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Gill110951 (talk | contribs) →Proposal to add some alternative conditional solutions: the arbitration is about misconduct, not about content |
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:::So the whole arbitration is about Rick Block's complaint about my conduct? [[User:Glkanter|Glkanter]] ([[User talk:Glkanter|talk]]) 06:58, 12 February 2011 (UTC) |
:::So the whole arbitration is about Rick Block's complaint about my conduct? [[User:Glkanter|Glkanter]] ([[User talk:Glkanter|talk]]) 06:58, 12 February 2011 (UTC) |
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:::: @Glkanter: I think @Woonpton is right, and also @David Tombe knows about these things through personal experience and probably has good advice on how to survive. @Rick Block has complained about your behaviour, you can see his complaint somewhere. [[User:Gill110951|Richard Gill]] ([[User talk:Gill110951|talk]]) 12:29, 12 February 2011 (UTC) |
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::::@ Richard. I note that implicitly you now take the "conditional formulation" (decide after the door has been opened) as the MHP to be presented. Your "simple plus symmetry" solution, better is called "conditional using symmetry" solution, as the characteristic thing of a simple solution just is not considering any conditional probabilities. This solution and the one using Bayes' have always been considered correct solutions. Let us not discuss your third option, as IMO it is not suited for Wikikpedia. As for the term "simple", let's keep it for the simple solution, the one not being a correct solution to the conditional formulation of the MHP. The article should mention it with the criticism. [[User:Nijdam|Nijdam]] ([[User talk:Nijdam|talk]]) 11:15, 12 February 2011 (UTC) |
::::@ Richard. I note that implicitly you now take the "conditional formulation" (decide after the door has been opened) as the MHP to be presented. Your "simple plus symmetry" solution, better is called "conditional using symmetry" solution, as the characteristic thing of a simple solution just is not considering any conditional probabilities. This solution and the one using Bayes' have always been considered correct solutions. Let us not discuss your third option, as IMO it is not suited for Wikikpedia. As for the term "simple", let's keep it for the simple solution, the one not being a correct solution to the conditional formulation of the MHP. The article should mention it with the criticism. [[User:Nijdam|Nijdam]] ([[User talk:Nijdam|talk]]) 11:15, 12 February 2011 (UTC) |
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@Glkanter. I'm sorry, I disagree. Jeff is a mathematician. He is interested in correct arguments just as much as correct answers. I find it annoying that he doesn't explain *why*, from his point of view, you *have* to compute a conditional probability, but I do believe that that is his point of view. I also know good reasons for having that point of you. Of course, mathematics can not ever tell you what you have to do. It has no legal or moral authority. But it can tell you what it would be wise to do. |
@Glkanter. I'm sorry, I disagree. Jeff is a mathematician. He is interested in correct arguments just as much as correct answers. I find it annoying that he doesn't explain *why*, from his point of view, you *have* to compute a conditional probability, but I do believe that that is his point of view. I also know good reasons for having that point of you. Of course, mathematics can not ever tell you what you have to do. It has no legal or moral authority. But it can tell you what it would be wise to do. [[User:Gill110951|Richard Gill]] ([[User talk:Gill110951|talk]]) 12:29, 12 February 2011 (UTC) |
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@Nijdam. I am tired about bickering about what is THE Monty Hall Problem. You and I disagree. I think there are many mathematizations and any decent mathematization of course allows many different decent solutions. You may call the approaches which I listed by any name you like. I think the names I gave will be understandable to everyone interested in MHP and active on these pages, not just to the people who share your point of view, which I find dogmatic and inflexible. [[User:Gill110951|Richard Gill]] ([[User talk:Gill110951|talk]]) 12:21, 12 February 2011 (UTC) |
@Nijdam. I am tired about bickering about what is THE Monty Hall Problem. You and I disagree. I think there are many mathematizations and any decent mathematization of course allows many different decent solutions. You may call the approaches which I listed by any name you like. I think the names I gave will be understandable to everyone interested in MHP and active on these pages, not just to the people who share your point of view, which I find dogmatic and inflexible. [[User:Gill110951|Richard Gill]] ([[User talk:Gill110951|talk]]) 12:21, 12 February 2011 (UTC) |
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many people say switching is better, but are still wrong
many people will say switching is better because your odds are 1/2. This is wrong of course, because your odds are 2/3. I think this particular confusion needs to be made explicit in the article —Preceding unsigned comment added by 76.126.238.69 (talk) 23:41, 9 February 2011 (UTC)
How about the information angle ?
