User:Martin Hogbin/Morgan Criticism

This page is to state my criticism of the paper by Morgan et al on the Monty Hall problem. Please do not modify the 'Martin's criticism' section but add brief comments in the 'Comments' section referring to the section numbers in the 'Martin's criticism' section. Please have longer discussions on the associated talk page.

Martin's criticism

The paper

The paper being referred to is: Morgan, J. P., Chaganty, N. R., Dahiya, R. C., & Doviak, M. J. (1991). "Let's make a deal: The player's dilemma," American Statistician 45: 284-287.

Just for brevity, I refer to the paper as simply 'Morgan'.

General

Why criticize this paper?

Before the publication of the paper by Morgan, the Monty Hall problem was a simple problem with a simple solution that nearly everybody got wrong and one that that Marilyn vos Savant answered correctly in her Parade column.

The Morgan paper changed all this by conjuring up an additional layer of pointless and obfuscating complication. As Rosenhouse put it in his article 'The Monty Hall Problem', they presumed to lay down the law regarding vos Savant's treatment of the problem'. After the Morgan paper, text books and other statisticians followed suit and the simple but incredibly unintuitive problem that most people got wrong became an obscure and complicated problem that most people could not understand.

This extra and unnecessary complication is required because Morgan insist that the problem must be treated as one of conditional probability. They claim that information is revealed about the probability that the player has initially chosen a car by the door that the host opens when he has a choice. This makes any solution more complicated that it otherwise need be.

Tone of the paper

I am not an academic but I have read many scientific papers in my career as a physicist, including ones that changed the way we see the world (such as Einstein's famous 1905 paper on relativity) but Morgan's paper is the most arrogant and patronizing that I have ever seen. I am not alone on this opinion, Rosenhouse refers to the paper as 'their bellicose and condescending essay' .

The paper starts, not by defining the problem clearly, not even by providing a solution, but with an attack on Marylin vos Savant who gave the correct answer to the question in her magazine feature earlier. Without further explanation it then goes on to describe vos Savant's and others' solutions as 'false'. Morgan et al (incorrectly, see below) refer to their own solution as 'an elegant solution that assumes no additional information' and its condescending tone continues with gems such as, 'That this problem so entertainingly demonstrates the pitfalls of conditional probability calculations and interpretations...'.

Misquotation of the problem statement

At the start of the paper Morgan describe the problem that they are addressing by quoting a question from the 'Ask Marylin' feature in Parade magazine. They give as a direct quote from the magazine:

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 are goats. You pick door No. 1, and the host, who knows what's behind them , opens No. 3, which has a goat. He then asks if you want to pick No 2. Should you switch?.

Other sources quote the same question 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, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Morgan give no explanation for this misquotation.

Apart from a few unimportant rewordings Morgan seem to have changed You pick a door, say No. 1 to You pick door No. 1 and more importantly, later on, they change the host,..., opens another door, say No. 3 to the host, ..., opens No. 3.

Significance of this misquotation

So, why does the misquotation matter? Morgan claim that the problem must be treated conditionally on the basis that a specific door is open by the host and the host's choice of door may give information as to the location of the door. For this argument to be valid the door opened by the host must be specified. Their version, the host, ..., opens No. 3 clearly does this but the original version, the host,..., opens another door, say No. 3 is rather more ambiguous. Were it, for example, to say simply, the host,..., opens another door the argument about information being revealed by which door was opened would be invalid as no door has been specified. The misquotation subtly biases the interpretation of the problem to a conditional one.

The question being asked

The original Parade problem statement was a letter from a member of the public to a popular, general interest magazine about a TV show. Much information is missing in the statement and this needs to be added for a solution to be provided. It seems most likely to me that the original questioner actually wanted to know whether, for a contestant on the TV show, it was better to swap doors or stick and by how much.

It is quite probable that the writer of the letter to 'Parade' actually wanted to to know the answer to the problem: You will be offered the choice of three doors, and after you chose the host will open a different door, revealing a goat. What is the probability that you win if your strategy is to switch. This is what Morgan describe in their paper as 'the unconditional problem' but not 'the problem at hand'. One might ask who is best qualified to interpret a question from a reader to regular column in a general interest magazine, a bunch of university professors or the regular host of the column to which the question was addressed, vos Savant.

In common with others, Morgan discuss the game rules and their possible effect on the solution and come to same conclusions that most others (including vos Savant) do, that the host always opens an unchosen door to reveal a goat and always offers the swap. These have become the standard game rules.

