By using the addition rule in a situation that is not mutually exclusive, you are **double‐counting**. One way of realizing that you are double‐counting is to use the classic theory of probability: List all the different outcomes when flipping a coin twice and assess the ratio of favorable outcomes to total outcomes (see Table 1).

There are four total outcomes. Three of the outcomes have at least one head; therefore, the probability of throwing at least one head in two flips is

To use the addition rule in a non‐mutually‐exclusive situation, you must subtract any events that double‐count. In this case:

The notation, therefore, for at least one favorable occurrence in two events is

*P*( *A*∪ *B*) = *P*( *A*) + *P*( *B*) – *P*( *A*∩ *B*)

which is read: The probability of at least one of the events *A* or *B* equals the probability of *A* plus the probability of *B* minus the probability of their joint occurrence. (Note that if they are mutually exclusive, then *P*( *A*∩ *B*)—the joint occurrence—equals 0, and you simply add the two probabilities.)

Example 1

The probability of drawing a spade is