Probability: A Lively Introduction

For centuries, mankind lived with the idea that uncertainty was the domain of the gods and fell beyond the reach of human calculation. Common gambling led to the first steps toward understanding probability theory, and the colorful Italian mathematician and physician Gerolamo Cardano (1501–1575) was the first to attempt a systematic study of the calculus of probabilities. As an ardent gambler himself, Cardano wrote a handbook for fellow gamblers entitled Liber de Ludo Aleae (The Book of Games of Chance) about probabilities in games of chance like dice. He originated and introduced the concept of the set of outcomes of an experiment, and for cases in which all outcomes are equally probable, he defined the probability of any one event occurring as the ratio of the number of favorable outcomes to the total number of possible outcomes. This may seem obvious today, but in Cardano's day such an approach marked an enormous leap forward in the development of probability theory.

Nevertheless, many historians mark 1654 as the birth of the study of probability, since in that year questions posed by gamblers led to an exchange of letters between the great French mathematicians Pierre de Fermat (1601– 1665) and Blaise Pascal (1623–1662). This famous correspondence laid the groundwork for the birth of the study of probability, especially their question of how two players in a game of chance should divide the stakes if the game ends prematurely. This problem of points, which will be discussed in Chapter 3, was the catalyst that enabled probability theory to develop beyond mere combinatorial enumeration.

In 1657, the Dutch astronomer Christiaan Huygens (1629–1695) learned of the Fermat–Pascal correspondence and shortly thereafter published the book De Ratiociniis de Ludo Aleae (On Reasoning in Games of Chance), in which he worked out the concept of expected value and unified various problems that had been solved earlier by Fermat and Pascal. Huygens’ work led the field for many years until, in 1713, the Swiss mathematician Jakob Bernoulli (1654–1705) published Ars Conjectandi (The Art of Conjecturing) in which he presented the first general theory for calculating probabilities.