Review of "Fifty Challenging Problems in Probability "

Fifty Challenging Problems in Probability, by Frederick Mosteller. ISBN 0-486-65355-2.

This very short book (88 pages) is exactly what its title says: in full, "Fifty challenging problems in probability with solutions". Of course, "challenging" varies in meaning depending on who's being challenged; these problems aren't aimed at any single audience, but the level varies from "interested layperson" to "bright undergraduate" -- though one of the problems is a lightly disguised form of what was until recently a famous open problem in pure mathematics, thrown in just for fun.

Several of the problems, particularly near the start, can be solved by mere calculation; in a few cases the author presents a purely "calculating" solution where shorter and (in my view) more insightful solutions are possible. The first of the three problems quoted below is an example; there's an easy solution in about one line of algebra, but there's one I like better that requires no algebra at all.

Several of the problems will be familiar to old hands. Several will not. I found that most of the genuinely challenging problems were ones I'd seen before.

The best way to get a feel for a book of this sort is to look at some of the problems. Here are a few nice ones.

A three-man jury has two members each of whom independently has probability p of making the correct decision and a third member who flips a coin (majority rules). A one-man jury has probability p of making the correct decision. Which jury has the better probability of making the correct decision?

Two urns contain red and black balls, all alike except for color. Urn A has 2 reds and 1 black, and Urn B has 101 reds and 100 blacks. An urn is chosen at random, and you win a prize if you correctly name the urn on the basis of two balls drawn from it. After the first ball is drawn and its color reported, you can decide whether or not the ball shall be replaced before the second drawing. How do you order the second drawing, and how do you decide on the urn?

How thick should a coin be to have probability 1/3 of landing on edge?

The solutions are good, though (as noted above) the author has a bit more tolerance for brute force than I have. They're explained clearly, with some useful insights. The tone is generally informal.

This is a Dover publication, and as such is cheap and decently produced.