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Help in understanding

I'm almost fed up with this discussion, but I'll give it another try with this example.

The possessions of a married couple are described by

m for the male

f for the female

The amount of real estate in the possessions are

rm for the male

rf for the female

For the male the real state part in his possessions is: rm/m, for the female: rf/f and in their total possessions: (rm+rf)/(m+f). They concern different quantities, and with prenuptial agreement the values in general differ. Even when coincidently the value of rm/m equals rf/f (rm/m=rf/f), one wouldn't consider the fractions to represent the same thing.

Now what about a couple married in community of property? Of course as a consequence the values of rm/m, rf/f and (rm+rf)/(m+f) are the same. Yet I wouldn't call them the same "thing". I.e. if someone asks for (the value of) rf/f, it is possible to calculate (rm+rf)/2m, but without mentioning that this gives the right value because of the community of property, I would consider this (as a way of reasoning) wrong.Nijdam (talk) 08:27, 2 March 2010 (UTC)

Firstly, thank you for your patience. I am finding it hard to work out exactly where we disagree but I can assure you that I am not being awkward, it is just that I cannot understand your point.

In your example, I understand that are referring to two different things. In the MHP, I fully understand that: the probability that the car is behind door 1 before the host opens a door, the probability that the car is behind door after the host has opened door 3, and the probability that the car is behind door 1 after the host has opened door 2 are all different things. But consider this analogy. Suppose we have a triangle. It has three sides, all of which are different things, we could make them different colours or number them. Now suppose that we are told that all three angles are the same. We can immediately say, by symmetry, and with no knowledge of geometry, that the three sides must be of equal length. Even though we may have numbered the sides, they now become essentially equal even though we may not know the length of any of them.

It is just the same in the MHP. Once we are told that the host chooses a legal door randomly, we know that the probabilities described above must be equal, without calculating any of them. Thus, in order to calculate any one of them, we may calculate any other. This approach is very common in mathematics and has solved many problems that otherwise might be intractable. Martin Hogbin (talk) 10:50, 2 March 2010 (UTC)

Again, it's not about the calculation, but about what has to be calculated. In your example, we know all sides are equal, but if you need to know the size of let's say AB, you have to calculate AB, and it is not sufficient to say BC = 10. You need to add: AB=BC, and because BC=10, also AC=10.Nijdam (talk) 11:26, 2 March 2010 (UTC)

Nijdam, you must have seen this technique used before. We want to know side AB. We first show that AB=BC=CA by a simple symmetry argument, using the fact that were told that the angles are equal. We can do this without knowing the length of any side. Next we calculate BC (maybe in the given question this is easier than calculating AB). Finally we state that length of AB. Martin Hogbin (talk) 11:58, 2 March 2010 (UTC)

Do you say anything different than what I say?Nijdam (talk) 12:22, 2 March 2010 (UTC)

I am not sure, but if you are happy with what I wrote above, why are you not happy with this? We wish to calculate the probability that the car is behind door 1 after the host has opened door 3, that is to say P(C=1|D=3), a conditional probability. (There I have said the word). If the host chooses a legal door randomly we can say by simple symmetry, without any conditional probability calculation, that P(C=1|D=3) = P(C=1|D=2) = P(C=3). We know P(C=1) = 1/3, thus P(C=1|D=3) = 1/3. Yes we have calculated a conditional probability but without worrying about which door the host has opened, because we know it makes no difference. Martin Hogbin (talk) 13:44, 2 March 2010 (UTC)

I assume you mean to say P(C=1|D=3) = P(C=1|D=2) = P(C=1). Better also to add P(C=1|D=1)=0. What you now formulate is nothing more than what Rick and I have been defending all the time against .... well, also against you. Notice that the problem asks for calculating P(C=1|D=3). Nijdam (talk) 23:34, 2 March 2010 (UTC)

Thanks for correcting my typo (I have now used H=3 etc as originally) . I cannot see what we disagree about. Let me restate what I say. We want to calculate P(C=1|H=3) (a conditional probability) but before we start our calculation we note (either on the grounds of symmetry or using information theory) that P(C=1|H=3) = P(C=1|H=2) = P(C=1). Having made this observation we need no longer concern ourselves with the fact that a specific door has been opened or even that a door has been opened at all. We can proceed to calculate our conditional probability exactly as if it were the unconditional initial probability. The fact that these values are the same is not a mathematical coincidence that we need to unearth by calculating each probability separately and then comparing them, it is a mathematical certainty, proved at the start of the calculation. Do you disagree with any of that? Martin Hogbin (talk) 10:00, 3 March 2010 (UTC)

No need to repete. We do not disagree (now), you disagreed with this before. This is what Rick and I and Kmhkmh have said all along. And I hope you see now, that although P(C=1) and P(C=1|H=3) have the same value, seen by calculation or by simple reasoning, they are different probabilities. Got it? And then you have to understand also why the "simple solution" and the "combined doors solution" fail as solutions for this version of the MHP.Nijdam (talk) 10:58, 3 March 2010 (UTC)

I do not see why the simple solutions fail. If we know that the (conditional) probability of interest P(C=1|H=3) must be equal to the original (unconditional) probability P(C=1), the we can proceed to use P(C=1) in our calculations. This is exactly what the simple solutions do. Martin Hogbin (talk) 11:17, 3 March 2010 (UTC)

Just once more: the simple solution is based on P(C=1) and the correct solution on P(C=1|H=3). They share the same value, yet are different probabilities. I won't repeat it any more. You have to understand the difference, or do you not want to understand it. Nijdam (talk) 12:43, 3 March 2010 (UTC)

You are making a non-point. When the question such as 'What is the probability that the car is behind door 1?' is asked it is asking for a calculation of the numerical value of the probability, not a philosophical exposition of the meaning of the word. I perfectly well understand that P(C=1|H=3) is a different entity from P(C=1) but we can show that they must both have the same numerical value. It is again common in mathematics, when two different expressions are shown to to have exactly the same value, to substitute one expression for the other.

The problem, in some formulations, clearly asks for the (value of the) probability that the car is behind door 1 given that the host has opened door 3, P(C=1|H=3), a conditional probability. This value is know to be equal to the value of the unconditional probability (which is a different thing) P(C=1). We can therefore calculate the value of the unconditional probability in order to obtain the value of the conditional probability, which is exactly what the question asks for. Martin Hogbin (talk) 19:29, 3 March 2010 (UTC)

What's your point?Nijdam (talk) 12:55, 4 March 2010 (UTC)

That simple solutions that calculate the value of the probability of winning by switching given that the player has chosen door 1 and host has opened door 3 by using the fact that this value is known to be equal to that of the unconditional probability of winning by switching are correct, when the host chooses a legal door randomly. Martin Hogbin (talk) 15:34, 4 March 2010 (UTC)

If the solution explicitly mention the use of that fact, and of course if you mean ... calculate the value of the conditional probability of winning by switching.... Nijdam (talk) 17:05, 4 March 2010 (UTC)

I have no need use the word 'conditional' if I clearly give the condition in the problem statement. Selvin says that the host chooses randomly when he has a choice and in his solution he says that the probability of the car being behind door 1 is not changed by the host opening door 3. The point therefore is that, although it is a conditional probability, we can calculate the (value of the) unconditional probability in order to find the (value of the) (obviously but not explicitly stated) conditional probability. Martin Hogbin (talk) 17:56, 4 March 2010 (UTC)

You seem to be rather allergic for the word "conditional. I hope you still remember I said, I do not insist on using this term in the starting section of the article, but I do insist on making clear the difference. Do you remember?Nijdam (talk) 16:32, 5 March 2010 (UTC)

I have no objection to the word at all, I am merely pointing out that it is not obligatory to use the word. I also have no objection to mentioning that the problem is one of conditional probability. I am trying to understand what you think is wrong with the simple solutions. They happen not to use the word 'conditional'. Is that the problem? They then calculate the (value of the) unconditional probability in order to find the (value of the) (obviously but not explicitly stated) conditional probability, because we know the values must be equal. What do they do wrong? Martin Hogbin (talk) 17:34, 5 March 2010 (UTC)

@Martin (or anyone else) - which published simple solutions would you consider do what you're suggesting, as opposed to calculating the unconditional probability of winning by switching without any comment as to whether this must also be the (conditional) probability of winning given that the player has chosen door 1 and the host has opened door 3 (using any method whatsoever)? -- Rick Block (talk) 19:46, 4 March 2010 (UTC)

The only source I can think of which mentions that the probability the car is behind door 1 is unchanged by the host revealing a goat behind door 3 is Selvin. There may be others. Vos Savant alludes to this in her response to Morgan. But does not doing this make other sources wrong? It still happens to be true. I have no objection to discussing the issue once the simple problem has been properly and convincingly explained. I cannot see that it helps anyone to introduce this complication before the simle problem has been understood. Martin Hogbin (talk) 17:34, 5 March 2010 (UTC)

It may well be the last I say about this. I'm absolutely stunned you still dare to say: It still happens to be true. As if all you have been taught is in vain. Opening a door by the host changes the probability into a CONDITIONAL probability.Nijdam (talk) 12:09, 7 March 2010 (UTC)

No one is claiming otherwise. As discussed, what is true is that the value of the conditional probability that the player will win by switching given that the player has chosen door 1 and the host has opened door 3, having chosen a legal door uniformly, is exactly equal to the value of the unconditional probability that the player wins by switching given just the game rules. Most sources do not state this fact but is is nevertheless true. Martin Hogbin (talk) 14:34, 7 March 2010 (UTC)

Shorter proof of conditional result

Looking back at the original article I find it not so bad at the moment, the only thing that annoys me is the huge section on the Bayesian approach. If one wants to find the conditional probability (under the extra assumptions of uniform randomness which pedantic teachers of Probability 101 add to the statement of the problem so that dull students can solve it routinely with the tools given in the course) then it is much more elegant to do it with Bayes theorem in the "posterior odds = prior odds times likelihood" version.

I suppose the player has picked door 1. I suppose that indepedently of the player's choice, the car is uniformly hidden behind the three possible doors. Suppose the quizmaster opens a door uniformly at random when he has a chance. Let C denote the random doornumber hiding the car, let Q denote the doornumber opened by the quizmaster

Prior odds:

Prob(C=1) : Prob(C=2) : Prob(C=3) = 1 : 1 : 1

Likelihood of location of car C=1,2,3, given data Q=3

Prob(Q=3|C=1) : Prob(Q=3|C=2) : Prob(Q=3|C=3) = 1/2 : 1: 0

Posterior odds:

Prob(C=1|Q=3) : Prob(C=2|Q=3) : Prob(C=3|Q=3) = 1/2 : 1 : 0

The posterior probabilities are therefore 1/3, 2/3, 0 since they must add to 1 and be in the just mentioned ratios.

This computation is easy to generatlize to the situation where the quiz-master may use other probabilities. As long as the initial distribution of the location of the car remains uniform, the posterior odds are

Prob(C=1|Q=3) : Prob(C=2|Q=3) : Prob(C=3|Q=3) = Prob(Q=3|C=1) : 1 : 0

Since Prob(Q=3|C=1) can conceivably be anything between 0 and 1, the posterior odds on the location of the car being door 1 to door 2 is anything between 0 : 1 to 1 : 1. All we can say is "it never hurts to switch".

Gill110951 (talk) 10:25, 2 March 2010 (UTC)

I had the same objections. Nijdam (talk) 11:29, 2 March 2010 (UTC)

Gill, is this correct:
In situation where the quiz-master may use other probabilities but the initial distribution of the location of the car remains uniform, the posterior probabilities are
Prob(C=1|Q=3) : Prob(C=2|Q=3) : Prob(C=3|Q=3)   =   [anything from 0 to 1/2],   [anything from 1 to 1/2],   0   (they always add to 1).
Okay? -- Gerhardvalentin (talk) 14:48, 2 March 2010 (UTC)

OK, I think that is exactly what I said. You have translated from the language of "odds" (probability ratios) to the language of probabilities. Since probabilities add to one, it is good enough to know the odds. The odds of 400 to 100 and the odds of 4 to 1 are the same, both odds mean exactly the same as the probabilities 0.8, 0,2 . Gill110951 (talk) 10:40, 6 March 2010 (UTC)

A question for all

There is a game show in which there are three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The player chooses a door randomly and, after the player has chosen a door, the door remains closed for the time being. The game show host, Monty Hall, then has to open any one of the two remaining doors that hides a goat and ask the player to decide whether they want to stay with their original choice or to switch to the remaining door. The host has probability q of choosing door 3 when the car is behind door 1 but this fact is not known to the contestant. Given only the above information and that the player has chosen door 1 and that the host has opened door 3. What is the probability of the player winning the car if they switch to door 2 from the perspective of the player?'

(mostly repeat from above) This is the problem that I claim demonstrates the difference between ignorance of the host's preference between two goats as a state of knowledge and simply not knowing the host's preference. My claim is this problem is mathematically identical whether we say the host has a probability q or say nothing about the host's preference and whether we explicitly say "from the player's perspective without knowing q" or not, and that the answer in all cases is 1/(1+p). The player doesn't know q, but knows the host might have a preference and by knowing she's picked door 1 and the host has opened door 3, the fact that the host might have a preference (although not the value) is within the player's state of knowledge. Somewhat perversely, this means a player who picks door 1 (without knowing the host's preference) has exactly a 2/3 chance of winning before the host opens a door, but after seeing which door the host opens has a chance between 1/2 and 1. The player's average probability is 2/3, but the question is asking for the conditional probability, not the average probability. I personally consider this a somewhat pedantic point and definitely WP:OR, but (for whatever reason) Martin seems extremely interested in other opinions about this. -- Rick Block (talk) 20:07, 4 March 2010 (UTC)

This is of course a demonstration of the new Blockian probability theory: If the value of the parameter q is unknown to you, just replace it by the parameter p, and continue to solve the problem in a classic way.

I'm with Rick, and I'm not in favour of applying the principle of indifference, or whatever principle there is to apply. If something is unknown, just say so, if you make some assumption, mention it.Nijdam (talk) 16:49, 5 March 2010 (UTC)

You seem to be agreeing that the question is, to say the least, unconventionally stated. But we have specified a parameter q, which could have a definite value. From the player's perspective you can, of course, chose to parameterise the host's unknown door preference with a new but unknown parameter p. Suppose q were equal to 0.2. What would be your answer then? Martin Hogbin (talk) 17:04, 5 March 2010 (UTC)

Principle of indifference

You seem to be saying that I insist on applying the principle of indifference. That is not so. So what would be your answer to this question:

There is a game show in which there are three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed behind the doors before the show. The player chooses a door and, after the player has chosen a door, the door remains closed for the time being. The game show host, Monty Hall, then has to open any one of the two remaining doors that hides a goat and ask the player to decide whether they want to stay with their original choice or to switch to the remaining door. The host has probability q of choosing door 3 when the car is behind door 1. Given only the above information and that the player has chosen door 1 and that the host has opened door 3. What is the probability of the player winning the car if they switch to door 2? Martin Hogbin (talk) 18:20, 5 March 2010 (UTC)

You haven't said what the initial distribution is, so assuming an initial distribution of p1:p2:p3 (where they sum to 1), the probability is p2/(p2+q*p1) - which (of course) simplifies to 1/(1+q) if the initial distribution is (or is assumed to be) uniform, and to exactly 2/3 if both the initial distribution and q are (or are assumed to be) "random". The question is much more complicated if the initial choice of door is random. -- Rick Block (talk)

So what is your answer? Martin Hogbin (talk) 22:04, 5 March 2010 (UTC)

I thought the above was already rather a complete answer. -- Rick Block (talk) 02:05, 6 March 2010 (UTC)

Of course you have Laplace behind you if you follow the principle of indifference, but you also run into self-contradictions and stupidity. Nowadays the principle of indifference is not so popular any more. If you are a full blooded subjective Bayesian then every single probability you talk about is just an expression of your subjective feelings, measured objectively by how much you are prepared to bet. So if you are prepared to bet even odds in either direction on the hosts, preferences, then you "have" probabilities of 50:50. But there's no principle saying you *HAVE* to be indifferent. Anyway, not everyone thinks it's so useful to treat all uncertainty in the way of a fundamentalist subjectivist. All knowledge is uncertain, all is sujective, everything is measured with subjective probabilities, all we have and all we need is Bayes theorem.Gill110951 (talk) 10:44, 6 March 2010 (UTC)

My point is that whatever you do, apply the principle of indifference, do not apply it, apply some real-world knowledge based on current or older TV game shows you must be consistent. There is no plausible justification for taking the car to be initially placed uniformly but the host's legal door choice to be non-uniform. This is Morgan's conjuring trick. Martin Hogbin (talk) 11:19, 6 March 2010 (UTC)

@Rick, it looks like three answers to me. For someone who is adamant that there is only one real answer to the MHP I find this surprising.

