I drew a tree diagram but I can't get the right answer listed below. Can anyone show me how to do this?
The New York State Health Department reports a 10% rate of the HIV virus for the “at-risk” population. Under certain conditions, a preliminary screening test for the HIV virus is correct 95% of the time.
a. If someone is randomly selected from the at-risk population, what is the probability that they have the HIV virus if it is known that they have tested positive in the initial screening?
b. If someone is randomly selected from the at-risk population, what is the probability that they have the HIV virus if it is known that they have tested negative in the initial screening?
The answer for a is 0.6786 and b is 0.0058
Let $T$ be the event that the randomly selected person tested positive, $H$ be the event that the randomly selected person actually has HIV. Then:
a) $P(H|T) = \dfrac{P(T|H)P(H)}{P(T|H)P(H) + P(T|H^c)P(H^c)}$
and
b) $P(H|T^c) = \dfrac{P(T^c|H)P(H)}{P(T^c|H)P(H) + P(T^c|H^c)P(H^c)}$
where $H^c$ is the compliment of the event $H$ (in words, $H^c$ is the event that the randomly selected person does not actually have HIV), and similarly for $T$ and $T^c$.
To help find some of these terms, I think the question expects you to interpret the statement that the screening is correct $95\%$ of the time to mean that: Given a random person with HIV, the test is positive with probability $.95$; given a random person without HIV, the test is falsely positive with probability $.05$. (This is quite a presumptive assumption and the question truly should have given more info about false positive readings and such since the "correct $95\%$ of the time" seems to be a bit ambiguous with interpretation). Anyway, with this, the statement "given that the randomly selected person has HIV, the test is positive with probability $.95$" is symbolically given by $P(T|H) = .95$. Similarly, the statement "given a random person without HIV, the test is falsely positive with probability $.05$" is symbolically given by $P(T|H^c) = .05$.
NOTE: In general $P(A|B) \neq 1- P(A|B^c)$!!! We are just forcing this coincidence here given the ambiguity of the "correct $95\%$ of the time" statement.
Can you finish from here...?