What's the Deal?

What’s the Deal?

He is no mean prophet who can indifferently guess. But yet there wanted not some beams of light to guide men in the exercise of their stocastic faculty, even as to this also. That accommodation of religion, and all its concernments unto the humours, fancies, and conversations of men, wherewith some of late have pleased themselves, and laid snares for the ruin of others, did shrewdly portend, what in this attempt of the same party we were to expect.

John Owen, 1662

Suppose a guy in a carnival booth offers to play a shell game with you for a dollar. If you give him a dollar, he will hide a pea under one of three shells, and if you guess correctly which shell has the pea, you will win three dollars, otherwise you will get nothing. You agree to play, so you give him a dollar, he hides the pea, and you make your selection. Then, instead of turning over the shell you selected (as you expected him to do), the guy turns over one of the other shells, revealing that it doesn’t contain the pea, and offers to let you change your selection. Should you switch?

In this circumstance it would probably be foolish to switch, because the guy presumably wants to avoid having to pay you the three dollar prize (just as you presumably want to win the three dollar prize). If your original selection was wrong, he would simply have turned over the shell you selected, revealing that it was empty, thereby avoiding having to pay. This is how he makes his money. His only plausible motivation for offering you the unexpected opportunity to switch from your original choice is that your original choice was correct and he wants to entice you into switching. Therefore, if and when this sequence of events occurs, it’s reasonable to infer that your chances of winning by switching are essentially zero.

However, some uncertainty may arise if the circumstances are altered just slightly. For example, suppose the booth has just opened, and a crowd is just beginning to gather. Before the carnival guy carries out the game he announces to the crowd that everyone should play because it’s a very easy game and you are very likely to win. Then, exactly as in the previous scenario, he hides a pea under one of the shells and you make your selection. Then he (unexpectedly) overturns one of the other shells, revealing no pea, and offers to let you change your selection. This is the same sequence of events as in the previous scenario, but in this case it’s conceivable that the carnival guy may actually be trying to help you win, to encourage other people to play, thereby “priming the pump” for future earnings. If indeed the guy stands to gain by your winning (this time), it might make sense for you to switch if he offers.

In the first scenario we have reason to believe the carnival guy wants us to lose, so if he (unexpectedly) offers us a chance to change our selection, we should stay with our initial choice. In the second scenario we may suspect that he wants us to win (at least this time), so we might be well-advised to switch if he offers us the opportunity. But in other cases we may be unsure of the carnival guy’s motivation. In general we may assume he wants to maximize his income, but this may actually be achieved by sometimes helping players win, not only to encourage on-lookers about their chances, but also to make the game more fun. So it’s reasonable to believe the carnival guy would create some extra drama by sometimes (but certainly not always) offering the switch, and he might modulate his decisions based on how they affect his overall earnings.

Now suppose we leave the carnival and go to a television studio where a game show is being filmed. The host has hidden a prize behind one of three doors, and goats behind the other two, and asks us to choose a door. We point to one of the doors. Then the host opens one of the other two doors, revealing a goat, and asks if we want to switch to the remaining unopened door. Should we switch?

This is essentially identical to the previous scenario at the carnival, and the realistic answer again depends on the host’s motivations for offering us the switch. Of course, it goes without saying that if the host was required to always reveal a goat and offer a switch (i.e., if the host has no freedom of choice in his actions), then we should certainly switch, because in that case the probability that the remaining unopened door holds the prize is obviously 2/3. But no real game show would work that way, and the stated question does not impose that counter-factual restriction.

In the absence of any restriction on the host’s freedom to decide his actions, we are faced with a problem of game theory, trying to infer from the given information the actual (or most reasonable, or most plausible) motivations for the host. As discussed above, if this was a carnival worker just trying to make money from the game, and we discounted the possibility that he is using us to “prime the pump”, we would normally infer that he wants us to lose, and that he would offer a switch only when our initial guess was correct. But a game show doesn’t charge contestants for the privilege of playing, and doesn’t make money from the contestant, it makes money from attracting the most viewers (to maximize advertising revenue). It is in the business of producing an exciting and entertaining show by (sometimes) giving away prizes.