Of course you know full well what I am getting at. Your three answers can be summarised:

1. Car placed uniformly, host chooses a legal door uniformly - answer 2/3
2. Car placed uniformly, host chooses a legal door non-uniformly - answer 1/(1+q), which is always at least 1/2.
3. Car placed non-uniformly, host chooses a legal door non-uniformly - answer indeterminate. Can be any value from 0 to 1.

In 1 above we consistently apply the principle of indifference.

In 3 above we consistently do not apply it and leave the unknown distributions as unknown or parameterise them, as Nijdam prefers.

Can you give me any logical rationale for choosing option 2. Martin Hogbin (talk) 11:13, 6 March 2010 (UTC)

I'm adamant that there's only one real answer? Now wait a minute. It seems to me you're the one who has insisted for over a year that the "simple" answer is the one and only necessary answer and that addressing the problem conditionally is merely an academic diversion from the true simple approach. What I'm actually adamant about is that the POV that the problem asks about the conditional probability must not be ignored.

You've asked for the logical rationale for choosing option 2 innumerable times before, and I've answered innumerable times. Is there something different you're expecting to hear this time? As I've said before, the reason Morgan et al. (and Gillman) address the problem this way is because this is how they interpret vos Savant to have addressed it (not THEIR choice, HER choice). This option (not coincidentally) also has the pedagogical value of exposing the difference between an unconditional solution (which will say 2/3 regardless of the host's preference) and a conditional solution.

Can you give me the logical rationale for asking the same question over and over again? -- Rick Block (talk) 20:58, 6 March 2010 (UTC)

That's beautiful. You're whole argument relies on YOUR INTERPRETATION OF how MORGAN INTERPRETS vos Savant. Despite all evidence indicating otherwise, including her discussion with Morgan. Glkanter (talk) 21:22, 6 March 2010 (UTC)

I think I should quit while I am ahead. Martin Hogbin (talk) 22:41, 6 March 2010 (UTC)

Discussion of some literature references

Pigeons are smarter than mathematicians

http://people.whitman.edu/~herbrawt/HS_JCP_2010.pdf

Are Birds Smarter Than Mathematicians? Pigeons (Columba livia) Perform Optimally on a Version of the Monty Hall Dilemma.

Walter T. Herbranson and Julia Schroeder Whitman College

Journal of Comparative Psychology © 2010 American Psychological Association 2010, Vol. 124, No. 1, 1–13

The “Monty Hall Dilemma” (MHD) is a well known probability puzzle in which a player tries to guess which of three doors conceals a desirable prize. After an initial choice is made, one of the remaining doors is opened, revealing no prize. The player is then given the option of staying with their initial guess or switching to the other unopened door. Most people opt to stay with their initial guess, despite the fact that switching doubles the probability of winning. A series of experiments investigated whether pigeons (Columba livia), like most humans, would fail to maximize their expected winnings in a version of the MHD. Birds completed multiple trials of a standard MHD, with the three response keys in an operant chamber serving as the three doors and access to mixed grain as the prize. Across experiments, the probability of gaining reinforcement for switching and staying was manipulated, and birds adjusted their probability of switching and staying to approximate the optimal strategy. Replication of the procedure with human participants showed that humans failed to adopt optimal strategies, even with extensive training.

[end of extract]

%%%%

If a pigeon is on a quiz-show, does it compute a conditional probability? Marilyn vos Savant/Craig Whitaker's question is not "what is the probability?", it is "what do you do". What action do you take. Decision theory. Game theory. (Is Monty Hall trying to cheat you?). The pigeons make mistakes and don't waste much time thinking. They learn from experience. Gill110951 (talk) 11:35, 6 March 2010 (UTC)

Something interesting at last. They should try it with Magpies. Martin Hogbin (talk) 13:43, 6 March 2010 (UTC)

I have just read the paper. It is fascinating, far more interesting that Morgan. This is the kind of thing we want in the article. Martin Hogbin (talk) 14:21, 6 March 2010 (UTC)

Isn't it great? Yes, next try it on magpies. Then dolphins. .... At some point intelligence will defeat youGill110951 (talk) 17:53, 6 March 2010 (UTC)

Mathematical Economics

The Journal of Economic Perspectives vol 1 nr 2 pages 157-163, Fall 1987

Published by American Economic Association

http://www.jstor.org/stable/1942987

Puzzles

Choose a Curtain, Duel-ity, Two Point Conversions, and More

Barry Nalebuff

In presenting economic puzzles, I have three goals in mind: some puzzles are chosen to stimulate research; others offer examples that will help undergraduate and graduate teaching; all should provide quality distractions during seminars. As usual, this feature begins with several speed puzzles, with answers provided at the end of the problems. Following are several longer puzzles, for which readers are invited - nay, challenged - to submit their own answers. The responses will be discussed in a future issue. I also encourage readers to share their favorite teaching problems and research puzzles (with answers, please). Please send your answers and favorite puzzles to: [Barry Nalebuff, Princeton.] Good luck.

Puzzle 1. Free to Choose

The TV game show "Let's Make a Deal" provides Bayesian viewers with a chance to test their ability to form posteriors. Host Monty Hall asks contestants to choose the prize behind one of three curtains. Behind one curtain lies the grand prize; the other two curtains conceal only small gifts. Once the costumed contestant has made a choice, Monty Hall reveals what is behind one of the two curtains that was not chosen. Now, Monty must know what lies behind all three curtains, because never in the history of the show has he ever opened up an unchosen curtain to reveal the grand prize. Having been shown one of the lesser prizes, the contestant is offered a chance to switch curtains. If you were on stage, would you accept that offer and change your original choice?

[ end of puzzle 1 ]

%%%%%%%%%%%%%%%%

[ fast forward to near end of paper ]

Answer to Puzzle 1

This puzzle is one of those famous probability problems, in which, even after hearing the answer, many people still do not believe it is true. You should always switch curtains. Here are two ways of understanding why. First, with probability1 /3 you picked the correct door . In this case, if you switch curtains you will certainly lose. With probability 2/3, you picked the wrong door. In this case, Monty Hall does not have a choice of which curtain to show you if he wants to keep the grand prize hidden. If you switch, you will certainly win it. Since your chance of initially picking the correct curtain was 1/3, if you never switch your chance of winning must remain at 1/3. In contrast, if you always switch, you win whenever your first choice was mistaken and this gives you a 2/3 chance of winning.

Second, before Monty Hall showed you anything, you had a 1/3 chance of having picked the correct curtain. Since two wrong choices are possible, Monty Hall can always open up one curtain and not reveal the grand prize. After he has done this, your chance of winning is still1/3, since you have learned no information about the door you chose. That leaves only one other unopened door which correspondingly must have a 2/3 chance of being the correct one (since probabilities always add up to one). Repeated experiments have shown that individuals make systematic violations from behavior predicted by von Neumann-Morgenstern expected utility theory. Since many if not most individuals choose to stay with their original choice, does this suggest we should look for alternatives to Bayes rule?

[ end extract ]

%%%%%%%%%%

Notice his mention of von Neumann-Morgernstern - every economist understands what he is talking about. And how he shows no interest whatever in computing a conditional probability. Note that he gives no literature references whatever. This is a speed-puzzle for people who are bored during a dull research seminar.

Gill110951 (talk) 11:55, 6 March 2010 (UTC)

Monty Hall for the Mathematical Association of America (Barbeau)

Edward J. Barbeau (2000) "Mathematical Fallacies, Flaws and Flimflam", a book published by the Mathematical Association of America, gives some history and the connections to related problems and many, many references. The infamous paper of Morgan et al is included in the list of publications, that's all. He presents as solution the simple statement "always switching gives you the car with probability 2/3". He doesn't write much more himself since the problem has already had so much attention (I have the impression that he finds it a bit boring, by now). All this seems to me to be a very reputable source for taking the simple unconditional problem and simple unconditional solution as perfectly legitimate and widely accepted. He includes in his list the prisoners' problem which of course is a problem of conditional probability! Gill110951 (talk) 12:46, 6 March 2010 (UTC)

Reality Check

The only difference between the two situations is whether the host had prior knowledge of where the car was when he revealed the location of a goat.

"In this version, the host forgets which door hides the car. He opens one of the doors at random and is relieved when a goat is revealed."

People seem to be forgetting that in both cases he reveals a goat. Once that goat is revealed, the chances that he revealed the car are 0.

So vos Savant was actually incorrect when she said it didn't matter what you did if the host didn't know where the car was, because other than the state of mind of the host the situations are identical.

More generally, if informally, stated: When answering a question of the form "If X occurs, what then?" you should assume a 100% chance of X occurring. —Preceding unsigned comment added by 71.63.76.128 (talk) 03:31, 24 June 2010 (UTC)

That is the surprising fact that this problem reveals. That it does matter whether the host know where the car is. Martin Hogbin (talk) 07:49, 24 June 2010 (UTC)

Which problems and solutions are conditional?

This is a continuation of a discussion from the main MHP talk page.

The Monty hall problem is one of conditional probability. We are given a game with a set of rules, which always must be the case and we are (in some formulations) given some conditions, which need not have been so but which we take to be the case in the example we are considering. In some problem statements the car is defined to be placed uniformly at random and the host is defined to choose a goat door uniformly at random, in other statements neither distribution is defined.

The unconditional case is simply that the producer places the car behind a door, player chooses a door, and the host opens an unchosen door to reveal a goat. The sample set consists of cases where all possible doors allowed by the rules have been chosen by the producer, the player and the host. Any problem based on applying conditions to this situation must start with the original unconditional sample set and condition it appropriately. Any solution to such a problem which does not do this is incomplete. Martin Hogbin (talk) 13:24, 20 July 2010 (UTC)

Is this your own opinion based on your WP:OR, or is it published somewhere? It might be helpful to use standard formal notation to clarify your point. The fully unconditional case is

1) $P({\text{win by switching}}\ |\ I)$

where I is the game rules. This is the probability of winning knowing the game rules. It is the probability of winning if the player decides whether to switch before initially picking a door.

The partially conditional case is

2) $P({\text{win by switching}}\ |\ {\text{player picks door 1,}}\ I)$

where the doors have possibly been renumbered so that the one the player chose can be referred to as #1. This is the probability of winning if the player decides whether to switch after picking a door, but before the host opens a door.

You cannot just renumber doors in probability problems. If you wish to do this you must provide a rationale as to why this can be done. Even then it is a short cut based on symmetry or the like. Martin Hogbin (talk) 14:58, 20 July 2010 (UTC)

Wrong! Of course you can "renumber the doors in probability problems" - or rename the variables, which is what you mean - any way you like. You just need to show that your argument is invariant with respect to the renaming, i.e. that the renamed conclusion equals the original conclusion applied to renamed variables.

That is exactly what I said. You need to prove that the doors can be renumbered without affecting the result. This is only true in the fully symmetrical case, in which case it also shows that the door opened by the host does not matter.

Sometimes this invariance is not obvious, and must be proven. Some other times it is obvious, and we say "without loss of generality, rename ...". In the MHP under the standard (K&W) expression this invariance is obvious, because BOTH the contestant and the host agree on the identity of (a) the door chosen by the contestant, (b) the different door opened by the host, and (c) the unique remaining third door. As these identities are perfectly established in the state of knowledge of both the player and the host, we are allowed to attach labels to them (i.e. "name them") any way we like, since the same names will apply to both the player and the host. Now, do you see any appeal to symmetry in all of this? What short cut are you talking about? glopk (talk) 15:51, 20 July 2010 (UTC)

You have appealed to symmetry. You have almost given the definition of it.

@Martin, where o where have I appealed to symmetry above? Please give the exact quote. glopk (talk) 23:36, 20 July 2010 (UTC)

The fully conditional case is

3) $P({\text{win by switching}}\ |\ {\text{player picks door 1 and host opens door 3,}}\ I)$

where the doors have possibly been renumbered so that the one the player chose is referred to as #1 and the one the host opened is referred to as #3. This is the probability of winning if the player decides whether to switch after both picking a door and then seeing which door the host opens. Are you suggesting computing probability #3 is somehow incomplete?

