| | Devlin's Angle |
Here, for the benefit of readers who have not previously encountered this puzzler, is what the fuss is all about.
In the 1960s, there was a popular weekly US television quiz show called Let's Make a Deal. Each week, at a certain point in the program, the host, Monty Hall, would present the contestant with three doors. Behind one door was a substantial prize; behind the others there was nothing. Monty asked the contestant to pick a door. Clearly, the chance of the contestant choosing the door with the prize was 1 in 3. So far so good.
Now comes the twist. Instead of simply opening the chosen door to reveal what lay behind, Monty would open one of the two doors the contestant had not chosen, revealing that it did not hide the prize. (Since Monty knew where the prize was, he could always do this.) He then offered the contestant the opportunity of either sticking with their original choice of door, or else switching it for the other unopened door.
The question now is, does it make any difference to the contestant's chances of winning to switch, or might they just as well stick with the door they have already chosen?
When they first meet this problem, most people think that it makes no difference if they switch. They reason like this: "There are two unopened doors. The prize is behind one of them. The probability that it is behind the one I picked is 1/2, the probability that it is behind the one I didn't is also 1/2, so it makes no difference if I switch."
Surprising though it seems at first, this reasoning is wrong. Switching actually DOUBLES the contestant's chance of winning. The odds go up from the original 1/3 for the chosen door, to 2/3 that the OTHER unopened door hides the prize.
There are several ways to explain what is going on here. Here is what I think is the simplest account.
Suppose the doors are labeled A, B, and C. Let's assume the contestant initially picks door A. The probability that the prize is behind door A is 1/3. That means that the probability it is behind one of the other two doors (B or C) is 2/3. Monty now opens one of the doors B and C to reveal that there is no prize there. Let's suppose he opens door C. Notice that he can always do this because he knows where the prize is located. (This piece of information is crucial, and is the key to the entire puzzle.) The contestant now has two relevant pieces of information:
1. The probability that the prize is behind door B or C (i.e., not behind door A) is 2/3.
2. The prize is not behind door C.
Combining these two pieces of information yields the conclusion that the probability that the prize is behind door B is 2/3.
Hence the contestant would be wise to switch from the original choice of door A (probability of winning 1/3) to door B (probability 2/3).
Now, experience tells me that if you haven't come across this problem before, there is a probability of at most 1 in 3 that the above explanation convinces you. So let me say a bit more for the benefit of the remaining 2/3 who believe I am just one sandwich short of a picnic (as one NPR listener delightfully put it).
The instinct that compels people to reject the above explanation is, I think, a deep rooted sense that probabilities are fixed. Since each door began with a 1/3 chance of hiding the prize, that does not change when Monty opens one door. But it is simply not true that events do not change probabilities. It is because the acquisition of information changes the probabilities associated with different choices that we often seek information prior to making an important decision. Acquiring more information about our options can reduce the number of possibilities and narrow the odds.
(Oddly enough, people who are convinced that Monty's action cannot change odds seem happy to go on to say that when it comes to making the switch or stick choice, the odds in favor of their previously chosen door are now 1/2, not the 1/3 they were at first. They usually justify this by saying that after Monty has opened his door, the contestant faces a new and quite different decision, independent of the initial choice of door. This reasoning is fallacious, but I'll pass on pursuing this inconsistency here.)
If Monty opened his door randomly, then indeed his action does not help the contestant, for whom it makes no difference to switch or to stick. But Monty's action is not random. He knows where the prize is, and acts on that knowledge. That injects a crucial piece of information into the situation. Information that the wise contestant can take advantage of to improve his or her odds of winning the grand prize. By opening his door, Monty is saying to the contestant "There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3. I'll help you by using my knowledge of where the prize is to open one of those two doors to show you that it does not hide the prize. You can now take advantage of this additional information. Your choice of door A has a chance of 1 in 3 of being the winner. I have not changed that. But by eliminating door C, I have shown you that the probability that door B hides the prize is 2 in 3."
Still not convinced? Some people who have trouble with the above explanation find it gets clearer when the problem is generalized to 100 doors. You choose one door. You will agree, I think, that you are likely to lose. The chances are highly likely (in fact 99/100) that the prize is behind one of the 99 remaining doors. Monty now opens 98 or those and none of them hides the prize. There are now just two remaining possibilities: either your initial choice was right or else the prize is behind the remaining door that you did not choose and Monty did not open. Now, you began by being pretty sure you had little chance of being right - just 1/100 in fact. Are you now saying that Monty's action of opening 98 doors to reveal no prize (carefully avoiding opening the door that hides the prize, if it is behind one of those 99) has increased to 1/2 your odds of winning with your original choice? Surely not. In which case, the odds are high - 99/100 to be exact - that the prize lies behind that one unchosen door that Monty did not open. You should definitely switch. You'd be crazy not to!
