The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser [Hardcover]

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 fifteen 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 fifteen 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.

People feel strongly that the answer the mathematicians have worked out is wrong and cannot be made right. 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 is obviously a fifty-fifty chance, and like so many obvious things, it is also wrong. Rosenhouse goes on to show several ways of calculating the problem, and he is good at explaining why you are twice as likely to win if you switch. 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. Many who first saw the problem in a _Parade_ magazine article by Marilyn vos Savant in 1990 weren't convinced, either. 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.

Believe it or not, this entire book is on the Monty Hall problem! The author, a mathematics professor, has analyzed this fascinating brain teaser from a variety of angles. After discussing the problem's history, he presents various attempts that have been made to understand it. The earlier attempts, including those by Marilyn vos Savant, tend to focus on logical arguments in order to arrive at the correct solution. But in order to solve the problem with mathematical rigour, the author uses some of the tools of his trade such as conditional probability and Bayes' Theorem. But that's not all. He also discusses a series of variants to the problem and proceeds to solve those as well. Finally, psychological and philosophical issues are also presented, partly in an attempt to understand why the human mind has been shown to have so much difficulty in solving this problem. The writing style is clear, friendly and authoritative, although some of the unfortunate editorial errors that the book contains may contribute towards slowing down a reader's attempts at following some of the author's arguments. Regarding accessibility, general readers can learn much from a good part of the main text because of the many clear explanations; however, several sections are fairly heavy with mathematics, a few of which can be rather challenging. Consequently, although anyone with an interest in this problem can benefit greatly from reading this book, math and science buffs are likely to glean the most out of it.

The "Monty Hall Problem" by Jason Rosenhouse is currently the best coverage of this important problem.

He covers the version of the problem as it was made famous in Parade by vos Savant, and also it numerous variations and generalizations, its history, its occurrence in various fields (psychology, philosophy, quantum theory), and he gives a rather extensive bibliography which will be of great use to the serious student. The depth of coverage varies depending on the topic. For example, the classical analysis is satisfyingly extensive, while the fringe areas (quantum Monty Hall, for example) are just touched upon, and then references are given in the bibliography.

The chatty tone of the text is such that it probably should be categorized as a "mathematics for entertainment" book. And as such, Rosenhouse has allowed himself literary license that one might not normally expect in a math book. For example, we have to wait to until page 42 before "probability basics" are actually discussed . Douglas Adams allusions aside, it might have been better to have given at least the classical definition of probability somewhat earlier. The definition is further developed in pages 84 - 88 when he excellently discusses the classical, frequentist, and Bayesian concepts of probability. This section I consider one of the best in the book.

Rosenhouse states that the book should be within the reach of any undergraduate math major. This is probably overkill. If you know what a binomial coefficient is and what it is used for, know the classical definition of a probability in terms of a sample space, and know how to sum simple series, then you should have no difficulties.

The great Hungarian mathematician Erdos, as Rosenhouse and others note, was a "victim" of the Monty Hall problem. As Erdos did work in the field of combinatorial mathematics and probability, this is significant. However, it must be emphasized that Erdos never actually attempted to solve the problem -- which takes all of about 1 minute to do if one writes down the sample space -- and which would admittedly would have been less than trivial for Erdos...No, Erdos was a victim because his intuition refused to accept the result, until somebody did a computer simulation and verified it for him. Hopefully, with the influence of this book (and others like it), this type of problem will find its way into high school textbooks so that future students of probability will develop a proper intuition.

Since this is a book that stresses math enjoyment, as mentioned earlier, Rosenhouse is allowed considerable license. Still I will point out a few things that bothered me:

On page 2, we are told that when physicist Paul Newman is interrupted in the 1966 Hitchcock move "Torn Curtain" by an impatient East German physicist who finishes off Newman's equations on a chalkboard that in reality "We don't finish off each other's equations." Well, physicists actually have done this, and rather famously so. So, quoting Max Born with regard to Oppenheimer: "In my ordinary seminar on quantum mechanics, he used to interrupt the speaker, whoever it was, not excluding myself, and to step to the blackboard, taking the chalk, and declaring: `This can be done much better in the following manner...' As Oppenheimer was a celebrity scientist during his lifetime, one can speculate that the script writer was aware of this anecdote.

On page 11, the famous "problem of points" of Pascal and Fermat is discussed. So the problem is, Alistair and Bernard are flipping a coin. Heads gives a point to Alistair, tails to Bernard, and the first person to 10 wins. The score is Alistair 8, and 7 for Bernard. If the game is stopped now, how should any prize be split? Rosenhouse correctly states that game will end after no more than 4 tosses...but it is a little bit too much license for my taste to claim, without explanation, that there are 16 possible scenarios. Drawing the tree diagram, we see there are only 10 real possibilities -- although each path of the tree is not equally probable, since the path lengths are different. Continuing this tree analysis, calculating .5 to the power of the length of a path gives a given paths probability and then summing each path's probability gives 11/16 for P(Alistair wins) and 5/16 for P(Bernard wins), and this agrees with Rosenhouse's result... However, Rosenhouse should point out that Fermat artificially allowed the game to continue even after a player had already won -- which is why he gets 16 possible scenarios instead of 10. Of course, Fermat included fictitious results in his calculation so that the paths would have the same length, and so by symmetry, the same probabilities.

An educational and entertaining read. Recommended.