The addition rule for probability with union and intersection
Sometimes a probability question gives us two ways for an outcome to count. We’ll use the addition rule to include outcomes that satisfy either condition and count the ones that satisfy both just once.
Use the addition rule for “A or B”
An event is a set of outcomes. The union contains every outcome in , in , or in both. The intersection contains the outcomes that are in both events.
When we add and , the intersection is included once in each probability. Subtracting leaves one copy of the overlap.
Example with two dice
Roll two fair six-sided dice independently. The ordered pairs are equally likely. Let mean at least one die shows , and let mean the sum is even.
For , there are rolls with a on the first die and more with a on the second die. For , both dice must be odd or both must be even, so there are rolls. The overlap is , , , , and .
When the events are mutually exclusive
Mutually exclusive events have no outcome in common, so and the union probability is . For instance, if a sample space has equally likely outcomes, with in and the other in , the union contains all . Its probability is .
Example with a two-way table
Choose one person at random from the 100 survey participants. The table shows each person’s favorite sport and how they were counted in the survey.
| Sport | Male | Female | Total |
|---|---|---|---|
| Football | 22 | 16 | 38 |
| Basketball | 13 | 8 | 21 |
| Other | 25 | 16 | 41 |
| Total | 60 | 40 | 100 |
The column total gives . The football row total gives .
For “female or other,” take the female column total of 40 and the Other row total of 41. The cell where the two meet contains 16 people, so we subtract that count once.
