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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%?

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  • $\begingroup$ 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 $\frac{0.25}{5+0.25}$. $\endgroup$ – barak manos Sep 13 '15 at 21:36
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We can breakdown the information into a table as follows:

enter image description here

You are given that the subject is HIV positive and there are 5.25 of them. Out of this group you want to know what is the probability of picking a male. This is simply calculated as 5/5.25 = 0.95 (to 2 d.p.)

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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 $p_{P}(M)=\frac{p(P \cap M)}{p(P)}$

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