Question

(1) Suppose that the rate of HIV infection in the population is 0.1%. An HIV test is 99.8% accurate when administered to infected individuals, and 99.9% accurate when administered to non-infected individuals. Suppose that this test is administered to Patient X, and has a positive outcome, indicating that Patient X is infected. What is the probability that Patient X actually is infected with HIV?

Answer 1

Let us denote the events

A : a randomly selected individual is actually infected with HIV

B : a randomly selected individual tested positive

Given

That implies,

Given that patient X is infected, the probability that patient X actually is infected with HIV

$=P(A|B)$

$=\frac{P(A\cap B)}{P(B)}$

$=\frac{P(A\cap B)}{P(A\cap B)+P(A^c\cap B)}$

$=\frac{P(B|A)*P(A)}{P(B|A)*P(A)+P(B|A^c)*P(A^c)}$

$=\frac{(0.998*0.001)}{(0.998*0.001)+(0.001*(1-0.001))}$

$=\frac{(0.998*0.001)}{(0.998*0.001)+(0.001*0.999)}$