Hello Everyone I am having trouble understanding how to use bayes theorem in this problem:
Suppose a physician assesses the probability of HIV in a patient who engages in risky behavior (unprotected sex with multiple partners of either sex, or sharing injection drug needles) as .002, and the probability of HIV in a patient who does not engage in those risky behaviors as .0001. Also suppose the Elisa test has a sensitivity (probability of having a positive reading if the patient has HIV) of .99, and a specicity (probability of having a negative reading if the patient does not have HIV) of .99 and does not depend on whether the patient has engaged in risky behavior. Let E stand for engages in risky behavior," H stand for has HIV," and R stand for positive Elisa result." Use Bayes Theorem to compute each of the following:
(a) P(H|E,R), (b)...Ill solve this on my own...
How do I use the theorem for a? Thank you in advance!
$$\begin{align} \Pr(H\mid E, R) =& \frac{\Pr(H, E, R)}{\Pr(E,R)}\\ =& \frac{\Pr(H, E, R)}{\Pr\left(H,E,R\right) + \Pr(\overline H,E,R)}\\ =& \frac{\Pr(R\mid H,E)\Pr(H,E)}{\Pr(R\mid H, E)\Pr(H,E) + \Pr(R\mid\overline H,E)\Pr(\overline H,E)}\\ =& \frac{\Pr(R\mid H,E)\Pr(H\mid E)\Pr(E)}{\Pr(R\mid H, E)\Pr(H\mid E)\Pr(E) + \Pr(R\mid\overline H,E)\Pr(\overline H\mid E)\Pr(E)}\\ =& \frac{\Pr(R\mid H)\Pr(H\mid E)}{\Pr(R\mid H)\Pr(H\mid E) + \Pr(R\mid\overline H)\Pr(\overline H\mid E)}\\ \end{align}$$
and it's your turn to substitute appropriate numbers in this formula.
If you practice much, actually you can solve this kind of question solely using definitions and formulae.