For some reason, Morgan then ignore the origin of the question and start to treat it like a question in a statistics exam. Rather than do as good statisticians should, and ask themselves what it was that the questioner actually wanted to know, they do not consider the player's state of knowledge in addressing the question but treat the question like a formal probability problem in which the questioner's statement is to be interpreted as literally as possible.

The significance of Morgan's treatment of the question

The problem that the original questioner most likely wanted answered was clearly an unconditional problem that had already been answered perfectly well by vos Savant.

The question being answered

After criticizing others for not answering the right question, Morgan then fail to make clear the exact question that they themselves are answering.

Initial distributions

Morgan do not make clear that they take the initial car distribution and player choice of door to be random. They also do not make clear that they take the host action to be non-random. From Bayesian perspective this is the same as not saying from which state of knowledge they are answering the question, a vitally important point. Choice of distribution is discussed in more detail below.

Door numbers

It is also not entirely clear from the Morgan paper whether their solution applies only to the specific door numbers quoted in the Whitaker question. Before giving their solution Morgan give their interpretation of the Whitaker question thus: 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.

Missing Host Knowledge

Morgan have also quietly dropped another important piece of information in their problem restatement. The original question had, '..the host, who knows what's behind them, opens Door 3...'. Why does the questioner tell us that the host knows what is behind the doors? This can only be to let us know that the host intentionally (and therefore always) reveals a goat. The omission allows Morgan to start by considering the more general case, in which the host may have chosen a goat door by chance. Not a bad idea but why not tell us this?

Morgan also seem to have overlooked the second letter from Selvin, who originally proposed the problem in the journal that Morgan published in, in which he makes clear that the host never reveals the car.

Significance of this lack of clarity

The Morgan argument proceeds on the basis that specific door numbers have been referred to in the question. This allows them to make the problem clearly one that is, strictly speaking, one of conditional probability.

On the other hand, we could take it that the original questioner was just giving door numbers as examples, purely for the purpose of clarity of explanation. In that case this should apply to all door numbers given. The question then becomes, The player has chosen a door, the host has then revealed a goat behind another door and the player is now offered the option to switch. This makes the problem one which might be regarded as unconditional or, at least, one where the conditional nature of the problem is insignificant.

This last interpretation is in accord with previous interpretations and solution of the problem, which are essentially that the Monty Hall problem is a simple puzzle which most people get wrong. It is also more directly equivalent to the three prisoners problem in this form.

The question that they have in fact answered

Morgan chose to interpret the original statement by Whitaker is a specific way. Although they do not make clear exactly what the question that they answered was, it is possible to work this out from their solution. The question that Morgan have answered would be along the lines of:

There is a game show in which there are three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show.

The player chooses a door randomly and, after the player has chosen a door, the door remains closed for the time being. The game show host, Monty Hall, then has to open any one of the two remaining doors and ask the player to decide whether they want to stay with their original choice or to switch to the remaining door.

Given only the above information and that the player has chosen door 1 and that the host has opened door 3. What is the probability of the player winning the car if they switch to door 2?

Conditions necessary to make the Morgan solution the only valid one

The full Morgan answer (the probability of winning by switching is 1/(1+q) is a valid answer to the above question but it still could be argued that we should take q to be 1/2 as we are given no information on its value in the problem statement. This, of course results in an answer (probability of winning by switching) of 2/3.

For the Morgan answer to be the only valid one we should add to the problem statement:

Given also that the host's probability of choosing door 3 when the car is behind door 1 is q

Some questions they have not answered

It is tempting to assume that the Morgan analysis applies to more general statements of the problem. In fact generalisation to anything other than the specific question stated above with those exact door numbers must be done with extreme care.

Consider the answer to this question:

There is a game show in which there are three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show.

The player chooses a door randomly and, after the player has chosen a door, the door remains closed for the time being. The game show host, Monty Hall, then has to open any door which hides a goat and ask the player to decide whether they want to stay with their original choice or to switch to the remaining door.

Given only the above information and that the player has chosen an identified door and that the host has opened an identified door and that the host has a known probability of opening a given door depending only on where the car is placed and which door the player initially chooses. What is the probability of the player winning the car if they switch doors?

Is the answer to this question dependent on the host's door preferences? It may surprise some people to know that it is not. It is always exactly 2/3, regardless of what policy the host might have regarding their choice of door when the player has originally chosen the car. This is all explained here.

