# How to Compute Probabilities Using the Addition Rule

You use the addition rule to compute the probability of the union of two events. Mathematically speaking, for events *A* and *B*, the addition rule states that

This shows that the probability of the union of events *A* and *B* equals the sum of the probability of *A* and the probability of *B*, from which the probability of *both* events is subtracted. Subtracting the probability of both events is necessary to avoid the problem of *double-counting*. This is shown in the following example:

Suppose that event *A* contains the elements 1, 2, 3 and event *B* contains the elements 3, 4, 5. The sample space contains the elements 1, 2, 3, 4, 5.

The corresponding probabilities are:

The union of *A* and *B* contains all the elements in the sample space:

As a result, the probability of *A* union *B* equals 1. (Recall that the sample space always has a probability of 1.) If you simply combine the probabilities of *A* and *B*, though, you will get a surprising result; they sum to 6/5, which is greater than 1.

This result occurs because the element 3 appears in both *A* and *B*:

The probability of 3 was counted *twice*, once in set *A* and once in set *B*, which accounts for the sum of the probabilities being greater than one. By subtracting the probability of the element 3, the correct probability of 1 is found.

The table shows the distribution of coffees (measured in pounds) the Big Bean Corporation produces during a given day.

Coffee Styles | Special Reserve Blend (S) |
Kona Hawaii Blend (K) |
Aromatic Blend (A) |
Total |
---|---|---|---|---|

Decaffeinated (D) |
0.12 | 0.80 | 0.22 | 0.42 |

Regular (R) |
0.24 | 0.12 | 0.22 | 0.58 |

Total |
0.36 |
0.20 |
0.44 |
1.00 |

If you choose a pound of coffee randomly from the daily output of the Big Bean Corporation, what’s the probability that it’s either the *Special Reserve Blend** (S) *or the *Regular* *(R)* (or both)?

In this example, you use the addition rule because you’re being asked to compute the probability of a union. You combine the probability of *S* with the probability of *R*, subtracting the intersection between them to avoid the problem of double-counting.

From the table, you can determine that *P*(*S*) = 0.36; that *P*(*R*) = 0.58;

Seventy percent of the coffee produced by Big Bean is the special reserve blend, regular, or both.

When two events *A* and *B* are *mutually exclusive* (that is, they can’t both occur at the same time), the addition rule simplifies to

For example, if you choose a pound of coffee randomly from the daily output of the Big Bean Corporation, what’s the probability that it’s either the *Kona Hawaii Blend (K*) or the* Aromatic Blend (A)*?

Because a pound of coffee can’t be both the Kona Hawaii Blend *and* the Aromatic Blend, events *K* and *A* are mutually exclusive. This means that you can use the simplified version of the addition rule: