Talk:Monty Hall problem/Arguments/Archive 3

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Game challenge

Start with an ace and two jokers. You're the dealer.

• Shuffle the 3 cards. Deal them face down left to right. The leftmost card corresponds to "door 1", the middle one "door 2" and the rightmost one "door 3".
• initial pick is random - I'll pick the card on the left (door 1). I now have a 1/3 chance of having the ace.
• Look at the card in the middle and then the card on the right (the two I didn't pick). Immediately turn over the other one if either is the ace. If you haven't done this, now you have a choice to make. You can comply with the MHP rules by turning either card over. It doesn't matter if I know how you're deciding, what matters is whether you decide (with preference p) or whether you're forced to take a random action (i.e. your preference p=1/2). To make the effect of this the most obvious (p=1), if you have a choice please turn over the rightmost card (door 3). If it doesn't matter how the host decides in this case and can't affect my initial 1/3 chance of having the ace, you should be willing to accept this method of deciding.
• host makes the offer to switch
• Player decides after you flip a card - I'll never switch, but simply vary my bet that I have the ace.

We've scrupulously followed the rules of the MHP (right?), so your claim is I have a 1/3 chance of having the ace and you have a 2/3 chance regardless of which card you turned over - i.e. you have a 2-1 advantage so should be willing to give me 1.5 to 1 odds if I never switch. I'll stay every time, but I'll bet a penny if you turn over the middle card (door 2) and $100 if you turn over the card on the right (door 3). Repeat at least 60 times and I'll do this for real money. My guess is I'll end up roughly $500 up even though we both know you'll win 2/3 of the time. My chances of having the ace change depending on which card you turn up, and by varying my bet I can make money if you give me anything better than even (1-1) odds. -- Rick Block (talk) 22:08, 27 March 2009 (UTC)

That is not an accurate representation of the MH scenario. You want to be the player in a case where the player knows the host strategy. In the MHP the host strategy is unknown. If we agree that the host strategy should be known only to the host (as is the case in the stated problem) then I am happy to play you as either the host or the player. Martin Hogbin (talk) 11:23, 28 March 2009 (UTC)

Where does it say the probability of interest is the probability from the player's perspective? As I read it, the question is about the probability given everything we know from the problem statement - which presumably the player knows as well, but in my view this is actually irrelevant (perhaps the player has no probabilistic skill whatsoever and can only guess). For example, if the problem statement said the host rolled a die and put the car behind door 1 if die=1, and door 2 if die=2, and door 3 otherwise and we asked what is the player's chance of winning by switching given which door the player initially picked and which door the host opens, we wouldn't "solve" this by assuming the player doesn't know how the host picked which door to put the car behind and start by assuming a 1/3:1/3:1/3 distribution.

I agree that there is nothing to say that the probability must be from the player's perspective but it must be based on the information given in the question and the host's strategy is not given. Thus in any game based on the problem the host strategy cannot be defined.

My view is that by saying the probability for the "MH scenario" is 2/3, we're saying it must be 2/3 for any variant that meets the stated conditions. This variant complies. The nature of a counter-example is to create an example that meets the stated conditions but doesn't have the claimed results. Since I can create a variant where the probability is not 2/3, but 1/2 or 1 depending on which door the host opens, either

• the problem is underspecified (this can be fixed by clarifying that the host must pick randomly or, where you seem to be going [a valid, but perhaps unusual direction], by clarifying that the player doesn't know the host's strategy) - this is exactly the same kind of quibble as the quibble that the host must make the offer to switch and must show a goat, or
• the answer that the probability of winning by switching given knowledge of which door the host opens is 2/3 is incorrect (in the sense that it is not always the case given only what is in the problem statement), or
• the problem is misstated and should be changed to clarify that the probability of interest is the unconditional probability (one way to do this is to force the player to decide before the host opens a door).

Pick one. -- Rick Block (talk) 18:05, 28 March 2009 (UTC)

It looks like neither of us will be winning any money just yet. I understand the argument the conditional probability of winning by switching is between 1/2 and 1. I cannot say that this is incorrect; it as the correct answer, but not to a question that anybody cares about. Martin Hogbin (talk) 22:23, 28 March 2009 (UTC)

Let's Clarify Morgan's Points

According to Morgan, does anybody's solution give a numeric result without the 'equal goat door' constraint?

According to Morgan, why would the 'Combining Doors' solution be a 'false' unconditional solution?

As there is no mention whatsoever of any 'host behaviour', or door numbers in Rick's 5 bullet points, why do we accept Morgan's criticisms that rely on them? Glkanter (talk) 23:34, 22 March 2009 (UTC)

From their conclusion:

In general, we cannot answer the question "What is the probability of winning if I switch, given that I have been shown a goat behind door 3?" unless we either know the host's strategy or are Bayesians with a specified prior."

We can get into the Bayesian thing if you'd like (my guess is you really don't want to do this), but what they're saying is to state a specific number you have to make an assumption about the host's strategy.

The combining doors solution is equivalent to their F5 that I quoted above.

Morgan's criticisms rely on interpreting the problem as a conditional probability problem. The fact that the doors are numbered is part of the problem statement, as is the potential for the host to have a preference for one door over another. Their criticisms match criticisms published in other sources as well, such as the Gillman paper referenced in the article. Here's a quote from the Gillman paper:

What is the probability P that you win if you switch, given that the host has opened door #3? This is a conditional probability, which takes account of this extra condition. When the car is actually at #2, the host will open #3. But when it is at #1, he may open either #2 or #3. The answer to the question just asked depends on his selection strategy when he has this choice—on the probability q that he will then open door #3. (Marilyn did not address this question.)

Two different math papers. Both say almost exactly the same thing. We accept both as reliable sources. -- Rick Block (talk) 00:19, 23 March 2009 (UTC)

Thanks Rick. I admire your patience and willingness to share your knowledge.

What if this is the question: "0. What is the aggregate probability of winning by switching - meaning across all players regardless of which specific door they pick and which door the host opens?"

What if we add 'the equal goat door' constraint to MvS' solution? Does an unconditional solution work then? If so, why is the Morgan conditional solution, which also requires the 'equal goat door' constraint, preferable to an unconditional solution?

Has Morgan really dis-proven the 'Combined Doors' solution? Or just 'dismissed' it?

Isn't there a 'rule' in Probability that says 'whatever is not constrained can (must?) be assumed to be random'?

This may be my last word on the topic: Given that there is no mention of any 'host behaviour' in any description of the problem (not the solutions), how does Monty's inevitable choice of losing doors affect the original 'chosen door = 1/3 probability'? Glkanter (talk) 01:54, 23 March 2009 (UTC)

Question 0 is the unconditional question, and the answer is 2/3 chance of winning by switching (no host preference required, any unconditional solution is just fine).

Adding the equal door constraint enables the unconditional solution to arrive at the correct answer and, therefore, to be used but it requires an argument of some kind like "Due to the natural symmetry of the situation, the conditional probability of interest must be the same as the unconditional probability. We can see the unconditional probability is 2/3 because ...". I'll note (my opinion) that this is a dangerous approach since people generally suck at recognizing when "natural symmetry" actually applies.

Morgan et al. don't directly say anything about the combined doors solution.

The rule in Probability is that whatever is not constrained may have any distribution (random is just one option).

Regarding your last question, let's try this one more time. The player's 1/3 chance at the beginning splits into the player's chances for the case where the host opens door 2 and the case where the host opens door 3. The 1/3 at the end (which is a conditional probability) is only a piece of the original 1/3 (half in the case where p=1/2). p (and 1-p) are the fractions of this 1/3 that end up in each case.

                   1/3           +          1/3         +         1/3    = 1
                   /\                       /\                    /\
                  /  \                     /  \                  /  \
                 /    \                   /    \                /    \
                /      \                 /      \              /      \
host opens:  door 2   door 3          door 2   door 3       door 2   door 3
              /          \             /          \          /          \
             /            \           /            \        /            \
          (1/3)p    +  1/3(1-p)  +   0       +     1/3     1/3     +      0 = 1

If p is 1/2 the two terms at the left are each 1/6, and when the host opens door 3 the (unconditional) chances are (1/6,1/3,0). To express these as conditional probabilities we divide by the sum, i.e. divide each by 1/2, which makes these (1/3, 2/3, 0). If the player is looking at two closed doors and one open door, this split has happened (in accordance with whatever p is). The upshot is that the original (unconditional) 1/3 and the resultant (conditional) 1/3 are never the same 1/3 - and the unconditional solution is talking about the 1/3 on the top line, not the conditional probability you can compute from the bottom line by dividing (1/3)p by ( (1/3)p + 1/3 ). Note that this turns out to be 1/3 if p is 1/2, but depending on p it might be anything between 0 and 1/2. -- Rick Block (talk) 14:55, 23 March 2009 (UTC)

Do you have a reference for your response to my 'Random rule' question? Or is that your opinion?

I've quoted Morgan on this (search for "natural symmetry" on this page). -- Rick Block (talk) 19:13, 23 March 2009 (UTC)

In the article, why not 'define the problem' as question 0 (above)? That's how I've always interpreted it, anyways. Who's to say? Then use the best unconditional solution.

Question 0 doesn't match any description of the MHP I've ever seen published anywhere. The whole point is to put the player in front of two closed doors and one open door and ask what the probabilities are - doing this is what makes it a conditional problem. -- Rick Block (talk) 19:13, 23 March 2009 (UTC)

If Morgan doesn't address the 'Combining Doors' solution, and 3 out of 4 sources do not include the 'equal goat door' constraint, then why not use this for the Solution, without any qualification? Glkanter (talk) 16:18, 23 March 2009 (UTC)

Why not say it's a conditional probability problem which most people answer assuming p=1/2? If we do this, then if someone asks about some stupid variant, like the "host forgets" one (which vos Savant has asked about) someone coming to the article here will see that it is a conditional probability problem and be able to figure out the correct answer. The combining doors solution says the answer to the host forgets variant is 2/3 (but it's actually 1/2) - vos Savant's comments about this are kind of interesting [1]:

Back in 1990, everyone was convinced that it didn’t help to switch, whether the host opened a losing door on purpose or not. Assuming a knowledgeable host who would always open a losing door, that was incorrect. (A knowledgeable host who opened a winning door on purpose wouldn’t have much of a show, would he?!)

Now everyone is convinced that it always helps to switch, regardless of what the host knows. But this is just as incorrect!

As I said above someplace - fuzzy thinking (specifically, treating a conditional problem as if it's unconditional) contributes to this. -- Rick Block (talk) 19:13, 23 March 2009 (UTC)

Without assuming any 'host behaviour' whatsoever, how is the 'Combining Doors' solution inadequate for the MHP problem via MvS as you understand it?

Here's a new can of worms...since 'host behaviour' is never mentioned in any 'non-variant' MHP problem statements (except occasionally to say he chooses goat doors equally), why can't we treat each instance of the game the same as flipping a coin? You know, the past results don't influence the probabilities of the next playing of the game? Glkanter (talk) 20:24, 23 March 2009 (UTC)

If we don't assume any host behavior we don't know how the original (unconditional) 1/3 splits when the player opens a door, so the chances of winning by switching are between 1/2 and 1, not 2/3 - so the combining doors solution is not correct. The issue is the question that is being asked. The question is not "what is the probability of winning by switching for all players who switch", but "given you're looking at two closed doors and one the host has opened, what is the probability of winning by switching". It's a conditional question.

Re worms - I may not be following what you mean. Do you mean treat the host's preference as 1/2?

If you want the problem to be unconditional, I think you have to change the question. Gillman's version is basically "decide before the host opens a door". Morgan's version is "host will open a door showing a goat, what is your chance of winning if your strategy is to switch" (implying a decision before the host opens the door). Grinstead and Snell's version compares players who unconditionally (i.e. decide before the host opens a door) switch vs. players who unconditionally stay. -- Rick Block (talk) 04:04, 24 March 2009 (UTC)

I just read MvS' response to 'the host forgets'. It violates one of the bullet points (premises), and needs no further discussion here. So Morgan doesn't address the 'Combining Doors' solution. I don't see how how it relies on any 'host behaviour'. The fact is, after revealing a goat the formula reads either 1/3 + (2/3 + 0) or 1/3 + (0 + 2/3) where the 1/3 on the left of the '+' is the chosen door and the numbers inside the parenthesis are the left and right unchosen doors. As both formulas are exactly equal to 1/3 + 2/3, I do not see this as conditional, regardless of any 'host behaviour'. Can you please provide a published source which either disproves the 'Combining Doors' solution, or explains why the 'Combining Doors' solution must be solved as a conditional problem? Glkanter (talk) 14:08, 26 March 2009 (UTC)

If you're so sure it's always 2/3 regardless of the host behavior why won't you put real money on it? -- Rick Block (talk) 14:04, 27 March 2009 (UTC)

We're playing cards with an Ace and 2 Jokers? We bet the same amount before each hand? I pick, you discard a Joker, and I get to switch every time? Sure, I'll play. I anxiously await your reply to my previous questions. Glkanter (talk) 14:37, 27 March 2009 (UTC)

No - your claim is the host never affects the odds so it doesn't matter if we bet before or after the host discards, or are you agreeing that the player's odds might change from 2/3 depending on some host behavior? I want to bet after as I described above. Regarding combining doors - it's clearly an unconditional solution and we've already talked about the references saying the problem is a conditional problem (Morgan, Gillman, Grinstead and Snell - there are plenty more if these aren't enough). -- Rick Block (talk) 15:28, 27 March 2009 (UTC)

I find this thread unproductive at this point. I will return to the main talk page and propose moving the 'Combining Doors' solution to a prominant and non-discreditted placement in the article. Glkanter (talk) 15:48, 27 March 2009 (UTC)

Words Have Meanings

I'd like to see the word 'variant' expunged from these discussions.

As it is used, it represents a new premise (or constraint) to the MHP. Any new premise represents a different problem than the specific MHP, with the 5 agreed upon bullet points.

I think it would be more precise to call these 'different problems', which intentionally share characteristics of the MHP. But, by definition, they are no longer the MHP. Glkanter (talk) 19:51, 28 March 2009 (UTC)

I agree that the wording 'Monty Hall Problem' is better used to refer to the simple (unconditional if you like) case. Martin Hogbin (talk) 21:25, 29 March 2009 (UTC)

Can we get this straight

I think that most people believe that Morgan's solution to the MHP applies only to the specific case where the player has chosen door 1 and the host has opened door 3. Who agrees with this and who disagrees? Martin Hogbin (talk) 21:28, 29 March 2009 (UTC)

Clarification Morgan's solution would also apply to a case where we were told that player has chosen door 1 and the host has opened door 2 but it does not apply if we are told that the player has chosen door 1 and the host has opened one of the other two doors. Martin Hogbin (talk) 08:53, 30 March 2009 (UTC)

Martin - The distinction between the conditional and unconditional situations here seems to confound you. Morgan et al.'s solution addresses the conditional case which is, as you say, player chooses door 1 and host opens door 3. Or, player chooses door 1 and host opens door 2. Or player chooses door 2 and host opens door 1. Or player chooses door 2 and host opens door 3. Or player chooses door 3 and host opens door 1. Or player chooses door 3 and host opens door 2. These are the conditional cases. Read the problem description. At the point the player decides, she will be in one and only one of these cases (not two, not all six - this is not a problem in quantum physics). Evaluating the probabilities in effect in one, and only one, of these cases is exactly what it means to evaluate the conditional probability. -- Rick Block (talk) 13:24, 30 March 2009 (UTC)

So let me get this quite straight. You agree that if we are told the player has chosen door 1 and the host has opened one of the other two doors the problem is not conditional and the probability of winning by switching is always 2/3? Martin Hogbin (talk) 18:18, 30 March 2009 (UTC)

Yes. The question as you phrase it is asking about the chances of winning given the host rules and that the player has picked door 1 (not also given which door the host has opened). However that's not how the problem is structured. The player can always see exactly which of the two doors the host opens. At the decision point, the player is standing in front of two closed doors and one open door. The player can therefore see which specific one of the two doors the host opened. There is no uncertainty about which door it is. If the player picks door 1 the question is NOT what are her chances of winning by switching given the host WILL OPEN (or has opened, if you must) an unknown one of door 2 or door 3, but what are her chances of winning given which specific door (one of door 2 or door 3) she has seen the host open with her own two eyes. Again, the distinction between the conditional and unconditional situations confounds many. -- Rick Block (talk) 18:51, 30 March 2009 (UTC)

I have no difficulty in distinguishing conditional from unconditional it is just that, from an earlier conversation with you, I got the impression that you believed that the answer to the question where the host had opened one (unspecified) of the two unchosen doors should be treated conditionally. Obviously that was a misunderstanding on my part.

It is interesting to consider what 'the probability of winning by switching' means in the Morgan context. It is a measure of what proportion of the games the player would win by switching if we repeat the game with the car being placed randomly each time but with the host having exactly the same door opening strategy and having opened the same door each time. Martin Hogbin (talk) 19:05, 30 March 2009 (UTC)

The earlier issue may have been around your (continued?) notion that it is possible to ask the "conditional" question about both doors the host might open simultaneously. Probabilistically, you're either asking an unconditional question (without being given the specific door the host opened - so might as well ask about the situation before the host opens the door) or a conditional question (given the specific door). You are correct about what Morgan et al. mean by "the probability of winning by switching". Again, this is exactly what is meant by the conditional probability (the probability given which door the player picked and which door the host opens). Do you see (yet) how this is the precise question that is asked? -- Rick Block (talk) 00:25, 31 March 2009 (UTC)

I think we have finally reached agreement. Morgan's solution is correct iff you take the question and the answer to have the meanings that I have given above. I am glad we have got round to the question that was asked. Morgan's assumption is certainly a possible and reasonable interpretation of the question but it is far from being the only one. This has been my main point all along. As I have said before, I am happy to accept that the Morgan paper as a reliable source on the subject of mathematics and probability; where I think they do not qualify so highly is in the subject of interpreting the question that Parade reader intended to ask and understanding how they might interpret the answer. As we have reached agreement on some things I will stop this discussion here and start a new section on how I think Morgan have got the question wrong. Martin Hogbin (talk) 08:38, 31 March 2009 (UTC)

Morgan's interpretation

Many people, including myself get the feeling that something is wrong with the Morgan solution to the MHP but find it hard to say exactly where the problem lies. None of Morgan's assumptions is unreasonable but in nearly every case there are others that could have been made. As usual, I am considering only the case that the host always opens an unchosen door to reveal a goat and always offers the swap. So how do Morgan et al manage to conjure up such a unintuitive answer? They do this by the way they interpret the question.

If we state the question in an impersonal form such as:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No 1 then another door opens, say No 3 which has a goat. After the door has opened you then have to decide whether you want to pick door No. 2. What is your probability of winning by switching?

This might well be interpreted to mean that the car is placed a random and the door is opened at random. With that assumption the answer is simply 2/3. There is nothing unreasonable about this assumption, as vos Savant said, we may well take the host to be acting simply as the agent of chance, and anyway the producer and the host have little to gain by adopting anything other than a random strategy.

Now consider the question in a more personal form:

Suppose you're on a game show, and you're given the choice of three doors: The producer has arranged for a car to be placed behind one door and goats behind the others. You pick a door, say No 1 then the host opens another, say No 3 which has a goat. He then says to you, "Do you want to pick door No. 2?" What is your probability of winning by switching?

Here we have two events that depend on the strategy of an individual. The answer is not 2/3 or even that given by Morgan but it depends on the probability that the producer has arranged for the car to be behind door 1. There is nothing unreasonable about this interpretation, we know that one person or another arranged for the car to be behind one of the doors and we have no idea what their method or motives were in this respect. The only real problem with this answer is that the solution is any number between 0 and 1 depending on the producer's car placing strategy - not very interesting.

What Morgan do is take the question and personalise the host, he has a strategy or a probability of taking certain actions if certain events occur. On the other hand Morgan tacitly take it that the car has been placed randomly. So hey presto! we now have an 'elegant' solution that gives a definite answer to 'the question that was asked' and proves everyone else wrong. Easy when you know how.

Now I agree that there is some justification for Morgan's assumption in that a host is mentioned in the question and a producer or stage hand is not, however, I suspect their their decision to take the question literally like this was motivated more by a desire to conjure up an 'elegant' solution than to answer the question that Whitaker probably intended to ask.

The further problem with Morgan's solution, which is relevant here, is the matter of what our readers will take the probability to meant. As we have agreed above it only refers to a case where the car is placed randomly but the host has a fixed strategy. Martin Hogbin (talk) 17:38, 31 March 2009 (UTC)

Martin - If the player's initial choice is random, it's equivalent to the producer (or whoever) randomly placing the car in the first place. If either one of these is random, then the probability of initially selecting the car is 1/3 regardless of which door the player picks (I've offered before, but would you like to see the math for this?). The "tacit" assumption that the car was randomly placed is an explicit assumption, clearly intended to match vos Savant's interpretation - as clarified by the explicit instructions she provided for the simulation she suggested (see [2]). Morgan et al.'s solution applies to a case where either the car was initially placed randomly or the player's choice was random and any case where the host is not constrained to pick randomly if the player initially picks the car (not just for a "fixed host strategy"). In particular, q (the host's preference for one door over another) might change every day, or might be constant - if the host is not constrained to pick randomly we simply don't know what q might be (or what it might depend on - perhaps the host opens the leftmost door on Mondays and the rightmost door on Fridays). There's absolutely no requirement that the player (or even the host!) actually know what value the preference parameter takes. From a problem definition viewpoint, not specifying this allows any number of (what Glkanter doesn't like to call) variants. If we don't specify this and then go on to say the answer is 2/3, the variants where q is not 1/2 are legitimate counterexamples that prove us wrong.

I think what you're basically saying is that it makes no sense not to assume q is 1/2. Sensible or not, it's a math problem - so if it's not given in the problem statement assuming it is wrong. This is important to point out for this specific problem since it is fundamentally a conditional probability problem. How the host decides which door to open in the case the player initially selects the car always affects the answer, whether you want it to or not. Saying the answer is 2/3 is saying p is 1/2. If this is not given then it is an unjustified assumption. Also note that it's not just Morgan et al. who have this viewpoint. Gillman as well. And Grinstead and Snell. And lots of others. It is in some sense a trivial oversight not to say the host picks randomly in this case. However, since it changes the answer from 2/3 to [0.5,1] it is probabilistically significant. -- Rick Block (talk) 19:09, 31 March 2009 (UTC)

Your first statement is wrong because it is given in the question that the player has picked door 1. There is no maths required. Given that the player has picked door 1, suppose that we have a producer who always puts the car behind door 1. You tell me what the probability of winning by switching is in that case? Now suppose that we have a producer who places the car behind door 1 with probability c, what is the probability of winning by switching then?

With regard to Morgan's assumption about random car placement can you tell me where in the paper they state that they have taken this to be the case?

I agree that there is no requirement for the player to know the host door selection parameter, that is because we are told that the host has opened door 3 In exactly the same way because we are told that the player has picked door 1 the manner by which the player picked that door is not important. We have been here before. Suppose the game were very simple and the player simply wins what is behind the door. Suppose that we are told that the player has picked door 1. What is the probability of winning. In that case it is clearly c (where c is the probability that the producer has put the car behind door 1). If we are told only that the player has randomly picked a door then the probability is, as you say, 1/3.

I am not saying that 'it makes no sense not to assume q is 1/2' at all, I am saying that there are four options. We can assume c is 1/3 (where c is again the probability that the producer has put the car behind door 1), or we can assume q is 1/2, or we can assume neither, or we can assume both. Morgan pick one of these possibilities as if it were the only one. How the host picks a door is completely unimportant in answering the question as given as we are told which was the door picked. Think of it as a conditional problem if you like, what is the probability of the player winning given that she picks door 1. It depends on the value of c.Martin Hogbin (talk) 21:57, 31 March 2009 (UTC)

You seem to be missing the main point of the paper, which (IMO) is that the problem is one involving conditional probabilities. Precisely what they assume for their solution is far less important than the basic approach - although (and I've said this numerous times) they assume exactly what vos Savant assumes, nothing more and nothing less (the probability of the initial pick being 1/3 is mentioned at the top of the first column on page 286 - and the bit about generalizations on p 287 clearly shows this to be a conscious assumption). The reason it's important to approach the problem conditionally is so that any assumptions are explicitly exposed. I believe it is the case that before their paper was published many solutions using an unconditional approach were touted as "the solution" with with most people not understanding that these solutions 1) directly address a slightly different question than the one that was asked (e.g. deciding to switch before the host opens a door - or any of the other unconditional versions we've discussed), and 2) if applied to the conditional question carried along an implicit assumption affecting the result.

