| Revision as of 09:58, 7 December 2012 editMartin Hogbin (talk | contribs)20,189 edits →The controversy continues← Previous edit | Revision as of 10:04, 7 December 2012 edit undoGill110951 (talk | contribs)Extended confirmed users8,007 edits →The controversy continues: media furore fine; academic quibbling doesn't belong thereNext edit → | ||
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| :::As my new section appears to have stirred up such a hornet's nest I have deleted it pending discussion. ] (]) 09:58, 7 December 2012 (UTC) | :::As my new section appears to have stirred up such a hornet's nest I have deleted it pending discussion. ] (]) 09:58, 7 December 2012 (UTC) | ||
| :::: I think the first part of the section on the media furore was excellent; but the "controversy continues" subsection completely out of place. ] (]) 10:04, 7 December 2012 (UTC) | |||
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Intro to Conditional Solution: biased host
I added to the introductory text to the conditional solutions the derivation of the conditional probabilities of 1/2 and 1 corresponding to known, maximal (in either direction), host bias. So the section now begins with an elementary derivation of conditional 2/3 by consideration of the expected outcomes of 600 repetitions in the standard situation, as well as illustration that if we have certain knowledge of host bias, we get different answers. The same motivation as many writers, e.g. Rosenthal, give for the conditional solution. Richard Gill (talk) 19:49, 26 November 2012 (UTC)
The section goes on to give three more derivations of conditional 2/3: by a conditional probability diagram, by a decision tree, and by extension of Vos Savant's table to 6 equally likely cases. I propose that the conditional probability diagram is deleted - it seems superfluous among all the other graphical and tabular methods, and it takes a huge amount of space. Any opinions on this? Richard Gill (talk) 19:54, 26 November 2012 (UTC)
- By all means! Delete all traces of any accessible conditional solution - it is of course merely an academic contrivance! We find your recent work here to be most excellent, although it is easily seen that the authors should perhaps read Misplaced Pages:Manual of Style/Mathematics.
- In case there's anyone clueless enough to miss the sarcasm, I'm saying no, the diagram should not be deleted. -- Rick Block (talk) 05:08, 27 November 2012 (UTC)
- Rick, you think the conditional probability diagram is more accessible than anything else? I really do wonder what "first time readers" find the most enlightening. Under the motto "less is more" the conditional probability part of the article would be more accessible (I think) if it was trimmed of superfluous fat, started with the a really good motivation of *why* a conditional solution, and clearly showcased the most accessible/appealing solution. Alternative derivations can then be relegated to a collection of "alternative approaches". (PS thanks for the tip regarding manual of style and thanks for the compliment, which I take as being seriously meant!)
- The article was reorganized by Martin a few weeks ago and this left the conditional probability parts rather disjointed. There was the big diagram alongside some other solutions but no explanation of where you "see" that the answer is 2/3 in that diagram. I have tried hard to remedy this, but I don't know if I succeeded (I added a row in the table, splitting 1/3 into two times 1/6).
- At the moment my feeling is that the discussion of the expected numbers of different outcomes in 600 repetitions, and the table of six equally likely cases extending vos Savant's table of three, are the most accessible derivations for people who never saw conditional probability before, can't read maths formulas, and are scared of arithmetic with fractions. If you have any familiarity with probability and conditional probability at all, then the decision tree is useful visual aid. Maybe I'm wrong. Maybe the big conditional probability diagram is a wonderful pedagogical tool for most readers. Richard Gill (talk) 15:55, 27 November 2012 (UTC)
- Richard, I did not (intentionally) change the organisation of the 'conditional' section, I moved it as it was.
