# HIV screening probabilities

Assume that we have the same amount of females and males and they go through a screening process. At the end it turns out that 5% men are positive, and 0.25% women are positive. If a person is picked randomly and that person is HIV positive what is the probability that he is a male?

Is the way to think about this and do it something along the lines of:

Firstly there are the same amount of females and males so the probability of picking a male is 0.5. Next, within that group of males, 5% are positive. So our probability of picking a HIV positive male is 2.5%?

• If a person is HIV positive, then the probability that he's a male is $\frac{5}{5+0.25}$ and the probability that she's a female is $\frac{0.25}{5+0.25}$. – barak manos Sep 13 '15 at 21:36

Let $M$: "The person selected is a man", $p(M)=p(\overline{M})=\frac{1}{2}$. Let P:The person is HIV positive. Then $p(P)=p(M)p_{M}(P)+p(\overline{M})p_{\overline{M}}(P)=\frac{1}{200}(5+0.25)$, and $p(P \cap M)=2.5$%. Hence the probability $p_{P}(M)=\frac{p(P \cap M)}{p(P)}$