Regarding your specific objections - if you're taking "say no 1" with regard to the player pick to mean that the only outcome of interest is when the player picks door 1 then why have you been arguing for weeks that the door the host opens is unspecified? Clearly door 1 and door 3 are meant to be illustrative choices, mentioned merely to put us into a conditional setting (so we can imagine the player standing in front of two closed doors and one open door and deciding whether to switch). Is this related to your continued claim that the Morgan et al. solution applies only if the player picks door 1 and the host opens door 3? Do you still not agree that the solution applies in any of the 6 conditional cases? I think virtually anyone reading the problem takes "door 1" and "door 3" to be a randomly selected case from among these 6 cases, with the expectation that the same results pertain to the other 5. It seems to me that this might be the point you're missing - that analyzing one of the 6 conditional cases and taking this to be representative of the answer in all 6 cases is entirely different than coming up with an "overall average" sort of answer. Any unconditional solution does the latter. Only a conditional solution does the former. -- Rick Block (talk) 00:18, 1 April 2009 (UTC)

Let me deal with your second point first. It is not totally clear whether the original questioner intended the question to refer to specific doors. I certainly believe that there is some doubt.

If we take it that the original question was meant to say something along the lines of, 'The player opens a door then the host opens another door', then we both agree that the simple unconditional solution applies and the Morgan paper is irrelevant.

If, on the other hand, we take it, as Morgan do, that the doors are to be specified, as in, 'The player has chosen door 1, the host has then revealed a goat behind door 3...' then we have agreed above that the probability of winning by switching depends on the hosts door preference. Do you also agree that in this case the probability of winning by switching also depends on the producer's initial placement of the car? If you agree this last point then we can continue to discuss the effect of Morgan's choices on the problem solution. Martin Hogbin (talk) 08:01, 1 April 2009 (UTC)

The point is: do we consider the average player of the game, who may decide before anything has happened that his chance of winning the car is 2/3 when switching. Or do we consider a player who made his initial choice and actually sees an open door with a goat. Not only do I think the latter is meant, but it is also the only interesting case.Nijdam (talk) 10:04, 1 April 2009 (UTC)

I am not sure why you keep asking me that question. I have made it quite clear that, if we take it that specific door numbers are given in the problem statement, then I am considering the probability after the host has opened a specified door. I agree that this makes the problem conditional and understand why it does so. As we have all agreed that if specific door numbers are not mentioned in the problem statement the problem is non-conditional, I am now talking only about Morgan's conditional interpretation of the question.

Not exactly. What we have all agreed is that is is possible to carefully craft a wording that logically asks an unconditional question. Where we disagree is whether this is even remotely reasonable and whether by doing so you've somehow made the Morgan paper irrelevant. Even if the doors are not numbered they have a physical location (left, right, and middle - for example), so they can always be distinguished and not numbering them achieves nothing. It is not the door numbers being in the problem statement that makes the problem conditional, it is letting the player decide to switch after a door has been opened. If the player is deciding after a door has been opened the player can tell which one she picked, which one the host opened, and which one is the other one. The question Nijdam asks is precisely the right question. Are we considering an "average" player of some type, or a player who can see two closed doors and a goat? I agree with Nijdam that the latter case is the only interesting one - and moreover would claim it's the only reasonable way to read the question. -- Rick Block (talk) 14:16, 1 April 2009 (UTC)

I am not sure what you mean by craftily worded. We either mention door numbers or we do not, thus:

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 door 1, and the host, who knows what's behind the doors, opens door 3, which has a goat. He then says to you, "Do you want to pick door 2?" Is it to your advantage to switch your choice?

is conditional and the answer depends on the host's door opening preference, but in this case:

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

the answer does not depend on the host's door opening preference. Whether you call it conditional or not I do not mind, I would call the 'condition' a null condition.

This has the form of a conditional question, and (as such) the answer does depend on the host's door opening preference. We don't know what it is, so the answer here is something between .5 and 1 (inclusive). -- Rick Block (talk) 01:57, 2 April 2009 (UTC)

However, none of the above affects what I have said. I agree that in the conditional case the probability of winning by switching depends on the host door opening strategy. Now the important point. Do you agree that the probability of winning by switching also depends on the probability with which the car was placed behind the chosen door? Martin Hogbin (talk) 11:12, 1 April 2009 (UTC)

Is the player pick random or not? This is the essential difference between the host preference and the car placement. The host preference affects the theoretical probability whether the player knows it or not. The car placement affects the probability computation only if the initial player pick is also not random (essentially meaning the player knows about it). You keep insisting that this is a problem of some sort with the Morgan et al. paper, but it's a completely different topic. They acknowledge it is possible to pursue this. They chose not to. Solutions that misread the question and present unconditional solutions are not choosing to ignore the potential host preference, they are (apparently unknowingly) answering a different question. -- Rick Block (talk) 14:16, 1 April 2009 (UTC)

No Rick you are wrong. If the car placement is not random it makes no difference whether the player knows about this or not, or whether their pick is random provided that the door that the player picks is specified. If the producer alway puts the car behind door 1 and we are simply told that the player has picked randomly, the probability that the player has picked the car is 1/3, but, if we are told in the problem statement that the player has picked door 1, what is the probability that she has picked the car? You tell me. It is not a hard question. Martin Hogbin (talk) 19:31, 1 April 2009 (UTC)

Martin - if the car is not placed randomly but the player picks randomly, the question "given the player has picked door 1 what is the probability the player has picked the car" basically negates the randomness of the player's pick. You can certainly ask this question but it has no relevance to the situation. It's like asking what the probability is of the car being behind door 1 if it's placed randomly but given the producers put it behind door 1 (??!!). Of course this probability is 1. But, it's a silly question. Reasonable questions here might be "what is the chance the player wins by switching given a non-random placement and a random initial choice?" (the answer is 2/3 if the host chooses between two goats randomly) or "what is the maximum chance the player wins by switching given a non-random placement and a knowledgeable initial choice?" (the answer to this involves three probabilities and examining how the player might maximize her chances of winning). Any question phrased "if the player picks, say door 1 ..." carries a VERY strong implication that this should be taken as a representative case (i.e. either the initial placement was random or the player's choice was random). -- Rick Block (talk) 01:57, 2 April 2009 (UTC)

We either take statements such as "if the player picks, say door 1 ..." to mean literally that the player opens door 1 or we take it to indicate a random choice. If we take it to indicate a random choice then the same applies to "if the host opens a door, say door 3 ...", we take this to mean that the host opens one of the unchosen goat doors randomly (unless there is only one). Alternatively we take the mention of a door number to mean literally that the mentioned numbered door is in fact opened, in which case the chances of winning by switching depend on the host's door opening preference and the initial car placement probability. You cannot have it both ways. Martin Hogbin (talk) 08:29, 2 April 2009 (UTC)

We take it to be a specific choice, but which one randomly selected. -- Rick Block (talk) 13:48, 2 April 2009 (UTC)

Rick, what is your answer to this very simple question? We have a producer who always puts the car behind door 1. A player chooses door 1. What is the probability that the car is behind the door that they have chosen? If you insist that it is important to know how the player chose door 1 then let is say that he rolled a die and divided the result by 2. Martin Hogbin (talk) 18:43, 2 April 2009 (UTC)

How do you interprete this game. You throw a die with a cup. And before you look at the result I have a peek and tell you whether it is below 3 or above 2 , say above 2, and let you guess, say 5. What will be your probability of guessing right? Nijdam (talk) 12:45, 2 April 2009 (UTC)

I interpret this to be asking P(correct guess|above 2). Possible outcomes are 3,4,5, or 6, so 5 is the correct guess with probability 1/4. -- Rick Block (talk) 13:48, 2 April 2009 (UTC)

I'm not surprised about your answer, Rick. The question is meant for Martin. Nijdam (talk) 17:21, 2 April 2009 (UTC)

I am not sure why you are still asking me questions about conditional probability. Perhaps you could explain what it is that you think I do not understand about it. I agree with Rick's answer.

We are now arguing about a very simple question. What is your answer to the question that I ask Rick above. Martin Hogbin (talk) 18:43, 2 April 2009 (UTC)

The conditional probability, given the player has picked door 1 is 1. Simple as that. Nijdam (talk) 09:32, 3 April 2009 (UTC)

(outindent) Martin - can you please explain what you're trying to accomplish here? -- Rick Block (talk) 13:38, 3 April 2009 (UTC)

Yes. I am trying to reach an agreement on what the mathematical facts are. I do not think that, in this respect, I am suggesting anything unusual or contrary to normal teaching in the subject. In particular I am trying to reach agreement on the player's probability of winning by switching in the following circumstances. If we take it that the player has opened door 1 and the host has opened door 3, with the usual rules, then:

If the producer places the car randomly and the host picks a door to open randomly (when he has a choice) the the probability of winning by switching is 2/3.

If the producer does not place the car randomly and the host does not open a door randomly then the probability of winning by switching depends on both the producer's car placement and the host's door opening strategy.

Do you agree? Martin Hogbin (talk) 17:39, 3 April 2009 (UTC)

I take it you wrote by accident "... the player has opened door 1 ...". The random car etc. implies that the conditional probability of winning by switching is 2/3. Let us not speculate about other situations. I specially put emphasis on "conditional", because the whole too lengthy discussion is about this, and you seemingly have forgotten to write this. Nijdam (talk) 18:59, 3 April 2009 (UTC)

Sorry, I meant the player has chosen door 1.

In both cases I meant the conditional probability given that the player has chosen door 1 and the host has opened door 3. In the first case the car was placed randomly and the host door opening strategy is random. In other words the host's strategy is to choose randomly between doors when he has a choice. In this case the conditional probability (that is to say, given that the host has opened door 3) of winning by switching is 2/3.

In the second case I mean that the car has not been placed randomly but according to the preference of the producer and the host does not choose randomly when he has a choice of doors but chooses which door to open according to his preference. In this case the conditional probability (that is to say, given that the host has opened door 3) of winning by switching is dependent on the producer's preference and the host's preference. Do you agree?Martin Hogbin (talk) 19:53, 3 April 2009 (UTC)

Of course, it's what Rick and I have written over and over. Nijdam (talk) 11:29, 4 April 2009 (UTC)

Neither of you has mentioned the effect of car placement before. Martin Hogbin (talk) 20:22, 4 April 2009 (UTC)

And, skipping ahead, do you Martin agree Morgan et al.'s solution is sensitive to both the car placement and the host preference, and that they explicitly, deliberately, and consciously assume the car placement is random, and that this (and their other assumptions, like the host always makes the offer to switch) exactly match vos Savant's published experimental procedure? -- Rick Block (talk) 12:03, 4 April 2009 (UTC)

In Morgan's paper the effect of non-random car placement on the probability of winning by switching is not mentioned anywhere. It is not explicitly stated to be random - if you believe this to be the case, please tell me where they say this. What the authors were conscious of when they wrote the paper I cannot say. I accept that on page 287 they mention generalizations which 'appear to be of less interest' including the 'possibility of nonuniform probabilities of assignment of the car to the three doors'. Considering that non-random car placement has just as much an effect on the result as non-random host door preference it seems odd to leave it as an afterthought.

What vos Savant says is irrelevant. Morgan start their calculation on page 285 where they discuss the more general scenario. There is no mention whatever at that stage that the car placement is taken to be random although the calculation has obviously assumes this fact with no justification or explanation whatever. Later, on page 286, the vos Savant constraints are added. It really is unfair to attempt to blame Morgan's failings on vos Savant. Vos Savant wrote a column in a general interest magazine and made reasonable assumptions in the circumstances, although these were not all made clear initially. Morgan et al were writing a paper for a learned journal and attempted a comprehensive solution to the question.

It does seem that you now agree that the Morgan paper only applies to the specific case where the producer has placed the car randomly but the host chooses with some not-necessarily-random probability which door to open. All other combinations produce different results. The only question that remains is how reasonable are the Morgan assumptions, considering all they achieve is to obfuscate a simple but very unintuitive problem, and how relevant they are to a WP article on the subject. Martin Hogbin (talk) 20:22, 4 April 2009 (UTC)

It is complete unclear to me what you're aiming at. Why bother about the placement of the car, as with the random placement it turned out to be difficult enough to understand. I thought we reached a point where you understood the "conditional nature" of the problem. It is not my intention to complicate the article with such considerations. It is about time the article gets a make-over. Nijdam (talk) 20:36, 4 April 2009 (UTC)

I have always understood the conditional nature of the problem although we still seem not to have reached agreement on exactly what formulations of the problem must be treated as conditional. My point is that the article does indeed need a makeover and the Morgan paper is a bad place to start as it addresses only one specific formulation of the problem.

In fact, can I just confirm that we agree on one other thing which is that the Morgan solution only applies if specific door numbers are mentioned in the problem formulation? Martin Hogbin (talk) 21:06, 4 April 2009 (UTC)

No. As we've discussed before lack of door numbers doesn't change anything since the doors can always be uniquely identified by their position on the stage. -- Rick Block (talk) 22:39, 4 April 2009 (UTC)

Perhaps I should have said that the question must identify the doors in some way, as in the player chooses the centre door and the host opens the leftmost door (but not as in the player chooses a door and the host opens another one) Do you accept that this is a necessary condition for the Morgan solution to apply. Martin Hogbin (talk) 23:16, 4 April 2009 (UTC)

No. And please stop with the word games. The only condition necessary is that the player be given the opportunity to switch after the host opens a door. This is what makes it a problem about conditional probabilities, which is what the Morgan paper addresses. You keep trying to imply it's an urn problem if we don't identify the doors (or something) but it's not an urn problem. The doors are inherently different, so unless you depersonalize the question (e.g. ask about all players who switch) if the player is choosing after the host opens a door it's a conditional probability problem. I think everyone agrees the player gets to pick after, and we want to imagine a player standing in front of two closed doors looking at a goat - this makes it a conditional probability problem. -- Rick Block (talk) 00:20, 5 April 2009 (UTC)

I am not playing word games I am making clear, as you once did, that a probability problem must be solved on the information given in the question. If information allowing us to identify the doors is not given in the question then we cannot use that information in the solution. The facts that the doors may have numbers on or that the player will know which door was opened are irrelevant as they are not given in the statement of the question. It is, if you like, still a conditional problem but the condition is that one of the host opened one of the two possible doors, we cannot say which. In this case the answer is 2/3.

If you want to discuss the relevance of Morgan's solution when the problem is viewed from from the player's perspective then I will be pleased to do so. 86.132.253.23 (talk) 08:46, 5 April 2009 (UTC)

Not everyone agrees with your conclusion that it's necessarily a conditional puzzle. And the player is not just "standing in front of two closed doors looking at a goat". Take a look at the Combining Doors solution. He's looking at the door he selected, the remaining door he did not select, and a goat. There's no condition. Glkanter (talk) 01:45, 5 April 2009 (UTC) Glkanter (talk) 01:58, 5 April 2009 (UTC)

Here's a quote from Feb 15, 2009 from your buddy, the subject matter expert: "By the way, I hope I haven't given the impression that I think the problem must be understood as conditional. As for Marilyn's exact statement, I think it's ambiguous enough on several fronts, and there's no reason to state with absolute confidence it should be conditional." Despite my request, he never explained this any further, and I'd be the last guy to twist his words, but this is what he wrote. Glkanter (talk) 02:09, 5 April 2009 (UTC)

That's a quote from user:C S, archived in Talk:Monty Hall problem/Archive 9. -- Rick Block (talk) 03:51, 5 April 2009 (UTC)

I had to re-read my posting 3 times before I realized the missing 'r', which changes 'you' to 'your'. Sorry for any confusion I caused. Glkanter (talk) 05:01, 5 April 2009 (UTC)

Martin - please explain to me why the Gillman paper says pretty much exactly the same thing as the Morgan et al. paper. They both conclude the chances of winning by switching for the Parade version of the problem (with the vos Savant assumptions) is not 2/3 but 1/(1+q) where they both say q is the unspecified host preference for one of the doors. -- Rick Block (talk) 22:39, 4 April 2009 (UTC)

Because that is correct. Provided of course that the host does have a preference and does not choose randomly and that the producer places the car randomly and does not have a preference. Martin Hogbin (talk) 23:15, 4 April 2009 (UTC)

Correction - provided only that the producer places the car randomly. 1/(1+q) is the solution whether or not the host has a preference. If the host chooses randomly (has "no preference") then q=1/2 and the answer simplifies to 2/3, but the answer is still 1/(1+q). This is a simplified form of (1/3)/(1/3 + q(1/3)) which directly matches the probability of interest (win 1/3 in the case the car is behind the "other" door when the only possibilities are this or the losing case with probability p(1/3)). Did you ever read my explanation to the anon poster at Talk:Monty Hall problem#How complicated is it???? -- Rick Block (talk) 00:20, 5 April 2009 (UTC)

Further correction. If you really want to be pointlessly pedantic then the probability of winning by switching always depends on the producer's car placement. If this is random then, of course, the probability is 1/3 that the car is behind each door and its effect can be ignored. The situation is exactly the same for the producer and the host, you can ignore the effect that their strategy may have or not. 86.132.253.23 (talk) 09:05, 5 April 2009 (UTC)

It's not pointlessly pedantic. I think you're continuing to miss the significance of this. If (assuming random initial placement) you say the answer is 2/3, you're saying q is 1/2 because the answer to the conditional question is always 1/(1+q). Combining this with the other conversation going on in this thread just above, saying the answer is either 2/3 or 1/(1+q) is also saying the initial car placement (or player's initial choice) is random - but this assumption matches how nearly everyone interprets the problem (and is not something Morgan et al. simply overlooked as you keep trying to suggest). Most people are unaware the host preference (which is usually unconstrained by the problem statement) directly influences the probability of interest. Saying I have a 2/3 probability of winning by switching is different than saying all players who switch have a 2/3 probability of winning (you do see the difference, don't you?). This is the point of the card game I proposed that you're unwilling to play with me where I'll have an overall 2/3 probability winning by switching but 1/2 or 1 any given time we play. What I keep hearing you say is my probability of winning, every time I play, is 2/3 but you'll only put money on it if you don't let me use the information about which card you flip (door the host opens) to adjust my bet. You're not stupid - you can see the probability will be 1/2 or 1 and will "average out" to 2/3. What I don't understand is why you then have such a hard time agreeing with me that the probability, per game, can vary between 1/2 and 1. -- Rick Block (talk) 16:20, 5 April 2009 (UTC)

I fully understand why the probability of winning is between 1/2 and 1. The point that I have been trying to make for some time now is that this solution only applies in certain formulations of the problem. Let us consider the problem with the usual rules and the conditional case - we must assess the probability of winning by switching after a door has been opened.

Firstly, it is trivially true to say the required probability is between 0 and 1 inclusive. We want to do better than that, in fact we want to give as narrow a range of probabilities as possible.

If we start with the case where the producer has not placed the car randomly and the player has not chosen randomly then the best we can do is still to say that the answer is somewhere from 0 to 1.

Now add the constraint that the producer has placed the car randomly but the host does not open a door randomly. If we now formulate the problem with identified doors then we both agree that the best that we can do is to say that answer is between 1/2 and 1 depending on what the host's (unknown) door opening strategy happens to be.

Now do the same but do not identify doors. We do not yet agree on this one. I say the probability is still, of course, between 1/2 and 1 but it is in fact 2/3. I am not sure if you still disagree.

Now add the constrain that the host must choose the door to open randomly. The probability is still, of course, between 1/2 and 1 but it is in fact 2/3. We agree on this.

Finally let us make the problem unconditional, the player decides at the start what they will do. Once again we agree that the answer is 2/3.

My point is that the answer of between 1/2 and 1 is only the best that we can do in some of the many possible formulations of the problem. It is easier to see just how restrictive the Morgan scenario is by considering the average results obtained by repeating the game many times. This has nothing to do with the conditional/unconditional issue, we can repeat the game with the relevant condition applied every time. Martin Hogbin (talk) 17:17, 5 April 2009 (UTC)

I think we're on the same page except for the bit about not identifying doors. My view is that the problem is either conditional, i.e. applies to a specific player standing in front of two closed doors and a goat (in which case the doors are inherently identified) or unconditional (applies only in some "average" sort of way). You seem to be thinking there's a third option - "conditional" but without specifying which door. This option is mathematically the same as the unconditional case - and seems pointlessly contrived. The Morgan et al. result can be easily observed, but requires keeping track of which door the player picks and which door the host opens per the #Excel simulation of difference between "random goat" and "leftmost goat" variants above. This is the criticism of the usual experimental validation as well. If you lump all 6 combinations of player pick and host door together, you're not keeping track of the data that might expose a host preference. This is fine if all you're interested in is the overall chance of winning by switching - but not if you want to know the probability of winning by switching given which door the host opens (i.e. this is not good enough if you want to decide whether to switch after the host opens a door). -- Rick Block (talk) 18:55, 5 April 2009 (UTC)

Maybe we are arguing about very little. The case when an unidentified door has been opened is the case that I have been referring to as having a null condition. You say it is mathematically equivalent to the unconditional case. It looks like we agree.

However, that brings us back to Morgan's interpretation of the problem as necessarily conditional. After misquoting the original question, Morgan state that '...the host has revealed a goat behind door 3', clearly conditional, but the question that Whitaker actually asked said, '... the host ... opens another door, say No. 3...'. We might well take the question to mean '...the host has revealed a goat behind another door', which we have just agreed makes the problem unconditional.

As the reliable source Seymann points out, we need to consider what Whitaker actually meant by the question. I very much doubt that he meant the number of the door that was opened by the host to have any significance. Martin Hogbin (talk) 21:31, 5 April 2009 (UTC)

Martin - Is the question what is your, Martin's, chance of winning by switching if you find yourself in this position (and you're allowed to switch after the host opens the door) or is it some overall average? When you, Martin, are in this situation you know which door you originally picked and which door the host opened. And, in this case, Morgan's solution applies. IMO, the whole point of the MHP is to put you in the player's shoes, at the point of the decision (i.e. after the host has opened a door). The absolutely standard way to make the odds 2/3 by switching is to include the random host pick constraint. This fixes everything. Why are you so reluctant to include this as a constraint? -- Rick Block (talk) 23:12, 5 April 2009 (UTC)

I have no objection to using the random goat door constraint as one of the justifications for the 'popular' solution.

I am sure that you agree that Morgan's paper is based on treating the question as a formal probability problem which must answered using only information given in the question. Taking this approach means that whether the player knows the host's strategy or not is unimportant, all that matters is what the host's strategy actually is, which we do not know. I agree that another, and possibly better, approach would be to put ourselves in the place of the player and answer from that perspective. My guess is that this is more like the question that Whitaker intended to ask. If we answer the question from the player's perspective then we can and must ask ourselves what we take the player's exact state of knowledge to be. This changes everything and essentially makes the Morgan paper irrelevant. Let us do it! Martin Hogbin (talk) 09:04, 6 April 2009 (UTC)

I've been waiting for (dreading) the return of the 'whose (who's?) point of view' question. I'm not sure who you would choose, if it was not the contestant. Why not choose the stage hand who parked the car who knows with 100% certainty when to switch? No, it can only be the contestant, and he has no way of knowing of any 'host behaviour'. Did you know there are actually laws in the United States that prevent the producers from sharing such knowledge with the contestant? Back in the 1950s, they did give contestants the answers. It became a huge scandal. To summarize, according to Morgan and Rick, the 1/3 vs 2/3 only holds true after Monty has revealed a goat with the 'equal goat door constraint'. Without it, it becomes, I guess, what, unknowable?. Take a look at the Combining Doors solution. Then extrapolate that Monty could have opened the other door. Still looks like 1/3 vs 2/3 to me.

You guys are missing the most important element in a proof: Common Sense. Glkanter (talk) 13:10, 6 April 2009 (UTC)

From the player's perspective, if she doesn't know the host's preference her chance of winning by switching is perfectly knowable - it's between 1/2 and 1 (and will average 2/3), and she should therefore switch. -- Rick Block (talk) 13:43, 6 April 2009 (UTC)

No, from the player's point of view, the probability of winning by switching is from 0 to 1, averaging out to 2/3. Or as we say in the worlds of statistics and common sense, simply 2/3. Martin Hogbin (talk) 12:58, 7 June 2009 (UTC)

From the player's point of view

As always, the solution depends on exactly what the question is. Let us assume the normal rules (the host always offers the swap and always opens an unchosen door to reveal a goat) and let us consider the particular case where the producer has placed the car randomly.

