Bayes' theorem is a statistical formula used to determine conditional probabilities. It describes the probability of an event based on prior knowledge of other events that have occurred. The formula calculates the probability of one event occurring given that another event has occurred. An example demonstrates using Bayes' theorem to calculate the probability that a ball drawn from an urn is from the first urn, given that the ball is red. The document discusses applications of Bayes' theorem in fields like marketing, medicine, puzzles, and decision analysis.