This suggests that the host is motivated to make the game as interesting as possible, and wants to avoid giving people obvious choices. For example, he certainly would not always offer a switch, since everyone would take the switch, giving them 2/3 probability of winning the prize. There would be little drama or interest in such a game, because neither the host nor the player really has any choices to make. The host would always be required to offer a switch, and the contestant’s best course of action would always be to take the switch. Everything is pre-ordained. In order for the game to be interesting, the host needs to act in such a way that, if and when a contestant is offered a switch, there is no consistent advantage to either switching or not switching. Thus it is at least plausible that the host of a game show wants to make the probability of winning the prize equal to 1/2 for either switching or not switching.

Interestingly, there exists a body of literature (albeit mostly in psychology journals) in which it is “proven” that it is not possible, by any strategy, for the host to accomplish this. The flaw in these “proofs” is that they consider only a very limited and artificially restricted range of strategies. It is actually quite easy to describe a simple strategy that accomplishes the intent, and it is arguably the strategy an actual game show host would use, although he might just apply it informally, doing the “card counting” (so to speak) in his head.

The simplest strategy would be for the host to offer a switch in every game when the contestant initially guesses right, and in half the games when the contestant initially guesses wrong. Using this strategy the host would offer a switch in 2/3 of the games. Another strategy would be for the host to offer a switch in 2/3 of the games when the contestant initially guesses right, and in 1/3 of the games when the contestant initially guesses wrong. (These contingent host choices could be made randomly on each game, or the host could alternate between helpful and hurtful offers, i.e., in a sequence of games, the host could wait for a game in which the contestant guesses right initially and offer a switch, then wait for a game in which the contestant guesses wrong initially and offer a switch, and so on. The sequence of helpful and hurtful offers could also be randomized by flipping a coin.) Using this strategy the host would offer a switch in 4/9 of the games. Still another strategy would be for the host to offer a switch in 3/4 of the games when the contestant initially guesses right, and in 3/8 of the games when the contestant initially guesses wrong. Using this strategy the host would offer a switch in 1/2 of the games. For each of these host strategies, when the contestant is offered a switch his probability of winning by switching would be 1/2. (The overall probability of winning, including the games in which a switch is not offered, would be 1/3.)

To formalize this, let us characterize a family of strategies by two fixed parameters, which we will denote by α and β, representing the probabilities that (or the fractions of games in which) the host will offer a switch when the contestant initially guesses right or wrong, respectively. Also let γ denote the (assumed fixed) probability that the contestant will switch when offered the opportunity. On this basis, the probability that the contestant will win the prize is 1/3 + γ(2β − α)/3. Thus if 2β exceeds α, the contestant should always switch, whereas if 2β is less than α the contestant should never switch. If 2β = α then the contestant’s overall probability of winning is 1/3 regardless of the value of γ, and the host offers a switch in (2/3)α of the games, and the probability of winning when offered a switch is [α + γ(2β − α)]/[α + 2β], which equals 1/2.

In 1990 a question from someone named Craig Whittaker appeared in the newspaper column of Marilyn Vos Savant (the world’s smartest woman) and provoked a flurry of irate comments and heated arguments which have continued to the present day in various venues. The question has come to be called “The Monty Hall Problem” (a name suggested by Steve Selvin when he proposed a similar question years earlier). The question presented in Vos Savant’s column in Parade magazine was

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?

Craig F. Whittaker, Columbia MD

This is essentially just the same question we considered previously. Of course there are many tacit assumptions the reader must make when considering a question like this. For example, we naturally assume the contestant prefers a car to a goat, the numbering of the doors is irrelevant, and so on. These sorts of natural assumptions are entirely justified. On the other hand, there is no justification for making the highly unnatural assumption that the host has no choice of actions. As noted above, making this assumption would trivially lead to the conclusion that contestants should always switch, giving them a probability of 2/3 to win the car. Such an assumption would be inconsistent with how any actual game show would operate. In fact, after reading Vos Savant’s column, Monty Hall participated in a demonstration, during which a contestant selected a door and Monty said “Too bad, you’ve won a goat.” The contestant objected "But you didn't open another door yet or give me a chance to switch", to which Monty replied "Where does it say I have to let you switch every time? I'm the master of the show.” In a later trial, the contestant initially guessed the door with the car, and this time Monty offered the switch, which the contestant took, resulting in another goat. Monty explained “I wanted to con you into switching there, because I knew the car was behind [the door you initially selected]. That's the kind of thing I can do when I'm in control of the game.”

