3. An Elisa test is a standard test for HIV. 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 specificity (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) P{H|E, R}

(c) P{H|E, R}

(d) P{H|E, R}.

The low probabilities even after a positive test led to the development of a more expensive but higher-specificity follow up test, which is used after a positive Elisa test before the results are given to patients.