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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. ... Read more

Customer Reviews (10)

The Monty Hall Problem
"Let's Make a Deal," with its popular host Monty Hall, was a staple of 1970s game shows, with squirming contestants trying to guess which of three doors concealed the grand prize. The contestant chooses a door, but does not open it. Monty opens a different door, always empty. The contestant faces a choice: stick with his original door or switch to the remaining door?

This mindbender is known as "the Monty Hall Problem," the subject of Jason Rosenhouse's witty new book. Rosenhouse, associate professor of mathematics at James Madison University, examines the problem from multiple angles, lucidly explaining why, though counterintuitive, the best choice is to switch doors.

//The Monty Hall Problem// is sure to elicit emotional responses--as the conundrum itself has for years among mathematicians, philosophers, physicists, and others. Rosenhouse acknowledges the provocative nature of the problem and faces it head on, thoroughly and convincingly explaining the math and logic of the problem and addressing several variations.

Although Rosenhouse's enthusiasm for the subject is infectious, readers whose education ended with Math 101 may find much of this book beyond reach, as it is thick with equations. But for those who aren't arithmophobes, //The Monty Hall Problem// is delightfully challenging.

Reviewed by Kelli Christiansen

monty-hall: a review
monty-hall is a great probability problem, this book does a good job at reviewing it, describing it and relating its history and other details

Brain Tease or Brain Torture
Full disclosure up front: I have not read the book, but have a point to make about the way it is being presented here.

Amazon is in business to sell books (and other things) and I have been a satisfied customer for almost the entire life of the company, but there is a truth in advertising issue here:

You do not need to buy this book in order to understand
(a) whether you would be better off, in the three doors game it discusses, by "sticking with your first choice, or switching to the other remaining door" and
(b) why.

You can go, as I just did, to Wikipedia and get a plethora of short or shortish explanations. The approach I found easiest to grasp was in a footnoted article there.

Suppose there were 100 doors instead of 3. You'd then have a 1% chance of picking the right one initially. The host "Monty" (the real Monty is still alive, by the way, and I wonder what he thinks of all this!) then shows you 98 doors with no prize. Clearly, you have a very good chance of getting the prize if you then switch and pick the one other remaining door rather than sticking with your first choice. But, it is actually a 99 out of a 100 chance, not a 98 out of 99 chance. This is because originally there was a 99 out of a 100 chance of the prize being behind a door you didn't pick, and that chance wasn't altered by your pick or the host then opening and showing you all but one of the remaining doors (and having doors HE opens always have no prize, at least that is they way this "paradox" is usually presented).

The same logic applies when there are only three doors instead of a hundred. You have a 1 in 3 chance of picking the right one originally, which means there is a 2 in 3 chance of the prize being behind one of the two doors you didn't pick. After "Monty" then opens one of those two and shows you that there is no prize there, the chances of the remaining door having the prize are also 2 in 3. You are better off switching.

The odds of getting the prize by switching are not as good when there are a total of 3 doors rather than 100, but you still have a 2/3 chance of getting the prize by switching and only a 1/3 chance by sticking with your original choice.

It is a tease to imagine playing the game, and pondering why there is any paradox at all.
You might wonder, as I did: Isn't there a 50-50 chance on the last go-round, no matter what, so how can it matter whether you switch or stick? Then you can figure out why this is actually not correct; actually, given that you were more likely to be wrong on first round, you are better off on the second round betting that you WERE wrong originally by switching to the last of the remaining doors you did not pick initially.
Or you can get stumped, as I did, and look up the answer.
This is a tease.

Having to decide if NOT figuring out the paradox has become so frustrating that you would pay $17.96 plus postage to end the frustration, and whether you want to blow that much of your Christmas budget and get the frustration-releasing answer only after waiting another week or two later for the book to come, is more like torture than tease.

For all I know this may be a wonderful book. Some other reviewers seem quite happy with it. But if so, it is not wonderful because you spend the whole book having fun, ala Agatha Christie, trying to find the solution yourself to one of the all-time "most interesting mathematical brain teasers before it is revealed to you."