Let us also take the case that we (the player) have chosen door 1 and just seen the host open door 3 to reveal a goat, one possibility of many.

What is the probability of winning by switching. Before we can answer that we have to clarify one further point about the precise problem formulation. It is easiest to make the necessary distinction by considering the average result of a player who chooses to swap after seeing the door opened in different circumstances.

If we repeat the game, with the car being placed randomly each time and with the same host who has a (unknown) goat door opening strategy, then the answer could be anywhere from 1/2 to 1, that is to say the average result could be, for example, that we would win by switching only 1/2 of the time. (In that case, if the same host were to open door 2 and we swapped we would win always)

If, on the other hand, we repeat the game with a new host each time, each with an different and unknown goat door opening strategy then the answer will also be between 1/2 and 1, but in just the same way that it will be between 0 and 1, it will be 2/3.

I am not sure how we phrase this distinction from the point of view of a single game, but it seems clear to me that we must answer from the point of view of a player who plays the game for the first time, thus we take it to be a new host. In other words, as this is our first game ever, we start with a new universe each time. Martin Hogbin (talk) 19:52, 6 April 2009 (UTC)

Let me ask this question, suppose the prize were $1200 cash, with nothing behind the other doors. Suppose you have chosen door 1 and just seen the host open door 3 and you make your choice of whether to stick or swap. Before the result is revealed the host offers you $700 for the prize that you have now chosen (the game rules are that he always makes this same offer whether you choose to swap or stick). What would you do: swap doors and take the prize, stick and take the prize, or take the cash? I know what I would do. Answer on the basis that you have never seen the game played before but know the rules. Martin Hogbin (talk) 20:02, 6 April 2009 (UTC)

I see that I have no takers on this one. I would swap and take the prize. Is there anyone who would do different? Martin Hogbin (talk) 09:15, 8 April 2009 (UTC)

No answers. Is that because people do not know the answer or dare not give one or just that this bit has now been forgotten? Martin Hogbin (talk) 17:14, 9 April 2009 (UTC)

Since The Contestant Can't Know Of Any Host Behaviour...

In MvS' version of the MHP, there is no reference to a Host Behaviour.

Any inclusion of a Host Behaviour as a premise creates a new puzzle, not the MHP.

It would be illogical for the Host to share this info with the Contestant.

In the United States, it would be illegal for the Host to share any such info.

The only logical 'point of view' for the problem is the Contestant's.

Any discussion of the Contestant's probabilities other than 1/3 vs 2/3 requires knowledge of where the car is. The contestant won't know this until after he has made his switch or no-switch decision.

So when Rick writes:

"From the player's perspective, if she doesn't know the host's preference her chance of winning by switching is perfectly knowable - it's between 1/2 and 1 (and will average 2/3), and she should therefore switch. -- Rick Block (talk) 13:43, 6 April 2009 (UTC)"

The fact that it will average 2/3 is the only salient point. And we already know this from solutions like the Combining Doors. Glkanter (talk) 11:23, 7 April 2009 (UTC)

Why does the forgetful host version (Monty opens a door without knowing there's a goat behind it, and it luckily turns out to be a goat) require a different solution method, or does combining doors work for this different problem as well, in which case the answer is the same? -- Rick Block (talk) 13:28, 7 April 2009 (UTC)

This is a new constraint, making it a different problem. It's a premise that Monty always shows a goat, so if his 'forgetfulness' ever led him to reveal a car, that's a different puzzle. But I'm curious, how would this 'forgetfulness' be evident to the contestant? He was told Monty would reveal a goat, and Monty revealed a goat. Glkanter (talk) 14:01, 7 April 2009 (UTC)

Yes, this is a different problem that vos Savant has discussed in other Parade columns. The contestant doesn't necessarily know (hmmm, just like the contestant doesn't necessarily know about a host preference), but the question is what is the probability of winning by switching given that the host randomly opened one of the two doors but (luckily) showed a goat. She says in this case it's 50/50. My question for you is why doesn't the combining doors solution work here. If it works sometimes but not others, I'm asking how we know when it works and when it doesn't. -- Rick Block (talk) 15:46, 7 April 2009 (UTC)

Rick, this is one of your favorite techniques for prolonging the discussion in order to avoid improving the article. Bring up something different, whether it's a card game, a forgetful host, an Excel simulation, anything but the MHP. I choose not to engage in such unproductive tangents. But I'm curious, how would this 'forgetfulness' be evident to the contestant? He was told Monty would reveal a goat, and Monty revealed a goat. Glkanter (talk) 17:44, 7 April 2009 (UTC)

I bring this up not to prolong discussion but to point out the problem with the approach. The approach is at best incomplete - so basing the article on this approach would make the article incomplete as well. Your failure to understand this is the only thing prolonging anything, and your favorite technique for prolonging the discussion is to simply ignore any and all arguments that expose the fallacy of your thinking. I'm trying to get you to see why and how the approach is incomplete, but I can't force you to understand it. On the other hand, as long as you don't I think your opinion about "improving" the article is essentially worthless.

Whether the forgetfulness is evident to the contestant is irrelevant. The question is what are the odds of winning by switching (not what does the contestant think the odds are). This is not a version I'm making up - it's a version vos Savant has discussed in her column (see [3]). It's directly relevant to the MHP, not a tangent. So, again, why does the combining doors solution work for the standard MHP but not this version? -- Rick Block (talk) 18:54, 7 April 2009 (UTC)

You wrote:

"The question is what are the odds of winning by switching (not what does the contestant think the odds are)."

The odds of switching are either 0 or 1. If the Contestant originally selected the car, then his odds of winning by switching are 0. On the other hand, if he has not chosen the car, the odds are 1. Because there are people who know exactly where the car is. But this is becoming increasingly ridiculous. If you're not solving the problem through the Contestant's POV, whose are you using? Glkanter (talk) 19:19, 7 April 2009 (UTC)

The odds according to what's known from the problem statement of course. You can pursue as many diversionary tangents as you'd like, but the question remains: why does the combining doors solution work for the standard MHP but not the forgetful host version? -- Rick Block (talk) 00:34, 8 April 2009 (UTC)

From the previously agreed bullet points:

• car is randomly placed
• initial pick is random
• host must make the offer to switch
• host must show a goat (and, hence, knows where the car is)
• player decides after the host opens a door

No one not associated with the show knows where the car was placed

There is no host behaviour other than to reveal a goat

After the goat has been revealed all one can do is generalize about the overall probabilities of 1/3 vs 2/3

A valid conclusion is reached: my chances of winning increase from 1/3 to 2/3 if I switch. Glkanter (talk) 02:14, 8 April 2009 (UTC)

And why does this solution work for the standard MHP but not the forgetful host version? -- Rick Block (talk) 04:03, 8 April 2009 (UTC)

Are you in agreement with what I wrote?

When the host forgets, does he reveal a goat? Glkanter (talk) 04:19, 8 April 2009 (UTC)

The only difference is the 4th bullet (host must show a goat). In this version the host happens to show a goat (and doesn't know where the car is). A goat is revealed. Why does the combining doors solution not apply? -- Rick Block (talk) 04:30, 8 April 2009 (UTC)

I'm not sure that still qualifies as a Probability question. How is this new bullet #4 explained to the Contestant? Does the Contestant still expect Monty to reveal a goat after he has chosen a door? Are you saying this happens one time? Does he sometimes forget and show a car? Does he always forget? Glkanter (talk) 04:46, 8 April 2009 (UTC)

This is Marilyn's answer from your link:

"Here’s one way to look at it. A third of the time, the clueless host will choose the door with the prize, and the game will be over immediately. In our puzzle, that didn’t occur. So we’re considering the two-thirds of the time when either: 1) You have chosen the door with the prize; or 2) The prize is behind the unopened door. Each of these two events will occur one-third of the time, so you don’t gain by switching."

So it's a completely different problem! Like I said, it has nothing to do with the MHP. Just another time waster. And you mis-stated the problem anyway. He doesn't 'happen' to show a goat. He randomly showed a goat. So bullet #4 becomes "Monty will reveal either a car or a goat". And if Monty reveals a car, bullet #5 doesn't even take place! What happens to bullet #3? How this has anything to do with a proof of the MHP is unknown. Glkanter (talk) 05:04, 8 April 2009 (UTC) Glkanter (talk) 09:48, 8 April 2009 (UTC)

And since the door is being revealed randomly, it doesn't even need to be the host selecting the door. It could be anybody. It could even be the Contestant. Hey, I know, let's have the Contestant reveal the 2nd door! Why, that would be just like a popular game called "Deal or No Deal"! You've heard of that, right Rick? And only a completely clueless person with no common sense understanding of Probability would be ignoramus enough to try to use Deal or No Deal in an effort to prove or disprove anything related to the MHP, right, Rick? Glkanter (talk) 09:44, 8 April 2009 (UTC)

Yes, it's a different but related problem. The problem statement is here. The reference to the standard MHP is unmistakable. The only difference is the host forgot which door hides the car, but does open a door showing a goat. You have an initial 1/3 chance of having selected the car. The other two doors have a 2/3 chance. At the point you're deciding whether to switch you're standing in front of two closed doors looking at a third open door. Surely the chance the car is behind your door is 1/3 and the chance it's behind the other two doors is 2/3. And, after the host opens a door (as you keep saying) 1/3 + 2/3 = 1. I understand Marilyn's explanation, but why doesn't the combining doors solution work? -- Rick Block (talk) 13:48, 8 April 2009 (UTC)

This is my final response ever to one of your edits. You have now clearly demonstrated, again, that you have no idea what you are talking about. Let's start with the bullets:

• car is randomly placed

    no change

• initial pick is random

    no change

• host must make the offer to switch

    not if he already revealed the goat, he won't

• host must show a goat (and, hence, knows where the car is)

    he will randomly reveal either a goat or a car (and, hence, he doesn't know where the car is)

• player decides after the host opens a door

    1/3 of the time (according to Marilyn), the player will not even be offered a switch

There are no teachings in Probability in which these would be considered the same problem, or where it would be considered appropriate to substitute the results of one for the other.

I've explained how the Contestant, or ANYONE NOT ASSOCIATED WITH THE PRODUCTION OF THE SHOW, will not know of any Host Behaviour, or the original placement of the car.

I've offered up the Combining Doors solution, which is in the article, and has apparently passed 2 (maybe 3) FA reviews as a source for the unconditional solution.

I've shown that the best the Contestant, or ANYONE NOT ASSOCIATED WITH THE PRODUCTION OF THE SHOW, can do is to generalize the 1/3 vs 2/3 to his single opportunity.

And it's been shown (by you!) that without the 'equal goat door constraint', the so-called conditional solution offers up something like "it's between 1/2 and 1, and averages 2/3" as it's best answer. Which, after all that horse hockey and complications, etc., is no better than the Combining Doors solution.

For all these reasons, I am again going to return to the MHP talk page and propose that the focus of the article be turned towards solutions along the lines of the Combining Doors solution. Glkanter (talk) 14:26, 8 April 2009 (UTC)

I'm sorry if this makes you angry. I'm not suggesting substituting the results, just the solution method. It seems like a simple enough question. I take it from your refusal to address it that you cannot explain why the combining doors solution works for the standard MHP but not for the host forgets version (and SHOUTING ABOUT IT doesn't make anything more clear for anyone). Since the question is the same in both cases (what is the probability of winning by switching) and the problems are so similar (only one detail different) it seems like the same method should be able to be used for both. Marilyn's approach for the forgetful host version is a conditional solution. She says "we’re considering the two-thirds of the time when either ...". The reason she does this is because the unconditional solution (2/3) is the wrong answer in this case. The answer to the question I'm asking (why doesn't the combining doors solution work) is because the question is a conditional question and in the host forgets version the conditional answer is different from the unconditional answer.

This finally brings us to the point I'm making about the standard MHP. Since the question in the standard MHP and this forgetful host version is the same (what is the probability of winning by switching), my claim is that the standard MHP is also a conditional question (if it's a conditional question when the host forgets it must also be a conditional question when the host remembers). The combining doors solution (or any other unconditional solution) is not directly answering the question. You can say it ends up with the right answer so who cares, but unless you know why it applies in one case and not another you're not understanding the problem. In fact, it only applies in the standard MHP if the host is also constrained to pick between two goats equally.

A solution to the standard MHP in the same style as Marilyn's solution for the forgetful host version would be something like:

Here’s one way to look at it. Half the time, the host will open door 2. In our puzzle, that didn’t occur. So we’re considering the half of the time when either: 1) You have chosen the door with the prize (and since we're considering half the time, your original 1/3 chance is divided in half and is now 1/6); or 2) The prize is behind the unopened door (in which case the host is forced to open door 3). The car is behind the unopened door one-third of the time but behind your door only 1/6 of the time, so your chances of getting the car double by switching.

This is a conditional solution as well and depends on the host opening door 2 and door 3 exactly half the time when the car is behind door 1. This is either assumed or must be given in the problem statement. If it isn't, then (just like the host forgets version) the conditional and unconditional solutions are different so none of the unconditional solutions (like the combining doors solution) work. -- Rick Block (talk) 19:18, 8 April 2009 (UTC)

Probabilities from the player's perspective

Strictly from the players perspective, what is the probability that she has originally chosen the car? Martin Hogbin (talk) 18:54, 9 April 2009 (UTC)

Formulation of the problem

I think we've reached a point where Martin tries to save the "unconditional" solution, by stating that the formulation of the problem permits an unconditional explanation.

Named door numbers

If in the formulation of the problem explicitly the numbers of the doors picked and opened are mentioned, the problem needs a conditional approach.

More general formulation

The problem may be formulated more general, without mentioning the door numbers explicitly, but indicating that a specific door is picked and a specific door is opened. Again conditonal.

The important distinction is whether the doors mentioned in the problem are identifiable. This could be done by means other that that of giving the door number, for example we could say the host opens the leftmost door or the player chooses the centre door but if we are going to do this we might as well give the doors numbers. The possibility of the answer not being 2/3 depends on the possibility that the host might have a non-uniform probability of opening doors when he has a choice. For it to have any meaning this probability must be expressed in terms of identifiable doors, as Morgan do.

We have all agreed that if the problem is stated in a way in which the doors are not identified, as in, the player has chosen a door and the host has opened another one, the condition is unimportant. Whether we refer to this case as conditional, unconditional, or having a null condition I do not mind, the important fact is that in this case the fact that the host has opened a door has no effect on the answer. Martin Hogbin (talk) 17:28, 7 April 2009 (UTC)

General formulation

It is possible to formulate a related, similar problem in such a way that the unconditional approach is valid. In that case the problem should address any player, regardless which door is picked and which is opened. Like: describe how the game is played and ask what the probability is for an average player to win the car (when switching).

Which one is "our" formulation? I'd say it is the first one, as door numbers are mentioned. If we interpret the naming of the door numbers as a way of expressing that any door numbers may be used, it is the second formulation. A strong argument in favour for either is that the problem says: suppose you are ..., meaning a specific player is intended and not the average player in general. Besides it is actually only an interesting problem if finally a door is opened and the player is confronted with this specific situation. Nijdam (talk) 13:05, 7 April 2009 (UTC)

The un-numbered door interpretation is all that is required for the Contestant to benefit from the 1/3 vs 2/3 advantage. All of your discussion about where the car is, and what to do when there are two goats is immaterial to the Contestant. He simply does not have any idea where the car actually is when he must make his decision. And Monty revealing a door does not change that. Because the Contestant can not know Monty's method. So whether he generalizes that it's 1/3 vs 2/3 overall, and makes his decision, or whether the Contestant for some reason says to himself "I don't know the host's preference, so my chances of winning by switching is perfectly knowable - it's between 1/2 and 1 (and I will average 2/3), I will therefore switch" (paraphrasing Rick Block, above), it doesn't make any difference whatsoever. Glkanter (talk) 14:19, 7 April 2009 (UTC)

What is the un-numbered door interpretation?Nijdam (talk) 16:30, 7 April 2009 (UTC)

Something along the lines of this:

"After the player has chosen a door, the probability it hides the car is 1/3. This probability is not influenced by the opening of a door with a goat by Monty, hence after Monty has opened a door with a goat, the probability the original chosen door hides the car is also 1/3. Because clearly the open door does not show the car, the remaining closed door must hide the car with probability 2/3. Hence switching increases the probability of winning the car from 1/3 to 2/3. Better? Nijdam (talk) —Preceding undated comment added 15:21, 6 April 2009 (UTC)."

But let's focus on the Contestant. He can not be aware of any Host Behaviour. Nor can he have any knowledge of where the car is located. So it makes no sense, and adds no value, to parse his decision beyond the generalized 1/3 vs 2/3. Glkanter (talk) 17:37, 7 April 2009 (UTC)

I believe that when the problem is viewed strictly from the player's perspective and state of knowledge, Morgan's 'goat door choice parameter' - p will evaporate. Martin Hogbin (talk) 18:23, 7 April 2009 (UTC)

I think that it is important to point out that there are at least two different and distinct interpretations in which the unconditional solution applies. The first is that we take the question to be asking about the overall probability of winning by switching, the totally unconditional case in which the questioner wants to know whether, in general it is best to swap or not, the decision being made at the start of the game. This is almost certainly what Whitaker actually wanted to know. When you think about it, most players must have a good idea whether they plan to switch or not at the start of the game and I cannot imagine a player saying to herself, 'Oh no, door 3 has been opened, I had better not switch now I have seen that'.

The other case is the one that, at first sight, appears conditional in that we consider that the decision will be made after a door has been opened but the doors are not identified. As we have all agreed (I hope) this has the effect of making the condition have no effect whatsoever, it is a null condition and the problem can be treated unconditionally.

You have agreed with yourself about this. I have not agreed, but have said repeatedly that if the decision is made after the door has been opened the player must know which door she picked and which door the host opened so this "formulation" of the problem is nonsensical. -- Rick Block (talk) 01:04, 8 April 2009 (UTC)

Rick, it was you who, quite rightly, told me that a probability problem must be solved on the information given in the question. If the information as to which door the host has opened is not given then it cannot be used or affect the answer, regardless of whether the player knows it or not. I thought we had agreed on this. Martin Hogbin (talk) 08:54, 8 April 2009 (UTC)

A third case, which I would also claim can be properly treated unconditionally, is where the host opens a goat door randomly. This again has the effect of making the 'condition' insignificant; no information can be revealed by this random act.

So, Morgan's version of the conditional solution applies only in the case that Whitaker is taken to ask the apparently conditional question, the car is placed randomly, the host has not chosen a goat door randomly, the doors are identified in the question, and, I believe, the problem is considered as a formal probability problem rather that strictly from the players perspective. As it has such limited applicability it should not dominate the article. Martin Hogbin (talk) 18:19, 7 April 2009 (UTC)

Doors not identified

I thought that we had all agreed on this but it would seem not. What are your answers to the following questions, assuming the usual rules that the host always offers the swap and always opens an unchosen door to reveal a goat and also assuming that the initial car placement is random.

We agree that if the problem states that, "the player has chosen door 1 and the host has opened door 3" then it should be treated conditionally and the answer (the probability of winning by switching) could be anywhere between 1/2 and 1 depending on the host's door opening preference.

What is your answer if the problem states, "the player has chosen door 1 and the host has opened one of the other two doors"? Note that the problem statement still clearly refers to the situation after the host has opened a door and we assume that the player can see which door this is. Who agrees that the probability of winning by switching is 2/3 in this case. Can we start with a simple agree/disagree response before we start any discussion on why. Martin Hogbin (talk) 09:10, 8 April 2009 (UTC)

Agree. Martin Hogbin (talk) 09:10, 8 April 2009 (UTC)

Mu (the case as described is nonsensical). -- Rick Block (talk) 13:54, 8 April 2009 (UTC)

It is perfectly simple English, what is nonsensical about it? Martin Hogbin (talk) 22:43, 8 April 2009 (UTC)

Nijdam, what is your answer? Martin Hogbin (talk) 17:15, 9 April 2009 (UTC)

I would have no answer, but a question: which door? Nijdam (talk) 19:54, 10 April 2009 (UTC)

But if you force me to answer, I give a conditional answer: If the host opened door 2, then ..., and if he opened door 3, then ... Nijdam (talk) 19:59, 10 April 2009 (UTC)

If you further insist, I might say: in both cases you win when switching with 2/3 chance.

One number please, you say. Me: 2/3 is the only one I can think off. Would you be satisfied? Nijdam (talk) 20:08, 10 April 2009 (UTC)

Nijdam, are you really trying to tell me that you would be completely unable to answer the question if it were posed in that way? The statement I give makes perfect sense. In fact in Krauss and Wang on page 7 they say something very similar, 'The corresponding formulation would be " Monty now opens another door and reveals a goat"'. It is clear from the context that they are talking about the player making a decision after the door has been opened. Martin Hogbin (talk) 20:12, 10 April 2009 (UTC)

There was simultaneous editing here, my reply only applies to your first answer.

So, Nijdam, do you agree that, if the door opened by the host is not identified, the probability of winning by switching is 2/3 and does not depend on the goat door preference of the host? Martin Hogbin (talk) 20:46, 10 April 2009 (UTC)

It would be pure theoretically. It would mean conditioning on the event "the door opened is not the door chosen", which is just the phrasing of one of the rules of the game. Because this event has probability 1, it is not really a condition. In fact it means the opening of the door didn't happen. Something like a player who is blindfolded. The theory permits to formulate such things, but the problem shows a different situation. In the show as we all know, and as a spectator may see with his own eyes, a door is opened and the player sees which one. That we have to calculate the conditional probabilities, is a reflection of the conditional answer I gave above. If you don't mention the door number, I'll give conditional answers, depending on the door numbers, but nevertheless conditional. It all seems to me a rather artificial effort of giving right of existence to the "unconditional solution", may be because people has strongly supported it, and are not willing to "give in". So I can say: before a door is opened by the host, the chance is 1/3 the player picks the door with the car. This wording has the same meaning as your formulation. Is it your opinion that the MHP asks for this probability? Nijdam (talk) 08:00, 12 April 2009 (UTC)

I do not seriously disagree with anything that you have said. The fact that the host is simply doing what he is bound to do by the rules does not change anything is agreed, that is why I have been consistently referring to this 'condition' as a null condition. I do not know if this is the correct technical term but it seems entirely appropriate in the circumstances. The fact that the player can see the doors is irrelevant. We must answer the question on the information that we are given, not on information that one of the characters in the problem may or may not have, see my comments on problem style.

I agree that what I am doing is an attempt to give a right of existence to the unconditional solution but ask you to consider that it is those who adhere too strongly to the paper by Morgan who are not willing to give in. Morgan is the only 'reliable source' to take such a strong line against the unconditional/simple solution. Many others treat the problem in an unconditional way but mention the issue of conditionality. The unconditional problem does have real existence and it can be extended, with a little intuition, to cover special conditional cases.

Your last question is a good one and it is the one we should all be asking. What does the MHP really ask? It is my opinion that the MHP does ask about the unconditional case. If we start from the most notable problem statement, the letter by Whitaker, then we should ask, 'What is it that he really wants to know?'. It would be a very good idea for us all to discuss that question and I would be happy to start a new section on this very important topic. Martin Hogbin (talk) 23:11, 12 April 2009 (UTC) [Re-signed]

Let me at least add to this that a lot of the strong adversaries of the simple solution, fail or refuse to understand that opening a door changes the initial probabilities into conditional ones, be it, in the usual case, with (only) for the chosen door the same values. Agree?Nijdam (talk) 11:16, 12 April 2009 (UTC)

I am not sure what you mean Nijdam, could you clarify please. Martin Hogbin (talk) 23:11, 12 April 2009 (UTC)

Devlin http://en.wikipedia.org/wiki/Keith_Devlin and many others, subsequent to Morgan's paper, have put forth the unconditional interpretation and solution as appropriate. http://www.maa.org/devlin/devlin_07_03.html Would you say he "fails or refuses to understand that opening a door changes the initial probabilities into conditional ones"? Glkanter (talk) 11:56, 12 April 2009 (UTC)

Well he is called Devlin, not God. The referred article is a column in a mathematical magazine, where mister Devlin gives his popular opinion. Not peer reviewed, so why bother with his ideas. Nijdam (talk) 17:11, 12 April 2009 (UTC)

Actually, 'mister Devlin' is most often referred to as Dr. Devlin. http://www.stanford.edu/~kdevlin/ Glkanter (talk) 19:15, 12 April 2009 (UTC)

His PhD certainly wasn't on a probabilistic subject. Nijdam (talk) 21:16, 12 April 2009 (UTC)

Whilst we are talking about the status of various papers we should see Morgan for what it is. Although it is published in a peer-reviewed journal it is not really a piece of current statistical research but more something of a novelty paper. The paper is of such a poor quality that I am surprised that it got past the referees. It also carries a 'health warning' in the form of a polite but significant critical commentary by Seymann at the end. Martin Hogbin (talk) 23:11, 12 April 2009 (UTC)

Why Morgan are right.

