The Pleasures of Probability

Description

[MR1329545](https://mathscinet.ams.org/mathscinet-getitem?mr=1329545)As a rule, probability textbooks—to which category the book under review belongs—present a gradual sequence of definitions and theorems, illustrating the theory with a wide range of examples which either involve the prototypical coin, dice and urn models, or more realistically deal with the gender of newborn babies, defects in mass-produced items, customer arrivals or opinion polls. Occasionally, paradox-generating problems are discussed, namely, problems in which the assumptions are not explicit, and modeling is expected to be part of the solution. The author chooses to present the subject in reverse order: there are no formal definitions or theorems in the text. These concepts gradually appear in prose when he discusses examples of all the types described above. Well-known paradoxes are abundant from the very beginning, because modeling awareness is a central theme of this textbook. Typical chapter names are 1. Cars, goats and sample spaces, 8. Baseball cards, the law of large numbers, and bad news for gamblers, 11. Breaking sticks, tossing needles and more: probability on continuous sample spaces. Conditional probability, independence, random variables, and the central limit theorem are all introduced with lengthy motivation and plenty of discussion. At the same time, all technicalities are avoided. Not a single infinite or nontrivial finite series is calculated, no differentiation or integration is present (most of the book deals with discrete sample spaces; in their infrequent appearances, densities are treated as approximations to histograms); the need for the concept of a joint distribution is hidden in two sentences in Subsection 8.1: (The coupon collector's problem). In this heuristic approach, the author elegantly manages to sidestep mathematical details all the way to the more advanced topics of Poisson processes, Markov chains and even Brownian motion. Other chapters include statistical applications and Monte Carlo simulations. Many will feel that the gentle and insightful nature of this book's account of probability theory justifies its name. Others would argue that only by first sweating through hard calculations can a student appreciate the deceptively simple foundations of the subject. Probably the readers to gain most from this book will be those who have already once gone through the "painful'' stage and may now use this book to look back and really grasp what had been going on, as well as students whose teachers have adopted its appealing arguments even in the more technical chapters of probability. Reviewed by [Eddy Mayer-Wolf](https://mathscinet.ams.org/mathscinet/search/author.html?mrauthid=121805)* * *The ideas of probability are all around us. Lotteries, casino gambling, the al­ most non-stop polling which seems to mold public policy more and more­ these are a few of the areas where principles of probability impinge in a direct way on the lives and fortunes of the general public. At a more re­ moved level there is modern science which uses probability and its offshoots like statistics and the theory of random processes to build mathematical descriptions of the real world. In fact, twentieth-century physics, in embrac­ ing quantum mechanics, has a world view that is at its core probabilistic in nature, contrary to the deterministic one of classical physics. In addition to all this muscular evidence of the importance of probability ideas it should also be said that probability can be lots of fun. It is a subject where you can start thinking about amusing, interesting, and often difficult problems with very little mathematical background. In this book, I wanted to introduce a reader with at least a fairly decent mathematical background in elementary algebra to this world of probabil­ ity, to the way of thinking typical of probability, and the kinds of problems to which probability can be applied. I have used examples from a wide variety of fields to motivate the discussion of concepts.