What
I cannot believe there is a book on this
"People feel strongly that the answer the mathematicians have worked out is wrong" (because it is)
On your first choice your probability of winning is Zero, because regardless of your choice Monty does not open your door.Monty by eliminating a wrong door has invalidated your first choice entirely for purely theatrical extension of the suspense.
Probability works when things are statistically random, when someone has special knowledge (knowing the empty door) it does not. You have a 50-50 chance, because that is the only random choice you are allowed to make.
If the question is wrong the answer wrong, it is not always about just checking the math.

A Detailed Analysis of an Intriguing Brain Teaser
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. ... Read more

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What a deal!
Did you know that mathematics is the onlyuniversal languagewe could expect intelligent aliens to understand? So says Ivan Moscovich, the designer of these puzzles and games which provide a learning experience that he calls "recreational mathematics." The title problem, one of dozens of puzzles within this collection, is the famous Monty Hall problem, which was popularized by Marilyn vos Savant in her Parade Magazine column. While being deceptively simple, its answer is surprising and counterintuitive. You would be amazed at how many people will not accept the correct answer! This puzzle alone, along with the solver's understanding of its solution, is worth the price of the book.

Interspersed among the puzzles in this collection are fascinating facts about such things as the number pi and how far a flea can jump at a single bound.The answer pages provide not only solutions, but explanations for the answers that use simple principles of mathematics. These puzzles are not just challenging and educational, but they are also a visual treat, since each puzzle has one or more colorful and beautiful illustrations. Some of these puzzles are very tricky, but that makes discovering the solution all the more rewarding. Enjoy!

Eileen Rieback ... Read more

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This short book explores the Monty Hall dilemma, a well known mathematical puzzle. The original problem, the controversy surrounding it and its solution are discussed. Further, the boundaries of the problem are expanded to consider prior knowledge and host intention. This book should be of interest to those who enjoy problem solving. ... Read more

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Puzzlers will feel fit to be tied - Dog Tied: If Fido is tied to a 10-foot-long rope, and his bone is 15 feet away, how is it possible that he can reach and enjoy his bone without breaking or stretching the rope? (And yes - the rope IS tied to something.) There's fun in finding the answer to this and other cool number-based problems. Dust off your mathematics and get solving. The intriguing enigmas include questions on interstellar communications, ancient geometry (Pythagoras and Plato), and even traffic patterns in "gridlock city." Or play the grasshopper jumping game. It's all fascinating. ... Read more

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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Monty Hall Problem, St. Petersburg Paradox, Two Envelopes Problem, Parrondo's Paradox, Three Prisoners Problem, Exchange Paradox, Necktie Paradox. Excerpt:The exchange paradox is a paradox within the probability theory and the decision theory which demonstrates problems that can arise when working with expected values . It can appear in different forms and hence under different names. Often it is presented in a form where two envelopes are used, and it is then called two-envelopes paradox . The setup The setup can appear in different forms using e.g. envelopes, boxes, and wallets. Using envelopes, it can be described as follows: In a game a player is presented two envelopes containing money. He is told that one envelope contains twice as much money as the other envelope, but he does not know which one contains the larger amount. The player then may pick one envelope at will, and after he has made a decision, he is offered to exchange his envelope with the other envelope. Cox discusses the apparent paradox and how it can be resolved. The problem Assuming that the player wants to maximise his gain, the crucial question is: Usually the answer to this question is made dependent of two circumstances, namely: Discussion The setup can be separated in two random events : In the following it is assumed that the selection of the amounts of money in the envelopes is realised as follows: A random, positive number X is chosen with some for now unspecified probability distribution. This amount of some currency units is put in one envelope in a symbolic way by writing it on a sheet of paper. Then the amount 2 X is put in the other evelope in the same way. When the player selects one envelope, it is assumed that he selects the amounts X or 2 X with a probability of 50 % each. The content of t... ... Read more

