User talk:Martin Hogbin/MHP - An open challenge

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Perhaps then you can explain to me why conditioning on the goat that is revealed is sillier than conditioning on the door that is opened. Martin Hogbin (talk) 22:35, 2 October 2012 (UTC)

Because we're told about the door that is opened and we can't help but discuss door numbers in some way or another in setting up a mathematical model of the problem. We're not told anything about the goats and we're obviously not supposed to start imagining that there's a boy goat and a girl goat and this should make a difference. Richard Gill (talk) 05:37, 4 October 2012 (UTC)

We are told that there are three doors and two goats. We are told that one of the doors is opened and that one of the goats is revealed. The doors were given numbers by vS but these have no meaning whatever to us. People tend to imagine the doors in a row and numbered from left to right but none of this is given in the problem statement. The point is that the doors are, in principle, distinguishable, regardless of whether they have numbers on them. The goats are also, in principle, distinguishable; goats are known not to be bosons. We imagine that the player sees the door that is opened and the goat that is revealed and that they know that there are two unopened doors and one unchosen goat. The only difference is that vS chose to give the doors numbers. She did this only to try to make the sequence of actions clearer, not because she wanted us to distinguish between doors that the host opened. This is something she later considered a mistake. The problem can be perfectly well described without door numbers.

You then say, "...we're obviously not supposed to start imagining that there's a boy goat and a girl goat and this should make a difference". I quite agree. As Seymann says, we should be looking towards the intent of the questioner in order to formulate the problem mathematically. It is quite clear from what vS says that, "...we're obviously not supposed to start imagining that the door number that the host opens should make a difference". Martin Hogbin (talk) 09:03, 4 October 2012 (UTC)

Exactly. If we chose door 1, and the car happened to be there, it would be equally likely for us which door was opened. In advance too, the car is equally likely, as far as we are concerned, behind any of the three doors. Hence: problem is symmetric in door numbers. Hence: whether or not we would win by switching is independent of the number on the door we chose and the number on the door which was opened. And obviously, 2/3 is the chance of winning by switching.

This is the simple solution and the conditional solution combined. You don't have to condition on door numbers because it doesn't make a difference. Which is obvious, and it's a matter of taste whether or not you bother to spell it out. In the maths classroom it's worth paying attention too, in a layman's intro to MHP not. However later in the survey for laymen of the literature on MHP you are bound to mention the issue.

The question is only: do you implicitly or explicitly ignore door numbers? It hardly costs any words to do it explicitly. ie, with motivation. So why not? Your solution is complete and you keep Nijdam quiet. You've shown in advance that doing conditional probability calculations is a waste of time, the result is a forgone conclusion. No need for Rick's lengthy and unituitive calculations.

This is literally, much ado about nothing. No dependence. No need to condition. Just say so, out loud. Richard Gill (talk) 04:10, 9 October 2012 (UTC)

So why then do some people consider a solution that conditions on the door that has been opened any more complete (and therefore necessary to have in the article) than a solution that conditions on the goat that has been revealed?

I can see what leads people to want to do this, but if they examine their preference critically they should see that conditioning on the door number opened is an arbitrary choice which they have, for no good reason, made themselves. It was clearly not the intention of neither W/vS or Selvin that we should condition on doors opened or goats revealed. Martin Hogbin (talk) 22:37, 9 October 2012 (UTC)

It is more complete because it shows explicitly that there is nothing to be gained by this conditioning. It is not an arbitrary choice. It is information which we are explicitly given by Vos Savant. If you don't use it then there is a gap in your argumentation. Maybe you can do better by using the further information which you've been given. Maybe you can do better still than 2/3.

Why it is necessary to have in the article: because a whole heap of reliable sources do it that way.

