EXAMPLE 29 Solving an “at least” problem

Using information in Example 28, find the probability that, in a random sample of 5 Americans ages 18–44, at least 1 of them smokes.

Solution

The phrase “at least” means that one or more of the five Americans smoke, so the probability we are looking for is:

Now, each of these events is mutually exclusive, meaning that our sample can’t yield exactly 1 American who smokes and exactly 2 Americans who smoke. Thus, by the Addition Rule for Mutually Exclusive Events, the above probability equals:

Calculating all these probabilities would take a while. So, we can use the probability of the complement we learned earlier to get us a shortcut. Note that “at least 1 American smokes” is the complement of “no Americans smoke.” Then, because the complement rule for probability is: , we get: