Introduction

In 1654 a gambler's dispute over how to fairly split the stakes of an interrupted game landed on the desks of two of France's leading mathematicians, Blaise Pascal and Pierre de Fermat. Their correspondence worked out, for the first time, the fundamental principles of what we now call probability theory. A century and a half later, after extensive research, Pierre-Simon Laplace published his monumental Théorie Analytique des Probabilités in 1812, laying the modern foundations of the subject -- including the Bayesian interpretation of probability that a later section of this chapter revisits.

What began as a way to settle gambling disputes has grown into one of the most widely applied branches of mathematics. Probability today prices life-insurance premiums, forecasts election outcomes, and describes the erratic motion of molecules in a gas. The everyday words we reach for -- chance, possible, probably, likely, odds, uncertainty, prevalence, risk, expectancy -- are all really informal stand-ins for the same underlying idea.

Our world runs on uncertainty. Every decision we make, from the trivial to the consequential, is made under some degree of it. Probability is the branch of mathematics built to measure exactly how likely an event is to occur -- and this chapter builds that measurement up rigorously, from the basic vocabulary of experiments and sample spaces, through the classical and axiomatic definitions of probability, to conditional probability, independence, total probability, and Bayes' Theorem.