Ways in which the question might be interpreted

So how else might the question be interpreted.

Without looking at the wording in detail

Before looking at the question wording in detail there a a couple of interpretations that do not require this

It is just a mathematical puzzle

The Monty Hall problem was known as a probability puzzle before Whitaker's question. Perhaps Whitaker was just trying to put the puzzle to vos Savant to see if she could answer it. In that case it would have been his undoubted intention to state the puzzle in its simplest form. Essentially the unconditional statement. Unfortunately the way he actually phrased the question meant that some people insisted that the problem was now conditional.

Mathematical puzzles are normally phrased to avoid any distracting complications. The problem was therefore intended to be phrased in its simplest (unconditional) form.

It was just an ordinary person asking a question to a popular magazine

The original question was not a question in a statistics examination, it was not a letter to a learned journal, it was not even a letter to a popular science magazine, it was a letter to a regular column by vos Savant in the popular general interest magazine 'Parade', which also contains items such as celebrity interviews and news on school sports. The question is presumed to have come from a member of the general public, with no special statistics knowledge or training. Studying every word of what he wrote is pointless. We must ask ourselves what was it that he actually wanted to know. Nobody could be better qualified to answer this question than vos Savant, and she clearly interpreted the question, with her own reasonable assumptions, in a non-conditional way. This might be how the question should be interpreted:

There is a game show, with these rules (and reasonable assumptions). Is it better to stick or switch, and by how much? Again an unconditional formulation.

The wording in detail

Here is the question again:

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

Suppose you're on a game show

What do these word mean? Suppose you are on a game show. This statement invites us to address the problem from the expected state of knowledge of a contestant on a game show. What would we expect this to be. There is no suggestion that the contestant has previously studied the show in detail. We would not expect the contestant to have any knowledge of where the car was initially placed, this would spoil the game completely. Similarly we would not expect the contestant to have any knowledge of the host's door choice policy. The best such a player could do would be to take both these things as uniform at random.

The host, who knows what's behind the doors...

Why are we told that the host knows what is behind the doors? The obvious intention of this wording is to let us know that the host is never going to reveal the car, as this would obviously spoil the show. There is no need to call this the 'vos Savant scenario', it is the obvious intent of the questioner.

...opens another door, say No. 3

Opens another door. This does not say 'opens door No 3'. It clearly means 'opens any one of the other doors, and just in case you are not clear what this means, an example of such a door might be door 3'. It is clearly not the intention of the questioner to consider specific numbered doors.

'You pick one of the doors and the host..., opens one of the other another doors'. Again this is an unconditional formulation.

Inconsistent choice of unstated distributions

Morgan claim to solve the problem using only information given in the problem. They do not do this as it is clearly impossible. There are three distributions that must be decided before a solution can be found; Morgan do this inconsistently. These distributions all relate to human decisions made at various stages of the game.

The choice of initial distributions is equivalent, in a more traditional (Bayesian) approach, to asking from whose state of knowledge are we answering the question. As stated above, the natural assumption, bearing in mind the words in the problem, Suppose you're on a game show..., would be that our answer is based on the likely state of knowledge of the contestant. They would not be expected to have any knowledge of either the initial car placement or the host's door opening policy.

Producer's initial car placement

At the start of the show, someone that I will call the producer places a car behind one of the doors. The problem statement gives us no information as to how this is done. For all we are told the producer could always place the car behind door 1. Morgan do not explicitly address this issue but it is obvious from their calculations that they take the car to be placed with equal probability behind any of the doors. This might be an application of the principle of indifference on the basis that neither we nor the player have any information on this distribution so we should take it as uniform at random. This is the same assumption that most others, including vos Savant, make.

It is reasonable to take it that the car is initially randomly placed but Morgan, who claim that they 'assume no additional information' should make clear this assumption and give a rationale for it. In fact Morgan make no mention of this assumption except at the end of the paper where, after all the criticism and calculation, they state that for those interested in studying the problem further, One possibility is... to allow non-uniform probabilities of assignment of the car to the three doors.

Player's choice of door

We are given no information as to how the player chooses a door. As it turns out this makes no difference to the solution in many cases but it would have been good practice for Morgan to have stated on what basis they took this choice to have been made. I assume that they took it as random by applying the principle of indifference again.

