Randomness

Throughout the ages, humankind has been faced with uncertainty. In the Old Testament, an individual's fate was sometimes determined by the drawing of lots. Today, lawyers argue the question of the reliability of DNA findings as legal evidence. Gamblers continue to win and lose fortunes on the outcome of a roll of a pair of dice in a game of chance, and meteorologists endeavor to accurately predict the chance of rain. The mathematics underlying these questions is, of course, probability theory.

Randomness is a book intended for a general audience that explains probability in an accessible way. The author develops the notion of chance and random events by describing the history of the subject. In the early chapters of the book, the author describes how dice and the casting of lots were used in various ancient cultures. I especially enjoyed the history of the types of dice used by these cultures. From this historical context, the author naturally develops the probability for the sum of two six-sided dice. The mathematics presented here is standard fare. Seeing it in its historical setting, however, the provides the reader a unique motivation for the study of the subject.

Historical themes are present throughout this little book. History is intermingled with probability theory, and often leads to lively and interesting discussions of the subject. For instance; a discussion of the history of the St. Petersburg paradox, a simple game of heads or tails in which the expected winnings for one of the players is infinite, leads the reader to a development of the concept of a fair game. This discussion, while considering the divergence of an infinite series, then turns to the law of large numbers, complete with its historical background.

A discussion of the (deep) philosophical question of determinism versus chance is the study of the middle chapters of Randomness. From the text: "Is a random outcome completely determined, and random only by virtue of our ignorance of the most minute contributing factors? Or are the contributing factors unknowable, and therefore render as random an outcome that can never be determined?" This question plagued sixteenth century philosophers who believed that all events were predetermined by God or by extrinsic causes determined by God. The discovery of Newtonian physics deepened the belief of scientists that everything about the natural world could be determined through mathematics.

This fervent belief that mathematical analysis would lead to universal laws prompted seventeenth century scientists to diligently investigate the earth and heavens around them. Naturally occurring quantities were studied and characterized. Scientists measured distances on earth, distances in space, orbits, and tides. However despite careful measurements and observations by the renowned scientists of the day, errors were continually discovered when quantities were re-measured. The frustration of random errors occurring in the work of these natural scientists becomes apparent while reading the numerous quotations the text includes. How daunting it must have been to these individuals when chance interfered with their study of "exact" phenomena. In this setting, the reader is presented with a motivation for considering the arithmetic average of a sample of measurements as a tool for reducing random error. With this background fully developed and entrenched, the reader is then treated to a historical development of the central limit theorem and the normal distribution.

Having clearly elucidated the normal distribution, Randomness then briefly describes the development of modern statistical methods. Most notable, in light of ever expanding computing power; is the brief history of random number tables and the difficulty of generating (pseudo) random numbers.

The text concludes with a nice treatment of some familiar probabilistic paradoxes. Of note is the discussion of the Prisoner's dilemma and its modern companion, the now-infamous Monty Hall Problem. These paradoxes are detailed and analyzed in a highly accessible manner, which is not an easy task, given the history of these paradoxes (also richly described in the text).

I found Randomness to be an engaging text on the fundamentals of probability theory. It provides a vivid historical account of the subject and thus fills a unique niche in the area of popular readings on the subject. I thoroughly enjoyed the author's lively style of writing and the posing of probabilistic questions in their historical context. The text could profitably be used as a supplemental reading for an introductory course in probability theory, or in a course in mathematics for the liberal arts student.

Randall J. Swift (randall.swift@wku.edu)is an associate professor of mathematics at Western Kentucky University. His research interests include nonstationary stochastic processes, probability theory and mathematical modeling. He is a co-author of the forthcoming MAA text entitled A Course in Mathematical Modeling.