- Hardcover: 208 pages
- Publisher: Oxford University Press; 1 edition (June 4, 2009)
- Language: English
- ISBN-10: 0195367898
- ISBN-13: 978-0195367898
- Product Dimensions: 9.2 x 0.8 x 6.3 inches
- Shipping Weight: 12 ounces (View shipping rates and policies)
- Average Customer Review: 3.9 out of 5 stars See all reviews (15 customer reviews)
- Amazon Best Sellers Rank: #1,231,609 in Books (See Top 100 in Books)
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The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser 1st Edition
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"Excellent survey...If one wants to see "The Full Monty," this is definitely the book to buy. Highly recommended." -- Choice
"Those intrigued by the original Monty Hall problem will find that this book is a superb source of variants of the problem, pays careful attention to the hidden assumptions behind the problems, and is written in a witty accessible style that never lapses into flippancy. This is a model of how to accessibly introduce mathematical material at an elementary level that is not a mere popularization of the material. A virtue of the book is that it goes beyond mere exposition to make some serious contributions to the discussion, including a proof that the strategy of switching at the last minute in the progressive version is uniquely optimal and a discussion of some philosophical treatments on the topic."--Mathematical Reviews
"...a masterful job of tracing the problem back to its origin...much more comprehensive and wide-ranging than the many articles on the subject that have dribbled out...Rosenhouse offers readers much to think about concerning the perplexing question of whether to stick or switch." -Science
"Rosenhouse is both entertaining and precise in his writing. He carefully makes the point that conditional probability is difficult to intuitively process, often because what is being conditioned on is not clear. The book is both informative and an entertaining journey for both those schooled in probability and those with little background in probability."--The American Statistician
"Overall, this book is an excellent example of how a problem that is understandable by all can be used to introduce key concepts in mathematics and probability. If you are already familiar with the problem, this book will make you think more deeply about the nature of chance, and what Rosenhouse describes as "the perils of intuition". If Monty Hall is new to you, then you have a choice: stick or switch? You may be surprised." -- Tom Fanshawe, Lancaster
About the Author
Jason Rosenhouse is an Associate Professor of Mathematics at James Madison University in Virginia.
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Top Customer Reviews
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.
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.
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