A First Course in Statistics, 12th Edition

3.3 Complementary Events

A very useful concept in the calculation of event probabilities is the notion of complementary events:

The complement of an event A is the event that A does not occur—that is, the event consisting of all sample points that are not in event A. We denote the complement of A by $A^c.$

An event A is a collection of sample points, and the sample points included in $A^c$ are those not in A. Figure 3.9 demonstrates this idea. Note from the figure that all sample points in S are included in either A or $A^c$ and that no sample point is in both A and $A^c$. This leads us to conclude that the probabilities of an event and its complement must sum to 1:

Figure 3.9

Venn diagram of complementary events

Rule of Complements

The sum ...