- Rick, as you know, I do believe that the 'conditional' solutions are 'merely an academic contrivance', but the proposal was to give them equal prominence so, as far as I am concerned you can edit them in whatever way you think showcases them well and makes them accessible to your intended audience. This is what Richard was trying to do. I asked Richard to think about the intended audience for this section and would make the same suggestion to you. Martin Hogbin (talk) 23:45, 28 November 2012 (UTC)
The example of 600 repetitions explains the "conditional approach" that is viable with a biased host, but never does address the direct approach. Rick's mapping is pointless and incorrect. The article must stop to be kept blurry and unclear. Gerhardvalentin (talk) 21:50, 29 November 2012 (UTC)
- What do you mean: the example with 600 repetitions never does address the direct approach? It tries to explain what conditional 2/3 means. The 600 repetitions are imaginary, not real. Later there is an equivalent description using 6 equally likely possibilities. Do you understand it? Richard Gill (talk) 06:42, 30 November 2012 (UTC)
- What do you mean by "the direct approach"? The simple solution? 600 repetitions ilustrates the simple solution too. Richard Gill (talk) 06:44, 30 November 2012 (UTC)
- The "600 repetitions", explaining all possible outcome, never belong to the section "Criticism of the simple solutions", where Rick did rearrange it. It belongs where it was, before Rick disarranged it. Gerhardvalentin (talk) 07:41, 30 November 2012 (UTC)
- Rick made some big changes. The "criticism" section is now a mess. He had better revise it, too. Richard Gill (talk)
- Is now cleaned up again. Richard Gill (talk) 08:11, 1 December 2012 (UTC)
- Rick made some big changes. The "criticism" section is now a mess. He had better revise it, too. Richard Gill (talk)
- The "600 repetitions", explaining all possible outcome, never belong to the section "Criticism of the simple solutions", where Rick did rearrange it. It belongs where it was, before Rick disarranged it. Gerhardvalentin (talk) 07:41, 30 November 2012 (UTC)
The problematic choice of goats
I believe much of the confusion this problem has generated has been caused by an incorrect choice of animal. If the goats are replaced by Schrödinger's cats, everything becomes much clearer. Rumiton (talk) 07:14, 28 November 2012 (UTC)
- Your suggestion is too late, a quantum version of the puzzle has already been proposed. Martin Hogbin (talk) 09:36, 28 November 2012 (UTC)
- Gosh, what are the chances of that? Rumiton (talk) 10:32, 28 November 2012 (UTC)
- Several different quantum versions. Ours (me and my friends') is the best (d'Ariano et al). Richard Gill (talk) 08:16, 1 December 2012 (UTC)
- Gosh, what are the chances of that? Rumiton (talk) 10:32, 28 November 2012 (UTC)
Source of confusion
The Sources of confusion section has this confusing sentence: "However, if a player believes that sticking and switching are equally successful and therefore randomizes their strategy, they should, in fact, win 50% of the time, reinforcing their original belief." It could be just due to English not being my native language, but I read that as going against the entire point of the paradox: that the player somehow suddenly has the intuitive 50/50 odds of winning at their last choice, should they simply choose to believe it so. I suggest that this be mended or expanded to better express whatever it is that it's supposed to mean. -Uusijani (talk) 10:56, 28 November 2012 (UTC)
- You are right, Uusijani. The article is a mess, never clearly saying what it purports to say. It explicitly should say that (50% x 1/3) + (50% x 2/3) gives "1/2". And in presenting the paradox it does not clearly say what Marilyn vos Savant herself did say concerning the correct "scenario" where the paradox arises. Gerhardvalentin (talk) 12:16, 28 November 2012 (UTC)
- It seems clear to me. If a player believes that there is no advantage in switching, and therefore randomises their strategy (in other words sticks half the time and switches half the time) they will win the car half the time. This will (illogically) reinforce their belief that it makes no difference whether you stick or switch. In this respect pigeons do better than humans.Martin Hogbin (talk) 23:31, 28 November 2012 (UTC)
The last sentence in the "Sources of Confusion" appears to be original research (it doesn't seem to have a reference), and it makes no sense. For example, if we had a game where switching would ALWAYS lead to the prize and not-switching would NEVER lead to a prize, and if someone believed it made no difference, so they randomly switch or not switch on a series of trials, they would win 50% of the time, but why would they not notice that they win 100% of the time when they switch and 0% when they don't? That simply makes no sense. That experience in no way confirms their prior belief that switching makes no difference. Quite the opposite, that sequence of trials shows that switching makes ALL the difference. So I don't believe this silly idea is a source of confusion. If someone can find a reference for that daft idea, then I guess it could be left in the article (with the reference added please), but if no one can find a reference, it should be deleted as original (not to mention dumb) research.My3scents (talk) 16:38, 3 December 2012 (UTC)
Intro to conditional solution
Some days ago, I wrote an intro to the conditional solution, motivating it (as many authorities do) by showing that if the host has a preference, then the conditional probability answer is different from 2/3. This intro has now got spit over two distant sections, I think by Rick. Please Rick do things properly or not at all. The "criticism" section is now a mess. Richard Gill (talk) 19:15, 30 November 2012 (UTC)
- The previous intro brought up the issue of biased host behavior - which is distinctly unnecessary to motivate the conditional solution. Most sources presenting conditional solutions do not (directly) bring this up. IMO, the biased host scenario is directly related to the criticism of the simple solutions - since these solutions produce the same "2/3 chance of winning by switching" result regardless of any host bias. The sources that are critical use the biased host variant to justify their criticism. -- Rick Block (talk) 01:10, 1 December 2012 (UTC)
- Many sources would disagree with you, Rick. Rosenhouse and Rosenthal use the host bias variant to explain and motivate the conditional solution. This forces people to realise that by ignoring door numbers they are implicitly assuming that the host's choice gives no information to the player.
- Traditional probability textbooks don't bother to motivate their approach, since in that context, one automatically figures out the conditional solution, because one works from first principles. First step: write down a probability model for the history of events leading up to the moment the player is asked whether he wants to switch. Second step: write down the probabilities of possible locations of the car given what the player has seen so far, using the definition of conditional probability. No need to be clever or creative. Just follow the rules and blindly (but carefully) calculate.
