Someone took an HIV test, and it came back positive. Then he was told by the doc

ID: 3073082 • Letter: S

Question

Someone took an HIV test, and it came back positive. Then he was told by the doctor that 999 out

of 1000 chance he got infected with the AIDS virus, since the HIV test produced a positive result

when the blood was not infected with the AIDS virus in only 1 in 1,000 blood samples. Assume that

on average 1 out 10,000 people who got tested is infected with HIV. Is the doctor’s reasoning sound?

Why? You could assume that the chance that a person who is infected with the AIDS virus tests

negative is approximately 0.

P(infected | P) = P(P | infected)*P(infected) / (P(P | infected)*P(infected) + P(P | not infected)*P(not infected))

P(P | infected) = 999/1000 = 0.999
P(infected) = 1/10000 = 0.0001
P(P | not infected) = 1/1000 = 0.001
P(not infected) = 0.9999

P(infected | P) = 0.999*0.0001 / (0.999*0.0001 + 0.001*0.9999) = 0.09083

The above calculated value indicates the probability of a person is infected from a virus given that the test is positive.

As this value is very low, doctor's reasoning does not sound correct.

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