The aim of this chapter is to present the definitions, formulae and results of probability theory we shall need in the main body of the book. Although we assume that the reader has had only a rather limited experience with probability theory and, if somewhat vaguely, we do define almost everything, this chapter is not intended to be a systematic introduction to probability theory. The main purpose is to identify the facts we shall rely on, so only the most important—and perhaps not too easily accessible—results will be proved. Since the book is primarily for mathematicians interested in graph theory, combinatorics and computing, some of the results will not be presented in full generality. It is inevitable that for the reader who is familiar with probability theory this introduction contains too many basic definitions and familiar facts, while the reader who has not studied probability before will find the chapter rather difficult.

There are many excellent introductions to probability theory: Feller (1966), Breiman (1968), K. L. Chung (1974) and H. Bauer (1981), to name only four. The interested reader is urged to consult one of these texts for a thorough introduction to the subject.

Notation and Basic Facts

A probability space is a triple (Ω, ∑, P), where Ω is a set, ∑ is a σ-field of subsets of Ω, P is a non-negative measure on ∑ and P(Ω) = 1.