How about first establishing what actual information is present after the host opens the door instead of playing with probebilities that become irrelevant once the door gets opened? Call it a methodological problem if you like. Probabilities of guesses can be mapped to presence and absence of information. The difference is that information based reasoning points clearly what's relevant and what's irellevant. Relevant is that player had new info which gives him 1/2 chance if he tosses a coin and chooses again at random. His first choice is irellevant since he didn't get any info about that choice, he only got infor that new choice with better ods is possible. If we believe that a random choice under uncertainty gives the best odds, then he will improve his chances only if he makes a new random choice, not a forced one.
Information-wise the 2 cases from initial state are merged when new information is provided - they don't exist anymore in the new state, they are indistinguishable. Same applies to the case with 1 mil doors. If player makes forced choice he remains in the prior state with much lower odds - he's not using new information. If he wants to use new information then he has to toss the coin again in order to realize the new state.
Another way to look at it is to realize that in an assembly only a random choice can select new configuration and all forced, non-random transitions are equivalent and confined to the same configuration.
His actual odds still improve from 1/3 to 1/2. ZeeXy (talk) 13:21, 3 November 2010 (UTC)
- (Attempt to) Pick wrong and increase the chances of switching right, or (Attempt to) pick right and not know until it's too late. Metaphysically speaking, I'd rather wing it. 70.15.11.44 (talk) 05:28, 4 November 2010 (UTC)
- When wondering where the car is, you shouldn't just use the hard information which you have in front of you, but also the likelihood that that information came to you under the different scenarios which concern you. You chose Door 1. The host is twice as likely to open Door 3 if the car is behind Door 2 than if the car is behind Door. 1. When the game is repeated many times, the car will be behind Door 2 twice as often as it is behind Door 1, within those occasions that you chose Door 1 and the host opened Door 3.
- Forget about probability, forget about information. This is about very simple arithmetic. Richard Gill (talk) 07:20, 5 November 2010 (UTC)
Wrong. If a host opens #3 you don't know if that's because you missed or guessed #1 and no imagination can help you. If you automatically switch to #2 that's equivalent to picking #2 in the first place -- he'd still open #3. Now what? :-) He could even let you keep switching till you turn blue. That's why he can be "generous" - coz you are at 50% ignorance and there's nothing that can help you. You could have started with 100 doors and he could have let you switch every time he closed a door and you'd still be in exactly the same state. That's why I said that information-wise prior states are merged as a result of the new info - they are indistinguishable, there is no observable difference. Your first 98 choices are irellevant for the new state -- you are still in a state of 50% ignorance. The only thing you can do to improve your chances in any 50% ignorance state is to toss a fair coin.
- Wrong, wrong. We are told as part of the problem statement that the quizmaster knows where the car is hidden, that he will always open a door revealing a goat, and that he will always offer us the opportunity to switch to the other still closed door. Richard Gill (talk) 13:22, 6 November 2010 (UTC)
Here's example with a real statistical ansamble
Say you were choosing among 3 presidential candidates and one got killed. Does that automatically make the one you haven't picked a 2/3 winner? :-) Then it has to apply to all other people who picked one of these 2. All these people are now real statistical ansamble - genuine massive sample of random choices with equal probabilities and you can clearly see that your particular initial choice is irellevant. Assume totally split election, every candidate having 1/3 before one got killed and assume no one really cares - everyone just wants to vote for the winner since then they get a coin if they voted for a winner. Suppose they all follow your automatic switch tactics and the 3rd voter set gets split equally. No one wins. What's the best chance of voting for the winner you have? 50% What the sole way to insure that someone does win and thereby 50% of you achieve the goal -- that no one does anything authometically but everyone tosses a coin. Why? Because fair coin toss is never 100% fair with finite number of tosses. Only the act of everyone tossing insures the winner. This is probably the best illustration how irellevant and blocking automatic switching is and how you do need to toss in order to actually realize the chance presented by new information.
