Complementary Events

Complementary events satisfy the following equations:

Example 1

If you roll a fair die, find the probability of rolling a 4 and the probability of not rolling a four.

Solution .

= Since not rolling a 4 is the complement of rolling a 4, we can get the probabilty of not rolling a 4 by the above rules. Thus,

= = =

Example 2

If the probability of snow is 0.1, find the probability of no snow.

Solution .

Using the complement rule, we get that the probability of no snow is 1 - 0.1 = 0.9.

Example 3 (refers to example four in the Addition Rule section)

If an employee is selected at random, find the probability that the person is not a manager and is not between 50 and 59 years of age.

Solution .

In example four of the Addition Rule section, we saw that

where denoted being between 50 and 59.

Since not being a manager and not being between 50 and 59 is the complement of being a manager or being between 50 and 59, we can use the complement rule. Thus, the probability we want is =