Bayes’s rule was applied to the problem of HIV testing in The American Statistician (Aug. 2008). In North America, the probability of a person having HIV is .008. A test for HIV yields either a positive or negative result. Given that a person has HIV, the probability of a positive test result is .99. (This probability is called the sensitivity of the test.) Given that a person does not have HIV, the probability of a negative test result is also .99. (This probability is called the specificity of the test.) The authors of the article are interested in the probability that a person actually has HIV given that the test is positive.
a. Find the probability of interest for a North American by using Bayes’s rule.
b. In East Asia, the probability of a person having HIV is only .001. Find the probability of interest for an East Asian by using Bayes’s rule. (Assume that both the sensitivity and specificity of the test are .99.)
c. Typically, if one tests positive for HIV, a follow-up test is administered. What is the probability that a North American has HIV given that both tests are positive? (Assume that the tests are independent.)
d. Repeat part c for an East Asian.