- Why start half way through the problem? Martin Hogbin (talk) 23:32, 1 December 2012 (UTC)
- Writers of probability texts start where they find it convenient and natural and sensible to start. It seems to me a very natural choice to start with (1) the car is hidden, (2) the player picks a door, (3) the host opens a door, (4) the player picks. Anyway, that's what everyone does, and it works fine. Next, since we will assume the player's choice is independent of the location of the car and later we want to condition on it anyway, we might as well consider the player's choice fixed. So we now have: (1) the car is hidden, (2) player picks door 1, (3) host opens a door, (4) player decides what to do. There is no point in making the story more complicated. We can't make it less complicated.
- Having decided the sequence of steps leading up to the player's choice and specifying the probability of each outcome of each step given the preceding history, we are in business to routinely calculate what we want to know: the probability the car is behind any particular door given everything that has happened up to the moment the player must decide. Richard Gill (talk) 08:06, 2 December 2012 (UTC)
- This may be what some sources do but it is incomplete. We are given a game in which the player can pick any of three doors. As we are given no information on how this is done we should take is as uniform at random. Does it matter how this initial choice is made? Of course it does. As you know, if the choice is made uniformly then the player's door choice and the location of the car are irrelevant. If the choice were not uniform, we would have to take account of these other two distributions (which, as it happens, we also take as uniform). So for the game, in general, the unconditional game, we must consider how the player initially chooses.
- This is what many sources do, it is completely standard and seems to me completely reasonable too. If you are using probability in the Bayesian sense to measure the beliefs of the player at the moment when he must make his decision, then you are interested in the probability that the car is behind door 2 given the player chose door 1 and the host opened door 3. How you, the player, chose your initial door is irrelevant. Richard Gill (talk) 14:26, 2 December 2012 (UTC)
- Some people (somewhat perversely) consider there to be two conditions implied by the question. These two conditions are: that the player initially chooses door 1; and that the host then opens door 3. The routine way to tackle this problem is to start with the sample space of the unconditional game and then to condition it according to the conditions that we wish to apply. To start half way through the problem with a hand-waving argument about starting where convenient is not acceptable if we want to be thorough. Martin Hogbin (talk) 10:14, 2 December 2012 (UTC)
- This is not perverse, it is eminently sensible. If you want to decide rationally on a course of action you need to know the probabilities of the different possible states of the world, given your information at the moment the decision is needed.
- The perverseness is in assuming that the problem intends to specify door numbers just because it mentions them as examples. We know it was not intended that way by either Selvin or vS. Martin Hogbin (talk) 19:12, 2 December 2012 (UTC)
- What Selvin or vos Savant intended is irrelevant - what's relevant is the literature which grew from their problems. It's also not perverse to introduce door numbers. Introducing door numbers is an eminently sensible way to solve the problem in a principled way. That's why all the probability texts do it that way. That's why Selvin wrote his second paper (because his first was sloppy, incomplete, and many people wrote to him and told him so). In his second paper he writes that his solution is based on the assumption that the host's choice is random. If you rely on this assumption, you need to use it, or your logic is wrong. He implicitly admitted that his first solution was wrong, by his own criteria. Richard Gill (talk) 07:26, 3 December 2012 (UTC)
- The perverseness is in assuming that the problem intends to specify door numbers just because it mentions them as examples. We know it was not intended that way by either Selvin or vS. Martin Hogbin (talk) 19:12, 2 December 2012 (UTC)
- There was no handwaving in my description of the solution. I told you explicitly that we assume that the choice of the player is independent of the location of the car and therefore that we can, and shall, condition on the choice of the player, since anyway we are only interested in the probability of possible locations of the car at the moment the player has to make his decision and given the information which he then possesses. Absolutely sensible, absolutely normal. Richard Gill (talk) 14:26, 2 December 2012 (UTC)
- Not exactly clear or simple. Why not start with all the possibilities and the exclude those that do not meet the specified conditions, just as you do with the host's door choice. Martin Hogbin (talk) 19:12, 2 December 2012 (UTC)
- Clear and simple if you've learnt some formal probability theory. Why not? Because it would be a complete waste of time. Conditioning on the choice of the host is not completely trivial. Conditioning on the choice of the player is completely trivial. Why? Because the choice of the host is dependent on the location of the car and the initial choice of the player. Some careful thinking and/or calculating is required. Richard Gill (talk) 10:31, 3 December 2012 (UTC)
- Not exactly clear or simple. Why not start with all the possibilities and the exclude those that do not meet the specified conditions, just as you do with the host's door choice. Martin Hogbin (talk) 19:12, 2 December 2012 (UTC)
- This is not perverse, it is eminently sensible. If you want to decide rationally on a course of action you need to know the probabilities of the different possible states of the world, given your information at the moment the decision is needed.
- This may be what some sources do but it is incomplete. We are given a game in which the player can pick any of three doors. As we are given no information on how this is done we should take is as uniform at random. Does it matter how this initial choice is made? Of course it does. As you know, if the choice is made uniformly then the player's door choice and the location of the car are irrelevant. If the choice were not uniform, we would have to take account of these other two distributions (which, as it happens, we also take as uniform). So for the game, in general, the unconditional game, we must consider how the player initially chooses.
