My account of the big creationism conference will resume shortly, but I really must take time out to discuss this article by Brian Hayes of American Scientist. He is discussing the Monty Hall problem, you see.
The story begins with this earlier article by Hayes. He was reviewing the recent book Digital Dice: Computational Solutions to Practical Probability Problems, by Paul Nahin. Having enjoyed Nahin’s previous book Duelling Idiots and Other Probability Puzzlers, I suspect this new one is worth reading as well.
The Monty Hall affair was a sobering episode for probabilists. In 1990 Marilyn vos Savant, a columnist for Parade magazine, discussed a puzzle based on the television game show “Let’s Make a Deal,” hosted by Monty Hall. The problem went roughly like this: A prize is hidden behind one of three doors. You choose a door, then Monty Hall (who knows where the prize is) opens one of the other doors, revealing that the prize is not there. Now you have the option of keeping your original choice or switching to the third door. Vos Savant advised that switching doors doubles your chance of winning. Thousands of her readers disagreed, among them quite a few mathematicians.
This is, of course, the way the problem is usually stated. I really must protest two things, however.
First, it needs to be stated clearly that Monty is guaranteed to open an empty door. This is plainly what Hayes has in mind in telling us that Monty knows where the prize is. The fact remains that this point is not completely clear in Hayes’ statement.
The more serious point is that we must have some piece of information regarding Monty’s method for selecting a door in those circumstances where he has a choice of doors to open (which happens whenever the contestant’s initial choice conceals the prize.) From Hayes’ statement we can conclude that if we play the game a large number of times, then a strategy of alwys switching will win two-thirds of the time, while a strategy of always sticking will win one-third of the time. The situation changes when we consider a single play of the game. Imagine that we initially choose door one and then see Monty open door two. What probability should we now assign to door one? We can conclude only that this probability is somewhere between one-third, and one-half, depending on Monty’s selection procedure.
The usual assumption is that Monty chooses randomly when he has a choice. Given this assumption we would say door one has a probability of one-third, and switching is plainly indicated.
So far, so familiar. Hayes begins his new essay with a story that will be sadly familiar to anyone who has engaged in serious discussion of the MHP:
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.
Heh. The controversy is definitely over, the existence of a few hold-outs notwithstanding.
Hayes goes on to give some specific examples from his correspondence, and gives a wink and gives a shout-out to the importance of agreeing on common assumptions about the structure of the game. He writes:
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.
Leaving out the most important assumption, alas.
At any rate, Hayes’ essay is very interesting and worth reading. But I nearly fell out of my seat when I read this:
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.
Oh for heaven’s sake! The mathematical community is unanimous and the problem is settled. Given the usual assumptions of the problem the wise course of action is to switch. Period. That is a fact, not an opinion. Anyone who says otherwise is wrong, wrong, wrong! Wrong in the same sense that it is wrong to say that 2 and 2 make 5. As for dissenters, well, they should not be stifled (because stifling people is rude), but they should definitely be treated as people who are confused. The whole point of mathematics is that problems get resolved to a deductive certainty. It is the great appeal of mathematics over other branches of science that when one of our problems is solved, it stays solved.
Believe me, I understand the problem is difficult and counter-intuitive. There are many plausible sounding arguments that can be made to defend different conclusions, and it can sometimes be difficult, even for mathematically savvy people, to distinguish the wheat from the chaff. But for all of that there is no controversy. There is no consensus opinion on the one hand with a handful of plucky dissenters on the other. There are only people who understand the problem and those who do not.