This is for newcomers to this discussion and anyone else who may find it interesting.

In the Monty Hall problem with the usual rules, that the host always offers the swap and always opens an unchosen door to reveal a goat, if we assume that the car was placed randomly then the probability that the player will initially choose the car is 1/3. As a player who swaps must always get the opposite of their original choice their probability of winning by switching must be 2/3.

A paper by Morgan et al states that, because the player decides whether to switch or not after the host has opened a door, the problem is one of conditional probability. The problem is described by Morgan in this way, 'To avoid any confusion, here is the situation: The player has chosen door 1, the host has then revealed a goat behind door 3, and the player is now offered the option to switch'. They then proceed to show that the probability of winning by switching is given by 1/(1+q) where q is the probability that the host will open door 3 if the car is behind door 1. How can the answer be different from that given by the simple argument at the start?

The argument at the start clearly shows that 2/3 of players who switch will get the car, how can it be that the player in the case Morgan describe does not have a 2/3 chance of winning the car? What if we do not consider all players who enter but only a sample of them? If we take a representative (or random) sample then the result must still be 2/3. However if we take a sample that is not properly representative of an average player who decides to swap then the answer may be different.

Although the player may choose any door, Morgan only consider players who have chosen door 1. If, as we have assumed throughout, the car is placed randomly this sample is representative of an average player and the result is still 2/3 for all players who choose door 1.

After the player has chosen door 1, the host may open door 2 or 3 (unless this reveals a car) however, Morgan only consider the case where the host has opened door 3. This makes the sample not representative of the average and makes the result no longer 2/3. Here is why. If the car is behind door 1 then the host can open either door 2 or door 3. We are not told how he decides which to open in this case and it could be, for example, that the host for some reason never opens door 3 if he has the option not to. This means that if the host has opened door 3 it must be because the car is behind door 2. If this is the case then the player will win with certainty if she swaps.

To sum up, because they only consider the case where a particular door has been opened Morgan do not take a representative sample of the of the overall or average case and they therefore get a different result. Martin Hogbin (talk) 20:05, 8 April 2009 (UTC)

Sorry Martin, may be you think: the last words will be the right ones, but this whole summing-up is fallacious. Nijdam (talk) 06:51, 9 April 2009 (UTC)

In what way? Martin Hogbin (talk) 07:20, 9 April 2009 (UTC)

Read what I wrote on the discussion page of MHP. Nijdam (talk) 15:18, 9 April 2009 (UTC)

I see nothing of relevance there. Surely the process of conditioning a sample space is precisely the same as taking a non-representative sample from it? Martin Hogbin (talk) 17:11, 9 April 2009 (UTC)

As no one else seem to want to contribute any intelligent argument to this subject, I will continue my explanation.

The sample taken from the from all possible the events in the unconditional case under the condition that the player chooses door 1 and the host opens door 3 is not always unrepresentative. It depend on the hosts door preference.

Let us take Morgan's case that q=0, which means that the host never opens door 3 unless they have to. Here, the fact that door 3 has been opened means that the car is behind door 2. So, to make that quite clear, if we consider the conditional case and the hosts door preference parameter q happens to be 0, we only consider cases where the car is behind door 2. Hardly a representative sample.

Now consider the case that q=1, the host always opens door 3 if he can. This means that the car is equally likely to be behind doors 1 and door 2 but not 3, again not a representative sample of the general case.

It the host happens to have an equal preference for both doors (q=1/2) then the car has a 1/3 probability of being behind any door and the sample is one that is representative of the overall or unconditional case. As we know, if q=1/2 the conditional and unconditional probabilities are the same. Martin Hogbin (talk) 13:07, 11 April 2009 (UTC)

Where to start? How about, they claim that the host's behaviour may provide information on where the car is. Then they give him a behaviour. Does the Contestant now know this? If so, hasn't this behaviour just become a premise of the puzzle? Well, that changes it so that it's not the MHP any longer. Talk about 'false solutions'! Glkanter (talk) 13:29, 11 April 2009 (UTC)

What you say is really what my section below is all about. In a formal probability problem, which is what Morgan have perversely taken Whitaker's question to be, anything not defined as random must be taken to be possibly non-random. Thus, as the host's behaviour is not specified in the problem statement to be random, we must consider the possibility that it is not random and that there may be some other has some process, such as the host having a preference for one door involved. In the style in which Morgan understand the question, what the player knows is not important, it is what we (as the answerers of the question) know, or do not know, that forms the basis on which the problem must be solved. What Morgan fail to do (amongst other things) is to consider the equally likely possibility that the producer has not placed the car randomly.

Some problems are often considered from the point of view or state of knowledge of one of the characters in the problem. It seems clear to me that Whitaker intended the question to be taken that way. In other words he was asking, from the point of view (state of knowledge) of a player on the show what would be their best estimate of the probability of winning by switching, which is 2/3 plain and simple. Martin Hogbin (talk) 17:04, 11 April 2009 (UTC)

Yes, 2/3, but the conditional probability, and not the unconditional one if that is what you mean by "plain and simple"!! —Preceding unsigned comment added by Nijdam (talk • contribs) 08:06, 12 April 2009 (UTC)

If we take a traditional view from the player's perspective in which we assume that the player has no information about the producer's or the host's behaviour then the 2/3 solution applies to both cases, which are indistinguishable. If we ask the question as to what the players best estimate of her probability of winning by switching is in the above case then all she can do is to assume the producer and host act randomly. In that case, even after an identified door, for example door 1, has been opened, her estimate of her probability of winning by switching remains at 2/3.

Before you tell me, I appreciate that taking the problem from the player's perspective and assuming she has no information about the game is not 'correct', it might be that she has some historical knowledge of where the producer usually places the car, but it is is traditional in these problems to assume no prior knowledge for the characters involved. Look at the Three_Prisoners_problem where it says 'Prisoner A, prior to hearing from the warden, estimates his chances of being pardoned as 1/3'. This estimate is not strictly justified as it could be that prisoner A has some prior knowledge of his likelihood of being executed but we traditionally take this not to be the case. The only way to resolve these ambiguities is to do what Seymann suggests and consider what Whitaker actually intended by his question. That would be a worthwhile discussion. Martin Hogbin (talk) 09:52, 12 April 2009 (UTC)

You know, this has bothered me all along. As I understand it, without assigning a host behaviour, Morgan says the probability of winning by switching is between 1/2 and 1, and that it averages 2/3. Or, is that only when the Contestant selects the car? No, that probability would be 0. So now we have Morgan saying 'it averages 2/3', and the unconditional solution saying 'it's 2/3'. I guess there's a difference from some standpoint, from a practical standpoint I don't see it. So, without his specious claim that the unconditional solution is false, (and someone better tell Devlin, http://www.maa.org/devlin/devlin_07_03.html as he seems oblivious to this), Morgan has accomplished nothing.

But wait. Don't ask me why, but he decides to ascribe a host behaviour. Now, the host is more likely to choose one door over another. Look, I'm a simple guy. I went to a state school. A land grant college at that. But they 'learned us' that a new premise describes a new problem. Was that true only on my campus, and not at Old Dominion University in Virginia? So how is it relevant, and not a Source of Confusion itself, to discuss these other problems when trying to describe the MHP paradox? Glkanter (talk) 23:35, 11 April 2009 (UTC)

I am not sure why you are arguing with me. I agree that from any real world standpoint the Morgan paper and its conclusion is irrelevant. I have demonstrated that on this page and on my analysis page, where at least one other editor seems to agree with the maths. There have been no significant arguments against my criticism of the Morgan paper on this page just a repeat of the, obviously untrue, claim that I do not understand conditional probability on the main talk page.

Morgan is right from a particular formal standpoint and given a set of arbitrary assumptions. I agree with you that the paper is awful and irrelevant to the real MH problem but is is not actually wrong. Unfortunately it is a paper published in a peer-reviewed journal and we therefore cannot completely ignore it although I wish we could. Martin Hogbin (talk) 01:15, 12 April 2009 (UTC)

My last edits were not directed at you. It just happens that you created a section called 'Why Morgan are right', and it seems like a good place to discuss why Morgan are NOT right. And like I said, this has bothered me all along. If it's a false solution, then why does someone like Devlin use it? And do I mis-understand the principal that new premises make new problems? I really want to know. So, I think Morgan is flat out wrong about the MHP. Published, to be sure. Peer-reviewed, maybe. Worthy of prominence in the MHP article? No way. On this, we agree. Glkanter (talk) 02:30, 12 April 2009 (UTC)

Yes, we do agree that Morgan's paper should not set the tone for this article. I also agree that does not even qualify as a reliable source in many ways. However where we may have to disagree is that the Morgan paper is technically correct in a limited formulation and style of the problem. My point was just this Morgan are right but in a way that is irrelevant. 86.132.253.23 (talk) 10:59, 12 April 2009 (UTC)

Well if you feel better in writing such comments down, ok. But remember: Morgan's is an article in a peer reviewed magazine, and Devlin just wrote in his popular column. Nijdam (talk) 20:33, 13 April 2009 (UTC)

See my comments about the Morgan paper at the end of the section above. It is essentially a 'novelty article'. Martin Hogbin (talk) 08:30, 14 April 2009 (UTC)

Problem style

This is a subject that I believe is central to many of the arguments on this page. Perhaps somebody who is an expert in the subject can clarify something. It seems to me that there are (at least) two styles of problem in probability questions.

Formal style

The formal style is to present a question that must be answered only on the information given in the question; what the characters in the problem may or may not know is irrelevant, as is any information that we may deduce from real life. In such problems it is necessary to specify whether that a certain selection or distribution is random or not. If such information is not explicitly given in the question then we must take it as unknown rather than random.

As an example, suppose this were the question:

A car is placed behind one of three doors. A person picks a door, what is the probability that they have picked a door hiding the car?

If we take this as a formal probability problem we should first observe that no information is given as to how the car was placed or as to how the player picks the door. We cannot assume that this is random, it could be that the car is always placed behind door 2 and the player always picks door 3, for example. If we take this problem to be a formal probability problem, the best we can say is the probability is from 0 to 1.

On the other if the question were to say:

A car is placed randomly behind one of three doors. A person picks a door, what is the probability that they have picked a door hiding the car?

or

A car is placed behind one of three doors. A person randomly picks a door, what is the probability that they have picked a door hiding the car?

The probability of picking the car is 1/3.

If in the case of solving a formal problem if we are to make assumptions, we must state them explicitly, we should not assume motives or strategies of characters in the problem (that is not to say that characters might not have strategise just that w should not assume what they are). So even if the problem were to be:

A car is placed behind one of three glass doors. A person picks a door, what is the probability that they have picked a door hiding the car?

The answer would not be 1 but would still be from 0 to 1 unless it is made clear that the person can actually see the car and will certainly choose it.

Traditional style

Traditionally probability problems have been presented from the point of view or state of knowledge of one of the characters in the problem. Normal human motives are also often traditionally assumed for these characters. For example in the 'Three prisoners problem' we have: 'Prisoner A, prior to hearing from the warden, estimates his chances of being pardoned as 1/3'.

If we have the same problem as before:

A car is placed behind one of three doors. A person picks a door, what is the probability that they have picked a door hiding the car?

and we treat it more traditionally we might say something like, 'It is natural to assume that the person was not aware of the way the cars were placed behind the door so their estimate of the probability that they have chosen the car would be 1/3'.

Here we put ourselves in the place of the player and assume the player's state of knowledge. Anything that we do not know e take to be random.

Can anybody confirm or expand on what I have said on problem style? Who disagrees? Martin Hogbin (talk) 14:50, 10 April 2009 (UTC)

Maybe the Contestant gains knowledge from the host's behaviour by accident.

Clearly, there is no explanation for how the Contestant gains knowledge of the location of the car when Monty opens a door revealing a goat. His behaviour could never intentionally be transmitted to the Contestant. I mean, that's got to be in the definition of a Game Show, right? In the US it sure is. http://en.wikipedia.org/wiki/Quiz_show_scandal

So, the only remaining way for Monty's action to give useful info is inadvertently. Maybe he walks past a goat, and somehow causes the goat to make a noise, and then opens the other goat door. Or maybe he picks up some goat crap on his shoe, and then opens the other goat door.

So my question is, is it common practice in Probability theory (or puzzle formulation) to factor in inadvertent acts? How about illegal ones? How does one assign a probability to this? Glkanter (talk) 02:28, 13 April 2009 (UTC)

This is the very first sentence of the article.

"The Monty Hall problem is a probability puzzle based on the American television game show Let's Make a Deal." (emphasis mine)

This is the very first sentence of the Wikipedia article "Quiz show scandals":

"The American quiz show scandals of the 1950s were the result of the revelation that contestants of several popular television quiz shows were secretly given assistance by the producers to arrange the outcome of a supposedly fair competition." (emphasis mine)

http://en.wikipedia.org/wiki/Quiz_show_scandals

So, does Morgan perhaps not understand the rules of a game show? Glkanter (talk) 02:45, 13 April 2009 (UTC)

What is interesting about this fact is that, if it were the intention to manipulate the results of the show, this could only be done by means of fixing the car placement, there is no strategy that the host can adopt to change the player's chances of winning. Martin Hogbin (talk) 09:16, 13 April 2009 (UTC)

What is your response to the various questions I ask above? Would you agree that Morgan's claim that the host's behaviour could provide information as to the whereabouts of the car is contrary to the normal (and legal) understanding of how game shows work? Glkanter (talk) 11:03, 13 April 2009 (UTC)

You need to start with the way Morgan set up the question, 'The player has chosen door 1 and the host has then revealed a goat behind door 3...'. Note that it is asked from the point of view of a third party observer. We are to answer the question on the information given. What the player knows in not important. It is only by setting up the problem in this way that Morgan can introduce the host preference into the problem at all. Morgan supporters need to remember this, you cannot say for example that the player must know which door has been opened so we must take that into account. Martin Hogbin (talk) 21:33, 13 April 2009 (UTC)

I should have included this, from the Wikipedia article. It's the problem statement in the first paragraph:

"A well-known statement of the problem was published in Parade magazine:

Suppose you're on a game show, and you're given the choice of three doors..."

So, it's stated that it's a game show, which means the Contestant can not be given any indication where the car is. And it says 'Suppose you're on a game show', which clearly states that this is from the Contestant's point of view. Why is Morgan still given any credibility at this point? Glkanter (talk) 22:42, 13 April 2009 (UTC)

I agree with you. Morgan massage the question (note that they even misquote the original question from Whitaker in their paper to make it closer th their own problem statement) to get an 'elegant' solution. Actually, I now think that Morgan are even more wrong than that. I will explain later. Martin Hogbin (talk) 08:37, 14 April 2009 (UTC)

Well, I love surprises.

But it's more than they're wrong. To me that implies their conclusion is wrong. Their entire premise is false. They completely disregard this simple statement: "Suppose you're on a game show". By definition then, the pov is the Contestant, and all the things that define a game show are (unstated) premises. Like, the host is not going to indicate to you (the Contestant) in any way where the car is. So, like I've been saying, the minute they begin describing host behaviours they're on to a different problem. And the Morganians ALWAYS invent a host behaviour when explaining their pov. Look at the article. Without host behaviours, Morgan would be about 3 sentences long. This is all they can say, "It's between 1/2 and 1, and averages 2/3". That, and 'The unconditional solution is false'. But that is clearly just an opinion that is not popularly held. See Devlin, and many others. Glkanter (talk) 12:09, 14 April 2009 (UTC)

I find it interesting that after 2 1/2 days, none of the Morganians have been motivated to respond to this section. And there's no requirement that they do so. Maybe they could just explain to me how I should interpret these 6 words (from Whitaker?), "Suppose you're on a game show...". Glkanter (talk) 15:43, 15 April 2009 (UTC)

A question

Starting with the standard MHP game rules (host always offers the swap and always opens an unchosen door to reveal a goat). Suppose that we take Morgan's situation, player chooses door 1 and host opens door 3. Now let us suppose that there is a population of equal numbers of hosts of two types: those where p=1 (always open door 3 where possible) and those where p=0 (never open door 3 unless forced to do so). Let us assume that our host was taken randomly from this population and is thus equally likely to be either type.

Here is the question. Given the above, what is the probability that given the host has opened door 3, the player will win by switching? I have already put my answer on another page. Who will give this conditional probability problem a go? Martin Hogbin (talk) 18:34, 14 April 2009 (UTC)

Conditional on the choice of door 1:

p=1
agG* 
gaG*
gGa
p=0
aGg
gaG*
gGa

hence: P(car=1|opened=3)= agG*/[agG* + gaG* + gaG*] = 1/3

Nijdam (talk) 19:54, 14 April 2009 (UTC)

I do not understand your answer. I presume you are saying that the probability that the player will win by sticking is 1/3? Thus the probability of winning by switching is 2/3.

Do you not find this answer interesting? Although we do not know the host behaviour, the probability works out to 2/3. Martin Hogbin (talk) 22:19, 14 April 2009 (UTC)

You did understand it well. I could have written it more extensively of course. What is so peculiar about it? We do know the host's behavior. Instead of a host with random behavior, it is a random host with a fixed behavior, equivalent more or less like to the flipping coin strategy. Nijdam (talk) 06:16, 15 April 2009 (UTC)

I agree with your answer but I think it is more general with that. I believe that the answer (probability of winning by switching) is 2/3 for any distribution of host behaviour that is symmetrical about 1/2. This includes a linear distribution of q, which I think is what Morgan get the answer ln(2) for. Do you agree? Martin Hogbin (talk) 10:18, 15 April 2009 (UTC)

No, ln(2) is the answer where 1/(1+q) is uniformly distributed, not q (Nijdam - this is in the Morgan et al. paper where they compute a specific number using a Bayesian method assuming an uninformative prior). -- Rick Block (talk) 13:35, 15 April 2009 (UTC)

Are you sure, Rick? Can you explain further please? Martin Hogbin (talk) 17:24, 15 April 2009 (UTC)

(outindented) The noninformative distribution is the uniform distribution for q. With this distribution the average probability of winning by switching is the average of 1/(1+q):

$\int _{0}^{1}{\tfrac {1}{1+q'}}f_{q}(q')dq'=\int _{0}^{1}{\tfrac {1}{1+q'}}dq'=ln(2).$ —Preceding unsigned comment added by Nijdam (talk • contribs) 20:32, 16 April 2009 (UTC)

Is that not what I said? We take q to have an equal probability (density) of being any number from 0 to 1 and integrate the probability of winning by switching (1/(1+q)) with respect to q. Martin Hogbin (talk) 00:27, 17 April 2009 (UTC)

Yes, you're right, I wrote it down explicitly, because it seems Rick has doubts.Nijdam (talk) 08:15, 17 April 2009 (UTC)

If we take it that Morgan took the unconditional distribution of q to be uniform then it seems to me that they fail to properly apply the condition that the host has opened door 3 in their calculation. Martin Hogbin (talk) 18:52, 17 April 2009 (UTC)

The prior distribution of q has nothing to do with the terms 'unconditional' and 'conditional', so no accusations of failure. Nijdam (talk) 23:24, 19 April 2009 (UTC)

That was not my point. First we assume a uniform prior distribution of q, we then apply the condition that the host has opened door 3. After we have applied this condition the distribution of q is no longer uniform. Do you not agree? 86.132.253.23 (talk) 22:17, 20 April 2009 (UTC)

You may mean the posterior distribution of q after the host has opened door 3. You may have a point there. The posterior is ${\tfrac {2}{3}}(2-t)$, and puts more weight, of course, on door 3. Nijdam (talk) 09:57, 21 April 2009 (UTC)

Yes, that is exactly my point. My original question of this section was a very simple example. Martin Hogbin (talk) 22:19, 21 April 2009 (UTC)

Well that's the problem with Bayesian approach, it doesn't mean it's wrong. Nijdam (talk) 08:56, 25 April 2009 (UTC)

I have not studied the Bayesian approach but I do not think that it is relevant to this issue. Morgan make an error in their calculation in that they fail to use the information in the door number opened by the host. They perform a calculation using the prior distribution of q whereas they should have used the posterior (after the host has opened a door) distribution. This is an error plain and simple however you look at it. Martin Hogbin (talk) 09:09, 25 April 2009 (UTC)

I think that I now know the calculation that Morgan should have done to calculate the player's probability of winning by switching if the prior distribution of the host's door opening parameter q is uniform. They should have had:

$1/(1+\int _{0}^{1}\ {f_{q}(q)}dq)=1/(1+\int _{0}^{1}\ {q}dq)=2/3$

I can show how I derived this if anyone is interested. Martin Hogbin (talk) 21:19, 29 April 2009 (UTC)

Another question

This is a question the first part of which I will answer myself.

A car is placed (not necessarily randomly) behind one of three doors, which are numbered 1,2, and 3. I propose to throw a fair die and if I throw 1 or 4 I will pick door 1, 2 or 5 door 2, 3 or 6 door 3. What is the probability that the door I am going to pick will have the car behind it?

My answer: 1/3.

Now I have thrown the die it has come up with 4 (door 1). What is now the probability that I have the car?

My answer: From 0 to 1 depending on the probability the car is behind door 1.

The question. How does getting more information give me a less good measure of the probability of winning the car?

Any answers? Martin Hogbin (talk) 19:13, 15 April 2009 (UTC)

Note that you implicitly assume the independence of car placing and throw. Now about you question. Why do you consider it a lesser good measure? It is a perfect measure, but the object of the measurement is unknown. The precise answer would be: the probability with which the car is placed behind door 1.Nijdam (talk) 20:40, 16 April 2009 (UTC)

Yes I do assume independence of the car placing and the throw of the die, it is hard to see how things could be otherwise but I guess I should have made this clear.

What puzzles me is that we go from being able to say that the probability of winning the car is 1/3 (a definite answer) to only being able to say that the answer is given by the probability that the car was placed behind door 1, which we only know is a number from 0 to 1 (an extremely unhelpful answer) but we have been given more information. This just seems wrong somehow. Martin Hogbin (talk) 00:36, 17 April 2009 (UTC)

When All You Have Is A Hammer, Everything Looks Like A Nail

Explain to me again how the original 1/3 probability of the Contestant choosing the car changes based on which goat Monty reveals?

The course titles for the Probability classes I took were 'Intro to Logic' and 'Symbolic Logic'. That's what is missing from the Morganian's arguments: Logic.

"Suppose you're on a game show..."

The MHP is a story problem. It can be solved using whatever technique is best. And being a story problem about a game show, the rules of game shows apply.

Since it's illegal in the US, there can be no 'host behaviour' known to the Contestant, which as per the article, quoted above, is how the problem is laid out. And the Combining Doors solution is not a 'false (unconditional) solution' as the Morganians interpret Morgan's paper. As there is no possibility of any 'host behaviour' known to the Contestant, there is no difference between deciding to switch before or after the goat is revealed. As if that mattered anyways. Glkanter (talk) 12:12, 19 April 2009 (UTC)

It is a conjuring trick and here is how it is done.

Firstly you misquote the original question, making, 'a door, say No 1' into, 'door No 1'.

You then rephrase the question so that it is not from the point of view of the player but is a formal probability probability problem. This means that we no longer consider the state of knowledge of the player but have to consider what actions are stated to be random in the question. If something is not stated in the question to be random then we must assume that it may not be. There are three choices in the problem, none of which is actually stated to be random in the Whitaker statement.

Next you quietly ignore two of the non-random choices, that of the producer placing the car and the player choosing the door. The reason for this is that otherwise the solution becomes rather dull. The probability of winning by sticking would then depend on the probability that the producer placed the car behind the door that the player chose. Of course, if the problem were considered from the players state of knowledge this would not be the case as the player does not know where the car was placed but taken as a formal probability problem where we are told that the player has picked door 1, the solution is obviously dependent on the probability with which the producer has placed the car behind door 1. He might always place the car behind door 1 for example. This effect is so obvious and boring, however, that it is discreetly ignored.

