Introduction

13.1 Introduction

The study of probability began as a way to measure uncertainty in random experiments. In earlier classes, you met the axiomatic approach to probability, developed by the Russian mathematician A.N. Kolmogorov (1903–1987), which treats probability as a function assigning a number to each outcome of an experiment subject to certain basic axioms. When all outcomes are equally likely, this axiomatic theory agrees with the classical theory, and using that relationship you computed probabilities of events for discrete sample spaces and learned the addition rule.

In this chapter we go deeper. The central new idea is conditional probability — the probability of an event given that another event has already occurred. It is the foundation for several important topics:

• Bayes' theorem, which updates probabilities based on new evidence.
• The multiplication rule of probability, which handles the probability of two events occurring together.
• The independence of events, where the occurrence of one event does not affect the probability of another.

Later we introduce a random variable and its probability distribution, learn to compute the mean and variance of a distribution, and study the Binomial distribution, which models the number of successes in a fixed number of independent trials.

Throughout this chapter, unless stated otherwise, we assume all experiments have equally likely outcomes. This assumption simplifies calculations and connects the axiomatic approach to the classical one.

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