Suppose we assume that 5% of the people are drug-users. A test is 95% accurate, which we'll say means that if a person is a user, the result is positive 95% of the time; and if she or he isn't, it's negative 95% of the time. A randomly chosen person tests positive. Is the individual highly likely to be a drug-user?
Marilyn's answer was:
Given your conditions, once the person has tested positive, you may as well flip a coin to determine whether she or he is a drug-user. The chances are only 50-50.
How can Marilyn's answer be correct?
The standard test for the HIV virus is the Elias test that tests for the presence of HIV antibodies. It is estimated that this test has a 99.8% sensitivity and a 99.8% specificity. 99.8% specificity means that, in a large scale screening test, for every 1000 people tested who do not have the virus we can expect 998 people to have a negative test and 2 to have a false positive test. 99.8% sensitivity means that for every 1000 people tested who have the virus we can expect 998 to test positive and 2 to have a false negative test.
The Times article remarks that it is estimated that about 2 in every 1000 college students have the HIV virus. Assume that a large group of randomly chosen college students, say 100,000, are tested by the Elias test. If a student tests positive, what is the chance this student has the HIV virus? What would this probability be for a population at high risk where 5% of the population has the HIV virus?
If a person tests positive on an Elias test, then two more Elias tests are carried out. If either is positive then one more confirmatory test, called the Western blot test, is carried out. If this is positive the person is assumed to have the HIV virus. In calculating the probability that a person who tests positive on the set of four tests has the disease, is it reasonable to assume that these four tests are independent chance experiments?