Conditional probability and Bayes' theorem

In everyday life, our knowledge of the past informs our predictions about the future. For example, if the team with the best record in a basketball league were about to play against the team with the worst record, we would likely estimate the chance of the first team winning the game to be higher than if we did not know that fact.

This same idea in the context of this chapter would be to calculate the probability of an event occurring after learning that another event has occurred. This is a conditional probability and it applies in situations where we learn information over time, which influences our evaluations of probabilities for subsequent events, which is important to machine learning, artificial intelligence, and many other fields.

Definition – conditional probability

For two events A and B where P(B) > 0, the conditional probability of A given B is as follows:

This is the proportion of the time A occurs...