Of course, Monty Hall’s approach to the game isn’t necessarily definitive, but in the absence of any explicit stipulation that the host of a hypothetical game show has no choice of actions, the reader is surely not justified in assuming it. Indeed we find anecdotal evidence that ordinary people confronted with this question are influenced by the possibility that Monty has a choice in his actions. For example, a woman named Anya, who decided the contestant should not switch, described her reasoning as follows: “You never know what Monty’s motivation is for revealing a door, so I would definitely stick with my first choice”. Surely many people base their answer on similar reasoning, perhaps without consciously realizing it, rather than making the unnatural and artificial assumption that the host has no freedom of choice.

Unfortunately, in her answer to Mr. Whittaker’s question, Vos Savant tacitly assumed the host has no free choice. Why did she make this assumption? Interestingly, she later wrote that

When I read the original question as it was sent by my reader [Whittaker], I felt it didn't emphasize enough that the host always opens a door with a goat behind it, so I added that to the answer to make sure everyone understood.

This is somewhat strange, for several reasons. First, although she says she added her assumption to her answer, she actually did not, as discussed below. Second, she says the question “didn’t emphasize enough” this assumption, whereas in fact the question did not mention or imply this condition (no host choice) at all. In fact, to the contrary, the problem statement asks us to consider a game show, and no game show would require the host to always do the same thing, resulting in a no-brainer for the contestants. Since Vos Savant recognized that Whittaker’s question did not imply that the host has no free choice, why did she assume the host has no free choice? Surely she should have answered the question that was asked, or else she should have modified Whittaker’s question to include the stipulation that the host has no free choice.

Admittedly when dealing with a puzzle columnist it’s unclear whether we should attribute the question to the columnist or to the reader who submitted it. If we attribute the question to Vos Savant, then the intent of the question is whatever Vos Savant says it was, but if we attribute the question to Mr. Whittaker, it is Vos Savant’s job to answer the question that was asked (not a different question of her own), and justify any assumptions that she has made in her answer.

The issue is even more clouded by that fact that, reportedly, the wording of the question as it appeared in the column was not identical to Whittaker’s original wording, meaning that Vos Savant edited the question herself. So, when she criticizes the wording of the question as it appeared in her column, who exactly is she criticizing? She had the opportunity to correct the statement of the problem if, as she later said, she did not think it clearly represented that question that she intended to answer.

Furthermore, as noted above, although Vos Savant says she clearly stated her assumption of no host choice in her answer (“to make sure everyone understood”), this is not true. Her answer to Whittaker’s question was

Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?

We can infer (or at least guess) from the stated probabilities (2/3 chance of winning by switching) that Vos Savant has assumed the trivial case in which the host has no freedom of choice and is required to always reveal a goat and offer a switch. However, her answer (including the million-door variant) does nothing to make this crucial assumption explicit, nor does it provide any justification for this assumption from the question that was asked, nor does she acknowledge that it is a highly unnatural and artificial assumption, and counter-factual to how any actual game show would operate. (Remember that Whittaker explicitly asked about a game show). It’s difficult to imagine what words in her answer Vos Savant thought provided the needed stipulation of what she assumed. Perhaps she is referring to the words “the host will always avoid the [door] with the prize”, but this would be a true statement regardless of whether or not the host always reveals a goat and offers a switch. What’s needed is a statement that she has assumed the host will always reveal a goat and offer a switch (and this should really be stipulated in the question). In a subsequent article, after the initial one had been severely criticized, Vos Savant did manage to smuggle in the required stipulation when she described the set of possible outcomes for the analogous shell and pea game. She wrote

I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you’ve chosen, we’ve learned nothing to allow us to revise the odds on the shell under your finger.

Thus only in follow-up article, and only with a two-word parenthetical remark “(and will)”, tucked inconspicuously and without justification into an analogy involving shells and peas, does Vos Savant finally explicitly acknowledge the counter-factual premise on which her answer is based. The best evidence that many of her readers didn’t understand the problem is the fact that so few of them pointed out that the innocent-looking “and will” totally changes the problem, and is inconsistent with the premise of the original question.

Is there any legitimate basis for arguing that Vos Savant’s assumption is somehow implicit in the question and/or the context? Some might argue that the assumption of no freedom of choice for the host leads to a more suitable question for this venue, in the sense that the answer with that stipulation can be derived by the rote application of simple probabilistic reasoning. In contrast, if we allow the host to have freedom of choice in his actions, the answer requires more sophisticated reasoning – although no more sophisticated than the kind we apply on a daily basis in real life. Making decisions often involves understanding the motivations of the actors involved in any situation. Of course, even for the simplistic “no host choice” question we tacitly base our answer on inferred motivations, such as the assumption that the contestant prefers a car to a goat. In the full version of the question, with the host allowed freedom of choice, we likewise must take the host’s motivations into account, but in that case the range of possible motivations is wider, and requires a more sophisticated evaluation.

