# Discrete Probability and Average-Case Complexity

## Abstract

Discrete probability is introduced as a method to estimate average cases, not to predict individual outcomes from a process subject to chance or unpredictability, like flipping a coin or rolling a die.

Basic definitions are given for the fundamental concepts: an experiment, a sample space, a probability function, events, independent events, conditional probability, random variables, and the expected value of a random variable. Expected value generalizes average value and is used for calculating the average-case complexity of algorithms.

Standard distributions, which are good models of many real-world processes including computer applications, are treated in some detail. These include the uniform distribution, the binomial distribution, and the geometric distribution.

This chapter ends with a proof that the average-case complexity of QuickSort is O(*n*lg*n*).