How many times are you renumbering these doors! I have no idea if it is complete or not if you keep moving the doors around. Can you do it again with real fixed numbers on the doors (unless they are indistinguishable). Martin Hogbin (talk) 14:58, 20 July 2010 (UTC)

It may be the case that I implies probability #3 is the same as probability #2 and/or probability #1, but since the decision point is after seeing which door the host opens the MHP ALWAYS directly asks about probability #3 while the simple solutions are addressing either #1 or #2. -- Rick Block (talk) 14:27, 20 July 2010 (UTC)

I can only answer your questions if you keep the doors still. Martin Hogbin (talk) 14:58, 20 July 2010 (UTC)

You do know what "without loss of generality" means, don't you? If you'd prefer, lets call the door the player chose door A and the door the host opens door C where A and C are both in {1,2,3}. In frequency terms, if we're talking about 900 instances of the show where the player's initial pick is random (uniform), probability #1 is what we observe for all 900 instances, probability #2 is what we observe for the (roughly) 300 instances where other players have initially picked the same door as the player we're talking about (they all picked door A for the same value of A as the player we're interested in), and probability #3 is what we observe for a specific subset of players (roughly 150 in the fully symmetric case) who all both initially picked the same door (door A, for a particular value of A) and see the host open the same door (door C for the same value of C as what the player we're interested in sees). Please tell me which of the above probabilities you're talking about, i.e. what is your sample set. -- Rick Block (talk) 15:26, 20 July 2010 (UTC)

Of course I know what "without loss of generality", the problem is that you have to prove this rather than just assert it. We all know that in the standard problem formulation we can let the host open door 3 without loss of generality. Why do you not just call the doors 1,2, and 3 like we have all been doing up to now. The doors are distinguishable so we can us whatever we like to distinguish them. Perhaps you could restate your case using door numbers. Martin Hogbin (talk) 17:30, 20 July 2010 (UTC)

Of course the doors are distinghuishable, but Rick is being careful to point out that each of his two "conditional" cases is in fact a set of decision problems: the "partially conditional" one represents the set of decision problems where it is known which door the player has selected, and the "fully conditional" one represents the set of decision problems where, in addition, the host has already opened the door. Re-numbering in this case is used to express each set using just one symbol. Equivalently, he could write:

"Partially conditional decision problem"

2) The set $\{P({\text{win by switching}}\ |\ {\text{player picks door i AND}}\ I):i\in \{1,2,3\}\}$

"Fully conditional decision problem"

3) The set $\{P({\text{win by switching}}\ |\ {\text{player picks door i AND host opens door j AND }}\ I):(i,j)\in \{1,2,3\}\times \{1,2,3\}\}$

and where, as I pointed out above (see "Wrong!") the choice of door numbering is (legitimately) arbitrary.glopk (talk) 17:56, 20 July 2010 (UTC)

What is a partially conditional problem? And why not just use ordinary door numbers?Martin Hogbin (talk) 21:51, 20 July 2010 (UTC)

Rick, you have the right idea, but use the wrong expressions to make your case. Indeed, the fully marginalized ("unconditional") case is:

1) $P({\text{win by switching}}\ |\ I)$

But the other cases to consider are best expressed as world states given the rules, namely :

"Partially conditional case" (I will retain your names for clarity):

2) $P({\text{win by switching}}\ {\text{AND player picks door 1}}\ |\ I)$

"Fully conditional case":

3) $P({\text{win by switching}}\ {\text{AND player picks door 1 AND host opens door 3}}\ |\ I)$

From the above it is obvious that the "unconditional" probability (1) is simply obtained by marginalization (i.e. summing) over player' picks and host's opening. (3) also yields in an obvious way the only probability the rational player must base their decision upon in a particular game:

4) $P({\text{win by switching}}\ |\ {\text{player picks door 1 AND host opens door 3 AND }}\ I)$

which is obtained by "conditioning" probability (3) with respect to all the non-goal variables (i.e. moving them to right of the '|'). glopk (talk) 16:12, 20 July 2010 (UTC)

The question remains what does Martin think he's talking about. There are three different decision points corresponding to three different formal probabilities and three different populations of players

1) fully unconditional, decision point is before the player has picked an initial door, applies to all players (this is what Carlton's "simple solution" addresses)

2) "partially conditional", decision point is after the player has picked an initial door but before the host has opened a door, applies to all players who pick the same door this player has picked (this is what vos Savant's solution addresses)

3) fully conditional, decision point is after the player has picked an initial door and after the host has opened a door, applies to all players who pick the same door this player has picked and see the host open the same door this player sees the host open (this is what the conditional solutions address)

@Martin - which of these are you actually talking about? -- Rick Block (talk) 17:28, 20 July 2010 (UTC)

Carlton's decision tree shows that his solution, while simple, is conditioned on a door being opened to reveal a goat after the contestant has chosen a door. That it is a 100% condition, is of course consistent with the probabilities not changing from 1/3 & 2/3. As per Selvin, vos Savant, Carlton, Morgan (corrected), Adams... Glkanter (talk) 10:46, 21 July 2010 (UTC)

This is what I am talking about:

Car hidden behind Door 3	Car hidden behind Door 1		Car hidden behind Door 2
Player initially picks Door 1
Host must open Door 2	Host randomly opens either goat door		Host must open Door 3
Probability 1/3	Probability 1/6	Probability 1/6	Probability 1/3
Switching wins	Switching loses	Switching loses	Switching wins
If the host has opened Door 2, switching wins twice as often as staying		If the host has opened Door 3, switching wins twice as often as staying

In the assumed problem statement we are told that the player has chosen door 1 then the host has opened door 3, however, the diagram above shows other options. It shows the host opening door 2, even though we know that he did not actually do this. Why? Because the knowing the doors that the host might have opened can be important in probability problems, although in this case it does not actually matter.

By the same token, we should show all the doors that the player might have chosen for the diagram to be complete. We all know that in this example it makes no difference but that is not always the case. Martin Hogbin (talk) 18:15, 20 July 2010 (UTC)

Man, this gives a new and literally graphic meaning to the word "sophistry" :-) @Martin, Rick has asked one very specific question, framing it in terms that are mathematically well defined in the context of the MHP under K&W. Care for answering him, instead of weaseling out with pretty pictures? I'll repeat the question for your convenience: which of Rick's three classes of decision problems are you talking about when speaking of the "unconditional case" at the head of this section? We really can't have an intelligent discussion unless we know what you are talking about, and there really are no alternatives other than those three: they account for all the relevant variables and all steps in the game. glopk (talk) 18:25, 20 July 2010 (UTC)

The diagram above is exactly what I am talking about. If Ricks questions do not relate to that diagram then they do not relate to my point. Martin Hogbin (talk) 21:51, 20 July 2010 (UTC)

In one version of this image, the caption under the "host opens door 2" column is "If the host opens door 3, this case has not happened". It's shown in this version not because it might be important what door the host opens, but to be a better match for vos Savant's table (both of these were in the same section of the article at one point) showing both the partially conditional and fully conditional views (the difference between these is actually the crux of the problem - everyone understands the former is 1/3, it is only the latter that confuses people). If you're looking for a fully exploded decision tree, the one in Grinstead and Snell should suffice. The question however remains the same. What is the decision point are you talking about - before the player makes the initial pick, after the player picks but before the host opens a door, or after both of these? If you prefer, an equivalent question is what player population are you talking about - all players, all players who pick some particular door (about 1/3 of all players if the car's location is uniformly random or the player's choice is random), or all players who pick some particular door and see the host open some other particular door (about 1/6 of all players if the games rules make the problem symmetrical)? These each correspond to exactly one of the three possible formal probability expressions. If you won't or can't say which of these you're talking about then either you're being deliberately obfuscatory or you're confused. -- Rick Block (talk) 20:03, 20 July 2010 (UTC)

It is exactly the full decision tree, as in G&S that I am looking for as the only complete solution. The ones given in the article fail for slight variations of the problem. You are not making any sense regarding player populations. In the normal formulation the player chooses uniformly at random. I am applying the condition that the player has, in fact, initially chosen door 1, because this is given in the problem statement. Martin Hogbin (talk) 21:51, 20 July 2010 (UTC)

@Martin, you are still not answering, or rather, you are giving TWO contradictory answers. If for your "unconditional case" you care for "the full decision tree, as in G&S [...] as the only complete solution", then the "unconditional case" is the decision problem:

1) $P({\text{win by switching}}\ |\ I)$

If, on the other hand you apply "the condition that the player has, in fact, initially chosen door 1", then your "unconditional case" is the decision problem:

2) $P({\text{win by switching}}\ |{\text{player picks door 1 AND}}\ I)$

where the choice of the door numbering can be arbitrary without loss of generality. Again, please give a straight answer: which of these two decision problems is what you call the "unconditional case"? YOU opened the second paragraph of this section describing an "unconditional problem". All Rick and I are asking is that you define it in unequivocal mathematical terms, and we have laid out the terms of choice for you. Can you please choose? If not, I ask with Rick: are you being deliberately obfuscatory or are you confused? glopk (talk) 22:55, 20 July 2010 (UTC)

My first paragraph is a little confusing. All I am saying is that any solution to what you call the fully conditional problem must consist of the full decision tree, as given in G&S, or an equivalent analysis. The solution given in the article to the (fully) conditional problem is incomplete.

Any argument based a reduced initial sample set, claiming to be valid without loss of generality, by symmetry, by door renumbering, or by any other logical means must be justified. Martin Hogbin (talk) 11:54, 21 July 2010 (UTC)

Editing break

The problem statement mentions the specific case of door 1 and door 3, which is the case analyzed by the conditional solutions. This is the primary justification for examining this case - i.e. because the problem statement says to. Under the standard interpretation, it turns out (and one might think the originator of the problem knew this) that the solution has no dependency on the specific pair of doors, which means this solution applies equally to any pair of doors. But even without saying this, a conditional solution that addresses the one and only case of player picks door 1 and host opens door 3 is complete because it is the case the problem description asks about. The other (usually, but not always, unstated) justification is that because the doors are numbered we can turn any particular case we'd like into a door 1 and door 3 case by renumbering the doors. This is an absolutely standard technique in probability analysis. What it means is NOT that we're assuming symmetry, but that if the player picks door i and the host opens door j leaving door k closed, then anywhere the solution says "door 1" it should be read as "door i" (meaning "the door the player initially picked, whatever door that is") and anywhere the solution says "door 2" or "door 3" should be read as "door k" or "door j", respectively. If the solution in the case of door 1 and door 3 has any dependency on the specific doors involved, as Morgan's 1/(1+q) does, then the dependency pertains to each specific case (not just the door 1 and door 3 case, but the door i and door j case - e.g. Morgan's q is any of six different host preferences, it is specifically the preference for door k over door j given the player has initially picked door i).

I have no objection to examining the specific conditions that the player chose door 1 and the host opened door 3 but the diagram above is not the way to do it. The correct way to do it is shown in G&S.Start with the complete sample set (based on the fixed game rules) and condition it based on the conditions (player and host choices) given in the problem statement.

Your bit about swapping door numbers is assuming symmetry with respect to doors, that is exactly what the term symmetry means, that you can change something (door numbers) but the answer (probability of winning by switching) stays the same. In the totally symmetrical situation, in which the host chooses a goat door uniformly at random, there is a symmetry with respect to door numbers (thus the simple solution is correct) in all other cases there is not, and the full tree is required.

I have given you a source (G&S) that recommends use of the full tree, can you give me any source that suggests swapping round door numbers? Martin Hogbin (talk) 17:47, 21 July 2010 (UTC)

I'm assuming what you're doing here is trying to reflect the criticism made against the unconditional solution back onto the conditional solutions. Let's contrast these. First, are there any sources that claim conditional solutions that only mention door 1 and door 3 are incomplete? If not, then from the perspective of the article it's just WP:OR (and from the perspective of this page it seems like arguing for the sake of arguing, since I truly believe you understand all this already). Second, presenting an unconditional solution without justifying the problem is symmetrical (which there are sources that criticize) directly addresses one of the first two formal probabilities (from way above at this point). At face value, they address the wrong problem (the decision point is not as described in the problem). Perhaps the authors of these solutions mean to address the fully conditional case, but because they (generally) say nothing about it their intent is ambiguous. Maybe they mean their solution applies to all cases because the problem is clearly symmetrical, but because they don't say this (and, even if they did, they should then justify why the problem is symmetrical) we simply don't know. These solutions certainly can be justified, but the fact is that they (at least typically) are not.

I am glad that you understand what I am getting at, perhaps you could explain it to Glopk. G&S state that they have resisted the temptation to reduce the tree size by making certain assumptions, in the interest of clarity. I suggest that we follow their example.

We have a source (Falk) that clearly states that if the host chooses a goat door randomly and the car is initially placed randomly the situation is totally symmetrical with respect to doors so we can ignore the door number opened by the host and the simple solution thus stands. I do not know of any sources that tries to explain the problem by swapping door numbers.

So, really, there's no contest here. On the one hand we have sourced, legitimate criticisms of the unconditional solutions. On the other hand we have an unsourced, BS criticism of conditional solutions. As Nijdam says, it's time for this to end. -- Rick Block (talk) 14:21, 21 July 2010 (UTC)

Quite the reverse, we have a source recommending the full tree solution for the conditional case and a source that describes the use of the symmetry argument to justify the simple solution as impeccable. We have no sources recommending swapping the door numbers round.

It is time for this to end when we reach a consensus. So far at least three editors agree with me. Martin Hogbin (talk) 17:58, 21 July 2010 (UTC)

@Martin, thanks for your non-reply. It is not acceptable and a clear indication of lack of either competence or good faith on your side. Further debating you about an "unconditional" solution to the MHP that you cannot or will not clearly define is pointless. I won't ask again, I have better things to do with my time than wondering about your unability to communicate in a fair and competent manner.

Further, your assertion that the door numbering in the MHP (under K&W) cannot be chosen arbitrarily without recourse to symmetry is plainly wrong, and again shows your lack of competence (or good faith). The MHP under K&W would not be symmetric without the explicit requirement that the host chooses randomly and uniformly between goat doors when two are available to open, but even without such requirement a door numbering could be chosen arbitrarily in any analysis without loss of generality, since such numbering only depends on the fact that the K&W statement unambiguosly assigns a unique "role" ("identity") to each separate door, and both the host and the contestant are equally aware of said roles. Because of such shared awareness, a one-to-one mapping is always well defined between any particular numbering that the contestant, the host, and anyone else care to choose. None of this is controversial, and your attempt to make a controversy out of it suggests a fondness for arguing for its own sake.

@Rick. I concur, this has to end. If mediation won't do it, I suggest that the entire content of this thread could be used for further escalation - as clear proof of incompetence or malice on Martin's part. glopk (talk) 15:50, 21 July 2010 (UTC)

I really have no idea what your complaint is but your threatening tone is not justified. I have conducted a civil discussion on the topic of the article on a talk page specially set up for that very purpose. Rick seems to understand perfectly what I am getting at as is shown in his reply above. He does not agree with me, that is why we are continuing to discus the subject. There is no WP policy concerning 'failing to agree with Glopk' that I am aware of. Martin Hogbin (talk) 17:11, 21 July 2010 (UTC)

Well before any lack of agreement with Glopk, the above paragraphs show that you can't even agree with yourself about what you mean. I am not threatening, I am requesting some basic intellectual honesty without which no conversation is possible. I asked you three times yesterday to state precisely what you meant, and always you attempted to weasel out with rethorical or even pictorial (!) artifices, never answering one simple, clear, well-stated question. Were I still assuming good faith on your side, the unescapable conclusion is that you are hopelessly confused. But now I strongly suspect you aren't, and that you refuse to give a precise mathematical statement of your "unconditional" case because that would put you in the unwelcome (for you) position to have to admit that it is not a solution of the MHP under K&W, but of a different and less well-specified problem. And therefore you continue to argue and hope to win the point by exhaustion. I may sound grumpier than Rick Block, but rest assured that we both understand what you write and what you fail to write. He called your BS above in as many words. This BS has been going on for too long, and its effect has been to put the FA-rated article on mediation and edit lock. Again, this has to stop. Please stop. glopk (talk) 18:20, 21 July 2010 (UTC)

I have no idea what you are talking about when you ask for a 'precise mathematical statement of your "unconditional" case'. The point that I wish to make has been understood perfectly well by Rick. Perhaps you could ask him to explain it to you. Martin Hogbin (talk) 21:35, 21 July 2010 (UTC)

You still have no idea? Honestly? OK, let's try one last time: YOU started this discussion saying (direct quote):

The unconditional case is simply that the producer places the car behind a door, player chooses a door, and the host opens an unchosen door to reveal a goat. The sample set consists of cases where all possible doors allowed by the rules have been chosen by the producer, the player and the host. Any problem based on applying conditions to this situation must start with the original unconditional sample set and condition it appropriately. Any solution to such a problem which does not do this is incomplete..

Rick immediately replied asking YOU to clarify what is the question asked in this "unconditional case" of yours, and laid out in mathematical form the only possible answers. I'll repeat his question in a form that entirely sidesteps any issue about door numbering. Here is the question:

Which of the following three sets of probabilities directly affects your "unconditional case", in the sense that they are the only probabilities a rational MHP player would consider in order to decide whether to switch or stick?

(1) The set of one element $\{\ P({\text{win by switching}}\ |\ I)\ \}$

(2) The set of three elements $\{\ P({\text{win by switching}}\ |\ {\text{player picks door i AND}}\ I):i\in \{1,2,3\}\ \}$

(3) The set of six elements $\{\ P({\text{win by switching}}\ |\ {\text{player picks door i AND host opens door j AND }}\ I):i\in \{1,2,3\},j\in \{1,2,3\}-\{i\}\ \}$

where $I\,$ denotes the rules of the game in the K&W statement .

