The Dice Problem-Then and Now

Abstract

Is it probable that probability gives assurance? Blaise Pascal (1623-1662) asked this question more than three centuries ago. His answer, it appears, was in the negative: Nothing gives certainty but truth [7, p. 268]. This question is not asked much anymore. And if it is, the answer is vastly different, embedded as it must be in definitions and theorems. Nowadays the answer to this question is a qualified yes. Despite his apparent lack of confidence in probability, Pascal was intensely interested in problems involving probability, and in fact is regarded by many as the inventor of probability theory (see [3] and [5]). An oft-repeated story goes like this. In 1654, Antoine Chevalier de Mere (1607-1684) presented Pascal with several gambling problems, among them the so-called dice problem. Pascal, in turn, commu? nicated these problems to Pierre de Fermat (1601-1665), and in the ensuing correspondence between Pascal and Fermat the theory of probability was born. Initially, probability was practical and experiential, rooted in games of chance and in insurance and annuities. Gradually the emphasis became more abstract and analytical, especially after the formulation of the Central Limit Theorem by Pierre-Simon Laplace (1749-1827). It was not until the 1930's, however, with the presentation by Jerzy Neyman (1894-1981) ([4] and [6, p. 310]) of a clear logic for the confidence interval approach to statistical inference that probability realized its potential for giving assurance to the man in the street. More recently, with the prevalence of survey sampling and the extensive use of computers in statistical research, probability has taken a turn again toward the empirical. My purpose in this paper is to look at the dice problem from several points of view: experimentally, analytically, and via the computer. In so doing, I intend to retrace in a way the various phases in the evolution of probability. First, a statement of the dice problem: In throwing two perfectly balanced dice, how many tosses are needed to have at least an even chance of getting a pair of sixes at least once!