Host's choice of door

The problem statement does not tell us how the host chooses which door to open when the player has initially chosen the car. Exactly as in the case of the initial car placement, neither we nor the player have any basis on which to decide this distribution. Logically we might apply the principle of indifference again and take the host's choice to be random. This is exactly what vos Savant said that she had done but, for some inexplicable reason, it is not what Morgan do.

Morgan start by assuming a non-uniform distribution which they characterize using a parameter 'p' (and for some reason a redundant parameter q which equals 1-p). Having done this they are then able to conjure up an 'elegant solution' that the player has a chance of winning by switching of 1 /(1+q).

Finally they go on to consider the question that they should have started with, the special case that p=1/2, meaning that the host chooses uniformly at random (from the doors hiding goats), in other words the standard Monty Hall problem as answered correctly by vos Savant. Morgan thus make the standard problem a special case of their artificially obfuscated version.

It is puzzling that Morgan do not take the host's door choice, when the player has initially chosen the car, to be uniform, bearing in mind that years earlier, in the same journal that the Morgan paper was published in, Selvin who originally proposed the problem in a letter made clear in his second letter that the host would choose uniformly at random when he had a choice of door.

Significance of the inconsistent choice of distributions

There are two possible consistent assumptions that Morgan could have made; that all distributions were non-random or that all were random. In the former case the question becomes uninterestingly impossible, for all we know the car might always be placed behind door 2 and player might always choose door 1.

The only interesting and consistent assumption to make is that all the distributions are uniform, including the host's choice of door to open when the player has initially chosen the car. In this case, although it might be argued that the problem is still one of conditional probability there is a simple and obvious symmetry which makes it quite unnecessary to treat it in this way.

It is clear that Morgan are solving the problem from the state of knowledge of someone who has no idea of the initial placement of the car but who does know the host's policy for deciding which door to open. This is very unlikely to be the state of knowledge of a player or a member of the the audience, neither is it strictly based only on the information given in the question.

Error in calculating the probability of winning by switching given a uniform prior distribution of q

This is perhaps the most surprising error in the paper. Morgan criticize other solutions because they do not 'use the information in the door number shown'. On page 286 they attempt to calculate the probability of winning by switching given a noninformative (assumed uniform) prior distribution of their parameter q. (This is equivalent to assuming that the host has been randomly chosen from population of different hosts, each of whom has a fixed probability of opening a given door when the player has initially chosen a car. In other words each of the possible hosts has a fixed value of q and there are equal proportions of hosts having any given value of q) They calculate this probability to be ln(2), about 0.693. A little thought shows by simple symmetry that this answer cannot be correct. Astoundingly, Morgan have failed to use the information in the door number shown. Before the host has opened a door the distribution of q is uniform but once door 3 has, in fact, been opened the distribution of q changes to one in which it is more likely that door 3 will be opened. Morgan should have based their calculation on the posterior distribution of q. When this is done the result becomes the expected 2/3. Indeed it has the value 2/3 for any distribution of the parameter q which has the expectation value of 1/2. (When though about the right way this is quite obvious).

A description of the correct calculation is given here[1]. This calculation has now been published in the May 2010 issue of 'The American Statistician' as a letter (from Martin Hogbin and Nijdam) to the editor, see [2]. This confirms that Morgan did indeed make an error in their calculation, in fact they have thanked us for correcting their mistake.

Significance of this error

This error is significant because, before the Morgan paper, the probability of winning by switching was simply 2/3. Morgan claim to show that this answer is not always correct and could be the mysterious 1/(1+q). Calculating the answer in the case of a plausible distribution of host door choice parameter (q) to be other than 2/3 adds weight to this argument, whereas the correct answer in this case is the expected 2/3 again.

Comment by Prof Seymann

Rather unusually, the paper by Morgan et al is followed by a comment by Prof Richard G Seymann. This appears to me to be politely critical of the paper.

It talks about the difficulty of problem definition which it follows with this statement:

Without a clear understanding of the precise intent of the questioner, there can be no single correct solution to any problem. Thus, with respect to the three door problem, the answer is dependent on the assumptions one makes about the intent of the one who originally posed the question.

Although Seymann does not explicitly say so, I take this to mean that Morgan did not answer the question that the original questioner intended to ask, or at the very least Morgan have not considered the various possibilities.

