# How to Compute Probabilities by Following the Complement Rule

Two events are said to be complements if they are mutually exclusive *and* their union equals the entire sample space. This is represented by the complement rule, which is expressed as follows:

*P*(*A** ^{C}*) = 1 –

*P*(

*A*)

*A** ^{C}* is the complement of event

*A*.

Here’s an example: Suppose that an experiment consists of choosing a single card from a standard deck. Event *A* = “the card is red.” Event *B* = “the card is black.” Events *A* and *B* are complements because *A* and *B* are mutually exclusive (no card can be both red and black). The union of *A* and *B* is the sample space (the entire deck, because all cards must be either red or black, so the union of *A* and *B* equals the entire sample space.)

Coffee Type | 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 |

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

The complement of event *D* (decaffeinated coffee) is event *R* (regular coffee) because all coffee must be either decaffeinated or regular, and no coffee can be *both*. You can find the probability of the complement of *D* as follows:

*P*(*D** ^{C}*) = 1 –

*P*(

*D*)

Referring to the table, you can see that *P*(*D*) = 0.42. Therefore, *P*(*D** ^{C}*) = 1 –

*P*(

*D*)

*= 1 – 0.42 = 0.58, which is equal to*

*P*(

*R*).