- Why start half way through the problem? Martin Hogbin (talk) 23:32, 1 December 2012 (UTC)
- PS This is the principled way to solve the problem from the perspective of probability theory. From that perspective, giving a simple solution is simply plain wrong. It only coincidentally happens to give the right numerical answer, and anyway, it does not target what the probabilist wants to know, according to his own "rule book". Note: this is nothing to do with Morgan and nothing to do with biased hosts. Selvin 1975b gave the conditional solution in response to complaints from correspondents that his initial 1975a solution was wrong (that is: the derivation was wrong). Amusingly though, he does not admit it was wrong, and in the same article reproduces the "combined doors" simple solution provided by Monty Hall himself. So he just includes the conditional solution to satisfy the nit-pickers. On the other hand he emphasizes that all his solutions use the unbiased host assumption. Richard Gill (talk) 08:14, 2 December 2012 (UTC)
- Satisfying nit pickers is fine but we must make sure that we pick all the nits. Martin Hogbin (talk) 10:14, 2 December 2012 (UTC)
- No. No own research. We have to survey the literature. It Martin Hogbin has problems with how modern scholars teach probability theory to their students, he had better get his objections published. If anyone takes any notice of them, then in about 10 years Misplaced Pages can start to take account of them too. Richard Gill (talk) 14:32, 2 December 2012 (UTC)
- There is no OR her it is standard practice to start with and unconditional case and the add in the conditions but I am happy to leave these solutions to those who feel that they are necessary.
- No. No own research. We have to survey the literature. It Martin Hogbin has problems with how modern scholars teach probability theory to their students, he had better get his objections published. If anyone takes any notice of them, then in about 10 years Misplaced Pages can start to take account of them too. Richard Gill (talk) 14:32, 2 December 2012 (UTC)
- Satisfying nit pickers is fine but we must make sure that we pick all the nits. Martin Hogbin (talk) 10:14, 2 December 2012 (UTC)
- PS This is the principled way to solve the problem from the perspective of probability theory. From that perspective, giving a simple solution is simply plain wrong. It only coincidentally happens to give the right numerical answer, and anyway, it does not target what the probabilist wants to know, according to his own "rule book". Note: this is nothing to do with Morgan and nothing to do with biased hosts. Selvin 1975b gave the conditional solution in response to complaints from correspondents that his initial 1975a solution was wrong (that is: the derivation was wrong). Amusingly though, he does not admit it was wrong, and in the same article reproduces the "combined doors" simple solution provided by Monty Hall himself. So he just includes the conditional solution to satisfy the nit-pickers. On the other hand he emphasizes that all his solutions use the unbiased host assumption. Richard Gill (talk) 08:14, 2 December 2012 (UTC)
- Popular sources for conditional solutions do seem to feel a strong need to motivate the approach. Especially since they have already, typically, explained some simple solutions.
- Oh well. Misplaced Pages is not a text book. We just survey what's out there: motivation is not relevant. Richard Gill (talk) 08:08, 1 December 2012 (UTC)
- Rick, I have challenged you on several occasions to give sources that say the conditional solutions are necessary in the case where the host is stated to choose evenly between legal doors. I think you may have found one, maybe two. Most other sources which give conditional solutions explain and justify them by considering the case where the host is biased or they do not mentioning the host's strategy. Martin Hogbin (talk) 23:30, 1 December 2012 (UTC)
- Just about every beginner's text book in probability theory gives the conditional solution without motivation and only for the symmetric case because it is the natural principled way to solve the problem from the point of view of probability theory. See Selvin (1975b). Richard Gill (talk) 08:18, 2 December 2012 (UTC)
- All taken from the same original source. Martin Hogbin (talk)
- Just about every beginner's text book in probability theory gives the conditional solution without motivation and only for the symmetric case because it is the natural principled way to solve the problem from the point of view of probability theory. See Selvin (1975b). Richard Gill (talk) 08:18, 2 December 2012 (UTC)
- That's what you say. Everyone who has studied probability theory can solve MHP quite on their own without any need for any source, and they'll almost all solve it the same way that Selvin did in his second, 1975b, note. It's the result of routinely writing down the natural assumptions and then routinely calculating the thing you have to calculate, from first principles. Richard Gill (talk) 14:32, 2 December 2012 (UTC)
But it was a good idea to put the "300 repetitions" into the conditional probability diagram, and take out my splitting into 6 equally likely cases! Richard Gill (talk) 08:13, 1 December 2012 (UTC)
- I do not think so, Why 300 and not 6 as I suggested? Martin Hogbin (talk) 23:31, 1 December 2012 (UTC)
- There are two common ways to make probability calculations intuitive, both of them are based on converting fractions to whole numbers, one goes for large numbers, the other goes for small.
- Large numbers: people think of probabilities in terms of what would happen in many (imaginary) repetitions. Emphasis on many. For this example, thinking of 300 or 600 repetitions is convenient. How many times would each outcome, roughly, occur? About 100 times this, about 100 times that. The story has to have large numbers to become realistic. Talking about what we would expect in 6 repetitions doesn't make sense. Anything could happen. We certainly wouldn't expect the four distinct probability 1/6 outcomes each to occur exactly once and the single probability 1/3 outcome to occur exactly twice.
