The ELISA test was approved by the U.S. government in the mid-1980s to screen donated blood for the presence of the AIDS virus. The test works by detecting antibodies, substances that the body produces when the virus is present, but it makes some mistakes. ELISA was designed so that when a given blood sample is in fact contaminated with AIDS, the test gives a positive result (that is, ELISA reports that in its opinion this blood sample has AIDS in it) 99% of the time, whereas when the blood being tested is not contaminated with the virus ELISA will announce a negative result 94% of the time. The prevalence of AIDS in the population of people who donate blood is thought to be about 1%.
Suppose someone comes in, donates blood, and the ELISA test comes out positive. Show that the probability the person actually has AIDS given this positive result is only about 14%!
The following hint might be useful.