Bayes' Theorem

Suppose we are interested in two events, A and B. In this case, event A might represent the event that a patient has appendicitis and event B might represent a patient having a high white blood cell count. The conditional probability of event A given event B is essentially the probability that event A will occur when we know that event B has already happened.

Formally, we define the conditional probability of event A given event B as the joint probability of both events occurring divided by the probability of event B occurring:

Note that this is consistent with the way in which we define statistical independence. Statistical independence occurs when the joint probability of two events occurring is just the product of the individual probabilities of the two events. If we substitute this in our previous equation, we have:

This makes sense intuitively because if we know that two events are independent of each other, knowing that event B has occurred does not change the probability...