Monty Hall Redux
How do you persuade yourself that a statement is true or an answer is correct? How do you persuade someone else? Various subcultures have evolved strategies and devices for dealing with these issues—scientific experiment, mathematical proof, trial by jury, consulting holy writ or the Guinness Book of Records. The trouble is, these methods are not perfectly reliable. Sometimes they convince people of a falsehood or fail to convince them of a truth.
In the July-August issue of American Scientist I reviewed Paul J. Nahin's Digital Dice: Computational Solutions to Practical Probability Problems, which advocates computer simulation as an additional way of establishing truth in at least one domain, that of probability calculations. To introduce the theme, I revisited the famous Monty Hall affair of 1990, in which a number of mathematicians and other smart people took opposite sides in a dispute over probabilities in the television game show Let's Make a Deal. (The game-show situation is explained at the end of this essay.) When I chose this example, I thought the controversy had faded away years ago, and that I could focus on methodology rather than outcome. Adopting Nahin's approach, I wrote a simple computer simulation and got the results I expected, supporting the view that switching doors in the game yields a two-thirds chance of winning.
But the controversy is not over. To my surprise, several readers took issue with my conclusion. (You can read many of their comments in their entirety here.) For example, Bruce Sampsell of Chapel Hill, N.C., wrote:
The initial statement of the problem as three doors with a prize randomly placed behind them does have a 1/3 probability of being the right choice prior to the opening of a door that doesn't contain a prize. Once a door is opened without the prize, the problem changes to two doors, each of which is equally likely to have the prize behind it. There is no advantage to switching doors.
It seems we are at an impasse. Neither the mathematical arguments given in my review nor the simulation results I reported there were enough to persuade Sampsell. On the other hand, Sampsell's assertion does not persuade me. How can we settle our differences? I could try further arguments and analysis; so could Sampsell. I could publish my simulation program so that others could run it for themselves; but my program could have an error or might be deliberately rigged to give the result I favor. There's an online version of the game created to accompany a John Tierney article in the New York Times; but in some quarters even the Times is not above suspicion.
Another correspondent, Ingrid Eisenstadter of New York City, put it this way:
So, it doesn't really matter that Mr. Hayes has written a program showing that you should switch doors after one has been eliminated, because it is based on the notion that one cannot reselect the original door, leaving arbitrary assumptions, word games and personal opinions. I would be happy to provide a similar computer program showing that the world is flat.
The mention of assumptions and word games introduces an important caveat. We can't expect to reach the same conclusions unless we all play by the same rules and understand the problem in the same way. A number of commentators—going back to the first wave of controversy in the early 1990s—have pointed out that certain assumptions are crucial to the analysis of the Monty Hall puzzle. In particular, it's important that Monty Hall must always open one door and offer the option of switching, and the door opened can never be the one initially chosen by the contestant, nor can it be the winning door.
Differences in the interpretation of the problem statement doubtless account for many of the continuing disputes over the Monty Hall puzzle. But there is a residuum of disagreement that is not so easily explained away. In another letter, David Lippmann of Austin, Texas, offers this analysis:
If Monty Hall always opens a door that does not conceal the prize and that the contestant has not chosen, there are only eight possible outcomes. Suppose that the doors are A, B and C and that the prize is behind C. The possible pairs of guesses are A followed by A, A followed by C, B followed by B, B followed by C, C followed by B, C followed by C, C followed by A, and C followed by C. Four of the second guesses are C, independent of what the first guess was. The probability that the contestant will win the prize is 1/2.
Lippmann and I apparently share the same assumptions about the game-show rules, and I even agree with his conclusion, in a limited sense: If the player chooses randomly whether to stay or switch, then the probability of winning is indeed 1/2. Nevertheless, I believe that Lippmann's analysis is incorrect, and a player who always switches doors has a two-thirds chance of winning. (The source of the error is that the eight cases do not all have the same probability. Monty Hall sometimes has two choices about which door to open, sometimes only one.)
The issue that concerns me here is not who is right and who is wrong about the odds of winning on Let's Make a Deal. The issue is how I can persuade anyone that my answer—or any particular answer—is correct. I have been stewing about this for several weeks, frustrated that I am powerless to communicate what I take to be a simple truth. But I've finally decided that what the episode demonstrates is the vigorous good health of the scientific enterprise.
Making progress in the sciences requires that we reach agreement about answers to questions, and then move on. Endless debate (think of global warming) is fruitless debate. In the Monty Hall case, this social process has actually worked quite well. A consensus has indeed been reached; the mathematical community at large has made up its mind and considers the matter settled. But consensus is not the same as unanimity, and dissenters should not be stifled. The fact is, when it comes to matters like Monty Hall, I'm not sufficiently skeptical. I know what answer I'm supposed to get, and I allow that to bias my thinking. It should be welcome news that a few others are willing to think for themselves and challenge the received doctrine. Even though they're wrong.
All the same, in the future I think I'll find some other example when I need to illustrate probability calculations.
The Monty Hall Probability Puzzle
The puzzle that has entered the lore of mathematics is not exactly the same as the game played on television. The situation in the mathematical puzzle is this: You are shown three doors and told that exactly one of them has a prize behind it. After you choose a door, Monty Hall (who knows where the prize is) opens one of the two unchosen doors, showing that the prize is not there. You are then offered a choice: Stick with your original door or switch to the remaining unopened door.