Question

There are 550,000 people in the US infected with HIV. Of these people, 275,000 are drug users, and
the rest are not drug users. The total population of the US is 250 million. There are 10 million drug
users in the US. The standard blood test for HIV infection is not always accurate. The probability that
someone who is infected with HIV will test positive for HIV is 0.99. The probability that someone who is
not infected with HIV will test negative for HIV is also 0.99. Answer the following questions, clearly
stating any assumptions that you need to make.

Suppose that a randomly chosen person takes the standard blood test for HIV, and the outcome of the
test is positive. What is the probability that this person is infected with HIV? Is your answer
surprising?

Answer 1

From the given data, the following Tables are calculated:

Table 1:

	Infected with HIV	Not infected with HIV	Total
Drug user	275,000	9,725,000	10,000,000
Not drug user	275000	239,725,000	240,000,000
Total	550,000	249,450,000	250,000,000

Table 2:

	Infected with HIV	Not infected with HIV	Total
Test positive	550,000 X 0.99 = 544,500	2,494,500	3,039,000
Test negative	5,500	249,450,000 X 0.99 = 246,955,500	246,961,000
Total	550,000	249,450,000	250,000,000

P(Infected with HIV/ Test positive) = P(Infected with HIV AND Test positive)/ P( Test positive)

                                                = 544,500/3,039,000

                                                 = 0.1792

So,

Answer is:

0.1792

Our answer is surprising because we expect a high probability for a person to be infected with HIV if the outcome of the
test is positive. But, we got the probability value of 0.1792 only.