Conclusion

By using a mixture of error, misquotation and inconsistency Morgan make the problem one that must be solved by using conditional probability. In doing so Morgan turn a simple well-defined mathematical puzzle that most people get wrong into an unexceptional undergraduate exercise in conditional probability. Let us consider the effect of changing their approach.

Apply the principle of indifference consistently

Doing this means that we must take the host choice to be random. This makes the problem arguably strictly conditional but with an obvious symmetry that makes the condition have no effect.

Take it that the questioner did not intend the door numbers to be significant

In other words take it that he meant the host,..., opens another door This is one of two interpretations of the question but one just as valid as Morgan's, where a door number is clearly specified.

This has the effect of removing the condition. At the start of the game the rules tell us that the host must open another door to reveal a goat. When he does just this, with no door being specified, there is no condition to consider. The problem can now be treated unconditionally.

Calculate the probability of winning by switching given a uniform prior distribution of q correctly

This gives us our expected answer of 2/3, which is inherent in the symmetry of the problem.

Interpret the question in a manner sympathetic to its origin

All Whitaker probably wanted to know was whether it was better strategy for a player on the game show to swap or stick and by what margin. This is clearly an unconditional problem and one that had been answered perfectly correctly by vos Savant.

Result

The problem now becomes what it always had been, a simple probability puzzle that most people get wrong.

What Morgan should have done

Morgan et al make a valid and interesting point, this is how they might have put it more plainly.

Their section 'To switch or not to switch' should have started with a statement explaining their approach, along the lines of, 'We will start by considering an interpretation of Whitaker's question referring to the case that the player is asked to choose after specific door numbers have been chosen but no information at all about the host behaviour is given. We will also assume that we have no prior knowledge of the game'.

It should have continued with, 'Because we are given no information about the distributions of the original placement of the car and players original choice of door we will assume these to be random'. They should also make clear at this stage that they take it that the host always offers the swap (although in the real game he never did). Referring to the distribution of the door opened by the host, they then might say, 'The host's behaviour is potentially more complex and of critical importance to the problem as it may depend on the location of the car. We will represent this by the parameters Pij etc'.

They could then go on to discuss the case where the host chooses any unchosen door randomly (including a clear description of what would happen if the host revealed a car) and show, as indeed they do, that in this case the probability of winning by switching is 1/2.

Next they could say, 'In the question we were told that the host knows what is behind the doors. We might take this to mean that he used this knowledge to never reveal a car, thus P22=P33=0 etc. We are given no information on the distribution of the host choice of door when the player has initially chosen the car, and indeed the player would not be expected to have any information on this distribution either, thus we should take the distribution of the host's choice to be random and set P12=P12=1/2, however we will start by considering the interesting possibility that this is not so (in other words we know that the host has a given preference for a particular door) thus P22 may have any value from 0 to 1'. They could then go on to demonstrate the interesting result that the player still cannot do worse by switching and note that this is a conditional probability problem. Finally they should set P12=P13=1/2, which might be properly called the vos Savant scenario, and confirm vos Savant's correct answer for the case that she clearly had in mind.

A more charitable view

Why did four professors of statistics write a paper with so much missing? The answer seems to me to be that they did not intend the paper to be altogether serious. It was essentially 'a bit of fun' or perhaps 'a novelty article'. After all, the paper is hardly groundbreaking research in the forefront of probability theory. Viewed in this light it all makes much more sense. It was intended to be a lighthearted look from a new angle at an infamous mathematical puzzle, poking a bit of gentle fun at previous 'attempts' to solve the problem.

The problem for Wikipedia is that it was published in a (lighter weight) peer-reviewed journal. This makes it a reliable source of the highest quality, whether the authors intended this or not. Unfortunately other sources, such as text books, took their lead from this paper, and Morgan's bit of fun took on a life of its own as it became the 'industry standard' solution. Let us see it for what it is, four academics having a bit of a laugh where the humour has maybe gone a little awry in places. In 1991 the problem did not need a peer-reviewed paper to solve it; this had been done long before.

Why the Monty Hall problem is not one of conditional probability

This section is not directed specifically at the Morgan paper but more generally against the argument that the Monty Hall problem is a problem of conditional probability. There are many different reasons why this is not so. I list and discuss them below.

What is meant by 'conditional' here?

By conditional here, I am referring to the claim made by some people that the specific door opened by the host must always be treated as a condition of the Monty Hall problem. In other words that we must consider it a condition of the problem which door the host opens.