Maybe you should try reading carefully my article in Statistica Neerlandica, and later writings on my home page [1]. I wrote all that stuff so I don't need to keep repeating it. Richard Gill (talk) 13:27, 10 October 2012 (UTC)

Of course, I agree that conditioning is not necessary at all but you are now treating the vs/W statement like an exam question wher we are given information because we are expected to use it. You know that information that is volunteered (the door number opened by the host) is not necessarily any more significant than information that is not openly stated (the goat revealed by the host).

We are told that the host reveals a goat (one of two goats). The host may have a preference for one goat and may reveal it whenever possible under the rules. In that case, from the frequentist perspective that Morgan have taken, the probability of winning by switching depends on the goat that has been revealed. Ther is absolutely no difference between goats and doors in reality. The only difference is that Vs chose to tell us a door number. In an exam we assume that we have been given information for a reason but, in real life, we have to think for ourselves.

Incidentally, Kraus and Wang make very clear why the door numbers are so important. Remember, vos Savant's question is a trick question. The specific door numbers are deliberately introduced in order to make you visualize the problem in a way which leads you instinctively to the wrong answer. Realizing that the door numbers are irrelevant is part of the solution. And a correct solution requires not just a correct answer but a correct argument for that answer.

The door numbers do make the problem harder according to K&W but ther is no suggestion that vS did this on purpose. She said herself that she did it only to make the events clearer.

K&W (psychologists) make it abundantly clear that naming the door numbers in the problem description is exactly what leads people to give the wrong answer and stick to it. Vos Savant's thinking (if any) is irrelevant. Moreover K&W make abundantly clear that the royal road to understanding why 50-50 is wrong is realizing that the door numbers (names) are irrelevant. Read them. Read them, carefully. Richard Gill (talk) 16:04, 13 October 2012 (UTC)

Part of a complete understanding of MHP is understanding where the 50-50 argument breaks down. Just like TEP we (emphasis on we - we editors...) know the right answer; the question is, what is wrong with the reasoning which gave the wrong answer?

Absolutely but that has little to do with which door the host opens. Nobody worries about that when they fiorst hear the question.

I think you have become so familiar with MHP that you don't recognise the steps which a newcomer has to make to resolve the paradox. Of course, the door numbers are irrelevant, so the combining doors solution is valid! It's valid because door numbers are irrelevant. And they are irrelevant because of symmetry.

On the contrary I am the one who wants to understand the position of a newcomer to this problem. I think most newcomers disregard the door number opened by the host completely, but that still does not help them get an answer. In fact we know that. None of the people who give correct answers ever mentioned the door number opened by the host until the Morgan paper.

There are three steps in getting to the bottom of MHP. One is realizing the answer is 2/3 not 50-50.

Agreed

The second is giving a complete argument for your answer. The degree of completeness is perhaps a matter of taste. Different readers have different needs. Editors of wikipedia have to stand above this, be aware of all needs, all complexities. The third step is understanding where the mistake lies in the intuitive but wrong reasoning which gives 50-50. Richard Gill (talk) 18:57, 10 October 2012 (UTC)

I would reverse your last two steps.

Degree of completeness is indeed a matter of taste. Mathematicians can get as pedantic as you like if they want to. My point, as argued above is that conditioning on the door number opened by the host is no more complete than conditioning on the goat revealed by the host. There is plenty you could condition on if you want to be perverse, but best not to be and keep it simple.

PS I just noticed you have given the complete argument yourself: "The important point to consider is whether any event that occurs after you have chosen your door but before you decide whether to swap or not gives you any information about the whereabouts of the car. Under the standard assumptions you know that Monty will reveal a goat, because he must do under the rules, you also gain no information from his choice of door when you happen to have originally chosen the door hiding the car because the host must choose randomly between the two doors available to him under the standard assumptions in that case. So, the host opening, say door 3 to reveal a goat tells you nothing you do not already know, thus your original odds of having chosen the car cannot change. In the standard version of the problem you have a 2/3 chance of winning if you swap. Martin Hogbin (talk) 16:02, 27 May 2012 (UTC)". But you should have said "tells you nothing you do not already know about whether or nor the car is behind door 1". It sure does tell us something about whether or not the car is behind door 3! The odds on the car being behind door 1 do not change. The odds on the car being behind door 2 or door 3 do change. You used symmetry (and you used Bayes' rule, odds form). This proof was given (incomplete) by Devlin; withdrawn by him after receiving criticism; it was completed (i.e. shown to be correct once a missing step is filled in) by Gill. Richard Gill (talk) 19:10, 10 October 2012 (UTC)