This is what I am talking about. This is the point that I and Rick have been asking you to explain for a day and a half now. Do you understand the question? If you do not, why are you still editing the MHP article? If you do, may we please have an answer? Which of the above 3 sets answers your "unconditional case"? glopk (talk) 00:26, 22 July 2010 (UTC)

I do not know what you mean by 'your "unconditional case"'. The unconditional case is well understood by everyone here. It means that the probability of winning by switching is to be determined given only the rules of the game. The door originally chosen by the player and the door opened by the host are not specified and could be any door permitted by the rules. I assume this is your option 1 above, it has only one value.

I have agreed that my opening paragraph was not particularly clear but I think you must be reading something into it that is not there. It is a was intended to be a non-contentious statement just stating that in conditional probability problems the usual procedure us to start with the full sample set and condition it by removing those elements that do not correspond to the stated conditions. Are you disagreeing with that? Martin Hogbin (talk) 08:04, 22 July 2010 (UTC)

Martin, this still not good enough. Do you "assume" it is option 1, or is that your answer? The whole point of contention (and of Rick's initial question to you) is that there is NOT one single "unconditional case" that "is well understood by everyone here". There are TWO: option 1 above, which is addressed by Carlton's "simple" solution, and option 2 above, addressed by vos Savant's solution. So, I ask again: Which of the above 3 sets answers the "unconditional case" you want to talk about in this Arguments page? May we at last have a straight, no "I assume", no finger-crossed, answer? Thank you. After you answer I'll be happy to discuss with you about the "usual procedure" that one follows for solving conditional probability problems (hint, it's not what you think). glopk (talk) 14:55, 22 July 2010 (UTC)

I am not sure why you are making such a meal of this, I have no particular point to make about the unconditional case. By 'unconditional' I mean with no conditions other than the game rules. This is your option 1 above. Martin Hogbin (talk) 20:48, 22 July 2010 (UTC)

Well, there you are Martin. In most of the discussion the unconditional case is considered the case where the overall probability of winning by switching is calculated, i.e. where the player is asked to switch before the host opens a door. Nijdam (talk) 21:41, 22 July 2010 (UTC)

Sorry Nijdam, I do not understand your point.

Maybe I misunderstood you, and did you only mean to say: case (1) mentioned above by glopk. But why, if that's the case, don't you simply say so? Nijdam (talk) 22:57, 23 July 2010 (UTC)

What is your answer to my point. How do you justify not showing all the doors that the player might have initially chosen according to the rules?

That's interesting, Rick. Nijdam says a lot of things. Usually they are suggestions that Wikipedia policy be violated in one or more ways. You *never* say a peep about those. Now, his suggestion that this has to end, and certain editors should be ignored is to be followed?

There is no validation whatsoever for favoring one view over another while editing the article. Any caveats warning that simple solutions are flawed are BS. They represent a POV, and that's a violation. Offer the various solutions, without editorializing or OR. Present all the controversies together, again, without choosing sides. Glkanter (talk) 15:46, 21 July 2010 (UTC)

@Martin (continuing from above - ignoring the intervening interjection) - What exactly are you trying to accomplish here? Is it that you think that the conditional solutions currently presented in the article are incomplete, and that the article should say this in some way? Or that you think the conditional solutions are incomplete and you want to argue this point here in hopes of gathering a consensus on this viewpoint? Or is it that you think the conditional solutions in the article should be replaced with some other conditional solution? Or is it something else? It would be really helpful if you'd just directly say what you're trying to accomplish. -- Rick Block (talk) 15:04, 22 July 2010 (UTC)

What you guys ignore is that Selvin and vos Savant give similar answers. Neither of them to your liking. So you try to substitute your editorial opinions, also known as OR, for the reliably published sources. Or, you choose other sources to disparage the sources you don't like. That violates NPOV. That ain't right. Glkanter (talk) 15:50, 22 July 2010 (UTC)

@Glkanter - this is yet another example of your long history of wp:disruptive editing. I won't comment further except to identify it. -- Rick Block (talk) 17:43, 22 July 2010 (UTC)

@Martin - please ignore all intervening attempts at disruption and answer the question I've asked. Thank you. -- Rick Block (talk) 17:43, 22 July 2010 (UTC)

I would describe your comment, Rick Block, as yet another in a long history of playing cat and mouse games with this article. Glkanter (talk) 17:51, 22 July 2010 (UTC)

Editing break 2

@Rick, I am trying to show that the rigid distinction that some people make here between the simple solutions (wrong, incomplete, answer the wrong question, not supported by sources, only answer the unconditional problem) and the probabilistic solution given (correct, complete, answer the question asked, supported by sources) is not justified. The preference shown by some editors here for a particular class of solutions is just their POV and is not supported by either the facts or the sources. I have shown that, if you want to be pedantic, the probabilistic solution is not complete either and that it fails for slight variants of the problem. In some ways it might be good to have a more complete solution in the article as per G&S but I am not insisting on this. What I would like to see is an end to uninformative and off-putting 'health warnings' for the simple solutions. Martin Hogbin (talk) 20:43, 22 July 2010 (UTC)

There's a lot going on here - sounds like 5 possibly separable things. I would suggest we talk about one at a time, starting with what seems to be the main point of this thread - i.e. "if you want to be pedantic, the probabilistic solution is not complete either and that it fails for slight variants of the problem". We most likely need to talk about the other things as well, but I'd like to stay focused on this one for the time being. Assuming you're OK with this I have a few questions. Is this (pedantic) point one that you want to be directly reflected in the article, or is this just a background "point of understanding" that you believe and want other editors to believe as well? Is there any source that directly makes this point (if so, what source exactly), or is it something you've thought of on your own (i.e. wp:or)? -- Rick Block (talk) 15:32, 23 July 2010 (UTC)

You asked me what I was trying to accomplish and I told you. I think there is an important point of understanding here, which that there are not just right and wrong solutions to the MHP but a range of solutions varying in rigour. Is this OR? All I can say is that G&S say that for reasons of clarity they decided not to make certain assumptions (that is to say use a reduced tree). This source therefore tells us that they consider the reduced tree to be less clear. It is quite clear from the full tree that, if the car is not placed uniformly, the reduced tree gives the wrong answer, just as, if the host chooses a goat door non-uniformly, the simple solution gives the wrong answer.

Regarding putting a more complete solution (the G&S full tree) in the article, I think to might not be a bad idea to put it in somewhere, if we want to be really complete. What do you think? Martin Hogbin (talk) 17:04, 23 July 2010 (UTC)

Thank for you saying what it is you're trying to accomplish. Your statement that there are a range of solutions varying in rigor is fine and I don't think anyone would disagree. However, interpreting what G&S say as meaning that they consider conditional solutions that specifically address the door 1 and door 3 combination "incomplete" is ridiculous. They don't say anything remotely like this. In fact, their own solution specifically addresses only the door 1 and door 3 combination. Any solution that addresses this specific combination can be used to address any other combination by renumbering the doors. We've discussed this above. Renumbering in this way does not assume the car is initially uniformly distributed, or that the problem is symmetrical - only that the doors are uniquely and consistently identifiable by both the host and the player. This is an important point of understanding that you are apparently not understanding at all. Saying conditional solutions are "incomplete" because they address this specific case is a completely baseless criticism. Equating this criticism of conditional solutions to the (published) criticisms of unconditional solutions is pure BS. -- Rick Block (talk) 06:18, 24 July 2010 (UTC)

G&S do indeed show a reduced tree for the conditional problem but they do so immediately after first showing the full tree for reasons of clarity. [Martin]

Agree. But what I'm saying they don't say is anything remotely like "conditional solutions that address only the door 1 and door 3 combination are incomplete". -- Rick Block (talk) 18:03, 25 July 2010 (UTC)

Your renumbering doors argument is exactly the kind of problem that arises when you do not consider the whole picture. Of course, it is possible to address the problem by renumbering doors but no source does so and it is prone to reaching false conclusions my missing subtle points. Just like the frequency approach it requires extreme care to avoid mistakes. Hand-waving arguments along the lines of, 'we can just renumber the doors' are not appropriate for such a contentious topic. Doors can only be freely renumbered for the fully symmetrical problem. In all other cases extreme care is required to reach the correct conclusion. If you really want to work through a well-defined door renumbering solution with me I would be happy to do so but I see no benefit in addressing the problem casually using that approach, especially as there are easier and more reliable ways of solving all variants of the problem. [Martin]

No source addresses the problem by renumbering doors? I completely disagree. Any source that solves the problem conditionally, and reaches an answer that has no dependency on the specific door that was chosen (or opened) is showing that the renumbering doesn't matter. This seems to be the point you are not understanding. If you randomly pick a specific example case, solve the problem conditionally, and your answer is independent of the case you picked, then your answer applies equally to all cases - which shows (does not assume) that the problem is symmetrical. You might critically say that this is typically not stated by those presenting conditional solutions - but it is completely obvious (especially since the problem statement generally selects the specific case and the conditional solution addresses this case). The issue with the unconditional solutions is that this happens the other way around. To claim an unconditional solution applies to the conditional question you have to first show the problem is symmetrical. Those presenting unconditional solutions usually don't show this (or even say anything about it) - moreover they typically don't say anything at all about one of the critical assumptions that is required for the problem to be symmetrical (i.e. that the host picks between two goats uniformly randomly). As Morgan says - one is tempted to conclude they do not understand that the conditional problem and the unconditional problem are not the same, and that 2/3 is the answer to the relevant conditional problem only if the host picks between two goats randomly. -- Rick Block (talk) 18:03, 25 July 2010 (UTC)

No source addresses the problem by renumbering doors. That is a simple matter of fact. Your erroneous OR on the subject does nothing to change that. Are you asserting that if you randomly pick a specific example case, solve the problem conditionally, your answer is independent of the case you picked? See section below.

Morgan now say that 'the answer is 2/3 period'.

Solutions based on probability theory with fixed door numbers are simpler to understand and less prone to subtle errors, which is why most of the sources use that approach. Using that approach it is quite straightforward to calculate the probability of winning by switching given a set of rules, a set of initial distrubutions, and any conditions to be applied. This is what I have done, with Nijdam, on my analysis page. I have not tried to add any of the results that I have found to the article because it constitutes OR (although some here might call it a routine calculation) but there is no doubt that mathematically correct answers to any well-defined problem can be calculated and there is no reason why such results should not help us organise the article, based on reliable sources. [Martin]

Do whatever you want for your own understanding. However, do not make baseless claims like "conditional solutions are incomplete". -- Rick Block (talk) 18:03, 25 July 2010 (UTC)

There is nothing baseless about that assumption. There are three doors and the player can pick any one of them. Any solution which does not show this is incomplete, despite your unsupported claim that the doors can be freely renumbered. Martin Hogbin (talk) 00:27, 26 July 2010 (UTC)

My argument is simply that anything less that the full analysis is a short cut, based on some kind of assumption. The probabilistic solution given on the article is a short cut based on the assumption that the door initially chosen by the player does not matter. The simple solution is a short cut based on the assumption that the door opened by the host does not matter. Even with the standard rules, in varying circumstances one, both, or neither of those assumptions might be valid. You are still pushing a POV based on neither what the sources actually say the sources nor mathematics. Martin Hogbin (talk) 10:41, 25 July 2010 (UTC)

Your argument seems to be based on a fundamental misunderstanding. That the door initially chosen by the player does not matter is not an assumption made by a conditional analysis - it is a result shown by the fact that the conditional analysis has no dependency on this door. What sources actually say (these are all quotes - I assume you know the references by now) are that an unconditional solution is "shaky", or "does not address the problem posed", or "does not quite solve the problem that Craig [Whitaker] posed", or is "incomplete", or is "false". This has nothing to do with my POV. The issue here is what these sources say conflicts with your POV that the unconditional solutions are just fine, and you are insisting the article reflect your POV rather than what these published sources have to say. -- Rick Block (talk) 18:03, 25 July 2010 (UTC)

Once again you claim that, the conditional analysis has no dependency on the door originally chosen by the player. This is an incorrect assumption on your part. See section below.

The answer to the conditional problem has no dependency on the door originally picked by the player?

Rick, you say '...the door initially chosen by the player does not matter is not an assumption made by a conditional analysis - it is a result shown by the fact that the conditional analysis has no dependency on this door', and 'Any source that solves the problem conditionally, and reaches an answer that has no dependency on the specific door that was chosen (or opened) is showing that the renumbering doesn't matter'. Are you really claiming that answer to the conditional problem has no dependency on the door originally picked by the player? Martin Hogbin (talk) 00:27, 26 July 2010 (UTC)

The answer to the conditional problem, given the usual interpretation of the problem (i.e. car is uniformly randomly placed initially and the host chooses uniformly randomly between two goats) has no dependency on the door the player initially picked - the answer is "2/3" chance of winning by switching. This answer is a constant, not affected by the initial player pick or the door the host opens. [Rick]

Yes, of course. Assuming the car is initially uniformly placed this is the fully symmetrical case. The probability of winning by switching does not depend on either the player's initial choice of door or the host's choice of door. The simple solution therefore solves this problem. Why do you need a solution that shows the host opening door 2? We know this does not happen.Martin Hogbin (talk) 08:47, 26 July 2010 (UTC)

You can change the conditions, and if you do then the answer will change as well. The most general conditional answer of the chance of winning by switching (given the player has picked door 1 and the host has opened door 3) is p(door 2)/( p(door 1)*p(host opens door 3|player picked door 1) + p(door 2) ). In the usual interpretation this is 1/3 / ( (1/3)*(1/2) + (1/3) ), resulting in 2/3. All of these values, including the final 2/3, are identical regardless of which door the player initially picks and which door the host opens. This is what I mean by "no dependency". -- Rick Block (talk) 04:39, 26 July 2010 (UTC)

I cannot understand what you have done here. Are you calculating the probability of winning by switching given that the player has initially chosen door 1 and the host has opened door 3 to reveal a goat?

What do you say is the probability of winning by switching given that the player has initially chosen door 1 and the host has opened door 3 to reveal a goat? Martin Hogbin (talk) 08:47, 26 July 2010 (UTC)

Martin - Do you not understand what is written at User:Martin Hogbin/Monty Hall analysis? In the very last section you have a general solution which is (ignoring many terms that necessarily have the value 1 when talking about the specific case of player has picked door 1 and host has opened door 3 under the rules that the host will necessarily reveal a goat) that the probability of winning by switching is

P(C=2) / ( P(C=1)*P(H=3|C=1) + P(C=2) )

This is identical to what I said above (which, BTW, I wrote before looking at your analysis page). Looking at this formula you can plainly see under the usual assumptions that the car is uniformly distributed, i.e. P(C=1) = P(C=2) = P(C=3) = 1/3, AND that the host has no bias, i.e. P(H=3|C=1) = P(H=3|C=2) = P{H=2|C=1) = P(H=2|C=3) = P(H=1|C=2) = P(H=1|C=3) = 1/2, then this formula will have the identical form and identical values regardless of which door the player picks and which door the host opens - allowing one to conclude (not assume) that the probability of winning by switching is independent of the door the player picks and the door the host opens.