I am not referring to the condition that the host opens a door to reveal a goat. Although this fact could be regarded as a condition, no one argues that it is necessary to treat this as a condition of the problem.

To put it anther way, where the problem is stated in such a way that we are to give a solution after the host has opened a door, I assert that the probability that the player has the car behind her chosen door after the host has opened a door to reveal a goat is trivially equal to the probability that the car was behind the chosen door before the host opened a door.

It is a mathematical puzzle

The problem originated from a letter by Selvin to the American Statistician journal. It was clearly intended to be a simple mathematical problem, along the lines of the 'Three prisoners problem', rather than a serious statistical question. The problem was later made famous by Whitaker's letter to 'Parade' and the subsequent solution by by vos Savant. Vos Savant sensibly treated the problem very simply, ignoring, or perhaps overlooking, the possibility that there might be a conditional element to the problem, in that the door number opened by the host could be significant. The letters that she received from her readers regarding her solution confirmed that her approach was correct, as they all argued that the answer (probability of winning by switching) was 1/2 and not 2/3. Nobody argued that the door opened by the host was of any significance.

Vos Savant treated the problem as a simple mathematical puzzle which most people would get wrong. In dealing with mathematical puzzles, there is a long tradition of not over-complicating the problem. Basically it is normal in such cases to make whatever assumptions are necessary to keep the problem simple. This is exactly what vos Savant did and it is the essence of the Monty Hall problem. Even if there are minor loopholes in the puzzle statement it is usual to overlook these.

Whitaker's question to 'Parade' is the definitive problem statement

Some people insist that, rather than take the problem to be a simple well-defined puzzle, we must address the question that Whitaker asked in 'parade', this is certainly the most notable problem statement, although the problem was originated by Selvin years before. Whitaker's actual words are given in Section 1.3 above.

Whitaker intended to ask the unconditional problem

What did Whitaker actually want to know? Did he want to know the probability of winning by switching given that the player has chosen door 1 and the host has opened door 3? Why on earth would anyone want to know the answer to this question? What about if the player has chosen door 2 etc? It is almost certain that he did not want that exact question answered, so what could he have wanted to know?

It is quite probable that all he wanted to know was the answer to the general, and clearly unconditional, question, For a player on the show just described, is it better policy to stick or swap, and by how much?.

Whitaker did not intend the door numbers to be significant

In his question, Whitaker says, 'You pick a door, say No. 1'. Does he mean to specify a particular door, as discussed above? This is most unlikely. Maybe he meant the numbers to be examples only and the player could have picked any of the three doors. This raises an interesting question of whether Whitaker intended the doors numbers to have any significance at all. Did he mean 'You pick a door, and I don't care which one, this is quite unimportant', or did he mean 'You pick a door specific door, and note its number, bearing in mind that this could be important', there is no way to tell. A similar discussion applies to the door opened by the host, which is, in fact, more important. Therefore, it is quite possible that Whitaker did not intend the door numbers to be significant. With no door numbers, which door is opened by the host cannot be a condition of the problem'.

The problem is symmetrical

Based on Whitaker's question, as explained above, whatever approach you take to the problem (Bayesian based on the player's likely state of knowledge, or modern probability theory with unspecified distributions) the only consistent ways to treat the problem are to assume that all unspecified distributions are non-uniform, in which case the answer is indeterminate, or to take it that they are all uniform, which of course means that, when the host has a choice of doors, he chooses evenly between them. Selvin's second letter to the American Statistician (which was published several years before the Morgan paper) and some later problem statements explicitly state that the host chooses randomly when he has a choice (there must be a reason for this). So long as the question is intended to specify significant door numbers, and ask for a solution after the host has opened a door, the problem can be treated conditionally, but must it be? There are several arguments that the problem can be properly treated unconditionally.

Random information is no information

Morgan in their paper state that (my emphasis), '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. Thus is the player having been given additional information, faced with a conditional probability problem. If the door to be opened is chosen randomly then the number of the door door opened does not give any information. Morgan's argument fails and the problem is not a conditional probability problem.

Just apply symmetry

Many mathematical problems can be difficult or even impossible to solve until a particular symmetry is noticed. In the Monty Hall problem in which the host opens a door randomly when he has a choice there is a simple and obvious symmetry in showing that whichever door the host opens the answer must be the same as there is nothing to distinguish between the two doors that he might open. The problem is symmetrical with respect to the door opened by the host. It is therefore quite obvious that the solution when the host opens door 3 is the same as the solution when the host opens door 2 is the same a the unconditional solution.