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Chapters: Monty Hall Problem. Source: Wikipedia. Pages: 446. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: The Monty Hall problem is a probability puzzle loosely based on the American television game show Let's Make a Deal. The name comes from the show's host, Monty Hall. The problem is also called the Monty Hall paradox, as it is a veridical paradox in that the result appears absurd but is demonstrably true. A well-known statement of the problem was published in Parade magazine: Although not explicitly stated in this version of the problem, the solution is usually based on the additional assumptions that the car is initially equally likely to be behind each door and that the host must open a door showing a goat, must randomly choose which door to open if both hide goats, and must make the offer to switch. As the player cannot be certain which of the two remaining unopened doors is the winning door, most people assume that each of these doors has an equal probability and conclude that switching does not matter. In fact, the player should switchdoing so doubles the probability of winning the car, from 1/3 to 2/3. When the above statement of the problem and the solution appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming the published solution was wrong. Some critics pointed out that the Parade version of the problem leaves certain aspects of the host's behavior unstated, for example, whether the host must open a door and must make the offer to switch. However, such possible behaviors had little or nothing to do with the controversy that arose (vos Savant 1990), and the intended behavior was clearly implied by the author (Seymann 1991). Other interpretations of the problem in which, for example, the host does not have to make ...More: http://booksllc.net/?id=6026198 ... Read more

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Monty Hall Problem. Let's Make a Deal, Monty Hall, Three Prisoners problem, Bertrand's box paradox, Quantum game theory, Deal or No Deal, Bayesian probability, Bayes' theorem ... Read more

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(Four Book Set) Innovator Ivan Moscovich's number puzzles are as spectacular looking as they are stimulating. Moscovich has been designing and inventing inspired puzzles, games, toys, teaching aids, and hands-on science museum exhibits for the past 50 years. "In collecting and creating puzzles", writes Moscovich, "I favor those that are more than just fun, preferring instead puzzles that offer opportunities for intellectual satisfaction and learning experiences, as well as provoking curiosity and creative thinking." For the Mastermind Collection, Moscovich has gathered many of his ingenious inventions as well as imaginative mind bogglers created by other remarkable puzzlemakers. This series covers a wide range of mathematical ideas, through both classic and original puzzles, games, problems and more. This four book series contains: The Hinged Square & Other Puzzles, The Monty Hall Problem & Other Puzzles, Leonardo's Mirror & Other Puzzles, and The Shoelace Problem & Other Puzzles. ... Read more

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Chapters: Monty Hall Problem. Source: Wikipedia. Pages: 308. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: The Monty Hall problem is a probability puzzle loosely based on the American television game show Let's Make a Deal. The name comes from the show's host, Monty Hall. The problem is also called the Monty Hall paradox, as it is a veridical paradox in that the result appears absurd but is demonstrably true. A well-known statement of the problem was published in Parade magazine: Although not explicitly stated in this version of the problem, the solution is usually based on the additional assumptions that the car is initially equally likely to be behind each door and that the host must open a door showing a goat, must randomly choose which door to open if both hide goats, and must make the offer to switch. As the player cannot be certain which of the two remaining unopened doors is the winning door, most people assume that each of these doors has an equal probability and conclude that switching does not matter. In fact, the player should switchdoing so doubles the probability of winning the car, from 1/3 to 2/3. When the above statement of the problem and the solution appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming the published solution was wrong. Some critics pointed out that the Parade version of the problem leaves certain aspects of the host's behavior unstated, for example, whether the host must open a door and must make the offer to switch. However, such possible behaviors had little or nothing to do with the controversy that arose (vos Savant 1990), and the intended behavior was clearly implied by the author (Seymann 1991). Other interpretations of the problem in which, for example, the host does not have to make ...More: http://booksllc.net/?id=6026198 ... Read more

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Chapters: Monty Hall Problem, Wayne Brady, Billy Bush, Big Deal, Carol Merrill, Bob Hilton, Trato Hecho, Jonathan Mangum. Source: Wikipedia. Pages: 69. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: The Monty Hall problem is a probability puzzle loosely based on the American television game show Let's Make a Deal. The name comes from the show's host, Monty Hall. The problem is also called the Monty Hall paradox, as it is a veridical paradox in that the result appears absurd but is demonstrably true. A well-known statement of the problem was published in Parade magazine: Although not explicitly stated in this version of the problem, the solution is usually based on the additional assumptions that the car is initially equally likely to be behind each door and that the host must open a door showing a goat, must randomly choose which door to open if both hide goats, and must make the offer to switch. As the player cannot be certain which of the two remaining unopened doors is the winning door, most people assume that each of these doors has an equal probability and conclude that switching does not matter. In fact, the player should switchdoing so doubles the probability of winning the car, from 1/3 to 2/3. When the above statement of the problem and the solution appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming the published solution was wrong. Some critics pointed out that the Parade version of the problem leaves certain aspects of the host's behavior unstated, for example, whether the host must open a door and must make the offer to switch. However, such possible behaviors had little or nothing to do with the controversy that arose (vos Savant 1990), and the intended behavior was clearly implied by the author (Seymann 1991). ...More: http://booksllc.net/?id=6026198 ... Read more