This is all very different from a situation in which there would be some underlying cause and your sampling is really just measuring it since then you would expect sampling to converge with a very high probability. Without underlying cause it takes infinite number of samples to realize your statistics and that's strictly historical and completely irellevant for a particular singular trial. That's the part people easily forget when they start deluding thelselves with abstract statistics -- it's irellevant for a particualr sample unless there's underlying cause which will ensure rapid convergence. That's the sole thing that makes sampling worth anything -- if there is underlying cause there will be fast convergence. Have to remind you that theoretical limit for a big number rule to apply is infinite number of samples.
Go tossing coins and see how long runs of equal values you are going to get and how huge deviations from imagined 1/2 you are going to get. ZeeXy (talk) 12:44, 6 November 2010 (UTC)
- Maybe it would be best if the participants in this discussion only made reference to reliably published sources, as a method of discussing changes to the articles. I'm sure there are various more appropriate forums elsewhere on the internet that welcome spirited debate on the mathematics and logics of the MHP. Glkanter (talk) 12:52, 6 November 2010 (UTC)
This another typical response to the MHP article. Everyone wants to know why/how/if the answer is 2/3 rather than 1/2. Martin Hogbin (talk) 10:36, 6 November 2010 (UTC)
- @ZeeXy - if you'd like to discuss the mathematics behind the problem, I'd suggest we move this thread to the /Arguments subpage. If you're suggesting a change to the article, please say what change and on what source (or sources) you'd base that change. -- Rick Block (talk) 15:16, 6 November 2010 (UTC)
- Rick, there shouldn't *be* an 'arguments' page. Arguing the math/logic of the MHP is no more appropriate for a Wikipedia article talk page than discussing the greatness of your favorite musical performer with like-minded fans. The 'arguments' page should be deleted, rather than encouraged. Talk pages are for discussing editing Wikipedia articles. Glkanter (talk) 15:31, 6 November 2010 (UTC)
- The point of the /Arguments page is to have a place for these sorts of discussions, which are not directly related to editing the article, to be held. It is more or less like the Wikipedia:Reference desk - but with a specific focus on the MHP. There's one for various other articles on controversial topics, like 0.999.... You are absolutely correct that THIS page is for discussing editing the article. If you strongly feel the need to see a community consensus about whether the Arguments page should be deleted, please open a discussion at Wikipedia:Miscellany for deletion. -- Rick Block (talk) 17:23, 6 November 2010 (UTC)
Thanks for the suggestion. I may do that. If its a Reference desk item, well, then it belongs at the Reference desk. Or, I could just take the page off my Watchlist... Glkanter (talk) 21:55, 6 November 2010 (UTC)
Recent overhaul and state of the mediation
First of all thanks to all who put effort in the recent overhaul, which from my perspective works well overall.
I minor nitpicking I'd have though is the (incomplete) quotation of Behrends in the sources of confusion section. It should be mentioned while Behrends considers both answers as correct he does consider them as 2 slightly different problems or questions at least.
Another I'd like to know is whether the conflicting parties in the mediation are happy with the current version (or at least can live with it) or whether we still have (major) disagreements and an potenial editing conflict down the line. --Kmhkmh (talk) 15:46, 14 November 2010 (UTC)
- I think I am reasonably happy with the article as it is now. I did not realise that the article was being actively edited during the mediation so I am assuming any edits made during mediation to be non-contentious ones.
- As you will see on the mediation page, I have suggested that we start discussion based on the article as it is now, rather than rewriting large chunks of it from scratch. I guess you support this proposal. Martin Hogbin (talk) 10:41, 9 December 2010 (UTC)
- Just out of curiosity, what was the mediation over? The MHP is a stats problem, which does not strike me as something prone to violent arguments. --Ludwigs2 17:39, 8 February 2011 (UTC)
- Best look through the last couple of years' talk pages. I do not think anyone wants a re-run. Martin Hogbin (talk) 22:02, 8 February 2011 (UTC)
- Just out of curiosity, what was the mediation over? The MHP is a stats problem, which does not strike me as something prone to violent arguments. --Ludwigs2 17:39, 8 February 2011 (UTC)
- Without looking through the archives, why is there a long list of references, but no in-line citations? Cla68 (talk) 00:07, 10 February 2011 (UTC)
- The article uses Harvard style referencing. There are plenty of inline references. -- Rick Block (talk) 00:37, 10 February 2011 (UTC)
- I'm pretty much happy with the present article. But I did some more OR in the direction of creating a synthesis between simple and conditional solutions, see [1]. Richard Gill (talk) 22:50, 10 February 2011 (UTC)
Proposal to add some alternative conditional solutions
I would like to see some more mathematical solutions to the conditional problem, just as there are various informal solutions to the unconditional problem. I think they all give additional insight into MHP. There exist at least three solutions which follow a simple chain of logical reasoning, and which can be converted step by step into equivalent mathematical formalism (this is a useful exercise for the beginning student of the formal probability calculus who has to learn how to connect the formalism with ordinary logical reasoning and insight into the structure of the problem), but which avoid calculations or formula manipulation. These are: simple plus symmetry, Bayes' rule, and symmetry plus simple.