Finally we leave the host choice as non-random. Hey presto, rather than an obvious result like that of the producer placing the car non-randomly we can do a little maths and come up with an 'elegant solution'. Martin Hogbin (talk) 21:44, 19 April 2009 (UTC)

Here is the same trick without the props.

'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'. What is the probability that you pick the car. Most people, Morgan included, would give the answer 1/3.

Now misquote.'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 door No. 1'

Rephrase, 'On a game show a car is placed behind one of three doors. Given that door 1 has been chosen, what is the probability the car is behind the chosen door?' Answer to this, now conditional, question: anything from 0 to 1 depending on the probability the car was placed behind door 1. Martin Hogbin (talk) 22:52, 19 April 2009 (UTC)

What is this about? Do I notice some frustration? Nijdam (talk) 23:25, 19 April 2009 (UTC)

There is some frustration with several things, mainly by lack of ability to improve the article. Because of the way that Morgan's solution and ethos is incorporated throughout the article it is hard to make small improvements that stand alone. The article really need rewriting to make a better and more convincing job of explaining the basic, unconditional if you prefer, solution. Even thinking of a suitable description for this solution that properly describes its status is not easy. The article is currently featured and I am reluctant to make major changes on my own, I would much prefer to convince other editors that such changes are necessary so that we can work together to improve the article.

I am also frustrated that several important issues that I have raised above on the subject of probability and that are very relevant to the MHP have been ignored. The above explanation is intended to convince people how unhelpful and irrelevant the Morgan paper is to the real MHP. Martin Hogbin (talk) 08:50, 20 April 2009 (UTC)

Well we do disagree than on many points. From the discussions with you it seems you got an understanding of the nature of conditional probability. This should lead to the notion that the MHP is inherently of a conditional nature. If the article is to be improved, it should mention the right solution first. Then it may pay some attention to the often heard "unconditional solution", and mention the flaw in it. These considerations are in my opinion the same as Morgan's, and so Morgan's paper is very relevant and helpful. And it is a shame hat people like Devlin have written cheap columns on subject they hardly understand. Nijdam (talk) 22:59, 22 April 2009 (UTC)

You say the MHP is inherently of a conditional nature; this is where we disagree. I accept that it can be made into a conditional problem by ignoring the question that Whitaker undoubtedly wanted the answer to, which is, 'Is it generally better to switch?' and choosing to answer the specific question where the player has chosen door 1 and the host has opened door 3 (or other specified doors). Even when you have made the problem conditional, if you take it that the host chooses randomly, the condition makes no difference, yes, it is a condition, but who cares?

I have to keep coming back to the point that the simple problem, regardless of whether it is treated conditionally or not, is the one that most people get wrong most of the time. Martin Hogbin (talk) 17:58, 24 April 2009 (UTC)

Martin - do you agree that the problem consists of a first step where the player selects a door and a second step where the host opens a door and that the point of decision is after the second step? If so, you're agreeing it's a conditional problem. One can argue that the solution to the conditional problem must be the same as the solution to the unconditional problem (which, if the player doesn't switch, is mathematically equivalent to a problem where the player selects a door and the host does nothing), but the fact that the conditional problem is equivalent to this intuitively obvious unconditional problem (iff the host opens a random door if the player initially selects the car) is NOT obvious, and should have some reasoning behind it. That's all we're saying here. The point is not to complicate matters, but to address the actual problem in a way that's sound and won't lead readers astray if they encounter slightly different problems (like the "host forgets" version). Given a firm grasp of the unconditional solution to the standard version, why doesn't this same approach work for the "host forgets" version? As Glkanter says, it's a different problem, but how should the reader know when an unconditional approach works and when it doesn't? This question apparently completely stumped Glkanter above (I pushed it to the point where he apparently got very angry). Teaching people shortcuts without teaching them when the shortcuts are applicable seems like a bad idea to me. -- Rick Block (talk) 19:24, 24 April 2009 (UTC)

I do not agree the point of decision is necessarily after the host has opened a door. This is clearly the case in Morgan's reformulation of the problem but it may well not be what Whitaker wanted to know. He may well have been asking the question of what the best policy would be for a contestant on the show. You keep talking about 'the actual problem' as though this is Morgan's restatement of Whitaker's original question. The actual problem, if there is one, is what Whitaker actually wanted to know. This may or may not be conditional problem, we have no way of knowing.

You still do not seem to have understood my point about formal versus informal problem treatments. In the first case we must take any information that is not provided as indeterminate in the second it is common to treat it as random.

Regarding your point about addressing the problem in a way that is sound, most of mathematics consists of shortcuts. If we were to address every mathematical problem completely rigorously (and mathematicians can get very rigorous) we would never get off the ground with most problems. In order to make any headway with a problem you have to ignore some of the details initially. These can be considered, if desired, later. Martin Hogbin (talk) 00:18, 25 April 2009 (UTC)

Rick made it completely clear. And even if Whitaker had another problem in mind - and we may never know - the way he formulated the problem, with the stages Rick mentioned, forces it to be answered with conditional probabilities. And I have the same concern as Rick, and even more, as I notice that teachers and students often make use of the MHP and just use the wrong simple solution to explain the problem, supported, I hope only until recently, by Wikipedia. Nijdam (talk) 08:51, 25 April 2009 (UTC)

Rick made his version of the question clear but nobody can claim that Morgan's (or Rick's) formulation is the 'real' MHP. If we take the Parade statement as the definitive problem statement then the definitive formulation is given by 'the precise intent of the questioner'. This fact cannot be disputed. It is standard practice in any discussion of probability that before a question can be answered it must be decided what it means, particularly if much detail is missing from the question statement itself. This view is made quite clear by Seymann at the end of the Morgan paper. Are you disputing that this is the correct way to address a question from a non-expert?

For educational purposes my suggestion is to first formulate the problem in a way that the simple solution applies, if you consider it essential, formulate it unconditionally for this purpose. After that apparently simple but intuitively exceptionally difficult problem has been solved and understood, the conditional problem and could be introduced. Treating the two together at the start only serves to obfuscate both issues. Martin Hogbin (talk) 09:48, 25 April 2009 (UTC)

I'm curious. If Whitaker had asked vos Savant about Deal or No Deal, and had casually referred to unique suit case numbers as in "say, suitcase 13", would the equivalent of Morgan's literal interpretation regarding "say, door 3" still be considered the 'last word' by the Morganians? Glkanter (talk) 02:35, 25 April 2009 (UTC)

Of course, because that way they can get an 'elegant solution'. Martin Hogbin (talk) 18:20, 26 April 2009 (UTC)

The question that no one will address

I have raised this issue before as I believe that it is central to this MHP argument. Replies to date have simple been along the lines of 'No it is not'. Firstly let me make clear this point has nothing to do with whether the problem is conditional or whether the doors are distinguishable. It is about problem style.

Let me start by asking two similar questions that have two completely different answers. I have given my answers. Does anyone disagree with them? I ask those involved in discussion on this subject to add their answers below or to state whether they agree or disagree. Please do not make any assumptions about the way this relates to the MHP at this stage, I just want to get absolutely clear that we agree before proceeding. Martin Hogbin (talk) 10:02, 25 April 2009 (UTC)

1) A car is placed behind one of 3 doors. What is the probability that it is behind door 1?

• Indeterminate. The answer is simply the probability with which it was initially placed behind door 1.Martin Hogbin (talk) 10:02, 25 April 2009 (UTC)

2) A car is placed behind one of 3 doors. What is the probability that a person who chooses randomly will choose the door with the car?

• 1/3 The door choice is defined to be random son the initial car placement probability is important.Martin Hogbin (talk) 10:02, 25 April 2009 (UTC)

To #1 my answer is: If I am to "assume you are on a game show", my answer is 1/3, otherwise "between 0 and 1, averaging 1/3". To #2, my answer is 1/3. I tried my best to follow your instructions. Glkanter (talk) 13:18, 25 April 2009 (UTC)

I wanted the questions to be treated like questions in a test on probability, where you would be expected to answer the questions based only on the information contained within them. You may like to think that question 1 refers to a game show but you are not expected to use any knowledge that you think you might have about game shows to answer it. Martin Hogbin (talk) 18:46, 25 April 2009 (UTC)

@2): Formally the relation between the choice of the person and the placemant of the car must be specified. If they are independent, the answer is 1/3. BTW: what do you mean by: son the initial car placement probability is important? Nijdam (talk) 15:47, 25 April 2009 (UTC)

Sorry, that was a typo, it was meant to be "so the initial car placement probability is unimportant ". Surely a statement that the person chooses randomly is sufficient because a random choice is by definition independent of all else. If you think that it is necessary to add that the person's choice is independent of the car placement to my question 2 I will do so. Apart from that do you agree with my two answers? Martin Hogbin (talk) 18:37, 25 April 2009 (UTC)

I suspect Martin means "so the initial car placement probability is unimportant". Martin - I find your approach here extremely annoying. Please just say what you mean and drop the little puzzles. -- Rick Block (talk) 17:59, 25 April 2009 (UTC)

I do not understand what you find annoying. This is the arguments page, the place for discussion about the MHP. My questions are not meant to be puzzles, I am trying to ensure that we all start on the same wavelength, if we all disagree fundamentally on some aspects of probability it makes it difficult to move forwards. The way a probability problem is worded is extremely important, pretty much all the argument concerning the MHP is about interpretation of the question rather than the maths which is fairly straightforward. Nijdam and Glkanter have given their answers perhaps you could do the same then I will explain my point. Martin Hogbin (talk) 18:37, 25 April 2009 (UTC)

What is annoying is your metering out of your point. Probability theory allows for extremely precise problem statements and there really aren't that many choices for the MHP. It's either talking about:

1. P(win by switching)

2. P(win by switching|player picks door i) (where i is, say, 1)

3. P(win by switching|player picks door i and host opens door j) (where i is, say, 1 and j is, say, 3)

Various wordings in English map to one of these. The wording should make it obvious which one of the above probabilities is being discussed. If it doesn't, it's simply poorly worded. Nijdam and I are both saying the "standard" MHP is meant to map to the third one (and this is backed up by numerous math sources that know the difference). This is the notable problem that most people get wrong. Yes, the answer to #2 happens to be a shortcut to the same answer (under certain assumptions) - but this is not the problem that is asked. Your continued insistence that Whitaker's words actually mean something different is simply absurd. -- Rick Block (talk) 00:52, 26 April 2009 (UTC)

I do not understand the term, 'metering out of your point'. If you mean that I am going through it rather tediously that is because I want to ensure that we all agree at the start. There is little point going through a long argument only to find at the end that you disagree because you had not agreed on one of the starting premises.

You give three possible questions that might be asked but these are clearly not complete problem statements. A problem statement must make clear the setup of the problem. In the case of the MHP that would include the game rules (which are not in dispute) and the basis on which selections are made (are they random, unknown, based on an informed choice, independent of one another etc?) and possibly other things. Without this information the questions that you ask are meaningless.

Regarding Whitaker's words, do you really think that the question he intended to ask was,' Given that a player has picked door 1 and the host has opened door 3, is it best to swap?'? Most people are not aware of the conditional issue (which given reasonable assumptions does not make a blind bit of difference). The event that made the problem notable was the response to vos Savant's answer (in which the conditional issue was ignored).

Finally you have chosen not to confirm your agreement (or otherwise) with my answers above. I am not sure what you think you have to lose by answering unless you fear that it will lead to a discussion that will show up the weaknesses in the current treatment of the problem. I and several other editors (probably the majority of those currently active) believe, for various reasons, that the simple/unconditional problem and solution should be given greater prominence within the article, but rather than rush in and make major changes we have decided to discuss the issues on this page; one specifically set up for that purpose. As you know, I am trying to show up weaknesses in the current MHP treatment and these are concerned with problem style. In order to explain exactly what I mean, it is easiest to give examples to see where any disagreement lies. There is nothing more sinister than that to my 'little puzzles'. Martin Hogbin (talk) 09:47, 26 April 2009 (UTC)

By "metering out your point" I mean going through it slowly, without divulging where you're headed. I find it annoying not because I think I have anything to lose but because it's inefficient. If we knew where you were going with this, Nijdam and I could probably help point out specifically where you're going wrong. If the only way you'll proceed is by gaining consensus at each step then fine - lacking any other information, the answer to your first question is p where 0<=p<=1. The second one is 1/3.

The probability statements I offer are the only different probabilities involving the problem setup we're talking about (#2 and #3 cover the cases of the the specific individual doors, if these probabilities might vary by specific door number). Formally, they depend on the background information typically denoted I, which includes the games rules and the other considerations you mention.

Regarding Whitaker's words, I think he intended to ask a question that maps to #3 above. If you're not still confounded by the distinction between the unconditional and conditional situations here, you're giving a very good impression of one who is. The question is not restricted to door 1 and door 3, but involves a specific player pick and some specific door opened by the host. This problem did not originate with Whitaker, so your seemingly intense focus on analyzing precisely what he meant is entirely misplaced. The problem is a reformulation of other, classic problems in conditional probability (specifically the Three Prisoners problem which itself is essentially a reformulation of Bertrand's box paradox). The entire point of the problem is to create a situation with two unknowns that have identical unconditional probabilities but different conditional probabilities. In the MHP it turns out the conditional probability of the player's door remains the same as the original unconditional probability, but the question is most assuredly asking about the conditional probabilities in effect given the host has opened a specific (known) door after the player's initial (known) selection. Do you really think the question is meant to map to #2 (or #1) above and that it's nothing but an elaborate trick question meant to hide your second question (sort of like those primary school problems like "assume a plane crashes exactly at the North Pole - where are the survivors buried?").

The response to vos Savant's columns included the two math papers we're talking about (Morgan et al. and Gillman) who both clarify that the problem is a conditional probability problem (i.e. maps to #3 above). The reason people typically get this wrong (and to be clear, this is simply my opinion) is because people are unaccustomed to dealing with conditional probabilities. You and several other editors have argued with expert opinions about this so long that most experts here (pretty much anyone involved in wp:WikiProject Mathematics) simply ignore you. This definitely does not mean anyone is agreeing with you. -- Rick Block (talk) 17:05, 26 April 2009 (UTC)

Rick, you seem fixated on conditional probability. I made quiet clear at the start that that was not what this section was about. Anyway, you seem to agree with my answers. Martin Hogbin (talk) 19:48, 26 April 2009 (UTC)

The point that I am trying to make is that, in between the two problem examples that a gave above is a range of possible questions and circumstances where the answer is not quite so clear. Take for example:

3) A car is placed behind one of 3 doors. What is the probability that a person who chooses a door will choose the door with the car? (The circumstances are such that we would not expect this person to have any knowledge of where the car had been placed.)

Who would say that the probability is 1/3?

Could we also say that the car placement was effectively random because the chooser had no knowledge of it?

Now what about if we apply the same logic to the host's choice of door in the MHP. The player has no knowledge of the host's door choice policy, thus it is effectively random. Martin Hogbin (talk) 19:48, 26 April 2009 (UTC)

This is the point where you start to go astray. Problem statements are meant to be clear and this one is not. If the initial placement is random then the answer is 1/3. If the player's choice is random then the answer is 1/3. If by the wording of the problem you mean both of these may be taken as not random (and you either mean these to be taken as random or you don't, or the problem is ambiguous) then this problem is the same as your first problem and the answer is anything between 0 and 1. What about if we pay attention to what the most reliable sources say about the MHP (and I'm including Gillman here, not just Morgan et al.)? Either the problem statement specifies the host's door choice policy includes the constraint that the host choose randomly if the player initially selects the car or it doesn't. If it does, then the answer is 2/3 chance of winning by switching. If it doesn't, then the answer is 1/(1+q) where q reflects the host's preference (which the player may or may not know). -- Rick Block (talk) 20:22, 26 April 2009 (UTC)

Yes, I understand that the right and proper thing to do is to state whether a particular choice is random or not but there is common informal style of probability problem where we assume something to be random if we are given no information. For example in this quote from the Three Prisoners Problem, 'Prisoner A, prior to hearing from the warden, estimates his chances of being pardoned as 1/3' it is assumed that the decision of whom to pardon is random because the prisoners do not have this information, but nowhere in the question is this stated. Before you rush off to fix the article please bear in mind that this approach is quite common in informal probability puzzles, what is not known is taken to be random. Martin Hogbin (talk) 23:30, 26 April 2009 (UTC)

@Martin: I'm not sure where you're heading, but whatever is said or assumed about all these distributions, the solution is based on CONDITIONAL probabilities. Nijdam (talk) 20:50, 26 April 2009 (UTC)

Yes I know. Martin Hogbin (talk) 23:30, 26 April 2009 (UTC)

Rick and Nijdam, what is your response my assertion that it is common in informal probability problems to assume that, where no information is given regarding a method of choice, that choice is to be taken as random. This is clearly the case in the three prisoners problem. Martin Hogbin (talk) 08:25, 28 April 2009 (UTC)

What is common is to assume the Principle of indifference - which is not quite the same as what we're talking about here. Assuming the prisoner's initial chance of being pardoned is 1/3 is like assuming the car is initially randomly placed. In the Three Prisoner's problem (at least in Gardner's version) the warden flips a coin do decide which name to give in the case prisoner A is the one being pardoned (it's not unspecified and assumed to be random). This is what's similar to the point under dispute in the MHP. The reason it's specified is because the 3PP is also a conditional probability problem and the answer depends on this constraint (and Gardner certainly knew this). -- Rick Block (talk) 01:56, 29 April 2009 (UTC)

Fine, so why do we not apply the principle of indifference to the host's door choice and take it to be random also? It is exactly the same as the car placement choice. Both choices are not defined to be random, both were made by people (the car placement choice by the producer or his agent and the door choice by the host) and the player has no knowledge of either. Martin Hogbin (talk) 17:09, 29 April 2009 (UTC)

Are we talking about probability problems in general, or the MHP as it is generally understood, or specifically the Parade version, or specifically Morgan et al.'s interpretation of the Parade version given vos Savant's subsequent clarifications? Certainly as the MHP is generally understood and given vos Savant's clarifications of the Parade version, the initial distribution of the car is meant to be taken as random. In a game show situation, even if the producers don't randomize the initial placement, the player's initial choice pretty much has to be a random guess so the easiest way to analyze the situation is to assume an initial random distribution (since a random guess from among three alternatives regardless of the actual distribution has a 1/3 chance of being correct). On the other hand the player's choice has no such randomizing effect on the host's preference. -- Rick Block (talk) 19:41, 29 April 2009 (UTC)

I guess that I am talking about a sensible interpretation of the parade statement and how Morgan should have interpreted it.

There are three people who make choices, the player, one I will call the producer, and the host. None of the choices made is specified to be random in the Whitaker statement. The principle of indifference should logically be applied to all of those choices or none of them. There is no logic to saying the producer acts non-randomly, the player chooses randomly and the host acts non-randomly. Martin Hogbin (talk) 21:29, 29 April 2009 (UTC)

As it happens, I believe that Morgan's analysis does not apply if he producer places the car non-randomly and the player picks randomly. See my analysis page Martin Hogbin (talk) 21:39, 29 April 2009 (UTC)

This is Morgan's Entire Argument Against the "Combining Doors" Solution

'F1' refers to the strategy to always switch, as the original choice has a 1/3 chance of being the car.

"...It just does not solve the problem at hand. F1 is a solution to the unconditional problem which may be stated as follows: "You will be offered the choice of three doors, and after you choose the host will open a different door, revealing a goat. What is the probability that you win if your strategy is to switch?" the distinction between the conditional and unconditional situations here seem to confound many, from whence much of the pedagogic and entertainment value is derived"

It's a funny thing. This is exactly how I interpret the problem. I believe that the unconditional solution DOES properly "solve the problem at hand." I reject the notion that the interpretations that are door number defendant are the only "true" interpretations. And the above paragraph in no way explains how the unconditional interpretation is insufficient.

There are countless equally reliable published sources that subsequent to Morgan have put forth the unconditional solutions such as Combining Doors as their only solution.

It's time to bring clarity and common sense to the Monty Hall problem on Wikipedia. Glkanter (talk) 16:05, 26 April 2009 (UTC)

From Wikipedia:Reliable sources# Scholarship: Academic and peer-reviewed publications are usually the most reliable sources when available. However, some scholarly material may be outdated, superseded by more recent research, in competition with alternate theories, or controversial within the relevant field. Reliable non-academic sources may also be used, particularly material from reputable mainstream publications. Wikipedia articles should cover all significant views, doing so in proportion to their published prominence among the most reliable sources.

The Morgan et al. paper is in an academic and peer-reviewed publication. As far as I know, it is the only such source specifically about the precise issue we're spent so much time talking about (I'm not sure Gillman's paper was peer reviewed - and the other sources in the article itself are either non-academic, not peer reviewed, or not about the issue we're talking about). Yes, there are countless published sources that put forth unconditional solutions, but these are NOT "equally reliable". Per the quote above from WP:RS, the article should cover all significant views in proportion to their published prominence among the most reliable sources. Again as far as I know, your view that the unconditional solution solves the problem at hand is supported only by sources that are less reliable (if not FAR less reliable) than Morgan et al. Taking the RS guideline literally, this view doesn't actually need to be mentioned in the article at all.

It's time to stop arguing against what the most reliable sources say. -- Rick Block (talk) 17:36, 26 April 2009 (UTC)

Morgan's paper is an arrogant, patronising, incomplete, inaccurate, sloppy (being somewhat charitable), novelty article that somehow managed to get into in a peer-reviewed publication. It even carries its own 'health warning' at the and in the form of a comment by Seymann. Academic and peer-reviewed publications are usually the most reliable sources... sure, but not this one.Martin Hogbin (talk) 19:21, 26 April 2009 (UTC)

You are entitled to your own opinion, but unless you can cite an academic, peer reviewed source that says something different then this one is the best source we have. -- Rick Block (talk) 19:54, 26 April 2009 (UTC)

@Martin: I'm quite shocked by your burst-out. It even puzzles me where it originates from. From the fact that you doesn't like the outcome? That's quite unscientific. Because the content is wrong? Then come with proof. But I can assure you, in such simple math an error would have been detected ages ago. Because it prevents you from using the simple solution? Well, you should not use it, as it is wrong. So?Nijdam (talk) 20:23, 26 April 2009 (UTC)

I think that all that I say is justified. Let me give some examples:

Incomplete - The article fails to make clear that its analysis is based on the assumption that the car is initially randomly placed. (It mentions in passing after the analysis that people might like to consider non-random car placement)

Inaccurate - I thought you agreed that Morgan's calculation regarding a uniform distribution of q was wrong in that they used the prior distribution.

Sloppy - They misquote the original question by Whitaker (let us assume by accident).

'Novelty article' - This is hardly an paper pushing forward the boundaries of knowledge of probability theory.

'Health warning' - Why was there a comment by Seymann immediately after the paper stressing the importance of problem definition? Martin Hogbin (talk) 23:09, 26 April 2009 (UTC)

Which of the following statements is more accurate?

Morgan has expressed his opinion that the unconditional interpretation of the MHP is not what Whitaker intended to ask.

or

Morgan has demonstrated, using a scientific method, that the Combining Doors solution to the MHP is incorrect.