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Chapters: Simpson's Paradox, Birthday Problem, Monty Hall Problem, St. Petersburg Paradox, Boy or Girl Paradox, Two Envelopes Problem, Three Prisoners Problem, Sleeping Beauty Problem, Bertrand's Box Paradox, Nontransitive Dice, Exchange Paradox, Borel-kolmogorov Paradox, Berkson's Paradox, Littlewood's Law, Necktie Paradox. Source: Wikipedia. Pages: 112. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: The Monty Hall problem is a probability puzzle loosely based on the American television game show Let's Make a Deal. The name comes from the show's host, Monty Hall. The problem is also called the Monty Hall paradox, as it is a veridical paradox in that the result appears absurd but is demonstrably true. A well-known statement of the problem was published in Parade magazine: Although not explicitly stated in this version of the problem, the solution is usually based on the additional assumptions that the car is initially equally likely to be behind each door and that the host must open a door showing a goat, must randomly choose which door to open if both hide goats, and must make the offer to switch. As the player cannot be certain which of the two remaining unopened doors is the winning door, most people assume that each of these doors has an equal probability and conclude that switching does not matter. In fact, the player should switchdoing so doubles the probability of winning the car, from 1/3 to 2/3. When the above statement of the problem and the solution appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming the published solution was wrong. Some critics pointed out that the Parade version of the problem leaves certain aspects of the host's behavior unstated, for example, whether the host must open a door and must make the...More: http://booksllc.net/?id=6026198 ... Read more

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Bayes' theorem. Bayesian inference, Monty Hall problem,Bayesian network, Bayesian spam filtering, Conjugate prior,Deism, Empirical Bayes method, Prosecutor's fallacy, Ravenparadox, Recursive Bayesian estimation ... Read more

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Two book set. THE MONTY HALL PROBLEM & OTHER PUZZLES: Puzzlers will feel fit to be tied - Dog Tied: If Fido is tied to a 10-foot-long rope, and his bone is 15 feet away, how is it possible that he can reach and enjoy his bone without breaking or stretching the rope? (And yes - the rope IS tied to something.) There's fun in finding the answer to this and other cool number-based problems. Dust off your mathematics and get solving. The intriguing enigmas include questions on interstellar communications, ancient geometry (Pythagoras and Plato), and even traffic patterns in "gridlock city." Or play the grasshopper jumping game. It's all fascinating. THE HINGED SQUARE & OTHER PUZZLES: A colourful square, dissected into four parts, with hinges marked in black. If you leave the blue piece fixed and swing the others around their hinges, a new shape will emerge. Can you guess just by looking what it will be? This is just one of the tricky geometrical gems that will make a puzzler's mind work overtime. Try drawing a set of variously shaped polygons using only a compass and a ruler (no measuring allowed!); figuring out which of two sculptures is bigger (logic alone won't give you the answer); and lots more. ... Read more

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High Quality Content by WIKIPEDIA articles! An optimal decision is a decision such that no other available decision options will lead to a better outcome. It is an important concept in decision theory. In order to compare the different decision outcomes, one commonly assigns a relative utility to each of them. Sometimes, the equivalent problem of minimizing loss is considered, particularly in financial situations, where then utility is defined as economic gain. "Utility" is only an arbitrary term for quantifying the desirability of a particular decision outcome and not necessarily related to "usefulness". For example, it may well be the optimal decision for someone to buy a sports car rather than a wagon, if the outcome in terms of e.g. personal image is more desirable even given the higher cost and lack of versatility of the vehicle. In case the decision outcome is subject to uncertainty, an optimal decision is maximizing the expected utility. The problem of finding the optimal decision is a mathematical optimization problem. In practice, few people are verifying that their decisions are optimal, but use more intuitive approaches to obtain decisions that are "good enough". ... Read more