Conditional probability for virus and test checking

Question

A person undergoes an HIV test which :

a)is totally accurate when tested on people with HIV

b)has 4% false alarm rate

The person tested positive to the test. This person belongs in a particular group for which, $1$ in $250$ people are expected to contract the virus.

What are the chances that this person has HIV ?

I have problem extracting the data but here is my try:

Let's say that we have 2 events

$S:sick$ and $P:positive$

So, $P(P|S)=0.96$ and $P(P'|S)=0,04$

We also get $P(P)=1/250$.

Am i correct? Any help would be appreciated.

Answer 1

$P(HIV|+) = \frac{P(HIV+)}{P(HIV+)+P(No\ HIV|+)} = \frac{.004}{.004+.03984} = .09124$

For me it helps to construct a table like the one below so I'm less likely to be confused by percentages of different categories :

Answer 2

No. According to a), $\mathsf P(P\mid S)=1$. According to b), $\mathsf P(P\mid\overline S)=0.04$. (This is if you equate having HIV with being sick, which is medically inaccurate, as you can carry the HIV virus but not exhibit AIDS symptoms; a more accurate term would be “infected”).