You say above that these conditions mean the simple solution therefore solves this problem. You're saying this incorrectly. These conditions mean the simple solution produces the correct numeric value. The problem asks for the probability P(win by switching AND player initially picks door 1 AND host opens door 3). The simple solutions do not address this probability. They directly address either P(win by switching) (e.g. Carlton's), or P(win by switching AND player initially picks door 1) (e.g. vos Savanat's), sometimes using statements that are misleading and incorrect (such as "because you already know one of door 2 or door 3 is a goat, the host showing you a goat behind door 3 cannot affect the probability the car is behind door 1"). This is what multiple reliable sources are saying. -- Rick Block (talk) 16:58, 26 July 2010 (UTC)

Yes, Rick, they are criticizing many other equally reliable sources. Including Selvin, the originator of the puzzle, who solves it with a simple solution, and vos Savant, who popularized the puzzle, and solves it with a similar simple solution. Including text like what follows from the Solutions section (and similar wording that appears elsewhere) favors one reliable group over the other, and therefore violates Wikipedia's NPOV policy, regardless of who may be 'right'.:

"There are multiple approaches to solving the Monty Hall problem, under varied and more or less natural additional assumptions, but all giving the end result that the player should switch. Those who always switch have a 2/3 chance of winning the car as long as initially all doors are equally likely to hide the car. A main point of difference concerns whether we should be interested in the overall probability that switching wins the car, or whether we should be interested in the probability that switching wins the car in the specific case that the player has chosen Door 1 and the host has opened Door 3. In mathematical terms, are we interested in an unconditional or a conditional probability? Selvin, the originator of the problem, presented solutions based on both approaches (Selvin 1975a, Selvin 1975b)."

This banal paragraph (after much contentious editing) does nothing to illuminate any aspect of the puzzle. It's purpose still appears to be only to bias the reader. Glkanter (talk) 18:57, 26 July 2010 (UTC)

Rick, if you look at what I have written earlier you will see that I have already agreed that, if the host chooses a goat door uniformly and the car is initially placed uniformly, the problem is fully symmetrical and the door originally chosen by the player does not matter. To put this another way, if the car is initially placed uniformly and the host chooses a goat door uniformly:

1. The probability of winning by switching is the same whichever door the player initially chooses. Thus we can say that a solution which shows only one initially chosen door is sufficient to get the correct answer and solve the problem.
2. The probability of winning by switching is also the same whichever door the host opens. Thus we can say that a solution which shows only one door opened by the host is sufficient to get the correct answer and solve the problem.

For some unexplained reason you assert that 1 answers the correct question and 2 does not. Can you actually give any reason why this is so?

Consider also the case that the car is initially placed uniformly and the host chooses a goat door non-uniformly:

1. The probability of winning by switching is not the same whichever door the host opens. Thus we can say that a solution which shows only one door opened by the host is not in general sufficient to get the correct answer and solve the problem.
2. The probability of winning by switching is not the same whichever door the player initially chooses. Thus we can say that a solution which shows only one initially chosen door is not in general sufficient to get the correct answer and solve the problem.

In other words, to properly cover the case which requires the two possible doors opened by the host to be shown you must also show the three possible doors that the player might have chosen. Martin Hogbin (talk) 23:49, 26 July 2010 (UTC)

Martin - you're approaching this all wrong. The usual question is whether you should switch, giving as an example case player picks door 1 and host opens door 3 (meaning the probability of interest is the fully conditional probability). A solution that considers this example case addresses the exact case specifically addressed by the problem statement. In the usual interpretation the car is uniformly placed and the host chooses uniformly between two goats, so it turns out the door 1 and door 3 solution applies to all other combinations of player pick and door the host opens - and this can be readily seen to be true by looking at the elements of the solution. On the other hand, a solution that ignores what door the host opens, or ignores both which door the player picks AND which door the host opens may or may not produce an answer that's relevant to the question of whether you should switch in the given example case. You don't decide what sort of solution you'll use based on what constraints are given in the problem statement, but rather based on what question the problem statement asks. The constraints may make the answer you get the same given other solutions, but saying a particular solution is "sufficient" because it happens to produce the correct numerical result is a very odd way of approaching things.

You response is based on nothing but your personal POV. Of course you base a solution to a conditional probability problem on the specified conditions, What else would you base it on? The standard way of dealing with a conditional probability problem is to start with the complete sample set, based on the problem without conditions, then eliminate the events that do not conform to the specified conditions, then renormalise to get the conditional probability. This is not something I have just made up, it is the standard procedure.

I have no objection to using a simple and obvious symmetry with respect to door numbers to simplify the problem. Because problem formulation that you have given is symmetrical with respect to door number, it is perfectly reasonable to show just one door chosen by the player, because it does not make any difference to the answer which door the player initially chooses. Similarly, it is perfectly reasonable to show just one door opened by the host because it does not make any difference to the answer which door the host opens.

If the host chooses non-uniformly, or the car is not initially placed uniformly, the above symmetry does not hold and the full solution is required.

Your other example conditions are simply straw man arguments - but even so, a solution that addresses a particular case (player picks door 1 and host opens door 3) can be used to determine any specific case by simply renumbering the doors involved (and, again, this doesn't mean assuming the problem is symmetrical). For example, if the case of interest is player picks door 2 and host opens door 1, then you replace "door 1" anywhere it appears in the solution with "door 2", etc. The logic of the solution remains the same, but the answer will reflect the other specific player pick and door the host opens. In other words, your assertion that such a solution is "incomplete" is baseless. -- Rick Block (talk) 06:37, 27 July 2010 (UTC)

In your example, the door that the host opens is not important so the specific door that the host opens does not matter and does not need to be part of the solution. Unless you want to be really pedantic, in which case you must show the whole story. Martin Hogbin (talk) 16:24, 27 July 2010 (UTC)

@Martin. Perhaps an analogy may help you understand Rick's last point on door renumbering. Replace MHP with "Pythagoras' theorem" and "door numbers" with "letters naming the triangle vertices". Would you agree that the theorem's proof on one particular ABC triangle needs not be repeated for all the other 8 letter-vertex assignments in order to be "a complete proof"? If there is no such need, is it because some "symmetry" holds? (Symmetry of what?). glopk (talk) 14:26, 27 July 2010 (UTC)

Glopk, I know what symmetry is and I understand that in the fully symmetrical formulation of the problem (car placed uniformly, host opens a goat door uniformly) it can be quite reasonably argued that the door originally opened by the player is not important by reason of symmetry with respect to door numbers, thus a solution need not consider which door the player originally chooses. In the completely symmetrical formulation it can equally well be argued that the door opened by the host is not important because of the same symmetry with respect to the doors, thus a solution need not consider which door the host actually opens. This is exactly what the simple solutions do, they do not distinguish between doors opened by the host. Martin Hogbin (talk) 16:24, 27 July 2010 (UTC)

Martin, please stop saying this. Here is the (a) simple solution: P(C=1)=1/3; P(C=3|X=1,H=3)=0; hence P(C=2|X=1,H=3)=2/3. Nothing there about symmetry! And do you notice how nonsensical it is! Nijdam (talk) 21:07, 27 July 2010 (UTC)

Yes, your attempt at the simple solution is indeed nonsensical. Here is my simple solution:

Comment from Nijdam

Complete symmetry with respect to the doors means that the door originally chosen by the player is unimportant and the door opened by the host is also unimportant therefore the answer (probability of winning by switching) to the conditional problem (player chooses door 1 and host opens door 3) must be the same as the answer to the unconditional problem. Therefore by solving the unconditional problem with a simple solution we get the answer to the conditional problem, by necessity and not by chance. Please tell me what is wrong with this simple logical argument. Martin Hogbin (talk) 21:51, 27 July 2010 (UTC)

Martin, please, do understand, it is not about what I or you consider to be a simple solution, but what sources consider to be the simple solution. And that's what I showed above. This, as you also called nonsensical, reasoning, is the simple solution, given by some sources, in words, instead of in formula. And this nonsense is what we, Rick, Kmhmh glopk, I, are strongy opposed. Nijdam (talk) 09:31, 28 July 2010 (UTC)

Martin, do you understand this???!!!Nijdam (talk) 19:38, 1 August 2010 (UTC)

Nijdam, I really do not understand what you are saying. You seem to agree that the simple solution is logical and also that it is given by some sources. What then is wrong with it. Martin Hogbin (talk) 20:02, 1 August 2010 (UTC)

I'll spell it out for you then: Above I wrote down what sources call the simple solution. I wrote it down in exact formula's, not in words, for better understanding. Nevertheless is it the simple solution named by the sources. I call it nonsensical, and you agreed. What more can I say? Nijdam (talk) 20:46, 1 August 2010 (UTC)

Do you agree that the simple solution solves the problem of calculating P(C=1|H=2 or H=3)? Martin Hogbin (talk) 21:23, 1 August 2010 (UTC)

The simple solution, in its different variants mentioned by the sources, does not say it calculates P(C=1|H=2 or H=3), hence it does not solve this problem. Nijdam (talk)

Morgan say that it does solve the problem where the contestant decides whether to switch or not before they have seen a door opened by the host. Martin Hogbin (talk) 09:14, 2 August 2010 (UTC)

Which is not P(C=1|H=2 or H=3) but P(C=1). P(C=1|H=2 or H=3) is the situation after the host has opened a door, but the player (for some reason - perhaps the player is blindfolded after picking door 1) can't tell which. -- Rick Block (talk) 14:30, 2 August 2010 (UTC)

Morgan refer to the probability of winning by switching (which is the complement of P(C=1) if the player decides whether to swap or not before the host has opened a door. The rules sill state that the host must open an unchosen goat-hiding door. The host can only open door 2 or door 3 under the rules so the probability of interest is P(C=1|H=2 or H=3). Martin Hogbin (talk) 14:44, 2 August 2010 (UTC)

@Martin. Do you not realize that you contradict yourself yet again? Several paragraphs above you agreed (after much much prodding) that the simple solutions address the problem P("win by switching" |I), with no conditions - i.e. regardless of the door selected by the contestant and the one opened by the host. And, as Morgan states, an intuitive way to express this "regardless" within the MHP scenario is to say that the contestant decides whether to swap or not before the host has opened a door. Now, the only way that probability theory allows for expressing P("win by switching" | I) ignoring the conditions is by marginalizing the conditions away. That is (indicating with S the door initially selected):

$P({\text{win by switching}}\,|I)=\sum _{c\neq s,h}\sum _{h\neq s}\sum _{s=1}^{3}P(S=s,H=h,C=c|I)\,$ .

where $h\neq s$ expresses the rule "host does not open the door selected by the contestant", and $c\neq s,h$ means "switching wins". Do you really not understand the difference between the above and P(C=1 | H=1 or H=3) ? And if you do not, I ask again, why are you still editing the MHP article? glopk (talk) 03:51, 3 August 2010 (UTC)

Yes, you are quite right Glopk, I forgot how Morgan described the unconditional problem, the player has not yet chosen a door.

This is not, however, terribly relevant to the argument. My argument has always been that, in the standard case, in which the host has no preference for any door, the problem is completely symmetrical with respect to door numbers (you have alraedy agreed this). It therefore follows that, in the standard case the numerical answer to the conditional problem (player has chosen door 1 and the host has opened door 3) must by symmetry be equal to the answer to the unconditional problem (no doors specified). Thus a solution to the unconditional problem must give the correct numerical answer to the conditional problem. Do you agree with that? Martin Hogbin (talk) 09:31, 3 August 2010 (UTC)

Comment from Glopk

@Martin. I do not know what your argument has "always" been, since you change them quite often (maybe you can ask the Martin of several paragraphs above, or the one below, to tell you what their arguments were). However, what I (glopk) have agreed to is that (1) In the "standard" (K&W) formulation of the MHP the choice of door labelling (naming, numbering) must be irrelevant to any proper solution, so long as it is understood that both host and contestant agree that the doors are distinct, and agree on their separate identities. (2) The "standard" (K&W) formulation of the MHP is characterized by a property, which we call "symmetry" (or "the host's lack of bias"), that the host chooses at random uniformly which door to open when the contestant initially selects the door hiding the car. (3) As a consequence of this "symmetry" the so-called "simple" solution computes a value for P("win by switching" | I) that is numerically equal to the correct one. (4) But rather than explicitly compute the marginalization I wrote above, the "simple" solution uses a "sketchy" (Rosenthal) logical shortcut. (5) In particular, some published versions of the "simple" solution do not state that this "symmetry" is an absolute requirement, thus being incomplete. (6) Finally, the "simple" solution does not address the problem as stated, which is to compute the optimal choice for the contestant when his initial selection, and the host's opening of a specific door, are given. This is a decision problem whose answer depends on the value of $P(C=c,\,H=h,\,S=s|I)$, whereas the "simple" solution computes (or attempts to compute) the marginal I have written above, thus answering a separate problem. Now, I bet you won't agree with almost any of these points, but please do not mischaracterize what I write. You have agreed yourself with point (6), though am sure wil continue to deny having done so. glopk (talk) 16:12, 3 August 2010 (UTC)

So we agree that the simple solution calculates P("win by switching" | I) Do you agree that in the symetrical case we can say, without computing any probabilities, that P(Ws| I) = PWs | I and S=s and H=h and C=c) for all legal values of h,s,and c?

"Without computing any probabilities"? No, I do not, or rather, I cannot. It is not an obvious result, and I am not blessed with Fermi's "intuito formidabile". But you could show me, or you could ask Glkanter, he seem to posses a remarkable imagination (or perhaps be possessed by it). glopk (talk) 00:00, 4 August 2010 (UTC)

But, just in case you'd like to see what the answer looks like by computing "some" probabilities (perhaps for when Glkanter is not there to help), here it is:

Martin's conjecture: $P(W_{s}|I)=?\,P(W_{s}|S=s,H=h,C=c,I),\ \forall \,(s,h,c)\in \{{\text{legal values}}\}$.

where, $W_{s}={\text{win by switching}}\,$

and, presumably, $\{{\text{legal values}}\}\equiv \{s:s\in \{1,2,3\}\}\times \{c:c\in \{1,2,3\}\}\times \{h:h\in \{1,2,3\}-\{s,c\}\}$.

Theorem: Martin's conjecture is false.

Proof: By definition,

$W_{s}\equiv \,(C\neq S)$,

and, trivially:

$P(C\neq S|S=s,H=h,C=c,I)={\begin{cases}0,\,c=s\\1,\,c\neq s\end{cases}}$,

whereas, under the K&W statement of the MHP, and assuming random uniform initial door selection by the contestant, by inspection it is:

$P(W_{s}|I)={\tfrac {2}{3}}$.

Therefore my answer is a resounding NO: I do not agree with your conjecture. Now, of course all the above does not tell us anything new about the MHP. But it does tell us a lot about Martin Hogbin's grasp of probability theory, an' it ain't pretty by a loooong shot. So, here I am again asking of Martin Hogbin: why is he still editing the Wikipedia article on the Monty Hall Problem? glopk (talk) 06:44, 4 August 2010 (UTC)

By 'legal values' I meant values that are permitted under the rules of the game. Martin Hogbin (talk) 07:14, 4 August 2010 (UTC)

The "presumably" was sarcasm, and my proof is correct: there are only values permitted by the rules of the game in my proof. But your silly conjecture is false in any case. Read the proof, do your homework, study, learn and please, stop hurting Wikipedia until you have learned enough to avoid such blunders. glopk (talk) 14:34, 4 August 2010 (UTC)

You love to make a meal out of small errors. I included the initial car placement by mistake, it should have been uniformly distributed. It seems you do agree that P(Ws | I and S=s and H=h) = P(Ws| I) for all legal values of h and s, given the car is initially placed uniformly at random and the host chooses a goat door uniformly. Is that correct? Martin Hogbin (talk) 15:23, 4 August 2010 (UTC)

Martin, drop the fork and the meal: you are wrong again, and I don't agree with this novel blooper conjecture of your - 3rd blooper in less than a week. However, I am done doing your homework for ya, I'll just give you a failing grade and leave finding the mistake as a reparation exercise for you. I have a question, though: are you thoroughly enjoying yourself in displaying your basic ignorance of probability calculus? glopk (talk) 08:39, 6 August 2010 (UTC)

If you do not wish to discuss the subject seriously that is fine with me. Martin Hogbin (talk) 09:19, 6 August 2010 (UTC)

If you do not wish to discuss the subject intelligently, I am not interested in having an inane exchange. It is not my purpose in life to correct your erroneous formulae, especially when you put them out with such a sophomoric self-confidence. Really, it's not that hard: crack open a real introductory textbook on probability and information theory (I recommend this one), spend a couple of week on its first chapters, make sure you can solve all the exercises at the end of each chapter. Then come back and we can talk about probability. glopk (talk) 16:58, 6 August 2010 (UTC)

Martin, afraid you are wrong on two counts:

1. Are you really sure you understand what symmetry is? It can always be quite reasonably argued that the numbering (or naming, or labelling) of the doors is not important, and symmetry has absolutely nothing to do with it. Once it is recognized that both the contestant and the host must agree on which door is initially selected, and which (different) one is opened, all solutions are formally equivalent modulo a renaming of the labels. Again, this equivalence is not "symmetry". Rather, it simply an expression of the trivial fact that the choice of names for the variables is always arbitrary and subject only to requirements of self-consistency. The term "symmetry" in the context of the MHP is properly used only to denote the "standard" case in which the host has no preference for any door, but the freedom of renaming the doors without loss of generality applies in all cases, including the non-symmetric ones.

