Testing for HIV: An article in New York Times some time ago reported that college students are beginning to routinly ask to be tested for the AIDS virus. The standard test for the HIV virus is the Elias test. It is estimated that this test has a 99.8% sensitivity and a 99.8% specificity. (99.8% sensitivity means: for every 1000 people tested who have the virus we can expect 998 people to test positive and 2 have a false negative test. 99.8% specificity means: 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).
a) The New York 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 that this student has the HIV virus? b) What whould this probability be for a population at high risk, where 5% of the population has the HIV virus? c) Suppose Jack tested positive on an Elias test. Another Elias test is performed and the results are positive again. Assuming that the tests are independent, what is the probability that Jack has the HIV virus?
Your questions are quite standard. They are actually discussed in The Power of Logical Thinking. Look for the Bayes' theorem. I am pretty sure that a search on google will directly land you on HIV examples.