Glkanter (talk) 15:56, 27 April 2009 (UTC)

Morgan and three other math professors expressed their collective opinion in a peer reviewed math journal that an unconditional solution of the MHP does not solve the problem that Whitaker asked, and this opinion is reflected by another independently published paper (Gillman) and in the Grinstead and Snell textbook (both referenced by the article). So, this is not just Morgan's opinion but the published opinion of at least 7 math professors (plus reviewers). -- Rick Block (talk) 18:34, 27 April 2009 (UTC)

Plus me!Nijdam (talk) 21:36, 27 April 2009 (UTC)

In what way does being a math professor qualify one to be the final arbiter of how to interpret what is universally agreed to be an ambiguous question? It seems presumptuous to me for ANYONE to state with 100% certainty what Whitaker DID NOT mean. And you've agreed above that this is what Morgan does. Without this 'presumption of what Whitaker is NOT asking', there is nothing wrong with the Combining Doors solution. And many math professors, subsequent to Morgan, along with millions of Parade Magazine readers, do NOT share his presumptive certainty. Glkanter (talk) 09:49, 28 April 2009 (UTC)

Although there are plenty of ambiguities in the Parade version of the problem statement, that the question is about conditional probability is not universally agreed to be one of them (in fact, basically the opposite). The fact that people have a very hard time correctly solving conditional probability problems is very well known - conditional probability confuses people. -- Rick Block (talk) 00:46, 29 April 2009 (UTC)

How do you know what it was that Whitaker actually wanted to know? It is quite likely that he actually wanted to know if it was generally better to swap than stick (the unconditional problem) and by what margin. Martin Hogbin (talk) 21:43, 29 April 2009 (UTC)

The question is not how do I know what it was that Whitaker actually wanted to know. I don't claim to know - what I claim is that the sources that know what they're talking about say that the question clearly asks about the conditional probability. The question is why do you think it is likely that he actually wanted to know if it was generally better to swap than stick? Are there any references that discuss these two interpretations that support this interpretation or is it simply your personal opinion based on the number of sources that present an unconditional solution (without saying anything about what they think the question means)? Let me turn this around - if you wanted to unambiguously ask the conditional question how would you phrase it differently? -- Rick Block (talk) 04:23, 30 April 2009 (UTC)

The reliable sources that you are talking about are statistics experts they are not experts in understanding the intended meaning of a question posed by a member of the public in a popular magazine. Morgan et al changed the question in two stages, first by misquoting it then by restating it so that it was clearly conditional. They then proceeded to answer it on that basis. The first to question whether that approach was justified was Seymann in his comment at the end of the paper.

Krauss and Wang do not take such a strong line over conditional probability. Although they eventually do refer to Morgan's result most of their paper ignores the issue of conditional probability. Although they do not expressly say so, K&W give the strong impression, to me at least, that the conditional issue is unimportant.

Regarding the original question, most likely Whitaker did not consider whether he meant the question to be taken conditionally or not. As we know, if the host chooses randomly, it makes no difference. If I thought that it mattered which door the host opened, I would phrase the question along these lines, '...the host has opened door 3...what difference would it make to my chances of winning by switching if the host had opened door 2 rather than door 3?'. How would you answer that last question?

Finally, Glkanter and I are not claiming to know what Whitaker meant, we are just pointing out, as Seymann did, that nobody else can claim to know for certain either, no matter what their academic credentials. Martin Hogbin (talk) 08:27, 30 April 2009 (UTC)

First, Whitaker did not invent the MHP - his was a slight rephrasing of a problem originally posed by Steve Selvin, or more likely (based on how similar it is), a rephrasing of Nalebuff's version of Selvin's problem. The problem didn't originate from a "member of the public" but from academia (specifically, from a statistics professor!). I think I've mentioned this before, but your focus on what Whitaker meant is misplaced.

Second, I'm not just referring to Morgan et al., but in addition Gillman, and Grinstead and Snell. If this is not enough, there are numerous others - Falk is another (you might find her book illuminating, see for example [4]).

You admit Krauss and Wang do not say the conditional issue is unimportant. Their version of the problem is constructed so that the conditional and unconditional answers are the same (i.e. they explicitly include the constraint that the host pick randomly if the player has initially selected the car). This by no means implies the problem is not asking the conditional question. You still have not a single source that says anything remotely like "the standard MHP is intended to refer to the general probability of winning by switching regardless of what door the host opens as opposed to the conditional probability". Until you find a source that says something like this your claim that this is the case is simply WP:OR.

That the question is conditional does not imply there's necessarily a difference between the answers given that the host opens door 2 or door 3. I don't know, but it seems you're still confused about this. If the answer is 2/3 given any door the player initially picks and any door the host opens, this doesn't mean the question is asking about the unconditional probability. Do you get this? The question is clearly conditional. The answer may not vary based on the specific door the host opens (and certainly most people would expect this to be the case), but assuming this is true and proceeding to "solve" the problem unconditionally is entirely different from showing this is true based on the problem statement. -- Rick Block (talk) 14:35, 30 April 2009 (UTC)

You keep referring to the question. If you do not take Whitaker's question to be the definitive statement of the problem that only makes it even less well defined. The problem clearly can be stated in a way that makes it unconditional.

I am not the least bit confused about conditional probability. Even if the problem is stated in a way that makes it conditional, if the host chooses randomly it makes no difference. I have explained before that maths can be made as hard as you like by making things more rigorous but it is never taught this way because it would put people off before they had made any progress. It is exactly the same with this article, we should start by treating the problem unconditionally then, after that has been full explained and discussed, we can talk about the conditional treatment. Martin Hogbin (talk) 22:26, 30 April 2009 (UTC)

By the way, does anyone know where I can find a copy of Selvin's problem statement? Martin Hogbin (talk) 22:27, 30 April 2009 (UTC)

Selvin's problem statement is referenced in the article. Nalebuff's, too. We don't agree and aren't going to, but as long as you stick to references (I will as well) I think we'll be fine. -- Rick Block (talk) 03:13, 1 May 2009 (UTC)

That is fine with me, I have always made it clear that I wish the article's quality to be maintained. On the other hand a certain degree of boldness may be required.

I have followed the links in the article but they do not seem to lead to Selvin's problem statement. Martin Hogbin (talk) 16:59, 1 May 2009 (UTC)

They are references, not links. Selvin's problem statement is in the Feb 1975 edition of American Statistician (volume 29, number 1, on page 67). Nalebuff's is in the Autumn 1987 edition of Journal of Economic Perspectives (volume 1, number 2, page 157).

Boldness is fine as well - but you should really get consensus for structural changes beforehand. -- Rick Block (talk) 18:34, 1 May 2009 (UTC)

Of course, consensus (but, as per WP, no vote counting)! Just look out for Rick Block's veto hammer! Glkanter (talk) 17:12, 2 May 2009 (UTC)

Frequentist approach

It is easiest to see the limitations of Morgan's solution using the frequentist approach. WP says, 'Frequentists consider probability to be the relative frequency "in the long run" of outcomes'.

So what we need to do is repeat the show and take the average for all players who switch (which we all know gives the uncondional solution of 2/3). No, say Morgan, we are only to consider the case where the player has chosen a specific door (say door 1) and the host opens a specific door (say door 3).

So we repeat the show but only consider cases where the player chooses 1 and the host opens door 3. Does that give Morgan's solution. Not necessarily. When we repeat the show what do we do? Do we put the car in the same place every time? Of course not that would be silly.

Maybe we could consider the actual repeats of the show where the producers and hosts act as they normally would. Unfortunately that does not necessarily give Morgan's answer. The producer might not place the car behind door 1 1/3 of the time. We have to place the car randomly for every show.

So we need to repeat the show withe the car randomly placed each time but with the same host. Unfortunately the host might change his policy for choosing which door to open from time to time. We need to keep a host who always has exactly the same door opening probability.

So, Morgan's solution applies only to the case where we repeat the show considering only cases with the same doors being chosen by the player and the host, the cars are initially placed randomly every time but the host has a fixed policy for choosing which door to open. Does anyone seriously believe that this was what Whitaker wanted to know? Martin Hogbin (talk) 20:19, 26 April 2009 (UTC)

I'm completely speechless. What do you suggest? Let us simulate the show, and then properly. For simplicity: "abc" means chosen a, car behind b, opened c.

112 1000x chosen door 1
113 1000x chosen door 1, opened 3, this is one possibility: conditional 1/3
123 2000x chosen door 1, opened 3, this is another possibility: conditional 2/3
132 2000x chosen door 1
221 3000x
223 3000x
231 6000x
213 6000x
331 2000x
332 2000x
321 4000x
312 4000x 

Comment? Nijdam (talk) 20:34, 26 April 2009 (UTC)

I do not understand what you are saying. Martin Hogbin (talk) 23:13, 26 April 2009 (UTC)

I think Nijdam is suggesting the numbers would be what we would find (but don't obviously correspond to any particular simulation?). For example, in 6000 times where the player picked door 1, the car was behind door 1 and the host opened door 2 1000 times, the car was behind door 1 and the host opened door 3 1000 times, the car was behind door 2 and the host opened door 3 2000 times, the car was behind door 3 and the host opened door 2 2000 times. Comparing the 2 cases where the player picked door 1 and the host opened door 3 shows a 1/3 probability of winning by not switching and a 2/3 probability of winning by switching.

What Morgan would say is that yes indeed for each trial you need to record the player's pick of door, the door the host opens, and the outcome. This is how you would discover whether the host has a preference for a particular door or not (or, alternatively, if the host is supposed to be picking randomly if the player has initially picked the car this is how you would verify the host is actually doing this). If you collected all this data you would see a 2/3 overall probability of winning by switching (which I think we all agree with), BUT you might see that (for example) if the player initially picks door 2 the host tends to open door 1 and if the player picks door 1 or door 3 the host tends to open door 2 (assuming the host and player are standing on the stage to the left of door 1 and the doors are laid out 1-3 left to right this seems like an entirely reasonable possibility - why walk across the stage to open door 3 rather than open the first door the host reaches?). If all you measure is the overall wins by switching you won't (can't) determine this. Asking about the conditional probability doesn't mean we're restricting the outcomes to those where the player picked door 1 and the host opened door 3, but that we're looking at each of the six possibilities of player pick and door the host opens in isolation. If the host picks randomly when the player initially picks the car these will all be the same, and they'll all show a 2/3 chance of winning by switching. If the host doesn't pick randomly (and, in reality, even if he's supposed to he might not), then these can vary anywhere between 1/2 and 1. -- Rick Block (talk) 00:49, 27 April 2009 (UTC)

Rick, you say,' Asking about the conditional probability doesn't mean we're restricting the outcomes to those where the player picked door 1 and the host opened door 3'. Of course it does, that is what conditional probability means. To solve a conditional probability question we only consider those events where the condition is met.

If our condition is 'player picks door 1, host opens door 3' then we only consider events where that condition applies. We could have chosen to apply a different condition, for example, 'player picks door 2, host opens door 1' but to get a conditional probability we must pick the chosen condition and apply it. Martin Hogbin (talk) 18:38, 27 April 2009 (UTC)

Martin - we analyze a specific outcome (like, say, player picks door 1 and host opens door 3) as a representative sample. The result applies to any specific choice of player door and any door the host opens. If the host is not constrained to pick randomly when the player picks the car, the chance of winning by switching if the player picks door 1 and the host opens door 3 is between 1/2 and 1, but it is the same for any door the player picks and any door the host opens. Do you really not understand this? -- Rick Block (talk) 18:50, 27 April 2009 (UTC)

Yes I do understand that. The specific example that Morgan pick (based on Whitaker's question) is that the player picks door 1 and the host opens door 3. They make this perfectly clear in their paper. I am well aware that they could have chosen the case where the player picks door 2 and the host opens door 1, or indeed any other legal combination. In all cases the probability of winning by switching is between 1/2 and 1, but this probability it is not necessarily the same for each combination.

Morgan actually consider the one specific case that the player picks door 1 and the host opens door 3 and they calculate the conditional probability that the player wins by switching, given these specific doors, as 1/(1+q) where q is the probability that the host will open door 3 given that the car is behind door 1. Had the specific case that Morgan chose to consider been that the player picks door 1 and the host opens door 2, the the probability of winning by switching would have been 1/(1+p) where p=1-q. In general p is not equal to q.

So what does it mean to a frequentist to say that the probability of winning by switching is 1/(1+q)? It means that if you repeat the experiment many times, applying the given condition, the player will on average win 100/(1+q) percent of the time if they switch. Martin Hogbin (talk) 20:08, 27 April 2009 (UTC)

Do I take it that everyone now agrees with this? Martin Hogbin (talk) 08:26, 29 April 2009 (UTC)

Do you? And, if you do, how can you not see that Whitaker's question asks about a player who finds herself in one (and only one) of the six possible conditional situations - mentioning the (presumably randomly selected) door 1/door 3 combination in order to make this absolutely clear? -- Rick Block (talk) 13:45, 29 April 2009 (UTC)

This is the question that Morgan answer, I have never disputed that. Whether it is what Whitaker actually wanted to know is another matter.

Do you agree that, from a frequentist perspective, to get a probability other than 2/3 we need to repeat the experiment considering only cases with the same doors being chosen by the player and the host, the cars being initially placed randomly every time but the host having a fixed policy for choosing which door to open? Martin Hogbin (talk) 17:01, 29 April 2009 (UTC)

Do I take it that you all agree with the above statement or does anybody think that it is incorrect? Martin Hogbin (talk) 16:59, 18 May 2009 (UTC)

It is not completely clear what you mean. If in the experiment the car is placed randomly, the player's choice is independent of the position of the car and the policy of the host is the "random policy" then when the experiment is repeated, in 2 out of 3 cases where a specific door is chosen by the player and also a specific door is opened by the host, the car will be behind the "other" door. To get another answer, at least one of the condition has to be different, but changing a condition doesn't logically lead necessarily to another answer. So where are you heading? Nijdam (talk) 17:37, 18 May 2009 (UTC)

Yes, I say above that the host has a fixed policy for choosing which door to open. For example he might always open door 2 if possible or more generally he might choose door 3 with a fixed probability of q. My aim is to show the very limited circumstances, from a frequentist perspective, in which the Morgan solution applies. To get Morgan's answer you must have random car placement, random player initial choice of door but the same host door choice policy every time, and the player and host must always choose the same doors. Do you agree with this? Martin Hogbin (talk) 19:18, 18 May 2009 (UTC)

I just noticed that I have made two contradictory statements above. It should be that either the player chooses any door but we only consider cases where she has chosen a specific door or the player is constrained to always pick a specific door. Martin Hogbin (talk) 19:27, 18 May 2009 (UTC)

From a frequentist perspective, what is required is keeping track of which door the player picks and which door the host opens. So, we don't just count how many switchers win and lose and how many stickers win and lose, but are able to look at each of the 6 combinations of initial player pick and host door independently. Depending on the host strategy, the answer to the conditional question might depend on the specific combination of player pick and host door or it might not. For a frequentist, the host's strategy doesn't have to necessarily be constant (and it certainly doesn't have to be predefined or conscious) - just observable over a large number of trials. This is entirely consistent with Morgan et al.'s interpretation - their q could be derived from observation. And, yes, the observed chance of a player winning by switching who picks door 1 and sees the host open door 3 might be different from the chance given any other combination. However, if we derived the door1/door3 q from observation we would also see the chance of a player winning by switching who picks door 1 and sees the host open door 2 would be 1(1+(1-q)) as shown by Morgan et al. -- Rick Block (talk) 01:00, 19 May 2009 (UTC)

My point was about what Morgan's solution of Pr(Ws|D3)=1/(1+q) means from a frequentist point of view. To get this answer you need to consider only the case that Morgan do, namely that the player has chosen door 1 and the host has opened door 3. Obviously other combinations could be considered and these would give different answers, as you say, if the player chooses door 1 then Pr(Ws|D2)=1/(2-q).

The other point to note is that the car must be placed randomly each time but the host must always have the same probability of opening door 3 when the car is behind door 1. Do you agree with this? Martin Hogbin (talk) 17:21, 19 May 2009 (UTC)

The only part of this I agree with is the assumption that the car is placed randomly (which is as much a background rule as the "assumption" that the host must open a door and make the offer to switch - your contention that this is a deficiency of the solution presented by Morgan et al. is simply ludicrous). Morgan et al.'s solution applies for any initial pick of door and any door the host opens (through renumbering of doors). A frequentist could measure any specific player pick (say door 2 rather than door 1) and any specific door the host opens (say door 1 rather than door 3), but would in all likelihood measure all iterations of the game but keep track by door the player picks and door the host opens (all 6 combinations). Any specific combination is addressed by Morgan et al.'s result - as well as the 3 pairs of combinations each involving a specific initial player pick. A frequentist wouldn't care whether the host's strategy is constant, only what it averages out to over a large number of trials. Whatever this average is would provide the expectation for a player in the same situation as the previous players (the host's preference is revealed by what the host has historically done). -- Rick Block (talk) 18:46, 19 May 2009 (UTC)

I am well aware that Morgan might have considered any legal combination of player and host door pick with similar results but I am asking the question as to what the specific example that Morgan deal with means to a relativist. Morgan are quite clear as to the situation that they are considering, they say, 'To avoid any confusion, here is the situation: The player has chosen door 1, the host has then revealed a goat behind door 3, and the player is now offered the option to switch'. Morgan then go on to calculate the probability of winning by switching for this specific case as Pr(Ws|D3)=1/(1+q) where q is the probability that the host will pick door 3 if the car is behind door 1. For the moment, I am just asking what this solution to this specific question means to a frequentist. Perhaps it would be easier if you tell me what experiment you think would need to be repeated to get this specific result (that is to say just considering the case where the player picks door 1 and the host opens door 3). In other words, what exactly does the statement Pr(Ws|D3)=1/(1+q) (where q is the probability that the host will pick door 3 if the car is behind door 1) mean to a frequentist?Martin Hogbin (talk) 22:14, 19 May 2009 (UTC)

A frequentist reading Morgan et al. would realize q (or it's inverse p) applies per initial player pick, and to be able to answer any player's question about what that player's chances of winning are at the point of deciding whether to switch would endeavor to collect the information I've described above (about all combinations of player pick and door the host opens). For players who have picked door 1 the trials of critical interest are the ones where it turned out the car was behind door 1; the proportion of times the host opened door 3 vs. door 2 in this case is the basis for estimating q (and also its inverse p). There are 3 pairs of p and q values (one pair for each initial player pick). If a player who has picked door 1 and has seen the host open door 3 asks the frequentist what are her chances of winning by switching, the frequentist would look at the relevant data (all prior cases where the player has picked door 1 and car turned out to be behind door 1), would figure out q, and would say "your chances of winning are 1/(1+q)" (this would be the same answer that would be given by an incurious frequentist who might reply with the sum of the number of previous switchers who won plus the number of previous stickers who lost divided by the total number of players but counting only players who initially picked door 1 and were shown door 3 by the host). -- Rick Block (talk) 03:00, 20 May 2009 (UTC)

You seem to be answering a more general question than the one that I am asking. For example if I have a loaded (unfair) six-sided die and I say to a frequentist that the probability of throwing a 6 is 0.5 (Pr(6)=0.5) they would take it to mean that 'in the long run', that is to say if the experiment of throwing the specific die in question were to be repeated a large number of times, 1/2 of these times would result in a throw of 6.

Morgan say that, given that the player has chosen door 1 and the host has opened door 3, Pr(Ws|D3)=1/(1+q) where q is the probability that the host will open door 3 if the car is behind door 1. They go on to discuss specific cases where q=0, 1, and 1/2. Let us consider just the specific case that the player has chosen door 1, the host has opened door 3, and q=1, where Morgan show that Pr(Ws|D3)=1/2. My question is simply, 'Exactly what experiment do we need to repeat for the player to win 1/2 of the time?'. Martin Hogbin (talk) 08:32, 20 May 2009 (UTC)

Why do you ask something for which the answer is obvious? This is the much discussed "rightmost door preference host" (see, for example, the results of the simulation I posted some time ago [5]). Tell the host to open the rightmost door whenever possible. Play. Players who pick door 1 and see the host open door 3 (or, equivalently, players who pick door 2 and see the host open door 3 or players who pick door 3 and see the host open door 2) will win about 1/2 the time. In this case what would you say is the chance of winning by switching for a player who has picked door 1 and has seen the host open door 3? -- Rick Block (talk) 12:38, 20 May 2009 (UTC)

There is no dispute as to the the chance of winning by switching for a player who has picked door 1 and has seen the host open door 3 when q=1. I agree with you and Morgan that the answer is 1/2. The question was, 'Exactly what experiment you need to repeat to get this result?'. You state the obvious conditions, that we only consider games where the player picks door 1 and the host opens door 3. But we also need to state that, when the experiment is repeated, the car must be placed randomly each time, not be replaced where it was before. In other words it is no use placing the car randomly once then repeating the experiment with the car in that same position. No doubt you will say that this is obvious, but it needs to be made clear. On the other hand, to get an answer (Pr(Ws|D3)) of 1/2 we must not have a host with a different policy each time that we repeat the experiment, we must always have a host who has the same probability of choosing door 3 when the car is behind door 1. All these conditions must be made clear in a frequentist interpretation.

As a separate issue, you are wrong when you say '...(or, equivalently, players who pick door 2 and see the host open door 3 or players who pick door 3 and see the host open door 2) will win about 1/2 the time. I have only specified q (probability of choosing door 3 when the car is behind door 1) to be 1. This does not give us any information about the host's probability of choosing door 3 when the car is behind door 2 or probability of choosing door 2 when the car is behind door 3; these can still have any value from 0 to 1. Martin Hogbin (talk) 13:42, 20 May 2009 (UTC)

I said to tell the host to pick the rightmost door. This makes q=1 for door 3 in the case the player picks door 1 and also the other cases I mention as well (so they are equivalent). And for the umpteenth time, a frequentist doesn't care if the host's policy is constant, only that it have an average over a large sample (which it will whether it's constant or not). If we observe over 1000 trials that q=.75 it doesn't make any difference (to a frequentist) if the host alternates between q=.5 and q=1 or uses q=1 3 out of 4 times and q=0 the 4th time. Is there any particular point you're trying to make here? -- Rick Block (talk) 21:55, 20 May 2009 (UTC)

Of course, if you change the question you get a different answer. It is quite possible to consider any form of host door preference, for example the host may prefer odd numbered doors, or maybe the nearest door to where he happens to be standing at the time. Having decided on some kind of door policy for the host it is then possible to assign probabilities of various doors being opened in specific cases.

For some reason you seem unwilling to address the question that I asked, which is the only question that Morgan actually deal with in their paper. In my question, 'The player has chosen door 1, the host has then revealed a goat behind door 3, and the player is now offered the option to switch'. Given this question only, can you explain exactly what experiment would need to be repeated to get the solution Pr(Ws|D3)=1/(1+q)? Martin Hogbin (talk) 08:26, 21 May 2009 (UTC)

How have I not addressed this already? Tell the host to pick the rightmost door. Play. If you don't care to use the equivalence given this setup between the door 1/door3 combination and door2/door3 and door3/door2 combinations then play enough times to have a meaningful sample of the door1/door3 combination to look at. Another way is to simply play (without telling the host anything). When there is a meaningful sample of the door1/door3 combination figure out q based on the cases where the player picked door 1. Verify (for the subset of these cases where the host opened door 3) that Pr(Ws|D3)=1/(1+q). I'm not trying to be difficult here - I really don't have any idea what point you're driving at. If you don't like this answer then please drop the Socratic approach and just say what you're trying to say. -- Rick Block (talk) 14:46, 21 May 2009 (UTC)

Fine, your second answer at least addresses the question asked, but you have still missed out some essential details. How is the car placed behind the doors each time? Do we use a method of generating a random number from 1 to 3? Do we leave it to the whim of the producer or stage hand? Do we always replace it behind the same door?

What about the host? I presume we keep the same host throughout, who has the same policy on door choice. Do we ask the host to toss a coin? Are we allowed to change the host for one with a different policy? Martin Hogbin (talk) 19:34, 21 May 2009 (UTC)

A frequentist who wants to know the probability of X does X over and over again until there's a meaningful sample size. So, a frequentist who wants to know Pr(Ws|D3) has to count only occurrences when the player picks door 1 and the host opens door 3. Use whatever rules you want, but for the probability to be meaningful you need to be reasonably consistent. Morgan et al. use the same rules vos Savant did (e.g. random initial car placement), which pretty much match how you'd really run a game show. For the probability to be meaningful you clearly want to use the same host (letting the host "naturally" decide in the case where the player initially selects the car) or assign the host a controlled policy (like secretly flipping a coin to decide which door to open if the player has initially picked the car) - if you assign a policy the specific host wouldn't matter.

But, you know all this. So, what's your point? -- Rick Block (talk) 01:45, 22 May 2009 (UTC)

At last we agree. My point is that Morgans solution only applies in a rather contrived case. The producer must place the car randomly. For some unexplained reason we do not take it that the producer chooses to place the car behind behind door 1 with probability c1 etc we assume that this is done randomly so that C1=C2=C3=1/3. Please tell me on what basis this assumption is made.

The host, on the other hand, is taken to open a door (in any given set of circumstance) with a fixed probability. Please tell me why we not assume that the host, like the producer, chooses randomly.

The frequentist view, in my opinion, makes this inconsistency particularly obvious. Martin Hogbin (talk) 09:03, 22 May 2009 (UTC)

Ah, so this becomes the same point you're driving at below. Let's discuss it in one place (below). -- Rick Block (talk) 13:24, 22 May 2009 (UTC)

Error in Morgan et al?