I really do not care whether you wish to use your own understanding of the word symmetry or not. At least one source (Falk) uses that term to describe the standard version of the problem.

Call it what you like, the clear, obvious, and indisputable fact remains that, for the standard problem, the initial choice of door by the player and the door opened by the host do not affect the answer. Martin Hogbin (talk) 08:14, 28 July 2010 (UTC)

1. This invariance of any and all solutions (whether "simple" or "conditional") with respect to the choice of door naming has absolutely nothing to do with whether the published "simple" solutions address the MHP (under the K&W statement), or whether the "conditional" solution is incomplete, as you allege. The reason why the simple solutions do not address the MHP is that, as the sources say and Rick has been repeating to you many many times, their answers are based on computing P("win by switching" AND I) using an unstated (or wrongly stated) argument based on symmetry, without addressing the actual question, which demands for P("win by switching" AND "door opened by the host" AND I). Everyone here recognizes that the simple solutions can be correctly restated so they are correct, but (A) Even when correct, they still address the wrong question, (B) We have reliable sources that say they address the wrong question, (C) There are no reliable sources asserting that the conditional solution is incomplete, as you allege, and (D) No amount of your or Glkanter's WP:OR can change (A), (B) and (C). glopk (talk) 00:12, 28 July 2010 (UTC)

I am surprised that you have never seen the technique used before in mathematics where two different problems are shown to be equivalent. That is to say it is shown that the answer to both problems must be the same. In that case a solution to one problem becomes a solution to the other.

Vos Savant justifies her answer by saying that she took the host to be acting as the agent of chance. This means that the host chooses evenly between goat doors and thus the problem is equivalent to the unconditional one. Falk describes the symmetry argument as 'impeccable'. Rosenthal says about the simple solution, 'It is actually correct, but I consider it "shaky" because it fails for slight variants of the problem'. By slight variants he obviously means that the host does not choose a goat door uniformly.

Please demonstrate that you are not pushing a personal POV by providing either:

1. A mathematical argument which shows that in the standard (K&W) case the answer to the conditional problem is not certain to be equal to the answer to the unconditional problem.
2. A source which says that a simple solution to the standard case is incorrect. Martin Hogbin (talk) 08:14, 28 July 2010 (UTC)

Martin - #1 is the wrong question. It's like asking someone to show that the answer arrived at by computing the length of a hypotenuse of a triangle using Pythagoras's theorem is incorrect for a problem where the triangle can be, but isn't shown to be a right triangle. Of course the numeric answer is correct. What isn't correct is using Pythagoras's theorem to arrive at the answer without showing the triangle is a right triangle. Similarly, no one has ever disputed 2/3 is the correct numeric answer in the fully symmetrical case. But this doesn't mean that any argument that ends up with the answer 2/3 is correct. Facetiously - someone could say the player's first choice has probability 1/3 therefore the player's second choice (after the host opens a door) has probability 2/3. Show me this isn't the right answer in the standard (K&W) case. The answer is correct, but the logic is faulty. In this facetious case, the logic is ridiculous. In the case of simple solutions to the MHP, the logic is sound when applied to a subtly different question but (confusingly) produces the correct answer to the question that is asked as well. One way to expose the logical error (in the facetious case as well) is to construct a slightly different question where the same logic produces an incorrect answer. The issue with the simple solutions is that they inherently (and implicitly) assume symmetry whether the problem is constrained to be symmetrical or not. Showing this issue requires changing the problem so that it is not symmetrical.

Any method that can be shown must produce the correct answer is a proper solution. It is not coincidence that the unconditional problem and the standard conditional problem have the same answer, it is because it is quite obvious to anyone that the door opened by the host does not matter. Martin Hogbin (talk) 18:06, 28 July 2010 (UTC)

You know the sources for #2 - Morgan, Gillman, Rosenthal, Grinstead and Snell, etc.

Thank you for quoting Rosenthal as a source which says that a simple solution to the standard case is incorrect. It clearly shows that your claims are based on what you would like the sources to say, rather than what they actually say. He says, 'It is actually correct but I consider it "shaky" because it fails for slight variants of the problem'. The clue is in the words It is actually correct. He then goes on to explain in what circumstances (that is to say non-standard formulations of the problem) the simple solution is incorrect. Perhaps you could show me where in the other two sources it actually says that that the simple solution to the standard problem is incorrect. Martin Hogbin (talk) 18:06, 28 July 2010 (UTC)

How about if you show you are not pushing a personal POV by providing a source that presents a simple solution that you consider correct. Perhaps one based on the symmetry argument. As Nijdam says above, "it is not about what I or you consider to be a simple solution, but what sources consider to be the simple solution". Per wp:synth the question is not whether you can create a correct simple solution, but whether such solutions are published. -- Rick Block (talk) 14:45, 28 July 2010 (UTC)

Let me start with vos Savant, she explained that her solution was correct because the host acted as the agent of chance. Then there is Selvin's first solution, he later explained that the host chose evenly between goat doors. Martin Hogbin (talk) 18:06, 28 July 2010 (UTC)

So, how would you paraphrase either of these solutions for the article, in a correct way? vos Savant's is in the article already. Here's what it says:

---

The solution presented by vos Savant in Parade (vos Savant 1990b) shows the three possible arrangements of one car and two goats behind three doors and the result of switching or staying after initially picking Door 1 in each case:

Door 1	Door 2	Door 3	result if switching	result if staying
Car	Goat	Goat	Goat	Car
Goat	Car	Goat	Car	Goat
Goat	Goat	Car	Car	Goat

A player who stays with the initial choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three. The probability of winning by staying with the initial choice is therefore 1/3, while the probability of winning by switching is 2/3.

---

The reference is to the Dec 1990 Parade column. This reference says absolutely nothing about the host acting as an agent of chance. What it does say to justify this solution (immediately before) is the following:

The winning odds of 1/3 on the first choice can't go up to 1/2 just because the host opens a losing door. To illustrate this, let's say we play a shell game. You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you've chosen, we've learned nothing to allow us to revise the odds on the shell under your finger.

So - are you saying the reasoning vos Savant is using here is that she is (silently) saying the host is acting as an agent of chance? Or is it that she thinks that knowing the host can and will show a goat means that doing so cannot affect the probability of the player's initially selected door hiding a goat (which is, BTW, exactly what Falk criticizes as a powerful intuition that isn't the reason this probability doesn't change when the host opens a door)? Since vos Savant says the latter in the same column as the solution I'm thinking the latter is the more justifiable assumption - but maybe that's just me. Moreover, can you explain how her table connects with the example about the pea? In the pea example she's clearly talking about the conditional probability given which shell the host has picked up. The table on the other hand is clearly not addressing either specific case. It appears to me that she's mixing and matching prior and posterior probabilities and (arguably) is confused. I believe I know what you're talking about as far as "agent of chance" goes and recently asked you for the specific reference. Did you find it yet? [Rick]

Yes vos Savant is using the posterior probability for the prior probability because she knows that knowing the host can and will show a goat means that doing so cannot affect the probability of the player's initially selected door hiding a goat if the host chooses a goat door uniformly. Falk confirms this logic is correct. Vos Savant did omit to say initially that the host chose a goat door randomly but she later made clear that this was the basis for her answer. [Martin]

You mean she's using the prior probability (the original 1/3) for the posterior probability (the conditional probability given the door the host opens)? Are you sure? It seems obvious to me that her table is only talking about prior probabilities given an (unconditional) strategy of switching. The car is equally likely to be behind one of the three doors. In two of these cases you switch and win (regardless of which door the host opens) and in one you switch and lose (regardless of which door the host opens). Her table is completely ignoring the conditional case. Her statement before the table perhaps attempts to justify that the unconditional and conditional answers must be the same - but using the very words Falk criticizes.

And, again, do you know the reference where she says she's considering the host to be an agent of chance? All the Parade columns are reproduced here. Search for "agent". I'm not finding it. -- Rick Block (talk) 05:03, 29 July 2010 (UTC)

I believe that she made that statement in her response to the Morgan paper. Martin Hogbin (talk) 09:16, 29 July 2010 (UTC)

The article does not include Selvin's solution. But if it did, how exactly would you put his disclaimer? You've argued endlessly that the simple solutions must not, under any circumstances, mention anything at all about how the host chooses between two goats. Are you backing off of this stance now? Or are you continuing to insist these solutions are correct even without such a disclaimer? Please make up your mind. -- Rick Block (talk) 18:46, 28 July 2010 (UTC)

I am not sure why you say, 'You've argued endlessly that the simple solutions must not, under any circumstances, mention anything at all about how the host chooses between two goats'. That is quite wrong. I have argued that the simple solution must not have a disclaimer that it answers the wrong question. It does correctly answer the standard question, as defined in the article, in which the host opens a random goat door. I have no objection to making that fact clear in the article.

Now perhaps you can show me where any source says that the simple solutions are incorrect for the standard problem. Martin Hogbin (talk) 19:27, 28 July 2010 (UTC)

Are you really claiming you're never said the simple solutions must not say anything about how the host chooses between two goats? Must be some other Martin Hogbin who wrote here: "There must be no mention of the fact that the host has a choice of door when the player has originally chosen the car, this choice should not be shown in any diagrams."

Maybe some clarification is needed here. The K&W problem statement quoted in the article makes clear that the host chooses a goat door uniformly at random (unless you want to argue that they did not specify 'uniformly'). I therefore see no need to repeat this fact in the simple solution section. It would naturally be expected that the solution would be to the stated problem. However, I have no objection to repeating this fact in the simple solution section if you wish, with clarification that the host opens a goat door uniformly, although I would want to word it 'is taken to open a door...' to include the case that no information on the host's door policy is given but the principle of indifference is applied. I do not believe that anyone else will object to doing this either.

My objection was to diagrams which show the choice of goat door as part of the solution and statements that suggest that the specific door opened by the host might be important or might affect the probability of winning by switching. I also object to any statement to the effect that the simple solutions do not address the probability of winning by switching after the host has been seen to open a specific door to reveal a goat. This is incorrect in the above case and I believe that the majority of editors and sources agree with me over this. Martin Hogbin (talk) 09:16, 29 July 2010 (UTC)

I've posted the sources, with quotes, that say the simple solutions are incorrect for the standard problem numerous times. Aside from the strongest worded one (i.e. Morgan et al) several are conveniently archived here. -- Rick Block (talk) 05:03, 29 July 2010 (UTC)

Rosenthal

Even Morgan do not actually say that in the case that the host opens a goat door uniformly the simple solution is incorrect. The other sources you have quoted in the past simply do not say what you claim. The most obvious example of this is Rosenthal. Martin Hogbin (talk) 09:16, 29 July 2010 (UTC)

Are you saying I've misquoted these sources? If we're talking about interpretation I think you are the one misinterpreting or simply denying what they're saying. For example, please explain what you think Rosenthal means by saying the simple solution he presents is "shaky". -- Rick Block (talk) 14:28, 29 July 2010 (UTC)

That is a tricky one. What can Rosenthal possibly mean when he says, It is actually correct but I consider it "shaky" because it fails for slight variants of the problem. This sounds like a school comprehension test:

1. Does Rosenthal say the simple solution is correct? Ans: Yes, he says, 'It is actually correct'.
2. Does Rosenthal consider the simple solution shaky? Ans: Yes, he says, 'I consider it "shaky" '.
3. Why does Rosenthal consider the simple solution shaky? Ans: He considers it shaky because, 'it fails for slight variants of the problem'. Martin Hogbin (talk) 17:40, 29 July 2010 (UTC)

Let's page forward a bit in Rosenthal's paper, shall we?

The original Monty Hall problem implicitly makes an additional assumption [emphasis in original]: if the host has a choice of which door to open (i.e., if your original selection was correct), then he is equally likely to open either non-selected door. This assumption, callously ignored by the Shaky Solution, is in fact crucial to the conclusion (as the Monty Crawl problem illustrates).

"This assumption, callously ignored by the Shaky Solution, ..." Hmmm. Doesn't sound like he's thinking this is a perfectly fine solution to me. In fact, he sounds downright critical. The whole paper is nothing but a criticism of the "shaky" solution presenting an alternative conditional approach that produces the correct answer, not just for the "original" (symmetrical) MHP, but for a number of slight variants (and completely different problems) as well. I am not misinterpreting when I say this source is saying the simple solutions are incorrect. You are taking one sentence out of context and inverting the meaning of the entire paper. -- Rick Block (talk) 04:59, 30 July 2010 (UTC)

Oh, please. Rosenthal is referring to the 'original Monty Hall problem' description he gave earlier in his paper. Rosenthal intentionally left the assumption out as part of his instruction methodology, to set a trap. Selvin didn't leave it out. Your personal interpretation of Rosenthal's paper simply reflects your POV. Not fact. You're inexplicably discounting the very sentence where Rosenthal makes his thoughts known. Glkanter (talk) 05:14, 30 July 2010 (UTC)

I forgot some questions and answers.

• What does Rosenthal mean by 'slight variant'? Ans: He means cases where the host does not choose a goat door evenly, such as in the 'Monty Crawl' problem.
• Does this include the case where the host is defined to choose uniformly between goat-hiding doors? Ans: No, This is the problem to which he says the simple solution is 'actually correct'.

The additional assumption referred to by Rosenthal is included in the standard version of the problem (K&W statement) given in the our article. It was also clearly stated in my challenge to you, which was to find a source which says that in the case that the host opens a goat door uniformly the simple solution is wrong. You still have not done this.

As Glkanter says, the biased opening of a goat-hiding door by the host is an interesting trap that can be introduced into the MHP by careful wording of the problem statement and used an a useful exercise for probability students. In the standard formulation, as given in the article, this trap is absent. The simple solution is correct for the standard problem as Rosenthal makes clear.

In fact I have never seen a problem formulation of the MHP where the biased host trap is present, except is special formulations used to demonstrate this possibility such as Mont Crawl or as is given on my Morgan criticism [1] page. Martin Hogbin (talk) 09:16, 30 July 2010 (UTC)

Rosenthal is using "correct but shaky" in exactly the same way we all are here - which is that this solution produces the correct numeric answer but is based on incorrect logic. If your standard for a solution being correct is only that it produces the correct numeric answer, then my facetious solution above is also correct. I will agree the simple solution is as correct as this facetious solution. They both produce the answer "2/3 chance of winning by switching" for a problem statement for which 2/3 chance of winning by switching is the only correct numeric answer. They both fail for slight variants.