It’s possible to invoke what might be called “test taker’s reasoning” here. If while taking a test we are confronted with a question that, interpreted literally, would require more time to answer than is allotted for the test, or would require an equivocal answer, we might infer that a simpler question was intended, and we will be credited by answering that simpler question, even though it is not the question that was actually asked. This kind of reasoning can be represented by a nested packaging of the question. For example, we can consider the following question:

Suppose you are reading Marilyn Vos Savant’s column in Parade magazine, and one of the questions is [fill in Whittaker’s game show question]. What will Vos Savant say is the answer to this question?

In this case we might agree that the answer is “switch with 2/3 advantage”, because we suspect that Vos Savant is likely to approach this question with simple-minded reasoning, by making the artificial assumption that the host has no freedom of choice. This is not to say that her answer is correct (it isn’t, for the question as stated), but merely that we can reasonably infer (or at least guess) that this is how she would answer the question, treating it as a trivial probably puzzle rather than as a more difficult problem in game theory and general reasoning.

In response to Vos Savant’s column many people objected to her answer, and many of those people argued that the real answer is that the contestant cannot infer any clear advantage to switching or not switching. We saw above that this is arguably the correct answer, since the host on a game show has free choice of actions, and presumably wants to present the players with uncertain choices (because a game in which the host always offers a switch and the contestant’s best answer is always to switch would be absurd). We described the simple host strategy to accomplish this, ensuring that when offered a switch the contestant has equal chance of winning the prize by switching or not switching. However, ironically, although most people get the right answer (no clear advantage to switch can be inferred), it is often claimed that they arrive at this answer by two compensating errors. First, it is claimed that most people make the same artificial assumption that Vos Savant made, i.e., they assume the host has no freedom of action. Second, it is claimed that, on this basis, most people incorrectly think the probability of winning when offered a switch is 1/2 for switching. So they arrive at what is arguably the right answer to the question that was asked, but for the wrong reason – or so it is argued.

Vos Savant published several letters critical of her answer, among them a letter from a mathematics professor (although not a specialist in probability or game theory), who wrote

You blew it! Let me explain: If one door is shown to be a loser, that information changes the probability of either remaining choice – neither of which has any reason to be more likely – to 1/2. As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and, in the future, being more careful.

The professor’s rationale consists of the single sentence “If one door is shown to be a loser, that information changes the probability of either remaining choice – neither of which has any reason to be more likely – to 1/2”. This is obviously a non-sequitur, but we can’t infer from his words whether he is assuming the host does or does not have any choice of actions, because his “reasoning” doesn’t invoke either assumption. His answer is consistent with host choice, but of course this doesn’t imply that his reasoning was validly based on host choice. It seems more likely that he was just clueless. Later the same professor said "I wrote [Vos Savant] another letter, telling her that after removing my foot from my mouth I'm now eating humble pie. It's been an intense professional embarrassment." Thus, whatever his initial “reasoning” may have been (if, indeed, he applied anything that can rightly be called reasoning), he disavowed it later and agreed with Vos Savant’s answer. Presumably he adopted Vos Savant’s unstated condition (no host choice), and was then persuaded that his answer under that condition was wrong. But he gives no hint of the justification for adopting that unstated condition, nor of any understanding of how that condition (or its absence) affects the answer.

In contrast, we noted above the case of Anya (who got a B in math, and is perhaps representative of the “general public’s lack of mathematical skills” lamented by the math professor), and who said she would stick with her first choice precisely because she doesn’t know Monty’s motivation for revealing a door. Some might claim that Anya’s reasoning is exceptional rather than typical. Vos Savant reported that, of the thousands of letters she received, only a few explicitly mentioned the “ambiguity” in the question. However, this doesn’t really tell us how many of those people were influenced (as Anya clearly was) by the possibility that Monty might exercise some choice.