Firstly, before anyone else tells me, this is OR. However this is not the article, it is not even the article talk page, it is the arguments page and it is certainly relevant to the MHP.

At the bottom of the first column of page 286, Morgan calculate the conditional probability of winning by switching given that the the player has chosen door 1 and the host has opened door 3 (Pr(Ws|D3)) on the basis of a uniform prior distribution of the host door opening strategy parameter q. In other words they consider the case that that the host initially has a equal probability of having any value of q from 0 to 1.

Morgan calculate Pr(Ws|D3) by integrating the probability of switching (1 / (1+ q)) over the interval 0 to 1.

Nijdam shows Morgan's calculation thus :

$\int _{0}^{1}{\tfrac {1}{1+q'}}f_{q}(q')dq'=\int _{0}^{1}{\tfrac {1}{1+q'}}dq'=ln(2).$

The problem that I see is that after door 3 has in fact been opened, which is the condition under which we are trying to calculate the probability, the distribution of q will no longer be uniform. Because we know that door 3 has been opened it is more likely that the host will have a value of q closer to 1 than otherwise. The prior distribution of q is uniform but we must use the posterior distribution, which is not.

Using some dubious mathematical reasoning, which I am happy to explain to anyone interested, I believe that the correct calculation of Pr(Ws|D3) is:

$1/(1+\int _{0}^{1}\ {f_{q}(q)}dq)=1/(1+\int _{0}^{1}\ {q}dq)=2/3$

So what is the relevance of this? It certainly weakens the case for Morgan to be considered the definitive article on the MHP but it also resolves a paradox in Morgan's result that had puzzled me.

If we take a uniform prior distribution of q this means that we also have a uniform distribution of p (= 1 - q). Morgan calculate Pr(Ws|D3)to be 0.693. By the same token we could calculate Pr(Ws|D2) to be 0.693. There is the paradox. Given some particular situation (that the prior distribution of q is uniform) the probability of winning by switching is 0.693 if the host opens door 3 or if the host opens door 2, yet we know that the probability of winning by switching given that the host opened either door 2 or door 3 is always 2/3 regardless of the hosts strategy. This is easy to verify given any single value for q. I know that probability can be surprising but I find this hard to believe. Morgan must be wrong.

Perhaps someone who know about these things would like to comment. Do you see a paradox in Morgan's result? Are Morgan et al wrong? Is my result correct? Martin Hogbin (talk) 14:07, 4 May 2009 (UTC)

I'm not an expert in Bayesian reasoning, but I think that after the host has opened door 3, information is revealed about q, hence the posterior density of q is 2/3·(2-q). It also reveals info about the position of the car, hence the conditional prob. of the car behind door 2 is: 1/(2-q). We can ask for the expectation of this prob. with respect to q (being log(2)), but also for the conditional expectation given door 3 opened leading to the answer 2/3 (numerically equal to your answer, but I don't understand your derivation). Further I don't follow your criticism, don't even know what you mean. Nijdam (talk) 22:21, 4 May 2009 (UTC)

We agree that information about q is revealed by the opening of door 3. Do you also agree that that to calculate Pr(Ws|D3) we need to use the posterior distribution of q? Are you saying that you get a numerical answer of 2/3 for Pr(Ws|D3) given a uniform prior distribution of q? If so, perhaps you could show be how you derive this.

Well, assuming P(D3|C1)=q, we know that P(C2|D3) = 1/(1+q) and f(q|D3)=2/3·(1+q). (I prefer to speak of C(ar) etc. rather than of Ws, which is not properly defined.) Hence its conditional expected value, given D3 is simply $\int {\tfrac {1}{1+q}}f(q|D3)dq={\tfrac {2}{3}}.$ Of course derived from the apriori uniform disribution of q. Nijdam (talk) 21:47, 5 May 2009 (UTC)

While I take a while to consider your derivation perhaps you could comment on the fact that you get a different numerical result from Morgan. Who is right? Martin Hogbin (talk) 08:32, 6 May 2009 (UTC)

As you will no doubt have concluded, I have no formal training in statistics. Could you please explain to me what f(q|D3) is. I presume your integral is to be evaluated from 0 to 1. Martin Hogbin (talk) 09:05, 6 May 2009 (UTC)

Yes, please explain this. I assume f(q) = 1/(1+q), but then what is the meaning of f(q|D3)? With regard to probability computations q is an unknown constant. -- Rick Block (talk) 14:12, 6 May 2009 (UTC)

Well, the prior density of P(D3|C1)=q is f(q)=1 on [0,1], after door 3 being opened (D3), the postrior density is f(q|D3)=2/3·(1+q). Could be written down more extensively of course.Nijdam (talk) 08:38, 7 May 2009 (UTC)

Regarding my criticism, let me try to explain what I mean in stages. We are considering the situation where there is a uniform prior distribution of q. Do you agree that in this case the prior distribution of p (=1-q) would also be uniform? Martin Hogbin (talk) 16:34, 5 May 2009 (UTC)

Of course. Nijdam (talk) 21:47, 5 May 2009 (UTC)

OK. Give the uniform prior distribution of q, what do Morgan calculate the probability of winning by switching to be given that the host has opened door 3?

Given exactly the same distribution of q, using Morgan's method, what would be the probability of winning by switching if the host has opened door 2? Martin Hogbin (talk) 08:16, 6 May 2009 (UTC)

ln(2) in both cases. -- Rick Block (talk) 14:12, 6 May 2009 (UTC)

Given the same distribution again, what would be the probability of winning by switching given that the host has opened one (unspecified) of the unchosen doors? Martin Hogbin (talk) 21:49, 6 May 2009 (UTC)

What information is revealed about q by opening door 3? The fact that the conditional probability, given a uniform prior distribution of p or q, is the same whether the host opens door 2 or door 3 is hardly surprising (since the only difference between doors 2 and 3 is their arbitrary door numbers). That it is not 2/3 reflects the nature of uniform distributions. That it is not the same as "the probability of winning by switching given the host opened either door 2 or door 3" is because it is a completely different question. Rather than "Morgan must be wrong" the correct conclusion would be that Martin does not understand. -- Rick Block (talk) 03:06, 5 May 2009 (UTC)

The fact that the host has opened door 3 means that it is more likely that the host has a preference for door 3, in other words the posterior distribution is biased towards 1. Martin Hogbin (talk) 16:34, 5 May 2009 (UTC)

[ Outindented ] Martin, careful, you are making a common beginner's mistake: confusing a parameter that a proposition depends on with the proposition itself. I used to recommend my students to always separate the two things using a different notation, e.g. uppercase for propositions and lowercase for parameters. I don't have the Morgan paper, but it's easy to reconstruct their argument from the info provided in the MHP article and here. I will use in the following the same notation style as in the "Bayesian analysis" section of the WP article.

Begin by adding the following assumptions to the background $I\,$:

(a) The host "prefers" to open one of the two doors that are available after the player chooses the door with a car behind.

(b) His preference is independent of the car's location.

(c) His preference is established before the game is played.

Now add to the model one additional proposition:

$Q:\,$ "The value of the host's preference for opening the door most to the right of the one chosen by the player is q."

where q is a parameter in $[0,1]\,$.

Then, using the notation of the "Bayesian analysis" section in the article, the likelihood $P(H_{ij}|C_{k},\,Q,\,I)$ for opening a door is written as:

$P(H_{ij}|C_{k},\,Q,\,I)\,\,=\,{\begin{cases}\,\\\,\\\,\\\,\end{cases}}$	$\,0\,$		if i = j, (the host cannot open the door picked by the player)
	$\,0\,$		if j = k, (the host cannot open a door with a car behind it)
	$\,q\,$		if i = k and j > i, (of the two doors with no car, the right one has preference q)
	$\,1-q\,$		if i = k and j < i, (of the two doors with no car, the left one has preference 1 - q)
	$\,1\,$		if i $\neq$k and j $\neq$ k, (there is only one door available to open)

Then, using the same argument as in the "Bayesian analysis" section, and noting that it is $P(C_{i}|Q,\,I)=P(C_{i}|I)\,$ because of assumption (b) above, we find that:

$P(C_{2}|H_{13},\,Q,\,I)={\frac {1}{1+q}}={\frac {P(C_{2},\,Q|H_{13},\,I)}{P(Q|H_{13},\,I)}}\,$,

where the last equality stems from the definition of conditional probability.

However, we are interested in $P(C_{2}|H_{13},\,I)$, the probability that the player wins by switching regardless of the host's preference. We can compute it by marginalizing the previous result with respect to all possible values of the nuisance parameter q:

$P(C_{2}|H_{13},\,I)=\int _{0}^{1}P(C_{2},\,Q|H_{13},\,I)\,{dq}=\int _{0}^{1}{\frac {1}{1+q}}\,P(Q|H_{13},\,I)\,{dq}\,.$

So here we come to Martin's issue: is $P(Q|H_{13},\,I)\,$ the same as $P(Q|I)\,$, the latter being the prior distribution for the parameter q? The answer is yes because of assumption (c): The host decides on the value q of his preference before the game starts, therefore $Q\,$ must be independent of $H_{13}\,$. Morgan's result of log(2) for the value of the marginal then follows by assuming a uniform prior $P(Q|I)=f(q)=1\,$. glopk (talk) 05:51, 7 May 2009 (UTC)

??????In your notation (I drop the superfluous I) I calculate the posterior density of Q given H13 as:

$P(Q|H_{13})={\frac {P(H_{13}|Q)P(Q)}{P(H_{13})}}=$

${\frac {P(H_{13}|C_{1},Q)P(C_{1}|Q)+P(H_{13}|C_{2},Q)P(C_{2}|Q)}{P(H_{13})}}=const\cdot (1+Q)={\tfrac {2}{3}}(1+Q)$

Nijdam (talk) 08:50, 7 May 2009 (UTC)

Nijdam - Isn't your denominator $\,P(H_{13})=P(H_{13}|C_{1})P(C_{1})+P(H_{13}|C_{2})P(C_{2})$, in which case $\,P(Q|H_{13})=1$? In English, $\,P(Q)$ is not q, but the probability that the host's preference is q - which is 1. -- Rick Block (talk) 14:43, 7 May 2009 (UTC)

No, Nijdam is right, and I confused causation with logical independence (and should really stop doing math past 11 pm). $P(Q|I)\,$ is the prior distribution for the parameter q, which is uniform (i.e., 1) in our case. More precisely, expanding $P(H_{13}|Q,I)\,$ as Nijdam does:

${\begin{aligned}P(H_{13}|Q,I)&=\sum _{i=1}^{3}P(H_{13},C_{i}|Q,I)=\sum _{i=1}^{3}P(H_{13}|C_{i},Q,I)\,P(C_{i}|Q,I)\\&=\sum _{i=1}^{3}P(H_{13}|C_{i},Q,I)\,P(C_{i}|I)=q\times {\tfrac {1}{3}}+1\times {\tfrac {1}{3}}+0\times {\tfrac {1}{3}}={\frac {q+1}{3}}\end{aligned}}$

we have:

$P(Q|H_{13},I)={\frac {P(H_{13}|Q,I)P(Q|I)}{\int _{0}^{1}P(H_{13}|Q,I)P(Q|I)dq}}={\frac {(q+1)\times 1}{\int _{0}^{1}(q+1)\times 1\,dq}}={\tfrac {2}{3}}(q+1)$

and plugging this back into my last expression for $P(C_{2}|H_{13},\,I)\,$ above:

$P(C_{2}|H_{13},\,I)=\int _{0}^{1}{\frac {1}{1+q}}\,P(Q|H_{13},\,I)\,{dq}\,.=\int _{0}^{1}{\frac {1}{1+q}}\times {\tfrac {2}{3}}(q+1)\,dq={\tfrac {2}{3}}$.

So, kudos to Nijdam, ashes for Morgan and glopk (talk) 16:22, 7 May 2009 (UTC)

So do we all agree now that Morgan have got the answer wrong? Martin Hogbin (talk) 19:33, 7 May 2009 (UTC)

No. I'm good with $\,P(H_{13}|Q,I)={\frac {q+1}{3}}$ but given the Bayes expansion of $\,P(Q|H_{13},I)={\frac {P(H_{13}|Q,I)P(Q|I)}{P(H_{13}|I)}}$ I think the denominator also works out to ${\frac {q+1}{3}}$ since $\,P(H_{13}|I)=P(H_{13}|C_{1},I)P(C_{1}|I)+P(H_{13}|C_{2},I)P(C_{2}|I)$. Again, $\,P(Q|I)$ (which I read as the probability that the host's preference is q) is an invariant constant equal to 1 (by assumptions b and c). On the other hand, I've cut down on my coffee lately and this isn't precisely my field. What am I missing here? -- Rick Block (talk) 19:39, 7 May 2009 (UTC)

Rick, the equations in my paragraph above answer the question "What is the player's chance of winning by switching, assuming ignorance of host's preference for one door versus another?". Seen in this light, the answer must be 2/3, since we fall back onto the classic interpretation of the MHP from the player's POV. Treating q as a nuisance parameter requires the marginalization (i.e. sum) of $P(H_{13}|I)\,$ over all possible values of q, each value weighted with $P(Q|I)\,$ (== 1 with our uniform prior). Try choosing a different prior for Q and see what happens to convince yourself. I don't have the Morgan paper, so I won't comment further on what exactly is computed there, and whether it is wrong or not (no ashes on Morgan yet). glopk (talk) 00:24, 8 May 2009 (UTC)

Morgan et al. show the probability of winning by switching, treating the host's preference for the door that has been opened as an unknown constant q (in the case the player has initially picked the car), is 1/(1+q). The paragraph in question is this:

“ The above solution engenders disappointment in some students, who feel that there should be a single answer for Pr(W_s|D3) not depending on unknown probabilities. This provides an excellent opportunity to bring in the Bayesian perspective. For instance, the noninformative prior in the vos Savant scenario [the normal MHP rules, but minus the restriction that the host pick randomly with equal probability in the case the player initially picks the car] makes this probability

$\int _{0}^{1}{\frac {1}{1+q}}dq=ln(2)\approx .693$

”

Perhaps this is what you mean, but we're assuming the host might have a preference for one door over another without knowing exactly what value this preference has (i.e. it is equally likely to be any value between 0 and 1). Q seems to be the nuisance here - why not just put q as part of the background information (it's a constant, not changed by the player pick or the door the host opens)? And, even using Q as a "proposition" (it's simply a true statement, so treating it as a proposition seems somewhat curious) why isn't using the Bayes expansion of the denominator valid? -- Rick Block (talk) 01:43, 8 May 2009 (UTC)

I will leave the maths experts to argue out the maths for the moment but I do not understand why you insist that the opening of a door by the host does not change the distribution of q. Of course this distribution is initially fixed, just like the players choice of door is, but in the conditional case, that is to say only considering cases where door 3 has in fact been opened, we have taken a non-representative sample from the original set, so the conditional distribution of q changes in just the same way that the probability that the player has chosen the car (sometimes) changes when the host opens a door. Martin Hogbin (talk) 22:22, 8 May 2009 (UTC)

In fact with a uniform prior distribution of q, if door 3 is opened, a value of q=1 then becomes twice as likely as q=0. If we assume the distribution of q remains linear and is normalized, we get Nijdam's posterior distribution of 2/3(1+p), which I now agree with. Martin Hogbin (talk) 23:26, 8 May 2009 (UTC)

We're talking about the conditional case where the host has opened door 3. In this case, the host has a preference q. I assume you're thinking that if q is 0 the host opens door 3 only if the car is behind door 2 and if q is 1 the host opens door 3 if the car is behind door 2 or door 1, so if the host has opened door 3 q is twice as likely to be 1 as 0. I guess the question is are we assuming a uniform distribution of q given the host has opened door 3, or a uniform distribution of q before the host opens door 3. I think this again gets back to the nature of the problem. At the point of the player's decision, it's given that the host has opened door 3. If at this point we assume we know nothing about q, then I'm not sure if we should take its distribution as uniform before or after the host has opened the door. If we take the distribution to be uniform after the host opens the door then I presume you agree with Morgan et al.'s result. If we assume q is uniformly distributed at the start of the game (before the host opens a door), then q is influencing which door the host opens so we can then use the fact that the host has opened door 3 to adjust our assumption about the distribution. I'm not sure which interpretation is more reasonable, although my inclination is to trust a peer reviewed paper. -- Rick Block (talk) 02:01, 9 May 2009 (UTC)

What you are saying is exactly my point, however there is absolutely no doubt that Morgan are considering a prior distribution of q. They say, 'for the noninformative prior...'. Any attempt by anyone, including the authors themselves, to claim any other meaning is farcical and would only server to discredit the claimant even further.

Regarding the issue of peer reviewed sources, I have always supported the use of reliable sources over OR in Wikipedia and for that reason I have kept my unsourced opinions off the article page. Authoritative opinion is not set in stone and can be changed. If the experts here can agree that there is an error in the Morgan paper I would like to ask someone of sufficient standing in the statistics community to support me in writing to the American Statistical Association with a view to getting a letter pointing out the error published. This may be a first for Wikipedia but I believe that it is fully in accordance with its policies and ethos. On refelection, I accept that my calculation may well be wrong and I understand Nijdam's calculation of the posterior distribution. The first thing that must be decided is whether we agree that there is an error in Morgan's paper. So far Nijdam and Glopkg seem to think that there is. Martin Hogbin (talk) 10:43, 9 May 2009 (UTC)

(outindented) I'm back, see you folks have been diligent with math. Let me put nit straight: I think Morgan didn't calculate the right probability. I gave you my calcuations in short, and in the more or less clumsy notation of glopk, in which happens to figure Q and q without some reasonable system. I write down the calculation in full. C is the door with the car, H the door opened by the host (I write for short C2 for C=2 etc.) and Q is the parameter such that:

$P(H3|C1,Q=q)=q$

Prior density of Q (uniform):

$f_{Q}(q)=1,q\in (0,1)$

Posterior density of Q, given H=3 (door 3 opened):

$f_{Q}(q|H3)={\frac {P(H3|Q=q)f_{Q}(q)}{P(H3)}}=$

$=const\cdot (P(H3|C1,Q=q)+P(H3|C2,Q=q)=$

$=const\cdot (q+1)={\tfrac {2}{3}}(1+q)$

Don't bother about the other terms, they're all in the (normalizing) constant. Then:

$\!\,P(C2|H3,Q=q)=$

$={\frac {P(H3|C2,Q=q)P(C2|Q=q)}{P(H3|C2,Q=q)P(C2|Q=q)+P(H3|C1,Q=q)P(C1|Q=q)}}=$

$={\tfrac {1}{1+q}}$

Hence, using the law of total probability:

$\!\,P(C2|H3)=\int P(C2|H3,Q=q)f_{Q}(q|H3)dq={\tfrac {2}{3}}.$

There is no logical or mathematical argument for integrating here with respect to the prior density of Q. It's an easy made error. On the other hand we may start with a different prior distribution and end with any number we like. That's the power of Bayesian analysis. Nijdam (talk) 23:46, 10 May 2009 (UTC)

Do you think that we should write to the American Statistician with this result? Martin Hogbin (talk) 08:20, 11 May 2009 (UTC)

Thanks for confirming the result. On the "clumsy" notation, I will just redirect you to to a list notable users of it[6] (and if you happen to be in that list and use a different notation, well, congratulations). On the difference between Q and q, as you are (by your own admission) unfamiliar with bayesian terminology, I will point to you that in frequentist (and Kolmogorov's measure-theory) P.T., Q would be called a continuous random variable, and q would be called one value of that random variable. Is the difference clear enough now, or do I have to repeat it very very slowly? glopk (talk) 16:12, 11 May 2009 (UTC)

Well. I wouldn't call it confirmation. And on the clumsyness, it mainly concerns the difference between Q and q. As you know so well the difference between Q and q, then why do you mix them up? Nijdam (talk) 16:32, 11 May 2009 (UTC)

If you say A, then I say A (admitting a mistake, and graciously confirming that A is correct), and then you repeat A, that is a confirmation too. But if you want to call it a potato, bon appetit. On the "mixing", can you please point to a specific equation I've written where the two symbols appear together improperly? Note that any symbol $P(\cdot )\,$ that contains $Q\,$ in the $(\cdot )\,$ is a function of q by the definition of the proposition (or random variable) Q. Is your $f_{Q}(q|H3)\,$ a case of bad "mixing" too? glopk (talk) 19:14, 11 May 2009 (UTC)

Morgan is still on the faculty of Virginia Tech (see [7]). I emailed him a while ago on a different topic and he did respond. BTW - you do realize this paper was published nearly 20 years ago? Before claiming this is a new result I would suggest checking to see whether it's already been published looking first at papers (and books) that cite Morgan et al. I haven't personally chased down every published reference to this paper. References (even just academic references) to the MHP are so voluminous that verifying this is a new result would actually take some time. -- Rick Block (talk) 13:57, 11 May 2009 (UTC)

Write Morgan and ask him if the error has already been reported and published.Nijdam (talk) 16:32, 11 May 2009 (UTC)

Am back too. Yes, given Rick's summary of that paper's paragraph, I do think it is in error, i.e. that it is wrong to marginalize the result using the wrong using the prior for $Q$. The simplest explanation is that if the player has to assume complete ignorance of the host's preference for one door when two are available to open, then he is in the same position as in the classic interpretation of the MHP. glopk (talk) 16:12, 11 May 2009 (UTC)

While you are at it, can anyone confirm that the same result applies to any prior distribution of q that is symmetrical about 1/2. My simple symmetry argument, which is what got me started on this, suggests that it should. Martin Hogbin (talk) 17:41, 11 May 2009 (UTC)

You're right, with symmetric prior we get for the posterior density of Q, given H=3 (door 3 opened):

$f_{Q}(q|H3)=const\cdot (q+1)f_{Q}(q)={\tfrac {2}{3}}(q+1)f_{Q}(q),$

because EQ=1/2, and hence const=2/3. Then, as:

$\!\,P(C2|H3,Q=q)={\tfrac {1}{1+q}}$

it follows:

$\!\,P(C2|H3)=\int P(C2|H3,Q=q)f_{Q}(q|H3)dq=\int {\tfrac {2}{3}}f_{Q}(q)={\tfrac {2}{3}}.$

Nijdam (talk) 20:38, 11 May 2009 (UTC) I forgot to mention that the only thing needed is the expectation of Q being 1/2.Nijdam (talk) 20:41, 11 May 2009 (UTC)

Or more complete, in general:

$f_{Q}(q|H3)=const\cdot (q+1)f_{Q}(q)={\frac {1+q}{1+EQ}}f_{Q}(q),$

and:

$\!\,P(C2|H3)={\frac {1}{1+EQ}}\left(={\frac {1}{1+\int qf_{Q}(q)dq}}\right).$

So the general form of your dubiously (your words) obtained formula. Nijdam (talk) 20:54, 11 May 2009 (UTC)

Well done and thank you. Martin Hogbin (talk) 21:41, 11 May 2009 (UTC)

Nijdam, I think we should write to the American Statistician with this result. Martin Hogbin (talk) 08:05, 12 May 2009 (UTC)

Isn't assuming a uniform prior before the host has opened a door effectively transforming this into the unconditional problem? I.e this makes the Bayesian computation the chance of winning regardless of which door the host opens (which everyone knows is 2/3) as opposed to the chance of winning given the host has opened door 3. BTW - when I contacted Professor Morgan some time ago he said he'd already said all he has to say about this problem. If you do want to pursue this I'd suggest you run it by some more folks from Wikipedia:WikiProject Mathematics. I am not an academic, but since this is a peer reviewed academic paper my understanding of the proper procedure to follow is to contact the journal. If the editor agrees the point has any merit whatsoever then presumably Morgan and his co-authors would be given an opportunity to formally reply. In an academic sense, agreeing here (which I don't) that this is an error means nothing at all. Furthermore, by Wikipedia policy (Wikipedia:No original research), even if this were in the article (which it's not) agreeing here that it's an error without a supporting published source would not mean the article should be changed. -- Rick Block (talk) 16:18, 16 May 2009 (UTC)

I have not read anything but the post by Rick Block at 16:18, 16 May 2009. I cannot imagine there is much benefit to writing the editors of a journal about this sort of thing. If there were general agreement on the talk page that there is an error in the source, we could simply choose not to use it. — Carl (CBM · talk) 17:14, 16 May 2009 (UTC)

@Rick No, I do not believe this is the same as the unconditional problem. We are considering the case where the host (or a population of hosts) has uniform prior distribution of q. That is to say the host has an equal probability of having any value of q from 0 to 1. We the only consider the conditional case that the host has in fact opened door 3. It turns out that the answer is the same as for the unconditional case, as you might expect from the symmetry of the situation.

