The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser by Jason Rosenhouse (PDF)

Description

Mathematicians call it the Monty Hall Problem, and it is one of the most interesting mathematical brain teasers of recent times. Imagine that you face three doors, behind one of which is a prize. You choose one but do not open it. The host–call him Monty Hall–opens a different door, always choosing one he knows to be empty. Left with two doors, will you do better by sticking with your first choice, or by switching to the other remaining door? In this light-hearted yet ultimately serious book, Jason Rosenhouse explores the history of this fascinating puzzle. Using a minimum of mathematics (and none at all for much of the book), he shows how the problem has fascinated philosophers, psychologists, and many others, and examines the many variations that have appeared over the years. As Rosenhouse demonstrates, the Monty Hall Problem illuminates fundamental mathematical issues and has abiding philosophical implications. Perhaps most important, he writes, the problem opens a window on our cognitive difficulties in reasoning about uncertainty.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The writing is phenomenal – Rosenhouse tells a fun story. If you like to read stories about scientists, mathematicians, and old philosophical debates then this should be perfect for you.I don’t want to spoil this for you, so I’ll step carefully here. Once you have seen the solution you’ll probably realize that the answer is simple. But so many smart people continue fighting to be right even when the answer is simple and provable. I like to tell people about the Monty Hall problem and see what answer they come up with. And it’s really fun to see how they’re still baffled so often when you show them the answer. The best part is that some of the most befuddled people are the ones you’d think would already know.If you haven’t ever seen the solution before then Rosenhouse really should get the first crack at telling you, because he tells a great story. Unlike Malcolm Gladwell and the Freakonomics guys, he’s telling the story of something that’s truly counter-intuitive. And he does it very well.

⭐This book is not aimed at someone with a casual interest in the basic version of the problem. The mathematician author has clearly enjoyed examining and explaining numerous variants — to quote a reviewer elsewhere “progressive versions, Bayesian treatments of the problems, computer simulations, quantum versions, information-theoretic representations, common cognitive fallacies associated with the problem, and much more”. So the book is aimed at a reader who enjoys brain-teaser style logic puzzles, is capable of following (sometimes heavy) undergraduate math, and is willing to follow a trail around very diverse topics. If that sounds appealing to you then you will enjoy the book.

⭐Probability can be one of the most counter-intuitive concepts our brains struggle with. This book by mathematician Jason Rosenhouse explores one of the most famous brain twisters of modern times and along the way walks the reader through the basic ideas of calculating chance in a lucid and entertaining manner. Not everyone with be comfortable with all chapters (the author admits this), but with its exploration of history, sociology, and psychology of the Monty Hall problem, there is enough in this book to satisfy everyone. Plus you probably can use it to win a few bar bets.

⭐Jason Rosenhouse has written what is a witty and informative book on the remarkable Monty Hall problem. Never has such a simply stated problem caused so much anger, confusion and irritation but so little understanding. Anyone who knows the problem will have gone through all the phases so amusingly described by Rosenhouse – denial, anger, bargaining, depression and acceptance.The problem is named “the Monty Hall problem” because of its similarity to scenarios on the game show Let’s Make a Deal, but was brought to nationwide attention by her column in Parade magazine. Vos Savant answered arguing that the selection should be switched to door #2 because it has a 2/3 chance of success, while door #1 has just 1/3. This response provoked letters of thousands of readers, nearly all arguing doors #1 and #2 each have an equal chance of success. A follow-up column reaffirming her position served only to intensify the debate and soon became a feature article on the front page of The New York Times. Among the ranks of dissenting arguments were hundreds of academics and mathematicians.Now the heat that that this problem has generated is itself an intriguing phenomenon. According to Wikipedia Vos Savant is apparently a relative of physicist Ernst Mach and wife to the inventor of the artificial heart (Jarvik). She was listed in the Guinness Book of Records at some stage as the person with the highest IQ. She is a long standing columnist for Parade magazine which, with all due respect to that august magazine, is not exactly something you’d associate serious mathematical discourse. Check it out for yourself: […]With that background and being a woman I think some people in the mathematical community actually played the woman rather than the problem and ultimately paid the price of being proved wrong in a rather ignominious fashion. I bet the people concerned don’t put it on their CVs.Because this problem is so simple to pose it seems accessible to professionals and amateurs alike. An amateur does not attempt to venture a public opinion on the Riemann hypothesis but the Monty Hall problem is something the man in the street feels he can offer an opinion on. Suffice it to say that probability is hard, and it seems we may not be built to intuitively perform accurate calculations be they classical or Bayesian. When I first came across this problem probably over 10 years ago I solved it along the lines of page 54 of the book (an exhaustive enumeration of cases where the goats are distinguished ) which seemed to me the most basic way to do it given that many combinatorial problems can be solved by applying appropriate labeling. I was surprised by Rosenhouse’s classification of this approach in “other approaches” as I thought it was the obvious way to do it. Just goes to show how wrong you can be.The book does a good job of teasing out Bayesian theory and much more. Rosenhouse takes the basic Monty and like all good mathematicians he fiddles with the assumptions to develop more and more general problems. That type of approach is useful training for mathematically inclined people. On page 62 Rosenhouse does a basic set of calculations which he says can be “quickly” determined. He’s right but I found it odd that he did not footnote the intermediate steps given the amount of time he spends on other steps which are no more and no less difficult. Given the sort of heat this problem has caused one would not want to lose someone for the want of two lines of explanation.These are minor quibbles. The book is very good and the generalizations particularly interesting.[…]

⭐The book is fine but the kindle version is plagued with typos – this is not how the new book should look like. I wonder if the author looked at it at all – it puts in doubt his thoroughness and attention to detail self-advertised so often in the book

⭐Rosenhouse is great!This approach helps you to grasp the whole problem even if you areNot a mathematician. I just kept to wonder how Pal Erdos managed tomiss the right solution.By the way the many times stressed importance of random choice in the case of twoGoat hiding doors by Monty is not absolutely unambiguous.T.Fenyvesi M.D.

⭐Decent explanation

⭐This book explains Monty on several levels and is accessible to those of us who haven’t studied math formally for a while.

⭐Il volume rappresenta una vera e propria summa dell’ormai classico problema di Monty Hall, derivato da un famoso gioco televisivo degli anni 60, e qui esaurientemente trattato tanto nella formulazione originale quanto nelle sue innumerevoli varianti. La (contro-intuitiva) soluzione del dilemma viene proposta dall’autore utilizzando molti fra gli strumenti disponibili, quali l’albero delle probabilità, il teorema (o regola) di Bayes, l’enumerazione di tutti i casi possibili, la simulazione con prove ripetute del gioco. A buon diritto, il testo “potrebbe costituire la base di partenza per un corso introduttivo alle probabilità discrete”.Per chi si è occupato a lungo del problema, la sua soluzione è ormai di una evidenza cristallina. Infatti, quando si sceglie la porta sbagliata (2 possibilità su 3) e poi si cambia, ovviamente si vince: perciò si vince in 2 casi su 3, cioè con probabilità 2/3. Mentre, l’unico caso in cui si perde, se la strategia è di cambiare sempre, è quando al primo colpo si sceglie la porta corretta, evento che ha probabilità 1/3.È peraltro abbastanza noto che la maggior parte di noi arriva molto lentamente a convincersi della soluzione corretta, se mai ci arriva. Invece, uno studio nordamericano del 2010 ha dimostrato sperimentalmente che i piccioni adottano la strategia ottimale di cambiare la loro scelta iniziale in un tempo minore rispetto agli umani. Considerata la dimensione del cervello di questi uccelli, il risultato è certamente sorprendente.

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⭐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.

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