Simple plus symmetry: by symmetry the probability that the car is behind door 1 cannot depend on whether the host opened door 2 or door 3. The unconditional probability was 1/3. Therefore the two conditional probabilities are equal to 1/3 too. Reference: Bell (1982).
Bayes rule: the odds that the car is behind door 1 (the door chosen by the player) are initially 2 to 1 against. Whether or not the car is behind door 1, the chance that the host opens door 3 is the same, 50%. (In the one case because if the car is behind door 1, the host is equally likely to open either other door, in the other case, because if the car is not behind door 1, it is equally likely behind either other door, and the host's choice is forced.)
Symmetry plus simple. Pretend for a moment that the player's choice is also completely random. After the host's action, we can refer to the doors as: door chosen by player X, door opened by host H, door remaining closed (to which the player may switch) Y. From the simple solution we know that the door hiding the car, C, is either door X or door Y, with probabilities 1/3 and 2/3 respectively. By symmetry, the triple of door numbers (X,H,Y) is a completely random permuation of the numbers (1,2,3), and C either equals X or Y, with probabilities 1/3 and 2/3, independently of which of the six permutations is the permutation (X,H,Y). This tells us that the specific door numbers are irrelevant to the player who wants to maximize his chance of getting the car. The actual numbers are completely independent of the relationship of C to X,H,Y.
Nothing is changed by fixing the value of X, X=1 say. Now there are just two permutations possible, (1,2,3) and (1,3,2). By symmetry they are equally likely, and this is independent of whether or not C=X.
Each of these alternative proofs has pedagogical value for students of probability and statistics since they use extremely valuable tools. I think they each give further insight into "why you should switch". Each of the proofs is intuitive, you don't need a formal mathematical training to appreciate the ideas used in them. All the proofs explain why the ordinary lay person is perplexed by the argument that the simple solutions are "wrong" and that you have to learn Bayes's theorem and formal probability calculus to solve MHP properly - because each of the proofs make clear in a different way why the ordinary lay person is completely right not to be too bothered about the specific door numbers. Each of the proofs ties in with Vos Savant's wording "say, Door 1", and "say, Door 3", since we see that the specific door numbers are indeed irrelevant to the chance that switching will win and to the decision process of the player.
Reliable sources: the various uses of symmetry go back to discussants of the Morgan et al. (1981) paper, especially Bell (1982). The use of Bayes' rule is promoted by Jeff Rosenthal (2006?) (who by the way is a prominent mathematician and probabilist, as well as a prominent popularizer of probability and statistics whose "popular" writings are appreciated both by lay persons and by experts). See also a prepublication by me, [2], which is based entirely on what I learnt from fellow editors on the MHP page. Richard Gill (talk) 09:16, 11 February 2011 (UTC)
- I think it is generally useful to understanding have several ways of looking at a problem covering all levels of understanding and interest. The only thing I would want to keep in mind is that the MHP is essentially a simple problem that most people get wrong so we should start with simple and convincing solutions> After that, something for the experts. Martin Hogbin (talk) 11:06, 11 February 2011 (UTC)
- Those are all great ideas, Richard. Too bad the Conditional solution section is so grossly polluted with variants, hypotheticals, OR, NPOV and UNDUE WEIGHT violations, and other assorted crap intended to diminish the simple solutions, but which serves only to confuse the reader.
- By the way, is that great authority, Jeff Rosenthal, the same one who says of a simple solution?:
- "This solution is actually correct, but I consider it "shaky" because it fails for slight variants of the problem. For example, consider the following:"
- "Monty Fall Problem: In this variant, once you have selected one of the three doors, the host slips on a banana peel and accidentally pushes open another door, which just happens not to contain the car. Now what are the probabilities that you will win the car if you stick with your original selection, versus if you switch to the remaining door?"