- Yes I know that, but 600 is not a large number it is a quite small and arbitrary number. We give the impression to our readers that if we repeated the game 600 times we would get the exact results shown, which is , of course, wrong. After a large number of plays the results would tend towards the fractions shown but there is nothing special about 600. To show a total of 600 plays with exact figures is misleading. Martin Hogbin (talk) 10:41, 2 December 2012 (UTC)
- The text talks about 300 now, and says "about 100 ..." and so on. Not misleading at all, and easy to undertand. Richard Gill (talk) 14:16, 2 December 2012 (UTC)
- Yes I know that, but 600 is not a large number it is a quite small and arbitrary number. We give the impression to our readers that if we repeated the game 600 times we would get the exact results shown, which is , of course, wrong. After a large number of plays the results would tend towards the fractions shown but there is nothing special about 600. To show a total of 600 plays with exact figures is misleading. Martin Hogbin (talk) 10:41, 2 December 2012 (UTC)
- Small numbers: when all distinct outcomes (cases) have the same probability, then probabilities are found by counting all cases favourable to the event in question. This is the "table of six" solution which TotalClearance discovered some weeks ago, and which I subsequently found in a number of notable literature references. I recall that Martin and Nijdam both thought it was completely crazy. But anyway, it is now in the article. Richard Gill (talk) 07:55, 2 December 2012 (UTC)
- The table of six would be a good idea if it were indeed a table of six equally likely possibilities, but it is not. It shows exactly the same event twice in order to make the table work.
- It shows one event of probability 1/3 split in two equal parts, e.g. by imagining a fair coin toss being added to the story. This makes the table work and it is both intuitively and mathematically correct. It helps to have a bit of imagination. Richard Gill (talk) 14:16, 2 December 2012 (UTC)
- If you think that adding a coin toss, the result of which is completely ignored, to the problem makes it easier to understand go ahead and add it. Martin Hogbin (talk) 19:40, 2 December 2012 (UTC)
- I did already add that, seems you didn't notice. Moreover I gave a number of authoritative sources. Seems the solution is useful to some people. It does require a bit of imagination. Richard Gill (talk) 07:20, 3 December 2012 (UTC)
- If you think that adding a coin toss, the result of which is completely ignored, to the problem makes it easier to understand go ahead and add it. Martin Hogbin (talk) 19:40, 2 December 2012 (UTC)
- It shows one event of probability 1/3 split in two equal parts, e.g. by imagining a fair coin toss being added to the story. This makes the table work and it is both intuitively and mathematically correct. It helps to have a bit of imagination. Richard Gill (talk) 14:16, 2 December 2012 (UTC)
- The table of six would be a good idea if it were indeed a table of six equally likely possibilities, but it is not. It shows exactly the same event twice in order to make the table work.
- Small numbers: when all distinct outcomes (cases) have the same probability, then probabilities are found by counting all cases favourable to the event in question. This is the "table of six" solution which TotalClearance discovered some weeks ago, and which I subsequently found in a number of notable literature references. I recall that Martin and Nijdam both thought it was completely crazy. But anyway, it is now in the article. Richard Gill (talk) 07:55, 2 December 2012 (UTC)
- So both approaches are now included. Richard Gill (talk) 07:55, 2 December 2012 (UTC)
- What we have is neither fish nor fowl. It does not show the results of a large number of games or an number of equally likely events. Martin Hogbin (talk) 10:41, 2 December 2012 (UTC)
- Martin, you clearly haven't read the relevant section recently. It's all there, it's all carefully explained. Hopefully other people than you will find it illuminating, even if you don't. Richard Gill (talk) 14:16, 2 December 2012 (UTC)
- What we have is neither fish nor fowl. It does not show the results of a large number of games or an number of equally likely events. Martin Hogbin (talk) 10:41, 2 December 2012 (UTC)
- So both approaches are now included. Richard Gill (talk) 07:55, 2 December 2012 (UTC)
Vos Savant, Morgan, and the media furore
I did not get much response last time I mentioned this subject but it is quite a large part of the subject and should have a section in the article. I will start one. I am not fussed about the section title. Martin Hogbin (talk) 12:38, 2 December 2012 (UTC)
I have added a section. What do you think? Do we have a useable picture of vos Savant anywhere?Martin Hogbin (talk) 14:21, 2 December 2012 (UTC)
- There is Vos Savant and the media furore. A completely different affair was Morgan's fuss about vos Savant, which is already covered extensively in the article, and which (IMHO) needs to be downplayed. Richard Gill (talk) 14:35, 2 December 2012 (UTC)
- But anyway - I like the new section. Richard Gill (talk) 15:26, 2 December 2012 (UTC)
- No picture of Marilyn vos Savant on the Misplaced Pages page devoted to her. Richard Gill (talk) 15:31, 2 December 2012 (UTC)
- No, I looked on the vS page too.
- I agree that the Morgan argument is not really media but vS did have a fairly long and public argument with Morgan which followed on from her 'Parade' arguments. This may be of interest to our readers. I did not think the vS/Morgan argument was worth a section on its own so I put it in the vS/media section. There is quite a lot about vS/Morgan in the review literature.
