious)? If you are going to be pedantic you should be consistent.

Regarding RfC, I hope my proposal will enable all of us to work together in peace. We keep the sloppy/obvious simple solutions separate from the complete/pedantic ones but we still show them all. Some of us, like me, want to help the genral public to understand this famous puzzle, other may want to cover more angles and Martin Hogbin (talk) 00:18, 4 November 2012 (UTC)

By the way I am still waiting for someone to tell me the fundamental reason that goat ID is less important that door number. (I agree of course that it is best to ignore both). Martin Hogbin (talk) 22:20, 31 October 2012 (UTC)

I think we have both already given you a plethora of excellent reasons.

The three doors are distinguished in Vos Savant's question, in the real world they are distinguished. Left, middle, right as seen by the audience. The player chooses a door. He does not pick one of three indistingushable marbles. The goats are not distinguished. Anyone who draws a cartoon of MHP distinguishes doors, but not goats.

If you set up a computer simulation experiment to help people learn MHP you have to distinguish doors. You don't distinguish goats.

Psychologically the facts thats we distinguish doors by spatial location and the primacy of visual cognition are the reasons according to Kraus and Wang that people jump to and stick to the wrong answer. K and W write explicitly "less is more": understanding that the absolute door identies are irrelevant, what counts is only the roles they play vis a vis car, goats, actions of host and player, is the key to unblocking your mind and solving the problem.

The goats' names, sex etc are not part of the story. The door numbers are part of the story and their distinguishability is precisely what creates MHP.

Suppose the doors were indistinguishable. You'ld pivk one. The host takes one out of the game. There are rwo left. They're indistinguishable so you don't know which one was the one you first picked. So now you have to pick one anew at random. 50/50 chance of gettng the car. Richard Gill (talk) 08:09, 1 November 2012 (UTC)

You have indeed given me a plethora of reasons. That in itself is a sign of weakness of argument; one good solid reason would be fine. I cannot agree that your reasons are excellent though so let us go through them one by one.

Firstly, can we agree that, from a Bayesian perspective, given only the information in the W/vS statement, neither door number, nor goat ID, nor the host's name, nor what the host says says makes any difference. We have no knowledge of any of these things and how they might affect the probability of interest.

For the purposes of this discussion, therefore, we must assume that we have some reason to believe that some unexpected events may be relevant to our decisions. From a frequentist perspective, we might say that we suspect that not all relevant prior distributions are uniform. If we do not agree this then there is nothing to talk about.

So now to the reasons:

Yes, of course the doors are distinguishable, they are regardless of the numbers. So are the goats. We may choose to distinguish between doors and not between goats but there is no logical reason to do that. Martin Hogbin (talk) 14:04, 1 November 2012 (UTC)

We can and must distinguish between the doors: left, middle right. Player chose left door. Host opens right door reveals a goat. Player must choose between left door and middle door. We don't distinguish between goats. Richard Gill (talk) 14:59, 1 November 2012 (UTC)

Not so, the problem statement does not tell us hos the doors are arranged. There are three doors and we pick what is behind one of them. There are two goats and one is revealed.

Everyone knows the doors are arranged left, middle right. We see all three, from the start. We only see a goat once, jf we are lucky. Richard Gill (talk) 14:55, 2 November 2012 (UTC)

Everyone imagines things like that but nothing in the problem statement tells us how the doors are arranged. But I am not claiming the doors are indistinguishable anyway. I am saying that the goats are distinguishable. It is the host who has to distinguish between the goats for the answer to change and he can see both goats. If the host has a preference for one goat and that is the goat the he did reveal (for he surely only reveals one of them) then there is no advantage in switching. Martin Hogbin (talk) 18:19, 2 November 2012 (UTC)

I slowly came to the realization that I was taking account of the goats as well as the doors. *My* chances of 50/50 as to which door the host will open when he has a choice is because of my lack of knowledge about how the host does it. The host may well prefer one goat to the other. My odds on door 1 hiding the car don't change when I see "which" goat the host reveals because I know nothing about the host's preference for one goat or another. Seeing the goat which I do see does not give me any information about the car's location since I don't know the goats in advance and I don't know Monty's behaviour rules in advance. I don't know anything about these things! That's the point. My "objective" (uniform) Bayesian prior probabilities take account of my *lack* of knowledge.

So I think that my careful solution takes care of both doors and goats, and for that matter, the day of the week and which tie Monty is wearing this evening. It takes care of everything since it uses only what we know and everything that we know. Like it or not, the player does distinguish doors. The irrelevance of door numbers for his decision whether or not to switch is a conclusion (something which can be deduced), not an assumption, if you are careful and consistent and principled with your reasoning. You think that's pedantry. It's what I'm paid for. Richard Gill (talk) 11:25, 3 November 2012 (UTC)

Absolutely. So can we agree never to mention the door number opened by the host again, never to include it in a solution, never to say that a solution that includes the possible doors opened by the host is stronger than one that does not. If you agree that then I will agree not to mention goats again.

