Probability: Formal

Abstract

Probability theory is a branch of mathematics that evolved from the investigation of social, behavioral, and physical phenomena that are influenced by randomness and uncertainty. This article begins its survey of probability theory with a discussion of the impact of A. N. Kolmogorov's framework for the mathematical understanding of probability and the role that his axioms had in transforming probability from a modeling art to a mathematical science. The survey then discusses the important notion of a random variable, and the tools for describing such variables, including means, variances, distribution functions, densities, and the fundamental property of independence. The article also discusses the law of large numbers—the mathematical result that underlies the popular ‘law of averages’ and the central limit theorem—the mathematical result that suggests why the ‘bell curve’ is present in many social and physical phenomena. The article concludes by reviewing sources of further information on probability theory.