The HIV test produced a positive result when the blood was not infected with the AIDS virus (false positive) in only 1 in 1,000 blood samples. So the doctor concluded Mlodinow has a 1 in 1,000 chance of being healthy, when his test came out positive.
The problem is, his doctor confused two things:
(1) the chance that Mlodinow would test positive if he was not HIV-positive, with
(2) the chances that he would not be HIV-positive if he tested positive.
The first was indeed 1 in 1,000.
But to determine the second, more information is needed. Mlodinow is white, heterosexual, and does not abuse drugs. According to statistics from the Centers for Disease Control and Prevention, about 1 in 10,000 heterosexual non-IV-drug-abusing white male Americans who got tested were infected with HIV.
Consider a population of 10,000 such people. About 1 in the 10,000 was actually infected with HIV. Assume that the false-negative rate (infected by HIV but the test came out negative) is near 0. That means about 1 person from the population will test positive due to presence of the infection (true positive). Since the rate of false positives is 1 in 1,000, there will be 10 others who are not infected with HIV but test positive anyway (false positive). The other 9,989 will test negative (true negative).
So there are 10 false positives and 1 true positive. Only 1 out of the 11 who test positive are really infected with HIV. The answer to the second question is 10 out of 11. Not 1 out of 1,000. There is a big difference. Some basic understanding of probability is really quite useful, both in daily life, and as a professional.
This case was in The Drunkard's Walk. It is really a fascinating and useful book.