Re other sources, Grinstead and Snell assume the host opens a goat door uniformly and still say the simple analysis "though correct, does not quite solve the problem that Craig posed". Please read this carefully. What they're saying is correct is the analysis, which is an analysis of a preselected strategy of switching vs. staying. And they're saying this analysis, even though "correct", is not an analysis of the MHP (even assuming the host opens a goat door uniformly).

The Lucas, Rosenhouse, and Schepler quote is quite clear: "any proposed solution to the MHP failing to pay close attention to Monty’s selection procedure is incomplete". The problem they start with is one where the host chooses uniformly between goats.

Morgan et al., in their rejoinder to vos Savant's reply to their article: "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 she places on the reader's question, it is still a conditional probability problem. One may argue that the information necessary to use the conditional solution is not available to the player, or that given natural symmetry conditions, the unconditional approach necessarily leads to the same result, but this does not change the aforementioned fact."

These sources are all saying the simple solutions produce the correct numeric answer, but using faulty reasoning (i.e. reasoning that applies to a different problem). It distinctly appears that you are steadfastly refusing to admit that this is what they say only because what they say does not match your POV. I have repeatedly suggested we treat this as a POV issue and have the article not take a stance on this. Your response has consistently been that this is not good enough - that we must instead have the article endorse the POV (your POV) that the simple solutions are correct and present the POV that they address the wrong problem or are in some way deficient as an academic quibble (if at all). To make any progress at all, I think you have to be willing to admit that the sources I'm quoting say what I'm claiming they're saying. This doesn't mean that I want you to agree that what they say is The Truth - only that you agree that they are criticizing the simple solutions. Can we start there? -- Rick Block (talk) 15:46, 30 July 2010 (UTC)

Not me. You're wrong on Rosenthal, as demonstrated above, and previously, you were shown by Martin and I individually to be wrong on the others. But I do agree that it's not the editors' place to referee this dispute via the article. Taking either side violates NPOV. Glkanter (talk) 16:10, 30 July 2010 (UTC)

@Rick, There really is no point in trying to discuss sources with someone who claims that a source which says of the simple solution, 'It is actually correct' really means 'It is actually incorrect'. Note that he says the solution is correct, not that it happens by some fluke to give the correct numerical answer. Perhaps this quote, with my emphasis will help you understand what Rosenthal meant, 'Despite all the publicity, most people have at best a vague understanding of why vos Savant's answer is correct, and the extent to which it does or does not also apply to variants of the problem'. I feel that trying to discuss the more subtle points in other sources would be somewhat pointless. Martin Hogbin (talk) 18:02, 30 July 2010 (UTC)

Please read the words "vos Savant's answer is correct" again. Rosenthal is not saying the entirety of her solution is correct, but that her answer (2/3) is correct. If you are so blinded by your POV that you are unwilling to admit these sources are criticizing simple solutions then I agree it is pointless to continue this discussion. -- Rick Block (talk)

Perhaps we should ask some totally unconnected independent editors, who will not need any knowledge of statistics or even mathematics, to look at the Rosenthal paper and answer this question, 'According to Rosenthal is vos Savant's solution, also referred to in the paper as the 'shaky' solution correct for the standard (that is to say, not a variant) version of the problem?'. Martin Hogbin (talk) 19:02, 30 July 2010 (UTC)

Or should we ask 'Is Rosenthal criticizing vos Savant's solution, also referred to in the paper as the 'shaky' solution?' You are trying to define "correct" so narrowly that you're completely missing the point. -- Rick Block (talk) 23:25, 30 July 2010 (UTC)

We should not ask that because it is not the basis of our disagreement. I agree that Rosenthal does make some criticism of vos Savant but the question we have been arguing about is is 'Does Rosenthal say that vos Savant's solution is correct for the standard problem?'. Let us ask what others think about that question. Martin Hogbin (talk) 11:33, 31 July 2010 (UTC)

And I have already agreed Rosenthal says the simple solution produces the correct numerical answer for the standard problem. How about "Does Rosenthal criticize the simple solution even though it produces the correct numerical answer for the standard problem?". -- Rick Block (talk) 13:04, 31 July 2010 (UTC)

Let us make this really simple. I say that Rosenthal says that the simple solution to the standard problem is correct. Do you agree or do we need to ask others what his very clear words mean? Martin Hogbin (talk) 13:36, 31 July 2010 (UTC)

Correct, unqualified in any way? No, I don't agree. The entire point of his paper is that the simple solution is "shaky", obviously meaning he finds the logic wanting. This is the same thing all the other references I've mentioned are saying. You're continuing to deny this point. You say above you're agreeing he makes "some criticism of vos Savant". Perhaps it might be helpful if you explained what you're agreeing his criticism is, exactly. -- Rick Block (talk) 14:01, 31 July 2010 (UTC)

I say that Rosenthal says that the simple solution to the standard problem is correct. Let us ask some others and see if they agree. Martin Hogbin (talk) 17:39, 31 July 2010 (UTC)

Discussion with Gerhard

Vos Savant's "Ask Marilyn" column clearly was implying a strict "no telltale bias whatsoever" and rejecting any bias, later she confirmed that this was the basis for her answer.

In Parade Dec. 2, 1990 she presented her game "three shells and a pea" as an argument. And, rejecting any "bias", Marilyn vos Savant wrote:

You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell...

And Falk confirms that any distinction between "unconditional" and "conditional" can only be reasonable if you are adopting a "biased host", some idiosyncratic behavior.

And Falk confirms that in Marilyn's "shell game"-example any distinction of "unconditional" and "conditional" is as superfluous and can only be reasonable if someone does "suppose she is biased, because of some idiosyncratic reason of her own, toward lifting one of the two shells (and you know about that bias)..."

Falk confirmes that any "conditional approach" can only make sense if some bias is supposed or known. Otherwise not. Gerhardvalentin (talk) 23:45, 28 July 2010 (UTC)

"Before" and "after" - excepted for some idiosyncratic bias - are totally meaningless arguments.

Not any "opening of a door" - if distributed uniformly at random, if there should be a choice - gives new information. Again: Just "opening of a door" can never provide for a new "condition", but only any supposed underlying "bias" in opening a door. Gerhardvalentin (talk) 00:06, 29 July 2010 (UTC)

Gerhard - the issue is that if the player and host agree which door is which, opening a specific door always gives additional information and always affects the conditional probability in accordance with how the host chooses. The host's bias or lack of bias is exposed. If you were to run an experiment and keep track of the success of switching (by specific pair of doors) you can experimentally determine the host's bias (for each pair of doors). If the bias is 1/2, then the chances will be the same regardless of which door the host opens - but by repeated observation you can observe this. What you're saying about "before" and "after" is true if (and only if) the doors are actually indistinguishable. They are distinguishable, so solving the problem on the basis that they are not is incorrect (whether you get the correct numeric answer or not). Knowing which door the host opens means the host's bias is accessible to you (whether you think you know it or not). -- Rick Block (talk) 05:03, 29 July 2010 (UTC)

Rick, this reply is a mixture of the incorrect and the incomprehensible. Your statement 'if the player and host agree which door is which, opening a specific door always gives additional information and always affects the conditional probability in accordance with how the host chooses' is clearly incorrect. If the host opens a goat-hiding door uniformly at random, the posterior probability that the car is behind the door originally chosen by the player (and the probability that the player will win by switching) is necessarily equal to the prior probability. Please tell me in what way the conditional probability Pws is affected by the host door choice is this case.

The rest of your statement seems to one expressing some kind of faith that only one type of solution is correct. Any solution which must give the correct probability of winning by switching is correct. Martin Hogbin (talk) 08:44, 29 July 2010 (UTC)

Rick, of course you can use the fractions 7/11, and easily transform it to 14/22, and then reduce it mathematically to 7/11 again. Do you find this approach a necessity? To "add a bias" and then to eliminate it again with the factor "q = 1 / 2" isn't a necessity either. Is it? And, in 2/3 of games - the host having the car and a goat (or a goat and the car) to chose from - you will not be able to detect a host's bias. Are you saying you though can manage that? Please tell me how. Regards, Gerhardvalentin (talk) 09:02, 29 July 2010 (UTC)

Gerhard - what I mean is that "probability" can be both analyzed and observed, and that the analysis and observation should match (as the number of observations approaches infinity). If the doors are indistinguishable the car MUST be uniformly distributed (there is no way to unevenly distribute a car behind three indistinguishable doors) and the host's selection among two goats must be uniform (for the same reason) so the simple solutions are entirely correct - and there is no possible observation that will show any different results. However, if the doors are distinguishable then things are much more messy. The car might or might not be uniformly distributed. We could watch the show over and over again and see whether the car is uniformly distributed or not. Most people (both reading the problem and publishing solutions) assume the car is uniformly distributed. This is what makes the player's initial chance of selecting the car "obviously" 1/3. Anyone designing an experiment to simulate the problem would naturally select an initial random location for the car and this experiment would match the analysis (that the initial choice has a 1/3 chance of being correct). This uniform distribution is what makes P(win by switching), i.e. the overall chance of winning by switching, as well as P(win by switching AND player picks door 1), i.e. the chance of winning by switching for players who have initially selected door 1, both 2/3. To observe these probabilities we'd count either all instances of the show (for the former), or just those instances of the show where the player initially selected door 1 (for the latter). These observations will match our analysis (assuming the car is uniformly distributed).

However, the problem asks about P(win by switching AND player picks door 1 AND host opens door 3) - a specific case where the player and host both know which door the player picked and which door the host opens. To observe this probability we'd count only those shows where these conditions occurred - i.e. only shows where the player initially picked door 1 AND the host opened door 3. This subset of shows will, by observation, reveal (in the limit) the probability of winning by switching in this case. In 2/3 of these shows the host will be forced to show the "other" goat but in 1/3 of these shows the host has a choice. The probability in this subset of shows is always a function of how the host picks between two goats - it is always 1/(1+p) where p is the host's actual preference between the unpicked doors if both hide goats. If the host chooses uniformly randomly, then the player will have a 2/3 chance of winning - and the observation will reveal that the host is choosing uniformly between two goats. But being able to distinguish the doors means the observation always reveals the host's preference between the remaining two doors whatever it is. If the host is choosing randomly between two goats the information revealed is exactly this (that the host is choosing randomly). In all cases, the fundamental piece of information being revealed is how the host is choosing. Knowing this allows you to update the probability of winning by switching. -- Rick Block (talk) 14:28, 29 July 2010 (UTC)

@Gerhard. To put Rick Block's point in formal terms, once player and host agree on the identities of the doors, the host's preference for which door he opens is a nuisance parameter that must be considered (under penalty of your analysis being incomplete) and then marginalized away after choosing a prior distribution for it. Or you could just ignore the issue, and like Martin Hogbin, deftly sweep it under the carpet while muttering "Principle of indifference" three times fast. Choosing which alterenative is philosophically more satisfactory is left as an exercise, but the article should mention that there are reliable sources which dislike such under-carpet sweeping. glopk (talk) 15:48, 29 July 2010 (UTC)

The article on nuisance parameters is quite informative, it says, 'Given these, the joint distribution of only the parameters of interest can be readily found by marginalizing over the nuisance parameters. However, this approach may not always be computationally efficient if some or all of the nuisance parameters can be eliminated on a theoretical basis.' In other words, 'why calculate the probabilities for the case that the host opens each possible door when it is quite obvious that it makes no difference which door he opens'. I might add that the principle of indifference is not required because in the standard formulation use in the article the host is defined to choose evenly between goat doors. Martin Hogbin (talk) 17:14, 29 July 2010 (UTC)

@Rick, @Glopk. Did I get that right? You are saying "it's a way to determine whether the host is biased or not"? - Pardon me, but that was already a foregone conclusion: A "biased host", and certainly an extremely biased host, telling the contents of all three doors altogether in one third of all cases, never has been subject of the famous paradox: "Should she switch? - Pws 1/2 or 2/3?". That the host - if he has the choice between two goats - will open a door uniformly at random, without any bias, is already set, as per M.v.S.'s statement. - And you say it's of importance to "calculate back" in order to determine the extent of the host's bias? Is it that what you are saying? Gerhardvalentin (talk) 18:31, 29 July 2010 (UTC)

@Gerhard - What I'm saying is that seeing which door the host opens reveals additional information beyond simply knowing that the host will always open a door. By observing a specific case (say player picks door 1 and host opens door 3) the host necessarily reveals whatever preference he has between doors 2 and 3 (if the player picks door 1). The host may be totally unbiased, or the host may not - and by observation the host's actual preference will be revealed. Our analysis should match the observation (we should at least not be surprised). If our analysis says the probability of winning must be 2/3 because we know the host will always open a door, we observe that the host always opens a door, but we observe only 1/2 of the players who switch win in this specific case then what? If on the other hand we compute the conditional probability, we see it is (1/3) / ( 1/3 + x*1/3) where x is the host's preference - and if our observation does not end up with the 2/3 answer we expect (from taking x to be 1/2) we know what's up (the host is not choosing randomly). -- Rick Block (talk) 01:27, 30 July 2010 (UTC)

Rick, thank you so much for your answer and for your patience. Rather difficult for my actual perspective/field. Let me explain: In my disc. with Nijdam in de:wp I supposed that those three objects (1 car, 2 nuts) have been placed behind three doors uniformly at random, and I supposed the player makes her choice uniformly at random, and I supposed that the host - if he has the choice - opens one of his two doors (both hiding goats) uniformly at random, also. And I run my routine (written in dtm-m) 100-million times, not only counting each possible outcome, but ongoing listing the results, numbered consecutively from 1 to 100 million. You will know the outcome, you can guess it, it was exactly as expected.
If you now are singling out any individual limited areas of the protocol, and analyzing them, there are - depending on the magnitude of the partial area - admittedly deviations to the overall result. But those individual analyses cannot correct the principle of balance, they thus just lead to wrong conclusions in evaluating of any "host's bias". So, for my view, in the said scenario they are unnecessary, useless and even misleading. My conclusion: If there is no programmed host's bias, and you already know that from the outset, do you really have to confirm this even in hindsight? Once more:
Player picks door 1 and host opens door 3: Switching wins 2/3 and staying wins 1/3. The same applies to any possible selection of door numbers: 1+3, 1+2, 2+3, 2+1, 3+2, 3+1.
If you suppose a biased host and he just isn't biased, you will give the wrong answer (regardless the question however of whether he really is biased. In my test he was not.)
If the host isn't biased at all, and you know that from the outset, seeing which door the host opens reveals additional information? Maybe misleading information. My conclusion: Your "q" only makes sense if you assume that the host could indeed be biased. Otherwise not. Or am I wrong? Regards, and thank you once more. Gerhardvalentin (talk) 10:53, 30 July 2010 (UTC)

@Martin. Are you debating yourself? A few paragraphs above you demanded that the Principle of Indifference be mentioned, but now you state that it is not required. On the "obvious" lack of need for introducing a nuisance parameter, yes, in the standard (K&W) case it is not necessary, since the host explicitly has no bias there. But this is NOT what we are debating about here. The ONLY point of contention here is whether the reasoning behind the "simple" solutions is sound when applied to the MHP as stated, that is Win by switching AND player has selected Door 1 AND host has opened Door 3, that is with distinct doors. And to show that the simple reasoning is unsound we show that it fails for a slight variation of the problem, i.e. by considering the presence of a host bias. This bias (Morgan's q) is the nuisance parameter we are talking about. Now it can be proved that the odds of winning by switching are never less than one under a uniform prior for the host's preference (or bias). But to prove this result you actually must formally compute the marginalization, there is nothing obvious about it. And just to nip in the bud your expected rejoinders: (1) To use a uniform prior for the host's bias is NOT the same as assuming that the host has no preference (i.e. that q = 1/2). Rather it is a statement on the player's ignorance about the host's preference. (2) The use of the uniform prior for the nuisance parameter is not necessary for the analysis to be complete - from a Bayesian perspective the only thing that is required is that a prior be explicitly used (though some Bayesians would additionally insist that it be a proper prior). (3) The use of a uniform prior for the host's preference can be justified by the MAXENT postulate but, again, it is not a necessary condition for the choice of a prior. glopk (talk) 03:39, 30 July 2010 (UTC)

It is really not that hard to understand. If the problem statement defines the producer to place the car uniformly, the player to choose uniformly, and the host to open a goat-hiding door uniformly, as the K&W formulation does, then the principle of indifference is not required. If, on the other hand, no information is given for these distributions, as in the Whitaker question, then we must decide whether we wish to to apply the principle of indifference or not. If we apply it consistently, we get the K@W formulation, if we do not apply it, the problem is insoluble. Do you dispute any of that?