There has been at least one published paper in a psychology journal (Krauss and Wang, 2003) arguing that most people when asked the question as posed in the Parade column assume Monty has no choice (even though the question doesn’t stipulate this unrealistic restriction), but the argument presented in the paper is utterly specious. They say

We argue that most of the criticisms of the standard version regarding its unstated assumptions are mathematically relevant [i.e., valid], but not psychologically relevant, because participants still assume the intended [sic] rules, even if those rules are not stated explicitly. Evidence supporting this claim comes from the observation that the vast majority of people wrongly [sic] regard the stay and switch choices as equally likely to result in winning. Let us give examples of how assumptions different from the intended [sic] ones would make this "uniformity belief" in the standard version impossible...”

They then consider three possible “non-standard” strategies, one in which Monty always opens the middle door if possible, one in which Monty reveals a goat only if the initial choice is the car, and one in which Monty’s placement of the car is unevenly distributed among the three doors. They remark that in each of these three cases the probability of switching or staying would not be equal, and then without further ado they announce that “In sum, when solving the standard version, in which the required assumptions are not made explicit, people seem to assume the intended scenario anyway.” Thus their argument is that if someone thinks the two remaining doors are equally likely to contain the prize, they must be assuming Monty has no choice. As explained above, this is untrue. Arguably the most plausible host strategy results in the contestant having equal probability of winning for switching or not switching (when a switch is offered). Therefore, the reasoning in the paper, supposedly proving that this is impossible, is specious. (This is reminiscent of J. S. Bell’s remark that “what is proved by impossibility proofs is lack of imagination''.)

The cited paper is unfortunately representative of the abysmal quality of the “literature” on this subject. Claims to have proven that people are not influenced by the possibility of Monty exercising choice are unfounded. (It might be interesting to write a psychology paper examining the mental processes of psychologists who have written papers examining the mental processes of people responding to the Monty Hall Problem.)

Having said all this, it’s undoubtedly true that many people, if asked the question with the clear stipulation that the host must always reveal a goat and offer a switch, will get the wrong answer. If asked to explain their reasoning, they might very well say something like “There are now two unopened doors, and each door is equally likely, so the probability is 1/2 for each door”. But this merely shows that most people are not good at word problems. They clearly have not even made use of the stipulated fact that the host must always reveal a goat and offer a switch. Most of them would give the same answer, with the same reasoning, with or without that stipulation. They don’t clearly grasp the implications of it, and may be influenced by a mixture of (perhaps mutually exclusive) considerations, including uncertainty about Monty’s motives. For example, here is how Anya’s husband described the deliberations in his household

If the game show host opened a door and asked you if you wanted to switch, you still don't know which door holds the car and which door holds the goat. Since there are only two doors left for consideration, you now have a fifty-fifty chance of guessing the correct door. After analyzing this far, my wife, Anya, who received a B in Technical Mathematics added: "Besides, you never know what motivates Monty to reveal a door in the actual game. I would definitely stick with my first choice."

Can we say that these people are assuming Monty has no choice? Clearly Anya is not assuming this, but it’s less clear whether her husband is assuming it. His “reasoning” is that after the host has opened a door and offered a switch “we still don’t know which door holds the car” – which is perfectly true – and then he says “since there are only two doors left, you now have a fifty-fifty chance”. Well, this obviously doesn’t necessarily follow purely from the fact that there are two doors left and we don’t know for sure which one holds the car. But it also does not follow that the husband is assuming Monty has no choice. Notice that the husband did not object to Anya’s comment referring to Monty’s motivation for revealing a door. If he understood that Monty has no choice, why didn’t he say “No Anya, Monty’s motivation is not involved, because he has no freedom of choice”? I would argue that he didn’t say this because he did not clearly rule out the possibility that Monty has some choice. Whether this influenced his judgment about the odds is unclear, but it’s certainly possible that he vaguely thought Monty was neutral in the sense that his actions result in about a fifty-fifty chance (which is actually true, as explained above for Monty’s optimum strategy), whereas Anya is slightly more cynical and leans toward thinking that Monty wants her to lose.

If they had been asked a question in which it was clearly stipulated that Monty has no choice, would they have correctly answered that you should switch? Ironically the paper of Krauss and Wang reported that, if people are asked a question in which Monty’s complete lack of choice was clearly stated, along with other clarifications, almost everyone answers that the contestant should switch in those circumstances. But in the same paper they claim that people answering the questions as posed in Parade are not influenced by uncertainty about the host’s freedom of action! A good check on the significance of such studies would be to determine how many people, after being indoctrinated into the “switching gives 2/3” answer to the question with no host choice, would then give the same answer to the carnival question at the beginning of this article.