I am happy to discuss this further on Wikipedia:WikiProject Mathematics. What is the best way to do this? I have no intention of changing the article based on my (and nijdam's) OR. We intend to write to the journal but even its letters policy is very strict. We may need some support from a more prominent figure in the statistics community to get a letter published. Martin Hogbin (talk) 21:15, 18 May 2009 (UTC)

I've already solicited comments at Wikipedia talk:WikiProject Mathematics#Allegation of error in a peer reviewed source (CBM's response above is likely in response to this). -- Rick Block (talk) 01:07, 19 May 2009 (UTC)

New section

@Martin: Ok.

BTW: The whole calculation comes down to (using Bayes and the rules of the game):

$P(C2|H3)={\frac {P(H3|C2)P(C2)}{P(H3|C2)P(C2)+P(H3|C1)P(C1)}}={\frac {1}{1+P(H3|C1)}}={\frac {1}{1+EQ}}$

Nijdam (talk) 15:20, 12 May 2009 (UTC)

I gather you are academic. I therefore suggest that you write something and email it to me at monty@hogbin.org Martin Hogbin (talk) 16:48, 12 May 2009 (UTC)

The above result is quite obvious when you think about it in the right way. It simply says that there is no difference between a single host with a particular value of q and several hosts with the same average value of q. Martin Hogbin (talk) 08:43, 22 May 2009 (UTC)

Well, here you have to be carefull, as Morgan's error shows. If hosts has some result w(q) depending on their parameter q, distributed as f(q), the average would be $\int w(q)f(q)dq$ and not w(Eq). Nijdam (talk) 16:20, 22 May 2009 (UTC)

Surely you have got that the wrong way round. Your formula above is w(Eq), if I understand your notation. Martin Hogbin (talk) 20:41, 22 May 2009 (UTC)

Sure is, that why your 'obvious' isn't that obvious. Nijdam (talk) 21:26, 22 May 2009 (UTC)

It is only obvious when you think of it he right way. I had this thought after after a conversation with Rick in another section. Say the car is behind door 1, this is the only time host door preference matters. Imagine one host with q=1/2. He will pick door 2 half the time and door 3 the other half. Now replace him with two hosts, one with q=1 (who will always pick door 1) and one with q=0 (who will never pick door 1) from which one is chosen randomly for each game. Door 3 will still be chosen 1/2 the time. If you could not see the hosts, there would be no way to tell that there were two hosts instead of one. In fact any combination of hosts that opened door 3 half the time would be indistinguishable from one host with q=1/2. Martin Hogbin (talk) 22:33, 22 May 2009 (UTC)

Why Morgan's error matters

It may seem that the error in the Morgan paper is only relevant to the special case of a prior uniform distribution of q but I believe that it has more important consequences for the validity of the paper.

Morgan's main error, in my opinion, is to inconsistently apply the principle of indifference. In the question that Morgan choose to consider, that asked by Whitaker, the car is originally placed behind one of the three doors by someone I will call the producer, a door is chosen by the player, and a door is opened by the host. In no case is the choice specified to be random and we are given no other information as to how the choice was made.

As we (and the player) have no information as to how the producer initially places the car, Morgan, quite reasonably, apply the principle of indifference and take it that the producer places the car randomly.

As the player has no information as to what is behind any of the doors, Morgan, quite reasonably, apply the principle of indifference and take it that the player chooses a door randomly.

As we (and the player) have no information as to how the host chooses which door to open, the principle of indifference should also be applied and we should assume the host chooses a door randomly (within the rules of the game).

With the above consistent assumptions, although it might still be argued that the probability is still conditional, there are many arguments to show that it does not matter which door the host chooses. It is a degenerate case, by which I mean it is a condition that makes no difference to the answer (probability of winning by switching), which is still 2/3.

What Morgan do is fail to apply the principle of indifference to the host choice and instead give it a parameter q. They then show that this leads to an answer other than 2/3. In order to add weight to this unexpected result they then show (wrongly) that even with a plausible prior distribution of host door choice parameter q the answer is still not 2/3. Martin Hogbin (talk) 15:04, 20 May 2009 (UTC)

Just a short comment. Whatever arguments you may use, the problem will always ask for a conditional probability to be calculated. As I far, far above remarked: as soon as an event has happened, the probabilities are conditioned. And the reliability of Morgan's paper is not jeopardised by this (minor) miscalculation. Nijdam (talk) 21:50, 20 May 2009 (UTC)

Perhaps we could drop this conditional/unconditional argument for the moment, that is not the basis of by complaint. The point is that, if you apply the principle of indifference to the host's door choice policy, the answer (probability of winning by switching) to the MHP becomes 2/3. If it makes you happy to insist that the problem is still conditional that is fine with me. I agree that there is a condition if a specific identified door is opened but it is a condition that makes no difference to the answer if the host chooses randomly . The point that I am making is that all this nonsense about q, the host door choice parameter disappears if the principle of indifference is applied to the host, conditional or not. Martin Hogbin (talk) 22:36, 20 May 2009 (UTC)

Nijdam, what is you explanation or justification for the fact that Morgan apply the principle of indifference to the action of the producer but not the host? Martin Hogbin (talk) 09:08, 22 May 2009 (UTC)

I wouldn't give much thought about this. To see the MHP as a probabilistic problem, it has to be modeled. In the original wording not all parameters are explicitly given. Yet it lies at hand to model the placing of the car as random. It doesn't bring much extra to introduce preference parameters. As for the strategy of the host, his preferences tell much about the conditional probability of winning by switching. It also makes clear the flaw in the simple reasoning. I would have done the same analyzing this problem.Nijdam (talk) 16:08, 22 May 2009 (UTC)

I have no problem with using the MHP as an example of conditional probability but if you are intending to use it in that way you should point out exactly the formulation you are using. There is nothing wrong with stating the problem with the usual rules and then adding, 'assume that the car is initially randomly placed but that he host's action is not necessarily random'. The MHP then becomes a good example of a the unexpected effect of a seemingly unimportant condition. This formulation, however, is not a natural assumption or a reasonable application of the principle of indifference and to not make the exact problem formulation that you wish to be answered clear at the start is unfair misdirection. To do what Morgan do and claim in a published paper that everyone else has got the problem wrong because they did not happen to agree with your particular unstated assumptions is not right either. Martin Hogbin (talk) 12:54, 23 May 2009 (UTC)

(continuing from the thread apparently about the same point, above) Martin - Morgan et al. analyze the same problem vos Savant did (including her clarifications in later columns, note in particular the experiment she suggested). She made it absolutely clear:

1) the car is initially randomly placed

2) the player randomly selects a door (before the host opens a door)

3) the host knows what is behind each door

4) the host always opens a door revealing a goat

5) the host always makes the offer to switch

Do you see anything in any of the Parade columns about the host's protocol for which door to open in the case the player initially selects the car? I don't. The entire set of columns is reproduced here [8]. Although they don't say it quite this directly, Morgan et al. are clearly using vos Savant's interpretation of the question; they are specifically not applying the principle of indifference to the action of the producer, but rather using vos Savant's assumptions. They acknowledge it would be possible to analyze the case where the car is not placed with equal probability behind each of the 3 doors. Why are you so insistent that this is an "error"?

The only mistake that vos Savant (who was writing for a popular general interest magazine) made was not to make clear in her initial explanation that she took the host to choose randomly. She later did make this assumption clear.

Morgan et al were writing a paper for a peer reviewed journal. They have no excuse whatever for not making their initial interpretation and assumptions perfectly clear. Nowhere in their paper do Morgan actually state that they take the car to be placed randomly, we are left to divine this for ourselves from a vague remark at the end of page 285 and a comment at the end of the paper stating that that other possibilities might be considered. To blame vos savant for Morgan's shortcomings is absurd.

Even if we conveniently assume that Morgan have incorporated vos Savant's initial stated assumptions in their problem formulation they chooses not to criticize them. They accept without comment the assumption that the car is randomly placed but then choose to take it that the host acts non-randomly, even though neither we nor the player can have any idea of the host's door opening policy.

In fact Morgan do make quite clear the question that they are attempting to answer; they (mis)quote Whitaker's original question at the start of their paper. They then restate the question later in the paper 'To avoid any confusion' . To not make clear their initial assumptions with a rationale for each is inexcusable in a scientific paper that claims to be the last word on the subject. On the other hand it is the only way to get an 'elegant solution'. Martin Hogbin (talk) 16:31, 22 May 2009 (UTC)

If this were a real game show, with the rules as above but not specifying the host's protocol in the case the player initially selects the car, I would actually be surprised if the host chose randomly in this case. The probability of winning the car after picking door 1 and seeing the host open door 3 is 1/(1+q₁) and it's 1/(1+p₁) if the host opens door 2, with q₁ + p₁ = 1 (with similar pairs of p₂/q₂ and p₃/q₃). Before going on the show I'd study the past episodes and try to determine if the host has a preference. If I had done this would you agree my chances of winning would be 1/(1+q₁)? Even if they hadn't done this, wouldn't the chances of winning for anyone who picked door 1 and sees the host open door 3 be 1/(1+q₁) (whether they realize it or not)? -- Rick Block (talk) 14:09, 22 May 2009 (UTC)

Real world advice for a prospective contestant on the show is another matter. Studying previous showings to see where the car was most often placed and what the host usually did would be a good idea. My guess would be that both the producer's and the host's door choices would be approximately random. The producer probably told a stage hand to 'place the car behind one of the doors' and the host probably made a quick arbitrary decision at the time, as he would have nothing to gain from doing anything else. The point that I am making is that both decisions were made by people in a manner we will probably never know. We have two sensible options: apply the principle of indifference to both choices and therefore take both to be random; take a more strict line that we have no information about either so we have to assume arbitrary probabilities for both. Unfortunately neither of these options makes a very interesting paper. Martin Hogbin (talk) 16:31, 22 May 2009 (UTC)

You continue to miss the entire point of the paper. It was not published in a vacuum, but in direct response to the nationwide controversy in the U.S. following the publication of vos Savant's columns. The authors were obviously using the same interpretation of the question as vos Savant - even referring to it as the "vos Savant scenario". In her third column she suggested an experiment using the words "Set up a probability trial exactly as outlined below" - if this is not clear I don't know what is. The point of the paper is that the question is conditional and that the probability of winning is therefore a function of the host's preference in the case the player initially selects the car (which is completely ignored by unconditional solutions). And, BTW, where exactly did vos Savant make it clear that she was assuming the host picked randomly (perhaps you're thinking of the letter she wrote in response to this paper). -- Rick Block (talk) 02:15, 23 May 2009 (UTC)

For some reason, you continue to defend the indefensible. Even if vos Savant had clearly stated that she took the producer to place the car randomly but the host to choose a door non-randomly and Morgan et al had stated that they were starting with these assumptions because they were those that vos Savant had used, Morgan's first obligation would have been to comment on the inconsistent assumptions and application of the principle of indifference made by vos Savant. This was a scientific paper intended for publication in a peer-reviewed journal. To attempt to blame its deficiencies on a popular media furore is bizarre.

Regarding vos Savant's statement that she took the host to choose randomly, I am referring to her statement that she took the host to be acting as the agent of chance. But vos Savant's assumptions are quite irrelevant, it is Morgan et al who professed to be writing the definitive paper on the subject; it is Morgan et al who should have made their initial assumptions clear.

In any case, regardless of how they got there, Morgan do in fact base their argument on an inconsistent application of the principle of indifference. If this principle is applied consistently their main argument disappears. Martin Hogbin (talk) 10:03, 23 May 2009 (UTC)

And, for some reason, you continue to miss the entire point of the paper which is that since the problem asks about a conditional probability an unconditional solution is not the correct approach (this point is supported by the Gillman, Grinstead and Snell, and Falk references in the article). The assumptions vos Savant made are exactly relevant because the paper is written as a criticism of her solution, which obligates them to solve the same problem she was solving. Her statement that she took the host to be acting as an agent of chance is in her response letter to the Morgan et al. article. Since we're simply repeating ourselves is there anything in particular you're trying to accomplish with this thread? If not, I'd suggest we agree to disagree. -- Rick Block (talk) 14:20, 23 May 2009 (UTC)

I really am staggered that you believe that it is alright for a peer-reviewed paper to be inconsistent in its approach to a probability problem just because it is responding to a magazine column, but if that really is your view then, yes, we will have to agree to disagree.

Once you treat the problem consistently the conditional solution gives the same answer as the unconditional one and, although the point could be made that the problem was still strictly speaking one of conditional probability, this would be a very uninteresting point and certainly one not worthy of a paper. Martin Hogbin (talk) 21:43, 23 May 2009 (UTC)

There is not such as a conditional solution and an unconditional one. It's merely a way of speaking. That's one of the big problems in the MHP: if you give the analysis in words, you easily get confused. There is the solution, and it is necessarily in terms of conditional probabilities. No principle of indifference of whatever can change this. And the paper of Morgan, for one, is to emphasize this. Nijdam (talk) 20:25, 24 May 2009 (UTC)

Let me make clear: Morgan doesn't (explicitly) mention the distribution of the placement of the car. It may be considered an omission, but in the "context" of the discussions that time, one could accept it as an implicit assumption. Nevertheless it would have been correct if Morgan had mentioned it. On the other hand it doesn't affect the issue of the paper: the conditional nature of the solution. Nijdam (talk) 20:08, 23 May 2009 (UTC)

My main criticism of Morgan is not that they do not state that they have taken the original car placement to be random, it is that they have taken the producer's actions to be random but the host's actions not to be. There is no justification for doing this, regardless of what vos Savant or anyone else did. Martin Hogbin (talk) 21:43, 23 May 2009 (UTC)

Starting from Morgan et al., it's a trivial extension to show that if the car is placed with probabilities c1, c2, c3 the chance of winning by switching if the player picks door 1 and the host opens door 3 is c2/(c2 + q*c1). The note at the end of the paper suggests this is of less interest. They could of course have expressed it in this form, although then your criticism would no doubt be that they solved a different problem than vos Savant. What is your reason for studiously ignoring Gillman who also says it's a conditional problem and the probability is 1/(1+q)? And Falk, who also says it's a conditional problem? And Grinstead and Snell, who also say it's a conditional problem? -- Rick Block (talk) 01:43, 24 May 2009 (UTC)

You seem to be trying to avoid accepting a simple error. In their published paper, Morgan et al apply the principle of indifference inconsistently. Nothing vos Savant did before the event and nothing you do after the event can change this. Morgan made a mistake. Martin Hogbin (talk) 08:45, 24 May 2009 (UTC)

(outindented) They made a mistake, in calculating the wrong posterior probabilities. They were sloppy in not mentioning the distribution of the car. But they certainly were not mistaken in varying the hosts strategy. In fact it is one of the merits of the article. Nijdam (talk) 20:32, 24 May 2009 (UTC)

Suppose that we start with Whitaker's question. Do you agree that neither the original distribution of the car nor the host's strategy is given in the question?

Assuming that you agree with the above, do you the accept that to answer the question it is therefore necessary to have some kind of rationale for dealing with the missing information? Martin Hogbin (talk) 22:35, 24 May 2009 (UTC)

Why start with Whitaker's question and not vos Savant's explicit experiment [9]:

One student plays the contestant, and another, the host. Label three paper cups #1, #2, and #3. While the contestant looks away, the host randomly hides a penny under a cup by throwing a die until a 1, 2, or 3 comes up. Next, the contestant randomly points to a cup by throwing a die the same way. Then the host purposely lifts up a losing cup from the two unchosen. Lastly, the contestant "stays" and lifts up his original cup to see if it covers the penny. Play "not switching" two hundred times and keep track of how often the contestant wins.

Clearly, the initial placement is random (as is the initial player choice). The host selection in the case the player initially picks the cup with the penny underneath is not mentioned (so could be anything). This version matches a game show where the contestant is told a car is randomly placed behind one of three doors but is not told anything about the host's strategy in the case the player initially selects the car (only that the host will always open a door and reveal a goat and make the offer to switch). The experiment as suggested measures the unconditional chance of winning, not the conditional chance given the player initially selected #1 and the host revealed #3. Unless you're arguing Whitaker's question asks the unconditional question (which I think you aren't) then I think you're agreeing vos Savant is not answering the appropriate question (which is the main point of the Morgan et al. paper). -- Rick Block (talk) 20:25, 25 May 2009 (UTC)

To answer your question. When you answer a question you start with the question, not someone else's attempt to answer it, especially if you think that attempt is flawed.

You seem determined to defend the Morgan paper at all costs. The primary purpose of the Morgan paper was not to attack vos Savant (or any of the others mentioned) but to give a correct solution to the problem that Whitaker asked, as Morgan themselves put it, 'at least partially put the problem to rest'. Surely you must agree with this?

So let me ask again, regardless of any other attempts including Morgan to answer Whitaker's question, do you the accept that it is necessary necessary to have some kind of rationale for dealing with the missing information? Martin Hogbin (talk) 21:32, 25 May 2009 (UTC)

The question is Whitaker's question, as interpreted by vos Savant - as extensively discussed following the publication of her columns. The rationale for dealing with the missing information is match the assumptions made by the popular discussions at the time (i.e. random initial car placement, host must make the offer to switch, host must reveal a goat). These are the assumptions nearly everyone made (and still makes). Morgan et al.'s contribution is the insight that the host's preference matters. Fully explicit versions (e.g. Krauss and Wang) now include the constraint that the host pick randomly if the player initially selects the car. Before Morgan et al. (and Gillman who you're still ignoring for some reason) pretty much no one specified this detail of the problem setup (although Martin Gardner's version of the Three Prisoners problem includes such a constraint). This is not a detail to be left to the principle of indifference. It is either a constraint on the host or it's not. In either case, the problem is a conditional problem and Morgan et al.'s solution is correct. -- Rick Block (talk) 22:36, 25 May 2009 (UTC)

If you can tell me where I can get a copy of the Gillman paper I would be happy to comment on it. I note, however, that it was published after Morgan and have assumed therefore that it just propagates Morgan's original error.

Are you saying that Morgan have intentionally chosen to answer the formulation where the car is placed randomly, the player chooses randomly but the host chooses non-randomly? There is nothing in the paper to indicate this. If this is indeed the case then perhaps somebody should publish a paper that actually does 'solve the problem at hand', which most people take to be Whitaker's actual question. Martin Hogbin (talk) 11:30, 26 May 2009 (UTC)

As I've mentioned before the Gillman paper is available (hard copy) in the 1992 American Mathematical Monthly volume 99, pp 3–7. First page preview is at jstor [10].

I do not have easy access to such things. I will try my local library to see if they can get a copy.

Yes, Morgan et al. intentionally chose to answer the formulation (widely discussed at the time) where the car is placed uniformly randomly, the player chooses randomly, the host's action in the case the player has initially chosen the car is not specified, and the specific probability of interest is the (conditional) probability of winning given the player has chosen door 1 and the host has opened door 3. What indicates this is the very first sentence of the Morgan et al. paper "In a trio of recent columns ..." which establishes the context as the problem vos Savant analyzed. Most people understand Whitaker's actual question is underspecified (for example doesn't say the host must make the offer to switch or that the host necessarily won't reveal the car) but make the reasonable assumptions. What most people don't understand is that because the question is about a conditional probability, the host's choice in the case the player has selected the car is critically important to the answer and using an unconditional solution technique masks this factor. -- Rick Block (talk) 13:51, 26 May 2009 (UTC)

Then perhaps somebody should attempt to answer the question that Whitaker actually asked. You say that the question is about a conditional probability, but that is only because Morgan set it up that way. If you take the host door choice to be random and you rephrase the original question as, '...the host, who knows what's behind the doors, opens another door, which has a goat', the essentially conditional nature of the problem disappears. Martin Hogbin (talk) 16:24, 26 May 2009 (UTC)

Martin, you took the words right out of my mouth.

Take the door #s out of Whitaker's question, which imho are there for example (clarity?) only. Even with the door #s, I would, and do, argue that Whitaker did not mean to ask specifically only about doors 1 and 3.

Recognize that the 'host behaviour' by definition of a game show, and by US law, cannot demonstrate any bias observable to the contestant.

Acknowledge the above statement regarding 'host behaviour' need not be stated as a premise, precisely for the 2 reasons given.

And there you have it, Morgan's paper holds no water. His statement that the unconditional solutions 'solve the wrong problem' is factually incorrect. Glkanter (talk) 19:14, 26 May 2009 (UTC)

Given the way Whitaker (and, originally, Selvin) phrased it, there is no doubt that the entire point of the question is to make the reader imagine the player is standing in front of two closed doors and one open door in a circumstance in which the location of the car is still unknown but the probability of the two doors at this point is not equal. Specific door numbers are used to make it absolutely clear that this is the problem (not to limit the result to these specific two doors, but using them as representative of any two specific doors). Martin's rephrasing is equivalent to the player knowing the rules of the game but picking whether to switch or stay before the host opens a door, which misses the entire point of the problem as does any unconditional so-called "solution". The problem is a conditional probability problem where the probability of interest is the probability given that the host has already opened a door, as opposed to the probability knowing that the host will open a door. That neither of you can apparently see that these are completely different questions does not alter that fact that they are indeed different questions. -- Rick Block (talk) 01:12, 27 May 2009 (UTC)

Rick, do you really believe that I do not know the difference between the conditional and unconditional formulations of this problem?

The point is that Whitaker's actual words are ambiguous. Morgan slightly changed them to make the question clearly conditional, I slightly changed them to make the problem unconditional. Why is their interpretation better than mine?

Of course, we can use our real world knowledge to imagine the likely scenario to which the question refers, but that is not what Morgan do, they refer to the 'stated conditional problem' and 'information not given in the problem'. It is quite clear that Morgan claim to be answering Whitaker's stated question. Once we move into the real world, we need to ask ourselves what the state of knowledge of the various characters is. As Glkanter has repeatedly stated, the player has no knowledge of the host's strategy and we must therefore take it that they are equally likely to open any legal door. Martin Hogbin (talk) 09:02, 27 May 2009 (UTC)

There is too much talking about conditional and unconditional. Rick has made it completely clear, but if anyone wants to know: unconditional is only the placement of the car: P(C1)=P(C2)=P(C3)=1/3 (or any other distribution if you like), the choice of the player P(H1), P(H2) and P(H3) (uniform??), independent (?) of the position of the car. What is here of interest? Nijdam (talk) 10:08, 27 May 2009 (UTC)

What is of interest is the way that Whitaker's original question is converted into a precise problem formulation. Apart from the game rules, which we all agree on, there are several other matters that must be decided before the question can be answered. The final problem formulation, and therefore the correct method of solution, depends on how we decide to deal with the information that is not provided in the original question. I would refer you again to the wise Prof Seymann's comments at the end of the Morgan paper.

If these matters are not of interest to you that is fine, perhaps you are only interested in the mathematics of the problem after it has been unambiguously formulated, but I am personally interested in the complete process from initial problem statement to final mathematical solution. Martin Hogbin (talk) 15:47, 27 May 2009 (UTC)

And, do you not agree that the paradox is that you're standing in front of two closed doors, don't exactly know where the car is, but the probability of the two doors you're looking at aren't equal? What Nijdam is saying is if you don't make it conditional, you're just putting a whole lot of extra words around "what is your chance of picking a car from behind three doors" - which is hardly paradoxical at all. -- Rick Block (talk) 01:22, 28 May 2009 (UTC)

I think you have completely misunderstood the MHP. Even when unambiguously stated as an unconditional problem, most people get it wrong. But this is a very interesting question that is very relevant to the article. What exactly is the Monty Hall problem. I think it is worth a new section just to at least understand how different people see it.

Okay, Martin, you made your position clear. But ... why al this criticism of Morgan. Morgan is, like me, mainly interested in the mathematical well-formed problem. And actually this well-formed problem is mostly addressed as the MHP. So if you want to follow the evolution of the problem, fine, but let's skip the mathematical discussion for the time being. Nijdam (talk) 11:38, 28 May 2009 (UTC)

I criticize Morgan because they set themselves up as the last word on the subject. That is to say, in my opinion, they claim to be providing the definitive solution to Whitaker's question, both the formulation and the maths. In my opinion they make serious errors in both. Martin Hogbin (talk) 18:34, 29 May 2009 (UTC)