EXAMPLE 4.21

HIV testing. Many people who come to clinics to be tested for HIV, the virus that causes AIDS, don’t come back to learn the test results. Clinics now use “rapid HIV tests” that give a result in a few minutes. The false-positive rate for a diagnostic test is the probability that a person with no disease will have a positive test result. For the rapid HIV tests, the Food and Drug Administration (FDA) has established 2% as the maximum false-positive rate allowed.⁵ If a clinic uses a test that matches the FDA standard and tests 50 people who are free of HIV antibodies, what is the probability that at least one false-positive will occur?

It is reasonable to assume as part of the probability model that the test results for different individuals are independent. The probability that the test is positive for a single person is 0.02, so the probability of a negative result is 1 − 0.02 = 0.98 by the complement rule. The probability of at least one false-positive among the 50 people tested is, therefore,

There is approximately a 64% chance that at least 1 of the 50 people will test positive for HIV even though none of them has the virus.