You picked the wrong quote from me there. More recently I worded things correctly. If you remember I wanted to make it part of the question.

Martin, I think you are getting confused! Just like Devlin. You almost gave the full conditionsl solution, slipped up, and got into a panic. Then later you wanted to write the solution into the problem. Now that's what I call perverse. There is nothing perverse in taking careful account of door numbers. This is not mathematical pedantry. It's clear thinking. What is perverse, is changing the problem by bringing in crazy issues like whether the goats are boy goats or girl goats and whether or not we can use that information. Have you read Kraus and Wang recently? Have you read my papers recently? It's about the sources, not about your idee fixe. Richard Gill (talk) 06:28, 12 October 2012 (UTC)

I am not sure what you mean here. My statement you quoted above is incorrect, I agree that I made a mistake there. Martin Hogbin (talk) 18:58, 12 October 2012 (UTC)

Do you agree that it is easy to correct your statement? And not by writing the answer into the question. No. By being careful in talking about information pertaining to whether or not the car is behind door 1 (binary issue). Not about whether the car is behind door 1, or door 2, or door 3 (ternary issue). Richard Gill (talk) 15:53, 13 October 2012 (UTC)

Yes, I agree. But, in addition, I saw a possible advantage in making that fact part of the problem statement. No one else seemed to like that idea so I dropped. By the way, I think you have missed an important point. See User_talk:Martin_Hogbin/MHP_-_Fixing_Devlin

To sum up: the door numbers are specified in the problem, to lure you into giving the wrong answer. The solution consists in seeing the problem without the numbers. The reason why they may be ommitted is because of symmetry. Richard Gill (talk) 20:20, 10 October 2012 (UTC)

By the way, there is no difficulty in taking account of boy or girl goat. Suppose we saw that the goat is a girl goat. Since I have no information about the sex(es) of the goats this does not change my probabilities. Vos Savant doesn't mention sex colour race age or whatever of the goats, so we don't bother to say that what she didn't tell us is irrelevant. It is useful to say that what she did tell us, about the numbers, is irrelevant. That's not mathematical pedantry. It's being clear, complete, careful. It's one sentence. No expensive words. What the hell is the problem with that? One way to get the right solution to MHP consists of realizing that the door numbers are irrelevant! We mustn't just give an answer, we must also give a reasoning. What counts is not the number 2/3 but a sound reasoning which brings us there. Richard Gill (talk) 06:33, 12 October 2012 (UTC)

Not about the article

I am not discussing what we should write in the article here, that is why it is in my user space. I know that no source mentions the goats so we cannot put that in the article.

What I am talking about is the logical necessity to consider the door opened by the host rather than the goat revealed.

From the frequentist perspective used by Morgan et al, if the host has a door preference then you must take account of the door that he has opened to get the probability that the player will win by switching. This applies regardless of whether the doors are numbered or not. The host opens one door to reveal a goat. If the host has a door preference and the door he opens is the one that he prefers to open then your probability of winning by switching will be 1/2. Do you not agree? That is the whole basis of their paper.

On the other hand if the goat revealed is the one that the host prefers then also your probability of winning by switching will be 1/2.

