Um livro intrigante sobre um problema aparentemente simples que gerou polêmicas e mais polêmicas. Alguns momentos de muita matemática que um leitor desavisado perde o controle de seu cérebro que devem ser superados para chegar no excelente capítulo sobre as questões cognitivas ou o porque nossa mente não é capaz de fazer boas análises de probabilidade e risco. As variantes do problema são interessantes principalmente a que entra dois jogadores pois parece muito o mercado de ações que cada um sabe sua posição e intui a dos outros. Para quem curte matemática, probabilidade e ciências cognitivas é uma ótima pedida. ( )

Rating: 3.5 bumfuzzled stars of five

The Publisher Says: 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.

My Review: I'd rate it higher if I understood it....

Twenty years ago, a brouhaha erupted in Parade magazine, of all unlikely places, about a probbility problem, of all unexpected things. It's an exercise in applied probability mathematics. Here's the famous statement of the problem:

"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to switch your choice?"

Okay, so the answer is, "Always switch." You'll win about 64% of the time if you always switch, vs 31% of the time if you DON'T switch. This has been demonstrated again and again and again and again since the problem surfaced in 1959 (under a different name). People are *still* arguing about it! People with advanced degrees in math are arguing against the mathematical proof! (Which reinforces my absence of respect for the mere possession of an advanced degree.)

This book contains formulae and equations, so the phobic should pass it by. Being barely numerate, I skipped anything that had italic x's or y's, curly brackets, extra-large parentheses, or other quick identifiers of mathspeak, and I did okay.

What did I learn? 1) Jason Rosenhouse has a sense of humor and a quick way with a zinger. 2) Always switch doors on "Let's Make a Deal." 3) Rein in my curiosity about subjects I don't grasp readily...getting books via InterLibrary Loan means one has to read them too quickly for comfort!

Should you read it? Probably not. It's not a subject of interest to most people. If it is of interest to you, make sure you have ample time to revisit the more baroque sections. And run run run like a bunny if you see the word "Bayesian!" That way mouth-breathing, drool-dripping, eye-crossing befuddlement lies! ( )

Here's the Monty Hall problem: You are playing a game show according to the following rules. There are three doors. One door hides a car, and the other two hide goats. You want to win the car. You choose a door -- say door 1 -- but do not open it. The host "Monty Hall" then opens one of the two remaining doors -- say door 2 -- to reveal a goat. (Monty knows where the car is and he intentionally reveals a goat. Whenever Monty has a choice of doors he chooses randomly.) You may now choose to claim the prize behind your door 1 or you may switch to door 3. Should you switch?

The answer is "yes": switching doubles your chances of winning. If this answer doesn't make you uncomfortable, then chances are you've either heard it before or you're not paying attention. For most people, intuition says that you begin with three equally probable options, and when the host opens a door you are reduced to two equally probable options. But intuition is wrong. In fact, intuition is frequently wrong on probabilistic questions. Mathematicians delight in imagining scenarios where the most intuitive solution is wrong, and by this measure the Monty Hall problem is unusually delightful.

Delightful maybe, but is the problem sufficiently delightful for a book-length treatment? Surprisingly, yes. Variations on the Monty Hall problem are legion: What if Monty doesn't know where the car is? What if he's only pretty sure? What if he likes to open door number 3? What if there are more than three doors? More than one car? More than one player? More than one Monty? What if some doors are more likely to conceal the car than others? Variations are so rich that Rosenhouse, a mathematics professor at James Madison University, thought that he could design an entire introductory course in probability using only variations on the Monty Hall problem. And darned if he doesn't make a convincing case for it. What's more, his book is engaging, challenging, and even surprising in more places than just the answer to the original problem.

But there's more: it turns out that the Problem has applications or parallels everywhere, from physics to contract bridge. Rosenhouse wraps up with discussions of the problem by economists, behaviorists and philosophers. Who knew?

The only complaint I have about the book is the audience. It looks like a book of popular mathematics, but it's not. Rosenhouse acknowledges this up front, warning the reader that "the equations get rather dense, I'm afraid," but adding, "an undergraduate mathematics major should not have much difficulty understanding anything I have presented." He's probably right about that: the shame is, he might have made the book more widely accessible with a little more hand-holding through the rough spots.

For instance, Rosenhouse presents a couple of inductive proofs, but doesn't explain even briefly how inductive proofs work. The math undergrads will get it, or at least the upperclassmen will. But other potential readers, say bright high schoolers in Mu Alpha Theta or even the freshman math undergrads, may never have encountered induction. Without some sort of preparation they are not likely to find the proofs convincing.

Still the book is a joy. Even if you've never heard of mathematical induction there's much to like. If you enjoy recreational math, if you don't mind pulling out pen & paper to follow an argument, and if you're willing to skip over a handful of bits where the "equations get rather dense" then you won't leave unrewarded. ( )