In the K&W formulation, the nuisance parameter q can be eliminated theoretically by applying an obvious symmetry thus improving computational efficiency, just like the nuisance parameter article suggests. Martin Hogbin (talk) 09:32, 30 July 2010 (UTC)

@Martin. You obviously haven't read my last paragraph in its entirety - can you please do that before responding (see Question in your glossary below). Computational efficiency is no object here: it's not like we do a Monte Carlo simulation of the problem to marginalize the host bias - the integration is trivially done symbolically (a.k.a. "by hand"). The "symmetry argument" is not obvious - in fact you manage to get it wrong in a plurality of ways above. It's exactly this lack of its obviousness that make the simple solutions incomplete in answering the completely specified (conditional, i.e. distinct doors) problem. glopk (talk) 16:24, 30 July 2010 (UTC)

My use of 'computational efficiency' was a lighthearted reference to exactly what you have mentioned, the human computer. It is quite obvious that if the host had opened door 2 to reveal a goat in the K&W formulation the answer would be exactly the same as if he had opened door 3 to reveal a goat. Fretting about which door the host opened in the K&W formulation is an obviously pointless waste of human effort. Martin Hogbin (talk) 17:41, 30 July 2010 (UTC)

Simple Terminology (an intermezzo in scherzo form)

As a public service, and to further the understanding of these pages, we are glad to provide a short abstract from Martin Hogbin's New Improbable Dictionary of Probability.

• Problem Equivalence: If what you see at the end is the same, two problems are equivalent. Thus, in the late Roman Republic, the problems of what to eat while walking barefoot from Bononia to Rome, or while riding from Neapolis to Rome, were equivalent, since both the Via Aemilia and the Via Appia ended in Rome. See also: Solution.
• Solution: The answer to a question, regardless of how it is attained. For example, observing from orbit that the Earth is not flat is a proper solution to the timeless question about the shape of the planet. So is the following syllogism: "The Earth is a giant pancake, pancakes are round, therefore the Earth is round.".
• Exegesis: The critical explanation of the sources. To be done with the utmost care. For example, consider Falk where she writes "The deep intuition behind the simple solutions to the MHP is attained by chewing mushrooms and peyote mixed with jaguar urine in a deep dark well south of Laredo, and it's just about as illuminating as doing math on a blackboard". The meaning of this paragraph, elicited only after careful contextual analysis of the words "deep, intuition, illuminating", is that "The simple solutions to the MHP reflect deep illuminating intuition."
• Symmetry: an emergent property of a system made of parts, such that it remain unchanged if the parts are exchanged. For example, a split pea soup is symmetric, since the same taste is attained by adding in the pot cold water to dry peas, or dry peas to cold water. Also, the sex chromosomes in humans are symmetrical, since XY and YX are the same pair.
• Question: Something to be answered with excruciating delay, all the while talking over the questioner's head. glopk (talk) 15:39, 28 July 2010 (UTC)
• Standard way of dealing with a conditional probability problem: Include the universe and a pony in your sample set, sweep under the carpet everything that you don't understand or like while muttering "Principle of indifference", leave the re-nornalization as an exercise for the curious reader.
• Correct: That which fits one's preconceived notion of right, regardless of context. For example, Rosenthal's qualification that the simple solution, although correct, is "sketchy", in no way detracts from its absolute correctness. See also: Solution. —Preceding unsigned comment added by Glopk (talk • contribs) 16:16, 1 August 2010 (UTC)

You know, if Martin and I are so goddam stupid, then many (how many? a preponderance? most? a majority?) of the Math professionals in the world are just as goddam stupid. Get over yourselves. Glkanter (talk) 15:41, 28 July 2010 (UTC)

It is not even amusing. Had it been cogent I might have complained that it was a personal attack. Martin Hogbin (talk) 18:08, 28 July 2010 (UTC)

Your comment is not even cogent. Had it been a personal attack I might have complained that it was amusing. glopk (talk) 18:39, 28 July 2010 (UTC)

MHP Random Host Door

This will probably be annoying, but I take issue with the idea that the fact that Mr. Hall chooses a goat-revealing-door randomly (v.s. intentionally) has any bearing on a contestant's wisdom in deciding to switch. There is a causal mechanism which is improving the contestants odds in the standard problem by switching. What is this mechanism? The the utilization of inforation. Why can the contestant no longer take advantage of information when it turns out that this accurate information was generated by a random process? Remember the nice chart of outcomes in the article:

Door 1---Door 2---Door 3

car-------goat------goat

goat------car-------goat

goat------goat------car

Say, I'm on the show. I pick door 1. Mr Hall is worried because he forgot which door the goat is behind. He guesses randomly and reveals a goat (and with it some very valuable information). Ok, he asks me if I want to switch. What would you do? Look at the chart. What would you do? Theres only one scenario (out of three) in which switching leaves me with a goat. I would switch. I haven't even looked at the conditional probabilities arguments, but I don't need to. Look at the chart. I think its irrefutable. Am I wrong?..think about this now... what if, when Mr. Hall asks if I want to switch, I ask him if he knows what car the door is behind and he's already revealed a goat. Why should his answer affect mine? What if he says he thinks he remembers, but he's not sure. He's 50% sure he knows where it is. What if before picking his door, he wispers the answer into the ear of a non-english speaking child who remembers what he said phonetically, then Mr Hall falls and bumps his head and has temporary amnesia... he picks randomly (amnesia and all that) revealing a goat. Do you switch now? What if, before you make the decision to switch, the child wispers the car's location back to him. Do you switch? What if he promises that he'll ask the child where the car is before the doors are opened. Do you switch? —Preceding unsigned comment added by BradleDean (talk • contribs) 20:14, 17 September 2010 (UTC)

What you have overlooked is that, if the host chooses randomly, he might a door to reveal the car. Martin Hogbin (talk) 21:58, 17 September 2010 (UTC)

The standard answer is that if the host always opens a goat door the switch gives 2/3 odds. The standard answer is that if the host opens a random door the odds are 50/50.

I agree that overall you end up with a 50/50 split, but there are really two situations. A) The host opens the car. Chance of winning 0. You have a goat. The other door is also a goat. B) The host opens a goat. I don't see why this is different than the normal 2/3 answer. You had a 1/3 chance of picking the correct door initially. Therefore the choice of both the other two doors is 2/3. The fact that 1/3 of that 2/3 had a 50/50 chance of being eliminated (but we know it was not) does not change this? Explanation? Gaijin42 (talk) 23:45, 31 August 2010 (UTC)

I'm going to take a crack at this one. Half of the times that you would win when switching (you've selected a goat), the host will 'blow up' the game by revealing a car (he acts randomly). That never happens when you select the car. So, 1/3 of the time you select the car. 2/3 of the time, you select a goat, but 1/2 of those times the host blows it all up => 2/3 * 1/2 = 2/6 = 1/3. So, when he hasn't blown it up, they're both 1/3, which means it's 50/50. (The remaining 1/3 is when the host reveals a car, ending the game.)

By the way, since the host is acting randomly, it could be *anybody* acting randomly to open a door, even the contestant. Instead of 3 doors, substitute 24 suit cases, and there you have Deal Or No Deal, and why the contestant on *that* show does not gain an advantage by switching. Glkanter (talk) 00:00, 1 September 2010 (UTC)

In the case where he hasn't blown it up, how is that different than the standard game? His choice being random and taking a goat, or deliberate and taking a goat seems like it should not matter. Your original choice was still 1/3 chance and the other 2 doors are 2/3. You are still taking the combined 2/3 door choice in the case where the host didn't blow it up? Gaijin42 (talk) 01:20, 1 September 2010 (UTC)

This isn't pre-rehearsed, sorry. The situations are different because in the real MHP, there are only 2 outcomes. You've selected a goat, and switching gives you a car, or vice-versa. We agree these 2 outcomes are not equally likely. With Random Monty, there are 3 outcomes: you selected the car and Monty reveals either goat; you selected a goat and Monty reveals a goat; or you selected a goat and Monty reveals the car. Using the formulas I described above, all 3 are equally likely outcomes. When he reveals the car, the game is over. When he reveals a goat, the 'reveals a car' 1/3 probability is eliminated, leaving the other two 1/3 (50/50) probabilities. Glkanter (talk) 01:37, 1 September 2010 (UTC)

Actually I think most of the scenarios ive read say that when monty picks the car the whole game starts over. But regardless, I was talking about the situation where monty did not pick the car. In that situation where monty randomly revealed a goat. Are you in the same situation as the normal game where switching gives 2/3 odds? 205.156.136.229 (talk) 19:32, 1 September 2010 (UTC)

We are not in disagreement. There are 3 equally likely outcomes in the Random Monty scenario. Monty reveals a goat 2/3 of the time: 100% of the times when you've selected the car (100% * 1/3), and 50% of the times when you've selected a goat (50% * 2/3). Eliminate the times he reveals a car (50% * 2/3), which he doesn't do in your scenario, and you're left with the car being behind either your door or the other door, with each having a 1/3 probability. They're each 1/3, that makes them equal, 50/50. Glkanter (talk) 21:54, 1 September 2010 (UTC)

In the "standard" version of the game where monty always opens a goat, there is a 1/3 chance of it being behind my door, and a 1/3 chance of it being behind the remaining door, yet the overall odds are 2/3 to switch. How is that still not the case when monty randomly opened a goat? My basic problem is this. Either randomly, or deliberately monty opens a goat. The randomness/deliberateness should not change any of the math. In the case where monty randomly opens a goat, you should still have 2/3 odds by switching. 205.156.136.229 (talk) 15:50, 7 September 2010 (UTC)

You say, 'In the "standard" version of the game where monty always opens a goat, there is a 1/3 chance of it being behind my door, and a 1/3 chance of it being behind the remaining door'. That is not correct. After Monty has opened a door to reveal a goat there is a 2/3 chance that the car is behind the remaining (not chosen by the player or opened by Monty) door. To see why this is so look at the 'Simple solutions' section in the article.

If Monty chooses a door randomly this changes things. He has a one in three chance of opening a door to reveal a car, it which case it is assumed that the game will be declared void or replayed. These void games will only occur if the player has originally chosen a goat and would therefore win by switching. The result is that some (1/2) of the games where the player would have won by switching are void reducing the chances of winning by switching to 1/2. Martin Hogbin (talk) 16:34, 7 September 2010 (UTC)

@205.156.136.229: What you're confused about is how to use conditional probability to look at each situation. In the "random Monty" scenario, the host has a 50/50 chance of opening either door 2 or door 3 regardless of where the car is. The original 1/3:1/3:1/3 chances split evenly into the "host opens door 2" case and the "host opens door 3" case. If the host opens door 3, the chances are 1/6:1/6:1/6 (which add up to 1/2, as they should - the other 1/2 of the time the host opens door 2). With me so far? But, we're not done with the "random Monty" analysis yet. We're also given that the host didn't reveal the car. This means we need to split the 1/6:1/6:1/6 chances further, into "host reveals the car" and "host doesn't reveal the car" subcases. If by opening door 3 the host reveals the car, the car is clearly NOT behind door 1 or door 2, so the chances here are 0:0:1/6. In the case we're actually talking about (host opened door 3 and didn't reveal the car) the chances are 1/6:1/6:0. We're splitting the original 1/3:1/3:1/3 into 4 cases - host opens door 2 and reveals the car (0:1/6:0), host opens door 2 and doesn't reveal the car (1/6:0:1/6), host opens door 3 and reveals the car (0:0:1/6) and host opens door 3 and doesn't reveal the car (1/6:1/6:0). The last case happens only 1/3 of the time (1/6+1/6), and if this is what has happened then the car is equally likely to be behind door 1 and door 2. Expressed as conditional probabilities in this case (divide each by the sum), the chances are 1/2:1/2:0.

In the usual problem the original 1/3:1/3:1/3 chances split unevenly. If the car is behind door 2 the host MUST open door 3 and if the car is behind door 3 the host MUST open door 2, but the host opens either door 2 or door 3 (presumably equally often) if the car is behind door 1. So the 1/3:1/3:1/3 chances split into two cases - host opens door 2 (1/6:0:1/3) or host opens door 3 (1/6:1/3:0). Given the host has opened door 3, the chances the car is behind door 1 are only half the chances the car is behind door 2. Expressed as conditional probabilities in this case, the chances are 1/3:2/3:0.

Whether the host acts randomly or deliberately does change the math. The real problem here is the belief that the host opening a door cannot change the player's initial 1/3 chance of having selected the car. This statement is simply not true. The player's door has a 1/3 chance of hiding the car BEFORE the host opens a door. AFTER the host opens one of door 2 or door 3, the chance the player's door hides the car is a conditional probability which is some portion of the original probability divided by the probability of being in the case of interest. In the usual problem this is 1/6 divided by 1/2, which (not exactly coincidentally) is 1/3. In the random Monty version, this is 1/6 divided by 1/3, which is 1/2. The same analysis applies to the other doors as well. In the usual problem the chance the car is behind door 2 (if the host opens door 3) is 1/3 divided by 1/2 which is 2/3. In the random Monty version it's 1/6 divided by 1/3, which is 1/2. -- Rick Block (talk) 20:27, 7 September 2010 (UTC)

THE IMPORTANCE OF RANDOMIZING YOUR CHOICES

As a contestant, of course I would rather pick from 2 doors to get a 50% chance to win than pick from 3 doors with only a 33% chance of a win. So after Monty opens a door and reveals one of the goats, then I must flip a coin to randomize my choice. I would only switch doors if the coin flip tells me to. (If I follow a strict rule and always stick with door one, Or if I follow a rule and always switch to door 2 that will not help, I still win only 33% of the time.)

Here is a similar riddle: same set up but there are 10 doors, 9 goats but still only one car. As before, after I pick Monty opens a door and shows me one of his goats. Obviously I have only a 1 in 10 or 1 in 9 chance of winning, so I complain that this is a lousy game. Monty is in a sympathetic mood and he says to me "your right this game is for chumps" Let's forget this game. I will just set it up for you so there are only 2 doors, you pick one and you have a 50% chance to win!" I say "Yes, yes yes." We even pay the game 10 times, and I win (approximately) 5 new cars. --The lesson to take home regarding the 10 door or the 3 door game is to Randomize your pick After Monty improves your odds by opening one of the doors. 76.168.154.252 (talk) 07:23, 22 November 2010 (UTC)