The writing of numbers on the doors makes no difference at all to these simple facts. Martin Hogbin (talk) 18:58, 12 October 2012 (UTC)

Now you're being facetious. Call the doors left, middle and right (as seen by the audience) if you prefer. Player chooses left hand door. Host opens right hand door revealing a goat. And moreover you've explained that it might well make a difference which doors are involved, if we were given information about the host's behaviour. But we aren't.

Most readers will approach the problem using objective Bayesian probability, ie probabilities determined by natural symmetries in the information we have been given, ie in the problem description. They therefore have an extremely good reason to state that the identities of the doors involved in a particular case are irrelevant. The probability of winning by switching is therefore 2/3, whichever door was first chosen by the player and whichever door was opened by the host. Left, right or middle. Agree? Richard Gill (talk) 15:47, 13 October 2012 (UTC)

Yes, of course I agree, that is where we started years ago but many editors, including you it now seems, think that we need to show a solution/explanation in which we show the two doors that the host might have opened. We have just agreed that it makes no difference which door the host opens so there is no need distinguish between them in our explanation. The player picks a door, then the host opens a legal door, then the player decides whether to swap or not, yet you still seem to be arguing that the explanation which shows that the host might have opened door 2 is somehow better/more complete/answers the question asked.

My point, which you have not yet addressed, is that there is just as much reason (or as little reason of you prefer) to distinguish between the goat revealed by the host as there is to distinguish between the door opened by the host.

To put it another way. If you accept Morgan's argument about the host's possible door preference then you must accept mine about the host's possible goat preference. It is easy for me, I do not accept Morgan's argument. What about you? Martin Hogbin (talk) 15:43, 14 October 2012 (UTC)

I don't accept Morgan's argument: I follow the ways most people think about MHP ((1) objective Bayesian probability based on symmetry; (2) strategic thinking - decision theory). I also don't lay down rules how MHP has to be solved. I do think it is very natural to think of doors as being distinguishable doors but goats as being indistinguishable. Moreover Kraus and Wang explain how the way ordinary people visualize MHP, led on by Vos Savant's words, leads to a fixation on door identities and hence to the intuitive 50-50. So I have addressed the point which you raise.

Secondly, I don't understand why you obstinately continue to obstruct any simple solution which does take account of the numbers (e.g. by giving the reason why they may be ignored) which therefore is more complete than a solution which doesn't do this! The simple solution tells you that always switching gives you the car 2/3 of the time. Add two words of argument ("by symmetry") and you have strenthened this to "and moreover there's no way to do better". That's more, and it's 's easy to do. So what's your hangup? Richard Gill (talk) 12:42, 15 October 2012 (UTC)

I do not think either of us can claim to know how most people think about the MHP. K&W mainly say that door identities confuse people and should therefore be ignored.

Goats and doors are distinguishable. We may choose not to distinguish between them in the MHP and we may (or may not) find it necessary to give reasons why we choose not to distinguish. If you find it necessary to explain why you choose not to distinguish between doors then you should also explain why you choose not to distinguish between goats. On the other hand, you might consider both obvious. There is a symmetry with respect to doors and a symmetry with respect to goats.

To put some (entirely subjective and made up by me) numbers to what I mean, I would call Carlton's 'Not your original choice' solution 90% complete and a solution which mentions the door number opened by the host and explains why we can ignore this, say, 92% complete, so yes, it is more complete but only by a tiny and arbitrary bit. Why does the solution not show the doors that the player might have chosen, or the doors which might have hidden the car? That might make the solution, say 95% complete (I have to leave 5% for pedantic mathematicians and the really perverse).

The cost of this 'extra completeness' is that it addresses an issue that pretty well no one thinks about until they are told of it. There is no record of any one of the thousands of letters received by vS saying, 'Ah, but what if the host had opened the other door?'. There were no solutions (to the W/vS question) mentioning the possible door numbers opened by the host until the Morgan paper. Obviously this is not what people think about, so we do not need to address it to help them understand the problem. Martin Hogbin (talk) 17:42, 15 October 2012 (UTC)