Probability Theory

Abstract

The subject which is presently called probability theory seems to have its origins in the 17th Century. Indeed, it was only in the 17th Century that the concept of luck or chance evolved from its classical interpretation in terms of divine intervention on behalf of a chosen (i.e., lucky) individual into its modern interpretation in terms of perceived randomness resulting from inherent uncertainty or imperfect information1. However, once this evolution took place, the calculation of probabilities quickly attracted the attention of several brilliant minds. To mention a few: Bernoulli, Huygens, and, somewhat later, de Moivre and Laplace. Although many ingenious additional calculations were made during the 18th and 19th Centuries, it was only after Lebesgue introduced his integration theory that probability theory as we know it today became possible. Indeed, what has become the standard model for probability theory requires the existence and understanding of countably additive measures. Thus, first S. Ulam, in 1932, and shortly afterwards A. N. Kolmogorov, in 1933, based their axioms, what are commonly called the Kolmogorov axioms, of probability theory on Lebesgue’s theory of measures and integration. Although it will mean that we will have to ignore many very intriguing and potentially exciting aspects of the subject (e.g., finitely additive probability theory and what is sometimes called geometric probability theory), I will restrict our attention in this article to topics which can be rigorously treated in terms of Kolmogorov’s axioms. In fact, I will present only a very sparse sampling of the topics considered by modern probabilists working in the framework of those axioms.