Okay, one last attempt at an explanation. Back to the three door version now. When Monty has opened one of the three doors and shown you there is no prize behind, and then offers you the opportunity to switch, he is in effect offering you a TWO-FOR-ONE switch. You originally picked door A. He is now saying "Would you like to swap door A for TWO doors, B and C ... Oh, and by the way, before you make this two-for-one swap I'll open one of those two doors for you (one without a prize behind it)."
In effect, then, when Monty opens door C, the attractive 2/3 odds that the prize is behind door B or C are shifted to door B alone.
So much for the explanations. Far more fascinating than the mathematics, to my mind, is the psychology that goes along with the problem. Not only do many people get the wrong answer initially (believing that switching makes no difference), but a substantial proportion of them are unable to escape from their initial confusion and grasp any of the different explanations that are available (some of which I gave above).
On those occasions when I have entered into some correspondence with readers or listeners, I have always prefaced my explanations and comments by observing that this problem is notoriously problematic, that it has been used for years as a standard example in university probability courses to demonstrate how easily we can be misled about probabilities, and that it is important to pay attention to every aspect of the way Monty presents the challenge. Nevertheless, I regularly encounter people who are unable to break free of their initial conception of the problem, and thus unable to follow any of the explanations of the correct answer.
Indeed, some individuals I have encountered are so convinced that their (faulty) reasoning is correct that when you try to explain where they are going wrong, they become passionate, sometimes angry, and occasionally even abusive. Abusive over a math problem? Why is it that some people feel that their ability to compute a game show probability is something so important that they become passionately attached to their reasoning, and resist all attempts to explain what is going on? On a human level, what exactly is going on here?
First, it has to be said that the game scenario is a very cunning one, cleverly designed to lead the unsuspecting player astray. It gives the impression that, after Monty has opened one door, the contestant is being offered a choice between two doors, each of which is equally likely to lead to the prize. That would be the case if nothing had occurred to give the contestant new information. But Monty's opening of a door does yield new information. That new information is primarily about the two doors not chosen. Hence the two unopened doors that the contestant faces at the end are not equally likely. They have different histories. And those different histories lead to different probabilities.
That explains why very smart people, including many good mathematicians when they first encounter the problem, are misled. But why the passion with which many continue to hold on to their false conclusion? I have not encountered such a reaction when I have corrected students' mistakes in algebra or calculus.
I think the reason the Monty Hall problem raises people's ire is because a basic ability to estimate likelihoods of events is important in everyday life. We make (loose, and generally non-numeric) probability estimates all the time. Our ability to do this says something about our rationality - our capacity to live a successful life - and hence can become a matter of pride, something to be defended.
The human brain did not evolve to calculate mathematical probabilities, but it did evolve to ensure our survival. A highly successful survival strategy throughout human evolutionary history, and today, is to base decisions on the immediate past and on the evidence immediately to hand. If that movement in the undergrowth looks as though it might be caused by a hungry tiger, the smart move is to make a hasty retreat. Regardless of the fact that you haven't seen a tiger in that vicinity for several years, or that when you saw a similar rustle yesterday it turned out to be a gazelle. Again, if a certain company stock has been rising steadily for the past week, we may be tempted to buy, regardless of its stormy performance over the previous year. By presenting contestants with an actual situation in which a choice has to be made, Monty Hall tacitly encouraged people to use their everyday reasoning strategies, not the mathematical reasoning that in this case is required to get you to the right answer.
Monty Hall contestants are, therefore, likely to ignore the first part of the challenge and concentrate on the task facing them after Monty has opened the door. They see the task as choosing between two doors - period. And for choosing between two doors, with no additional circumstances, the probabilities are 1/2 for each. In the case of the Monty Hall problem, however, the outcome is that a normally successful human decision making strategy leads you astray.
Finally, just to see how well you have done on this teaser, suppose you are playing a seven door version of the game. You choose three doors. Monty now opens three of the remaining doors to show you that there is no prize behind it. He then says, "Would you like to stick with the three doors you have chosen, or would you prefer to swap them for the one other door I have not opened?" What do you do? Do you stick with your three doors or do you make the 3 for 1 swap he is offering?