An article in the New York Times some time ago reported that college students are beginning to routinely ask to be tested for the AIDS virus. The standard test for the HIV virus is the Elias test, which tests for the HIV antibodies. It is estimated that this has a 99.8% sensitivity and a 99.8% specificity. A 99.8% sensitivity means that, in a large-scale screening test, for every 1000 people tested who have the virus we can expert 998 people to test positive and 2 to have a false negative test. A 99.8% specificity means that, in a large-scale screening test, for every 1000 people tested who do not have the virus we can expert 998 people to have a negative test and 2 to have a false positive test.

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?

What would this probably be for a population at high risk, where 5% of the population has the HIV virus?

A scientist designed a medical test for a certain disease. Among 100 patients who have the disease, the test will show the presence of the disease in 97 cases out of 100, and will fail to show the presence of the disease in the remaining 3 cases out of 100. Among those who do not have the disease, the test will erroneously show the presence of the disease in 4 cases out of 100, and will show there is no disease in the remaining 96 cases out of 100.

What is the probability that a patient who tested positive on this test actually has the disease, if it is estimated that 20% of the population has the disease?

What is the probability that a patient who tested positive on this test actually has the disease, if it is estimated that 4% of the population has the disease?