I think you are also moving a little towards my slightly crazy suggestion that the lack of information provided by the host's actions is better being part of the problem, even though we cannot do that here because there are no sources which say that. Martin Hogbin (talk) 15:16, 3 November 2012 (UTC)

Not really, the player has no interest in the doors themselves, the player is interested in the prizes. The player really picks one of three possible prizes at random. They must not know what prize they have originally picked. This is achieved by placing the prizes behind doors. Martin Hogbin (talk) 14:15, 1 November 2012 (UTC)

Yes really, the player has to pick the left door, the middle door, or the right hand door. Richard Gill (talk) 15:00, 1 November 2012 (UTC)

The exact same problem can be described without doors. But in any case, I am not saying that we cannot distinguish between doors, I am only saying that there is just as much reason to distinguish between goats.

Suppose that the host had a goat preference. This is probably more likely than a door preference in reality; people often have their favourite animals, they rarely have favourite doors. Now the probability of winning (assuming we have some knowledge of this preference) depends on the goat revealed. Martin Hogbin (talk) 17:51, 1 November 2012 (UTC)

We don't have any knowledge concerning any preferences of the host. Our assignment of probabilities reflects our ignorance concerning the location of the car and our ignorance of how the host chooses a door to open, if he has a choice. This includes whether or not the host's choice depends on goat characteristics. Richard Gill (talk) 15:02, 2 November 2012 (UTC)

Of course we don't have any knowledge concerning any preferences of the host. That is what I say above. In that case I am trivially correct in asserting that the significance of the goat revealed is the same as that of the door opened, because the both have zero significance.

For the door opened by the host to matter we either have to have some knowledge of the host's preferences or are considering the frequency of winning if the game is repeated with a host who has the same preferences for every repeat. I am not actually arguing that the goat revealed is important only that it is as important as the door opened. There are no likely circumstances in which the door opened by the host is important and the goat revealed is not. Martin Hogbin (talk) 18:28, 2 November 2012 (UTC)

What you say is correct: what you say can be deduced, in a principled way, from what we have been told. It can't be assumed. You are jumping to conclusions. Fortunately for you, correct conclusions. Richard Gill (talk) 11:38, 3 November 2012 (UTC)

As a mathematician (and as someone who cares about correct logic), I find it useful and important first to formalise a problem then to make correct deductions. I separate these two steps rather carefully. That is what mathematicians are for. You muddle the two steps. That's what everyone does in ordinary life. Richard Gill (talk) 11:41, 3 November 2012 (UTC)

Deduced, not assumed: The article still says "Many sources add  [...]  the assumption that the host chooses at random which door to open if both hide goats, often but not always meaning by that, at random with equal probabilities." (e.g. K&W and others.)
IMO it could help to show that this is no "additional assumption", but (inevitably?)  deduced  from the context? Gerhardvalentin (talk) 12:06, 3 November 2012 (UTC)

Richard, the same arguments apply to the goat ID. You can argue that it is quite obvious that goat ID and door number opened by the host are both unimportant, or you can argue that we need to explicitly include them in our calculations and then prove that they are irrelevant. What exactly is it that tells you that goat ID is unimportant but that door number opened by the host must be proven to be irrelevant. Martin Hogbin (talk) 15:09, 3 November 2012 (UTC)

The goats are a very interesting part of the story, everybody remembers them. Most people, apart from here, would remember the fact that there were two goats but not remember which door numbers were opened.

So what? We only got to see one goat. We got to see three doors. When we tell the story to other people we help them imagine the situation by giving the doors names. We don't give the goats names and don't have to. Richard Gill (talk) 15:02, 1 November 2012 (UTC)

So what? We know there are two goats. Are you claiming that we can only distinguish between what we can see? That is a bizarre claim. Martin Hogbin (talk) 17:40, 1 November 2012 (UTC)

I am taking account of the fact that I know nothing about the host's goat preference. I am taking account of all the information we have been given. Including everything we know about the goats in the story. And the day of the weerk. My prior knowledge means that I may behave and reason as if the car is hidden uniformly at random and as if the host opens a goat door uniformly at random from among his legal choices. It follows from these *two* probability assumptions that if I chose door 1 and the host opened door 3, then *my* chance is 2/3 that the car is behind door 2. These are facts. Whether or not you like them, whether or not you find them interesting. Richard Gill (talk) 15:13, 2 November 2012 (UTC)