- The vS/Morgan section also helps explain what all the Simple/Conditional solutions are all about to the general reader who is not interested in the maths at all. I end the section by saying that Morgan consider conditional solutions to be necessary even when the host is defined to choose evenly, which is the position of the conditionalists, and you I believe. I do not think I say anything contentious or biased. Martin Hogbin (talk) 18:01, 2 December 2012 (UTC)
- You misunderstand my position. The fact is that different solutions get different conclusions under different assumptions. The numbers 2/3 may be the same but what it means is different. 2/3 of what? My opinion is that it is up to the consumer to choose. The simple solutions and the conditional solutions talk about different things. The decision theoretic solutions yet again. If you want to deduce the probability of winning by switching given you chose door 1 and the host opened door 3 you have more work to do and need to make more assumptions than if you want to deduce the overall probability of winning by switching. I do not say you have to go for one or for the other. I say you legitimately can go for one or the other. The consumer does need to be correctly informed. And false logic has to be exposed.
- The vS/Morgan section also helps explain what all the Simple/Conditional solutions are all about to the general reader who is not interested in the maths at all. I end the section by saying that Morgan consider conditional solutions to be necessary even when the host is defined to choose evenly, which is the position of the conditionalists, and you I believe. I do not think I say anything contentious or biased. Martin Hogbin (talk) 18:01, 2 December 2012 (UTC)
- If your solution is implicitly relying on a fair host assumption but you do not use it explicitly, then your logic is sloppy, your argument, at best, in incomplete. Thesimple solutions don't use this assumption and don't need it either. They draw a weaker conclusion than the conditional solutions. These are mathematical facts of life, like it or not. Richard Gill (talk) 07:16, 3 December 2012 (UTC)
- Do you agree that the Morgan argument forms some kind of introduction to the conditional solutions? It was how things happened historically with the vS question.
- I agree that the Morgan argument can be used as an intro to the conditional solutions. It's a good way to motivate the general reader, and is used by various authorities precisely for this purpose (Rosenhouse, Rosenthal, write popular accounts of the MHP story, and are anxious to motivate and to educate their readers). Historically: look at Delvin 1975a, 1975b. Devlin insisted in 1975b that his solution is based on a "fair host" assumption (or from the Bayesian perspective, from symmetry in our prior knowledge about the hosts 's behaviour). And he presented the conditionsl solution, showing where he uses this assumption. His earlier 1975a unconditional solution did not use that assumption. Clearly, he was criticized for that.
- Misplaced Pages is not a textbook. Our task is to inform, not to educate. I agree that the conditionalists do a poor job in explaining the rationale behind their approach (both in the literature, and here on wikipedia). The preferred motivation seems to be "you have to do it that way, because that's how we mathematicians/probabilists/... do it". A very sad state of affairs. That's why I wrote my own paper on the subject. Richard Gill (talk) 10:14, 3 December 2012 (UTC)
- That is the most important thing, to get the article right. Let me know if I am going too far or you think I am pushing my POV there. I want to add a bit more about what vS did get right, along the lines suggested by Gerhard. Martin Hogbin (talk) 09:40, 4 December 2012 (UTC)
- Looks OK to me. But I wouldn't say "false proof", I would say "incorrect proof". And I wouldn't say "false simulation" but "inappropriate simulation". But going back to her proofs, which are not formal mathematical proofs, but informal arguments: I would not say that her arguments are incorrect, either: they are correct arguments for showing that the unconditional probability of winning by switching is 2/3. She simply shows no interest in the conditional probability of winning by switching. Richard Gill (talk) 12:19, 4 December 2012 (UTC)
- Those were not my words and I agree that they a bit strong, I really do not want to start up the whole 'conditional' argument again. I would suggest the wording, '...given a certain assumption, her answer is correct; her methods of proof however, are not'. This is again pretty much a direct quote from Morgan but a little less inflammatory than Nijdam's wording. Martin Hogbin (talk) 14:07, 4 December 2012 (UTC)
- Inflammatory or not, it was bad use of English. Richard Gill (talk) 06:33, 5 December 2012 (UTC)
- Then let's quote Morgan verbatim. Nijdam (talk) 10:33, 5 December 2012 (UTC)
- Inflammatory or not, it was bad use of English. Richard Gill (talk) 06:33, 5 December 2012 (UTC)
- Those were not my words and I agree that they a bit strong, I really do not want to start up the whole 'conditional' argument again. I would suggest the wording, '...given a certain assumption, her answer is correct; her methods of proof however, are not'. This is again pretty much a direct quote from Morgan but a little less inflammatory than Nijdam's wording. Martin Hogbin (talk) 14:07, 4 December 2012 (UTC)
