what is the probability that the person actually has AIDS?
If we administer the test to a randomly selected American (who may or may not have AIDS) and the person receives a positive test result, what is the probability that the person actually has AIDS? This probability is
The idea behind Bayes’ Theorem is that we can find P(AIDS | POS) by thinking as follows. A person will receive a positive result (POS) if the person receives a positive result and actually has AIDS—that is, (AIDS ∩ POS)—or if the person receives a positive result and actually does not have AIDS—that is, . Therefore,
This implies that
This probability says that, if all Americans were given an AIDS test, only 38 percent of the people who would react positively to the test would actually have AIDS. That is, 62 percent of Americans identified as having AIDS would actually be free of the disease! The reason for this rather surprising result is that, since so few people actually have AIDS, the majority of people who would receive a positive result are people who are free of AIDS and who erroneously test positive. This is why statisticians have frequently spoken against proposals for mandatory AIDS testing.
In the preceding example, there were two states of nature—AIDS and —and two outcomes of the diagnostic test—POS and
. In general, there might be any number of states of nature and any number of experimental outcomes. This leads to a general statement of Bayes’ Theorem.
Bayes’ Theorem
Let S1, S2,…, Sk be k mutually exclusive states of nature, one of which must be true, and suppose that P(S1), P(S2),…, P(Sk) are the prior probabilities of these states of nature. Also, let E be a particular outcome of an experiment designed to help determine which state of nature is really true. Then, the posterior probability of a particular state of nature, say Si, given the experimental outcome E, is
where
We illustrate Bayes’ Theorem in the following case.