One half percent of the population has a particular disease. A test is developed for the disease. The test gives a false positive 3% of the time and a false negative 2% of the time. (a). What is the probability that Joe (a random person) tests positive? (b). Joe just got the bad news that the test came back positive; what is the probability that Joe has the disease?

Answera) 0.035b) 0.14Step-by-step explanation:Let J be the event that Joe has the disease. Let The be the event that Joe's test is positive. Pr(J) = 1/2% = 0.5/100 = 0.005Pr(J') = 99.5% = 0.995Pr(T|J) = 98% = 0.98 since 2% if the time if a person having the disease is omitted ("false negative ")Pr(T|J') = 3% = 0.03 since there are 3 false positivesa( Pr(T) = us the probability that Joe tests positive Pr(T) = Pr(T|J)* P(J) + Pr(T|J')*Pr(J') = (0.98*0.005) + (0.03*0.995) = 0.00049 + 0.02985 = 0.03475 = 0.035b) Pr( J|T) = Pr(JnT) / Pr(T) = (Pr(T|J)*Pr(J)) / Pr(T) = (0.005*0.98) / 0.035 = 0.0049/0.035 = 0.14