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

Person infected with HIV = 550,000 / 250,000,000 = 0.0022

	Positive	Not Positive	Total
HIV	0.002178	0.000022	0.0022
Not HIV	0.009978	0.987822	0.9978
Total	0.012156	0.987844	1

P (HIV|Positive) = 0.002178 /  0.012156 = 0.1792

Results is very surprising that only 17.92 % which are showing positive results have actually HIV.

Please revert back in case of any doubt.