No paper or text book can ever insist that there is only one solution to any given mathematical problem

Many text books make statements along the lines of 'this is a problem of Euclidean geometry' or 'this is a problem of number theory', just as the Morgan paper says, 'Thus is the player ... faced with a conditional probability problem' but I have never seen any other books or papers which say that their solution is the only possible way in which the problem can be solved, indeed such a statement in a mathematical book or paper would be seen as one of extreme arrogance. How can the author possible know what other solutions might be possible. What statements of this kind normally mean is the problem can be solved or is easily solved by a given method.

Any claim that vos Savant's solution is 'false' is simply absurd. It fails in the contrived and unrealistic case in which the host chooses non-randomly, just as Pythagoras' theorem fails for non-right-angled triangles. But, in the only logical case, in which the host chooses randomly, it gives the correct answer, and not just by chance. Apart from the two reasons given above why the solution is correct, we might also try to find some other objective criteria to tell us whether her solution is valid. Does it generalise to more than three doors? Yes. Is it a special case of a more general solution? Yes, even Morgan admit this. It is therefore a valid solution to the special case of interest.

Other papers and text books talk about the Monty Hall problem as being a problem of conditional probability. I think that we should assume that these speak from the traditional mathematical viewpoint that conditional probability is a good way to approach the problem, especially in a more general case. Only one paper has the remarkable arrogance to call all solutions other than their own false. There is never any compulsion in mathematics to solve a more general problem than the one set.

Some 'false' arguments that the Monty Hall Problem must be conditional

The word 'false' is used in this title with humorous intent. Even in my user space it seems a little rude to call other people's opinions 'false'. This in not intended to cause offence but it does demonstrate how Morgan's use of the term in a published paper might have appeared to vos Savant. Apologies to anyone here who is offended.

Opening of a specific door by the host reduces the sample space so it must be a condition of the problem

This argument is essentially a tautology. The sample space is not provided by the problem itself, it must be set up by the user by including all events that are considered significant, that is to say might affect the probability of interest.

The sample space used by Morgan is AGG2 AGG3 GAG2 GAG3 GGA2 GGA3 (Where the letters indicate what is behind each of the three doors and the number indicates the door opened by the host, A=Auto G=Goat )

If it is given that the host opens door 3, this sample set reduces to AGG3 GAG3 GGA3, therefore the event that the host opens door 3 reduces the sample space and is therefore a necessary condition of the problem according to the argument.

It is perfectly possible to set up a sample space DAGG2 DAGG3 DGAG2 DGAG3 DGGA2 DGGA3 XAGG2 XAGG3 XGAG2 XGAG3 XGGA2 XGGA3 (Where the letter D indicates that the host says the word 'door' and X indicates that he does not use this word) If it is given that the host opens door 3 this reduces to: DAGG3 DGAG3 DGGA3 XAGG3 XGAG3 XGGA3

If it is also given that the host says the word 'door' the sample space is reduced further to: DAGG3 DGAG3 DGGA3

Therefore the event that the host says the word 'door' is a condition of the problem.

It is also possible to solve the symmetrical problem by setting up the sample space A G (consisting of only two events where the letter represents the item originally chosen by the player)

When the host opens door 3 the sample space remains the same and by using either of the arguments above the probabilities associated with each event remain the same. There are no conditions. The problem is unconditional.

The sample space is not defined for us in any statement of the problem, it up to the solver to set up the sample space to include only events that are significant and that might be conditions of the problem.

Opening of a specific door is mentioned in the problem statement

It is not clear that it is the intent of the questioner is to identify a specific door opened by the host but even if we take this to be the case it does not mean that we must take it as a condition. It is often possible to put up a case that an event mentioned in the problem statement must be taken as a condition. For example we are told that two of the doors hide goats. Maybe the host always opens the leftmost legal door if the booby prizes are goats and the rightmost legal door if they are sheep.

We cannot take every event mentioned in the problem statement to be a condition; we must use our judgment to decide which events to consider

Opening of a door is 'part of the experiment'

One of the few valid points that the Morgan paper makes is, 'The modeling of conditional probabilities through repeated experimentation can be a difficult concept for the novice, for whom careful thinking through of this situation can be of considerable benefit'. The crucial issue in solving this problem by experiment is the exact way the experiment is devised. There has been much discussion and disagreement in the article talk pages on what urn problem best represents the Monty Hall problem.