Why does this subject lead to so much argument? I suspect it’s because there is more than one Monty Hall Problem, and there has developed a “battle of wills” between people who want to define what the real Monty Hall Problem is. The discussions have a certain Alice-in-Wonderland quality:

“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.” “The question is,” said Alice, “whether you can make words mean so many different things.” “The question is,” said Humpty Dumpty, “which is to be master—that’s all.”

Some people want to define the Monty Hall Problem as the simple probability puzzle that results from assuming the host has no choice. This puts it on the same level as Bertrand’s Box puzzle, the Boy or Girl puzzle, and so on. These people insist that the intended question includes the stipulation that the host has no choice, and the fact that this stipulation was absent from the actual question is of no importance. Some even go so far as to claim that anyone who raises the possibility that the host has a choice, or that readers may be responding at least in part (some perhaps without even realizing it) to the possibility that the host has a choice, is intentionally obfuscating for the basest of motives. An example of this attitude is found in a recent discussion, where a participant declared

I am convinced that most arguments about the rules stem from people who got the answer wrong (1/2) trying to justify their incorrect answer by pointing out possible ambiguities in the question.

Note the reference to 1/2 as “the incorrect answer”. On the other hand, some people believe the Monty Hall Problem refers to the question as it was actually posed in the Parade article, and they are interested in whether we can find the optimum answer to that question (as discussed at the beginning of this article). For example, Jan Schuller writes

The classic solution of the Monty Hall problem tacitly assumes that, after the candidate made his/her first choice, the host always allows the candidate to switch doors after he/she showed to the candidate a losing door, not initially chosen by the candidate. In view of actual TV shows, it seems a more credible assumption that the host will or will not allow switching. Under this assumption, possible strategies for the candidate are discussed, with respect to a minimax solution of the problem. In conclusion, the classic solution does not necessarily provide a good guidance for a candidate on a game show. It is discussed that the popularity of the problem is due to its incompleteness.

Still other people would define the Monty Hall Problem to be the overall set of issues highlighted by this episode, including the multiple distinct questions that have been posed, how the slight changes in wording can change the answer, the variety of approaches that can be taken, the distinction between probability puzzles and game theory problems, the misleading conclusions that can be drawn from inappropriate mathematical models and simulations, the application of ambiguous contextual information, the relevance of actual facts to hypothetical circumstances, the psychology of how ordinary people answer word problems, the test-takers strategy, and so on.

Another factor that seems to separate people in discussions of this subject is differences in the perceived difficulty of the probability puzzle with no host choice. To some people, the answer in this case is rather trivial and self-evident, similar to innumerable other elementary probability puzzles, and of no particular interest. To other people, this little puzzle is a uniquely profound conundrum, defying all human intuition, revealing a fundamental deficiency in the workings of the human mind. When these two types of people talk to each other about the Monty Hall Problem, there are bound to be deep divisions and mutual misunderstandings. Those who find the answer to the simple probability puzzle (no host choice) to be trivially obvious, and the assumption on which it’s based to be artificial, unrealistic and unjustified, tend to be more interested in the more challenging and realistic problem with host choice.

Of course, there are also people who genuinely have difficulty with the no-choice puzzle, just as they are fooled by other similar elementary probability puzzles, such as Bertrand’s Box puzzle and the Boy or Girl puzzle. Incidentally, Bertrand’s Box puzzle provides a good illustration of why we must distinguish between contingent facts and necessary facts in a problem statement. When we are told that someone reaches into a box and draws out a gold coin, this is understood to be a contingent fact, i.e., something that happened, not something that must happen. It’s also possible that the person could draw out a silver coin, in which case the trial would be excluded from the relevant sample space, which affects the probabilities. Likewise when we are told that the game show host offers us a switch, we cannot assume that this must always happen. We need to exclude from the sample space the cases when we are not offered a switch.

Another possible source of confusion is the odd fact that the arguably correct answer to the question in Vos Savant’s column happens to be the same as the most common incorrect answer to the question Vos Savant believes was intended by Mr. Whittaker. As a result, when someone says the answer to the question posed in Parade is that there is no clear advantage to switching doors, it’s easy to jump to the conclusion that they are giving the wrong answer to the question Vos Savant tacitly assumed was intended, rather than the (arguably) right answer to the question that was actually asked. The situation is complicated still further because most people aren’t mathematically sophisticated enough to clearly distinguish between the two questions, nor to articulate all the factors that influence their own reasoning in arriving at their answers.

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