July 7, 2009
If you are old enough, you remember the sensation that the Rubik''s Cube caused all the world over in 1980. No one is still alive that remembers the 1880 fad for the analogous two-dimensional "Fifteen Puzzle," which had 15 numbered blocks within a four by four container and you were supposed to arrange them numerically. Mechanical puzzles can make storms like these, maybe because you can solve them over and over again, but it isn''t often that word puzzles produce such fads.
True, the Zebra Puzzle, a reasoning exercise consisting of 15 seemingly unconnected statements that if regarded together the right way make a logical whole, was popular in 1962. Once you solved it, however, that was that.
The Monty Hall Problem entered the public consciousness in 1990 and has been completely solved, but because the solution is so counterintuitive, it is still on the minds of many.
One of those minds is that of Jason Rosenhouse, an associate professor of mathematics who has written "The Monty Hall Problem: The Remarkable Story of Math''s Most Contentious Brain Teaser" (Oxford University Press).
"My original idea for this book," he writes, "was that an entire first course in probability could be based on nothing more than variations of the Monty Hall problem." Indeed, some of the chapters here are full-power mathematics, with unknowns x, y and z, summation or conditional probability symbols, and complicated equations choked with parentheses within brackets, and more. Math phobics won''t get far with such stuff, but there is enough other material here, along with different explanations of the basic puzzle, that will be of interest to anyone who likes recreational mathematics in even the slightest degree.
It is fun to read Rosenhouse''s remarks on his career.
"Like all professional mathematicians," he writes, "I take it for granted that most people will be bored and intimidated by what I do for a living." After dining in a restaurant with friends, he knows that some will find it funny to assign to the mathematician the task of dividing up the check. He used to say, "Dividing up a check is applied math. I''m a pure mathematician," but since people were puzzled that there were any such categories, he uses, "Better have someone else do it. I''m not good with numbers."
He likes seeing mathematicians in movies: "I feel cheated if that character and his profession are presented with any element of realism." He doesn''t mind that when people learn he is a mathematician, and must have a PhD in the subject, they assume he is very smart, and he finds that he enjoys it when they defer to his opinion when mathematical or scientific topics are brought up.
"Unless, that is, we are discussing the Monty Hall problem."
People feel strongly that the answer the mathematicians have worked out is wrong and cannot be made right. The Monty Hall problem gets its name from the host of the game show "Let''s Make a Deal" which was first broadcast in 1963. It has to do with a set up that Monty Hall would use on the show.
Here is the problem: You are Monty''s contestant, and he presents you with three identical doors. One hides a car, which you want, but the other two doors hide goats, neither of which you want. You pick a door, but instead of opening it, Monty opens one of the other two doors. Monty knows, of course, where the car is and where the goats are, and he only opens a door that shows you a goat; in the case where you happened to pick the door hiding the car, he chooses one of the two remaining doors randomly. So then you have one door open with a goat, and two doors unopened, including the one you picked. Monty now says he will give you a choice: You can stick to the unopened door you originally picked, or you can switch to the other unopened door. So, do you stick or switch?
It''s so obvious! Two remaining doors, one with a prize, and no way to tell which is which. Obviously, the odds of picking the car, whether you stick or whether you switch, make no difference -- it''s fifty-fifty. It is obvious, and like so many obvious things, it is also wrong.
Rosenhouse writes, "You can share with a college class the glories of the human intellect, the most beautiful theorems and sublime constructs ever to spring forth from three pounds of matter in a human skull, and they will dutifully jot it all down in their notes without a trace of passion. But tell them that you double your chances of winning by switching doors, and suddenly the swords are drawn and the temperature drops ten degrees." We did not evolve to solve mathematical problems, and Rosenhouse reflects, "Probabilistic reasoning is just not something that comes naturally."
Rosenhouse goes on to show several ways of calculating the problem. Essentially, Monty is giving you extra information when he opens that door with a goat behind it. You had a one third chance of picking the door with the car to begin with, and if you have picked that door and switch, you lose. But you also had a two thirds chance of picking a goat to begin with, and (under the conditions of the problem), if you picked a goat and switch, you can only switch to the door hiding the car.
Don''t worry if the summary in this review isn''t convincing. Rosenhouse describes many different solutions, explaining each one, and if the recommendation to switch still doesn''t "feel" right, it is refreshing to realize that this is a problem that lends itself to experimentation. There are Web sites that allow you to play a simulation of the game over and over, or a friend can pretend to be Monty, an ace can simulate the door with the car, and two jacks can simulate the doors with goats. Do the experiment over and over, switching every chance you get, and the results will come closer and closer to winning 66 percent of the time, just as if when you toss a coin over and over the resulting heads or tails will hover around 50 percent.
If it takes a while for you to come around to the conclusion that switching is the better strategy, don''t worry. You are in good company. Paul Erdös, who is on anyone''s list of the top mathematicians of the 20th century, was given the problem by another mathematician, who was soon sorry for having done so "... because it was my experience that people get excited and emotional about the answer and I end up with an unpleasant situation." Erdös was eventually convinced by experimentation, but he was still not satisfied with the result.
Nor were many who first saw the problem in a Parade magazine article by Marilyn vos Savant in 1990. Rosenhouse prints some of the responses to her article, many of them from mathematicians and many of them withering in their disapproval of her correct analysis that switching is the best policy by a factor of two. He is embarrassed by the vituperative nature of some of the professional voices in opposition.
If this puzzle were not puzzling enough, Rosenhouse goes through many variables of the problem and its effects in different schools of thought (including quantum dynamics), because there is a huge amount that has been written about it.
Rosenhouse says he could write a second book with material he has reluctantly left out of this one, and this one covers: What if there are four doors? What if there are n doors? What if there is another player playing against you, or what if Monty opens any of the three doors randomly? It covers the history of the problem and the similar problems that went before it, and it covers the psychological causes of sticking or switching, and studies that show how people tend to stick in all the cultures so far tested. Best of all, for this reader anyway, it made the previously counterintuitive strategy of switching feel a little more sensible.
Rob Hardy is a local psychiatrist who reviews books for a hobby. His e-mail address is firstname.lastname@example.org.