Agreed. My argument is not that goat ID is important but that it is as important as the door number opened by the host. My personal opinion is that both have zero importance but there is no logical reason to make one important and the other not. Martin Hogbin (talk) 15:02, 3 November 2012 (UTC)

I do not follow your argument here. K&W say that the door numbers are a distraction, I agree with that but they do not say that the door numbers are important or that we must distinguish between the door that the host might open. Martin Hogbin (talk) 14:15, 1 November 2012 (UTC)

K&W say that solving the problem entails realizing that the door numbers are unimportant. They present experimental results which confirm this. You are so familiar with MHP, that you no longer understand how newcomers think and why they make the wrong answer. Richard Gill (talk) 15:05, 1 November 2012 (UTC)

I am not proposing that we present a solution based on goats, I am just pointing out that, if we want to be pedantic we must take account of the goat revealed. Martin Hogbin (talk) 17:44, 1 November 2012 (UTC)

I don't want to be pedantic. I want clarity and precision. I want to see clear thinking. I don't want to see a muddle. I don't care which solution anyone prefers, I don't think one is better than another. Dfferent sequences of logical deductions give different conclusions, starting from different assumptions. The consumer can choose.But the consumer should not be misinformed. Richard Gill (talk) 08:15, 2 November 2012 (UTC)

Fine. In what way do you think I want to misinform the reader? Martin Hogbin (talk) 15:00, 3 November 2012 (UTC)

I don't think you *want* to misinform the reader. I do get the impression that you don't see that there is a difference between giving a logically correct argument that the chance of winning by switching when you have chosen door 1 and the host has opened door 3 is 2/3, and giving a logically correct argument that the chance of winning by switching when you have chosen door 1 is 2/3. (The issue is not whether this difference is important or interesting: just whether or not there is a difference). Richard Gill (talk) 20:00, 3 November 2012 (UTC)

PS You haven't given us a fundamental reason why the door numbers are irrelevant. You know they are not part of the solution so it is true that it turns out they are irrekevant. You say "clearly Vos Savant, Selvin etc. meant us to... ". Well, that's your opinion. And anyway it's irrelevant since the question is what reliable sources do with the numbers. Some take them carefully into account, some don't. My proposals for Selvin-fixed and simple solution fixed show how it is easy to add-on the sentence or two which deals with the door numbers, for those who might have worries about them. For those who didn't have worries, the sentence or two confirms their intuition. Isn't thats nice for them? The "hard part" remains the first step - understanding what the simple solution says. It certainly says that initial intuition ("it doesn't matter") was wrong. I think this is a win-win way to approach MHP. Richard Gill (talk) 08:18, 1 November 2012 (UTC)

You still seem to have fallen for Morgan's conjuring trick in which they use a bit of clever misdirection to pull a rabbit out of a hat. By treating the door number opened by the host differently from that opened by the player they make a bogus case for making the door number opened by the host being an essential condition of the problem.

We could add a sentence somewhere saying that the actual door numbers are meaningless and were added by vS to help with the problem description and that they convey no information relevant to calculating the probability of interest (or of you prefer deciding whether to swap or not) to the reader and thus can be completely ignored. I am not sure where we should put that but I would not object to it, provided it makes clear that we can ignore the door numbers always. We can ignore the number of the door originally chosen by the player and the door number opened by the host. I still tend to think that this statement should occur after we have actually ignored the door numbers.

I am sure you know this well-known story. A mathematics professor is giving a lecture when a student asks, How did you get that result from the line above?'. The professor immediately replies, 'It is obvious!'. Then he stops, scratches his head, and walks out of the lecture theatre. Twenty minutes later he returns, says, 'Yes, it is obvious', and continues the lecture.

That is exactly how it is with the door number opened by the host. To most people it is intuitively obvious that it makes no difference which door number he opens. You as an expert have given the issue some considerable thought and concluded that it does indeed make no difference.

We do know vS's thoughts on the door numbers because she tells us that she added them herself and that she now considers this to have been a mistake. I think we can come to the same conclusion about Selvin also. When the door letter opened by the host was brought up in a letter he immediately made clear that the host would choose uniformly at random thus making the door letter unimportant. Martin Hogbin (talk) 09:26, 1 November 2012 (UTC)

I am not interested in the Morgan et al paper, it is awful. Vos Savant's and Selvin's thoughts are unimportant too. MHP lives its own life. Lots of people have written about it and our task is to report what is in the literature, not to present our own theories about MHP. Moreover, Selvin's response is bad. We are not told in advance that the host chooses uniformly at random. If we are arguing carefully and consistently from a Bayesian position of ignorance, then either of the host's choices are for us equally likely. He may well be making his choice according to a totally deterministic, systematic rule. We don't know. If he is using such a rule, he could for us be just as well using the opposite rule.

