The University of California at Berkeley's Wellness Encyclopedia (1991) contains the following statement in its discussion of HIV testing: "In a high-risk population, virtually all people who test positive will truly be infected, but among people at low risk the false positives will outnumber the true positives. Thus, for every infected person correctly identified in a low-risk population, an estimated ten noncarriers [of HIV] will test positive" (p. 360). Suppose that you have a friend who is part of this low-risk population but who has just tested positive.

a. Using the numbers in the statement above, what is the probability that your friend actually carries the virus?

b. Understandably, your friend is upset and doesn't believe that the probability of being infected with HIV isn't really near 1. After all, the test is accurate, and it came out positive!

Explain to your friend how the quoted statement can be true even though the test is very accurate both for people with HIV and for people who don't carry it. If it's easier, you can make up numbers to put in a table to support your argument.