- "This solution is actually correct, but I consider it "shaky" because it fails for slight variants of the problem. For example, consider the following:"
- Maybe you could clarify for me what the English term, 'actually correct' means and what bearing 'it fails for slight variants of the problem' has as to the 'actually correct'-ness of that simple solution as a technique for solving the Wikipedia article's subject, which is the symmetrical MHP? Glkanter (talk) 12:00, 11 February 2011 (UTC)
@Martin: yes indeed. My proposal would be to place these solutions in place of the present formal mathematical proof via Bayes' theorem. That proof can be replaced by a reference to the article Bayes theorem where it already figures as an example.
@Glkanter. Yes, Jef's words are incomprehensible. What is wrong with something which is right? What is the relevance that "it" fails for a different problem? I think Jef was using the word "solution" in two different sences without realizing it. The "answer" (2/3) is correct but the "argument" is not. Because the same argument gives the same answer, 2/3, for a different problem Monty Crawl, where 2/3 is the wrong answer. And the argument is not even applicable to Monty Fall. But why don't you ask him yourself? So I guess this was a momentary lapse. But I don't have e.s.p., so I can't tell what was going on in his mind. I can only guess. Richard Gill (talk) 19:05, 11 February 2011 (UTC)
- *I* don't have any problem comprehending his words. Nor do I find any 'lapse'. The only reason I bring it up, again, is because you have repeatedly agreed with Rick Block and others that those words constitute a 'criticism' of all simple solutions. That's an unsupportable conclusion by you people. His only point is to show why his 'discovery', or whatever, has utility in some other applications, which the simple solutions, of course, would fail in. This (these?) other method(s) is the whole point of his paper. It is in no way a 'criticism' of simple solutions as a solution to the symmetrical MHP at all. Hopefully nonsense like this will be exposed in arbitration, don't you think? And wouldn't it be nice if the Conditional solution section, which also talks about variants for 90% of the time was cleaned up to make room for your suggestion? Glkanter (talk) 19:18, 11 February 2011 (UTC)
- If you are expecting arbitration to settle the longstanding content disputes on this article, you will probably be disappointed. Content disputes are outside ArbCom's remit, and while sometimes they try to help resolve a content dispute by giving disruptive editors a "time out," as a rule they won't address the content dispute directly. Woonpton (talk) 05:40, 12 February 2011 (UTC)
- @Glkanter: I think @Woonpton is right, and also @David Tombe knows about these things through personal experience and probably has good advice on how to survive. @Rick Block has complained about your behaviour, you can see his complaint somewhere. Richard Gill (talk) 12:29, 12 February 2011 (UTC)
- @ Richard. I note that implicitly you now take the "conditional formulation" (decide after the door has been opened) as the MHP to be presented. Your "simple plus symmetry" solution, better is called "conditional using symmetry" solution, as the characteristic thing of a simple solution just is not considering any conditional probabilities. This solution and the one using Bayes' have always been considered correct solutions. Let us not discuss your third option, as IMO it is not suited for Wikikpedia. As for the term "simple", let's keep it for the simple solution, the one not being a correct solution to the conditional formulation of the MHP. The article should mention it with the criticism. Nijdam (talk) 11:15, 12 February 2011 (UTC)
@Glkanter. I'm sorry, I disagree. Jeff is a mathematician. He is interested in correct arguments just as much as correct answers. I find it annoying that he doesn't explain *why*, from his point of view, you *have* to compute a conditional probability, but I do believe that that is his point of view. I also know good reasons for having that point of you. Of course, mathematics can not ever tell you what you have to do. It has no legal or moral authority. But it can tell you what it would be wise to do. Richard Gill (talk) 12:29, 12 February 2011 (UTC)
@Nijdam. I am tired about bickering about what is THE Monty Hall Problem. You and I disagree. I think there are many mathematizations and any decent mathematization of course allows many different decent solutions. You may call the approaches which I listed by any name you like. I think the names I gave will be understandable to everyone interested in MHP and active on these pages, not just to the people who share your point of view, which I find dogmatic and inflexible. Richard Gill (talk) 12:21, 12 February 2011 (UTC)