Help for this question please and have details For the combi

For the combined ELSIA-Western blot blood test for HIV positive status, the sensitivity is about 0.999 and the specificity is about 0.9999.Consider a high risk group has a positive lest result. Find the positive predictive value of the test.

Solution

Let

D = has disease
T = test positive

Here, note that

P(true positive) = P(D) P(T|D) / P(T)

As

P(T) = P(D) P(T|D) + P(D\') P(T|D\') = 0.10*0.999 + (1-0.10)*(1-0.9999) = 0.09999

P(true positive) = 0.10*0.999 / 0.09999 = 0.99909991 [ANSWER]