There is no precise 'experiment' defined by the problem statement. We must set one up depending on what we think the important issues in the problem are

Comments

Please put brief comments here and continue longer discussions on the talk page

This is less positive than it should be:

"Similarly we would not expect the contestant to have any knowledge of the host's door choice policy."

The expectation on a game show is that the contestant WILL NOT, CAN NOT, and DOES NOT have such knowledge. Glkanter (talk) 14:25, 13 January 2010 (UTC)

It could be argued, although I agree that this is not within the spirit of mathematical puzzles, that the player might have studied previous shows and found out that the host tends to choose the leftmost possible door, for example. Martin Hogbin (talk) 15:07, 14 January 2010 (UTC)

I don't think so. I provided a link to 'Whammy/Press Your Luck' with just such an instance. It was an aberration to the common understanding of a game show. And immediately resolved. Should every shell, urn or dice problem make considerations for shady operators? That would seem much more likely. I understand you're trying to be open-minded and inclusive here. It's just that those things simply do not happen on game shows. They get pulled off the air in advance, or the show gets re-taped. Nobody watching a game show has any thgoughts that the contestant is any more informed than what is explained in the rules of the game. There's no fine print that 'maybe viewers/contestants pick up host biases'. Glkanter (talk) 16:18, 14 January 2010 (UTC)

Yes,I agree with you but am just speaking in more conservative terms. Martin Hogbin (talk) 17:27, 15 January 2010 (UTC)

I really like this careful critique of Morgan et al. Splendid. I support your conclusions entirely. Most people (maths professors) I talk to don't know the Morgan et al. paper, have never heard of Morgan et al., (nor Selwyn for that matter) and the problem they call the Monty Hall problem is the problem stated by Marilyn vos Savant. The earlier history is not known. Most ordinary people I talk to (and I do occasionally talk to ordinary people) vaguely know the Marilyn vos Savant story, if they have heard of Monty Hall at all. Gill110951 (talk) 15:27, 2 February 2010 (UTC)

Thanks for you kind words. Any suggestions for improvement would be welcome. Martin Hogbin (talk) 16:34, 2 February 2010 (UTC)

Martin, please could you have a look to Morgan's error?  –  Am I wrong?  Kind regards, -- Gerhardvalentin (talk) 22:45, 15 September 2010 (UTC)

Martin, my comment is: this is useless, pointless, and sometimes erroneous criticism. And it does not help in getting a better understanding of the dispute. Nijdam (talk) 22:32, 10 August 2011 (UTC)

Solutions

As we discussed there is the "full MHP", which in my opinion is the formulation as most people will understand it, in which the contestant is offered the possibility to switch after the host has opened a door. This, as you admitted, is not solved by the simple solution. The simple solution reads: the chance is 1/3 you initially hit the car, hence when switching you hit the car with chance 2/3. Actually I do not know to which formulation of the MHP this is a solution. It is a true statement, and it shows what on the average the chance is to get the car for contestants who accept the offer and switch, but I do not know any acceptable formulation of the MHP to which this is a solution. So, please, can you formulate such a version of the MHP? Finally there is the erroneous so called "combined doors solution" (and equivalent variants) which is still in the article. Why didn't you remove them? Nijdam (talk) 22:48, 10 August 2011 (UTC)

Nijdam, the contestant is offered to switch to the host's second door after the host has opened one of his two doors showing a goat. Should the contestant stay or switch? That's the full MHP. And the "total solution" says she should switch in this one given game, this will give her the car in 2 out of 3. The "total solution" of A. Gnedin shows that, without using "given" or ungiven door numbers, and without using conditional probability, and even without using probability at all. Proven by A. Gnedin. Did you read it?

What is your point? Will you give her other advice than to switch? Can you proof that switching will hurt in this one given game? If you "suppose" to have additional information on the actual location of the car, then such supposition is without relevance in this one given game and to the decision asked for. You might be doing conditional probability calculi if you like. And you can show all of that maths to students of maths. But without effect to the MHP. Relevance? You will never be able to give her better advice than "always" to switch, i.e. to switch in "every game", i.e. to switch especially in this one given game the question is about. Did you read the sources? Regards, Gerhardvalentin (talk) 10:01, 12 August 2011 (UTC)

Gerhard, You do not understand much of logic, do you? Nijdam (talk) 14:06, 12 August 2011 (UTC)