- Regarding solutions which rely on the fair host assumption, I do understand your position but you do not seem to understand mine. I will try to explain on my talk page. Martin Hogbin (talk) 09:54, 3 December 2012 (UTC)
- You already tried, but you did not convince me. I explained why, but you didn't accept my explanations. How about we leave it at that (agree to disagree)? Richard Gill (talk) 17:11, 3 December 2012 (UTC)
- If you like. Martin Hogbin (talk) 09:40, 4 December 2012 (UTC)
- You already tried, but you did not convince me. I explained why, but you didn't accept my explanations. How about we leave it at that (agree to disagree)? Richard Gill (talk) 17:11, 3 December 2012 (UTC)
The controversy continues
I'm not particularly happy with this title. It suggests a continuation of the controversy between vos Savant and the readers who refuse to belief her. This section, however, introduces the mathematical and logical criticism on vos Savant. Nijdam (talk) 10:57, 5 December 2012 (UTC)
- I reverted uncoordinated edit. Please respect the corporate endeavor. Gerhardvalentin (talk) 13:21, 5 December 2012 (UTC)
- How about the title "Controversy continues"? After all it is a different controversy, now. And how about a link to the section where this criticism is discussed at length, further down the page? Richard Gill (talk) 08:13, 6 December 2012 (UTC)
- Or even, 'A new controversy'. It was not my intention to try to mislead readers into thinking that the Morgan issue was a continuation of the original disbelief in vS. Martin Hogbin (talk) 09:13, 6 December 2012 (UTC)
- Call it: "A more serious controversy". Nijdam (talk) 11:06, 6 December 2012 (UTC)
- Nijdam, I think your latest addition is a bit strong and would better in the 'Conditional' section of the article where it could be explained more fully. I am also not sure 'more serious' is justified, 'more academic' might be better, but I am happy with Richard's "Controversy continues". This does not imply that it is the same issue continuing. Martin Hogbin (talk) 16:51, 6 December 2012 (UTC)
- Call it: "A more serious controversy". Nijdam (talk) 11:06, 6 December 2012 (UTC)
- Or even, 'A new controversy'. It was not my intention to try to mislead readers into thinking that the Morgan issue was a continuation of the original disbelief in vS. Martin Hogbin (talk) 09:13, 6 December 2012 (UTC)
- The text had "and suggested a false simulation as a way of experimental verification". I can't find any statement like that in the paper of Morgan et al. So I deleted this remark.
- Anyway, there is nothing wrong with the simulation. It certainly isn't "false". And it illustrates exactly the point which Marilyn's arguments prove and that she wanted to make, namely that always switching gives the car 2/3 of the time. Morgan et al. proved that there is no way to do better. In that sense they added something interesting and important to Marilyn vos Savant's analyses.
- A different point is that one can use the same simulation to study the conditional probabilities. It simply suffices to study the success rates (when always switching) separated out over the possible doors opened by the host. Morgan et al. already mentioned this in their paper. Richard Gill (talk) 19:09, 6 December 2012 (UTC)
- Yes, the section is meant to be about the controversies which surround the problem. I have written it is a balanced way which does not one side or the other but gives the general reader a feeling for the strong feelings that this problem seems to generate. Some people may find this interesting even if they are not interested in the maths. There is a place to address the actual point raised by Morgan and this is in the conditional section. Martin Hogbin (talk) 19:51, 6 December 2012 (UTC)
Question: what is the stupidest way to "sell" conditional solutions? Answer: how it is done in the article. I think that if you want to "sell" the conditional approach you need to present it in a positive way as adding something new and interesting to the simple solutions, complementing them and strengthening them. If you present it by saying that some professors of mathematics found some problems with vos Savant's solutions, that is the surest way to make sure that no-one will take any notice of it at all. Richard Gill (talk) 19:18, 6 December 2012 (UTC)
- I agree. Some kind of introduction to conditional solutions that indicates what conditional probability is, why it is necessary in some cases, and how it could make an unexpected difference to the answer if the host is biased for example seems like a good idea to me. Rick, at least, objects to this introduction occurring before the solutions themselves because, I presume, he sees this as reducing the impact of the solutions. I think the opposite way; a good introduction makes them more appealing and accessible. Martin Hogbin (talk) 19:51, 6 December 2012 (UTC)
I have two observations on the addition of this "Controversy" section to the present article (which is in addition to the preexisting "Criticism of the simple solutions" section).
- I find it remarkable that a section about "furore" does not clearly cite it's sources, and the subsection on "Controversy" appears to contain errors of attribution.
- Item: "Acting purely as the agent of chance", attributed to vos Savant, looks like what Richard G. Seymann wrote, "viewed as nothing more than an agent of chance", in a comment published with the Morgan, et al. paper. Is there a different, un-cited source in which vos Savant says this?