I am interested in logical consistency. I want to see a logical deduction from reasonable assumptions to an unavoidable conclusion. That's because I'm a mathematician, that is what my job is all about, that is an important part of a mathematician's role in science. But you are still muddling frequentist and subjective probability, assumptions and conclusions. Tell me what you want to assume, and why, and then we can discuss what follows from those assumptions, what doesn't. Don't keep changing the rules as you go along. You've just said that Vos Savant should have asked a different question, and that Selvin should have written down some further assumptions. But our task is to survey the literature about the question which Vos Savant actually did ask. That's the question which opens the article. That's the question which made MHP famous. You want to rewrite the question first, and then answer the question which you think she should have asked. Richard Gill (talk) 15:19, 1 November 2012 (UTC)

Richard, we have just gone round in a big circle. Years ago I, and several others, said that door numbers were completely irrelevant. In other words, the problem statement could be stated as:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door and the host, who knows what's behind the doors, opens another door ,which has a goat. He then says to you, "Do you want to change to the unopened door' Is it to your advantage to switch your choice?

We do not care which door the player originally chooses or which legal door the host opens.

I am not the least bit muddled about frequentist and subjective probability. My personal preference is to take a subjective view of the problem. Everything we have no knowledge of we must take as uniform. Having done this, what possible reason is there for considering the door number opened by the host as a condition of the problem. We know, right from the start, by definition, that is unimportant. Martin Hogbin (talk) 14:59, 3 November 2012 (UTC)

The door numbers are mentioned by Vos Savant in the problem statement. They are mentioned at the start of the wikipedia article. They appear in every graphical illustration of the problem. Like it or not, lots of readers will have them in their minds.

I agree that they can be ignored from the start. Not by definion. Not by rewriting the question so that no numbers are mentioned any more. No, by deduction. Because we are taking this Bayesian point of view and using uniform distributions for our total lack of information on where the car is hidden and how the host opens doors.

I don't understand your phrase "a condition of the problem". I think you are confusing the word condition, as a synonym for "assumption", with the word condition, as in "conditional probability".

Why do you object to some people wanting to know the chance of winning by switching given the door they chose and the door opened by the host? And why don't you want to tell them it's 2/3 because of the simple solution plus symmetry? Why do you think giving this information would confuse other readers? Richard Gill (talk) 20:24, 3 November 2012 (UTC)

No I do not mean assumption when I say condition. I mean, why must we go through the rigmarole of considering both doors that the host might have opened and then conditioning on the one he did, in fact, open? We know the answer before we start. Martin Hogbin (talk) 01:18, 4 November 2012 (UTC)

I have no objection to some people wanting to know the chance of winning by switching given the door they chose and the door opened by the host? It is quite intuitively obvious to most people that the answer is the same regardless of which doors are used but for the pedantic I have no objection to making this explicit later on in the article. You are still intend on explaining the blindingly obvious at the expense of the two things that confuse everyone. Martin Hogbin (talk) 01:22, 4 November 2012 (UTC)

No I am not intent on explaining the blindingly obvious. I *am* worried sometimes how well you understand the subtle issues (for instance: why Devlin's argument was wrong (Nijdam) / incomplete (me) and how to fix it. I am also worried that if we do not have some examples of conditional solutions in the beginning of the article, the end of the article will be dominated by academic pedants who will fill the pages with un-insightful formulas and spread propaganda that the simple solutions are "wrong". I am worried that such polarization will also distract attention from yet other approaches to MHP (the strategic-thinking approach, for instance). I want to protect future generations of probability students from Morgan-like terror of their dull teachers. I want the article to make clear that simple and conditional solutions blend together seamlessly, they don't conflict. I want the part of the article which explains simple solutions not only to be simple intuitive and attractive but also to logically correct. Otherwise you are scoring an "own goal", giving fuel to the formalists. I want the article to reflect the richness and diversity of ideas springing from this simple puzzle. Richard Gill (talk) 07:11, 4 November 2012 (UTC)

Perhaps I am missing something. What is missing from Devlin's solution? Martin Hogbin (talk) 21:13, 4 November 2012 (UTC)

Yes you are missing something. I'm tired of this. Devlin's correspondents pointed out there was a missing step in his argument. He retracted it. People like yours truly have fixed it using Bayes theorem, odds form. Richard Gill (talk) 08:38, 5 November 2012 (UTC)

Just to be sure that we are not arguing at cross purposes, perhaps you would be kind enough to tell me what the missing step is. Martin Hogbin (talk) 09:27, 5 November 2012 (UTC)