- M. vos Savant (1991), "Marilyn vos Savant's reply", Letters to the Editor, The American Statistician, vol 45, p. 347. Well worth reading! "Pure probability is the paradigm, and we published no significant reason to view the host as anything more than an agent of chance who always opens a losing door and offers the contestant the opportunity to switch, as Seymann states". Marilyn is saying that her MHP is intended to be a pure probability puzzle which is to be solved in the classical ("objective Bayesian") fashion by allocating equal probabilities to events which can be judged "equally likely" under obvious symmetries. It is not to be solved by speculations on the psychological mechanisms of door choice by a real host in a real show. Richard Gill (talk) 09:26, 7 December 2012 (UTC)
- Item: "Morgan et al responded by saying that ... a conditional solution was still strictly necessary", does not accord with what they wrote in their rejoinder published with Seymann's comment: "Certainly the condition p = q = 1/2 should have been put on via a randomization device at this point. It could also have been mentioned that this means that which of the unchosen doors is shown is irrelevant, which is the basis for solving the unconditional problem as a response to the conditional one." Is there a different, un-cited source in which Morgan, et al. contradict themselves saying, in a response to vos Savant, that a random agent of chance still requires a strictly "conditional" solution?
- Morgan et al. answered vos Savant on the same page. In particular they say "One of the ideas put forth in our article, and one of the few that directly concerns her responses, is that even if one accepts the restrictions that she places on the reader's problem it is still a conditional probability problem". So: they insist that Whitaker's question requires a conditional probability solution, though of course they admit that it can be easily obtained from the unconditional solution plus symmetry. And we have that solution later in the article. Richard Gill (talk) 09:26, 7 December 2012 (UTC)
- More broadly, the article appears to have taken a turn in the opposite direction from that indicated by two of the three closing statements in the recent RfC.
- Closing statement by I Jethrobot: "Long discussions of how a limited number of sources have criticized vos Savant's approach seem unwarranted."
- Closing statement by Churn and change: "Criticism of Savant's solution does not meet our neutral-point-of-view policy."
- Closing statement by Eraserhead1 does not specifically address the issue of criticizing solutions.
I fear my prediction, that framing the RfC in terms of the structural ordering of solutions would not settle the problems of non-neutral point of view and undue weight, has been realized. I encourage contributors to reconsider the wisdom of playing fast and loose with the sources, and of devoting major portions of the article to discussion about what amount to quibbles over ambiguities and formalisms or, mostly, poorly contextualized observations about solving generalizations of host behaviors. ~ Ningauble (talk) 20:14, 6 December 2012 (UTC)
- Dear Ningauble, the fact remains that there is a large literature devoted to quibbles over ambiguities and formalisms. These quibbles are discussed at length, critically, in what seem to me to be the authoritative reliable sources on MHP: Jason Rosenhouse's book, Jef Rosenthal's book chapter. What Misplaced Pages calls tertiary sources. (Primary equals original research articles, secondary equals research surveys concerning the latest developments written by researchers in the field, tertiary equals standard textbooks, written by reputable academic authors and published by reputable mainstream publishers). The wikipedia page surely has to report the existence of these quibbles, in a neutral way. Misplaced Pages can not take sides. It should surveys the disputes neutrally so that the reader can judge for themselves.
- Secondly, the fact remains that almost every student who meets MHP in their introductory statistics class is going to meet the conditional solution there, presented in a matter of fact and direct (constructive) way, "look, this is how we can now solve this famous problem", usually taking it absolutely for granted that the question of whether or not the player should switch should be answered by calculating the probability of the different possible locations of the car given the information which the player has at the moment they are asked whether or not they want to switch.
- So like it or not, the article has to present both popular solutions suitable for ordinary folk to discuss in casual conversation at a pub or a party, and academic solutions suitable for students of probability theory, where naturally both a higher level of precision in use of concepts, and rigour in argumentation with them, is required.
- My guess is that readers of wikipedia and editors of wikipedia come roughly equally divided from these two communities. Notice that since the RfC the content of the article is little changed, what has been going on is reorganisation of the material that was already there. The editors you quote apparently don't like to read criticism of simple solutions. Well, they don't have to. It is now well separated from the solutions themselves; the two kinds of solutions are well separated as well, so people who are happy with simple solutions don't have to struggle with unfamiliar concepts or with arithmetic with fractions. Richard Gill (talk) 09:48, 7 December 2012 (UTC)
- It does seem that I have accidentally sparked off a re-run of the original debate. This was most certainly not my intention, which was to show some of the issues which made the problem so infamous. I would be quite happy to delete or move some of the 'Furore' section if you think it will prevent the original argument from re-starting. The Vos Savant sections are,I think, not so contentious. The fact that thousands of people wrote in telling vS that she was wrong is a major part of the background to the problem.
- To repeat, this section was intended to be an introduction to the history of the problem, which will be of more interest to some people than the mathematics, rather than a detailed discussion of the issues involved.
- Some suggestions I would make are: remove the Furore section entirely, or move it to later in the article. Remove or move just the 'Controversy continues' section. Rewrite the section to have very little mathematical content and to concentrate on the media, social, and psychological issues. I would go with any of these. what do you both suggest? Martin Hogbin (talk) 09:50, 7 December 2012 (UTC)
- As my new section appears to have stirred up such a hornet's nest I have deleted it pending discussion. Martin Hogbin (talk) 09:58, 7 December 2012 (UTC)
- I think the first part of the section on the media furore was excellent; but the "controversy continues" subsection completely out of place. Richard Gill (talk) 10:04, 7 December 2012 (UTC)
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