A random person is selected from a large population in which 5% are users of a dangerous illegal drug.A drug test that correctly identifies users 99% of the times and nonusers 95% of the time (this percentage is called "specificity" of the test)is administered to this individual and gives a positive result.What specificity would actually be required to conclude that the probability that this individual is actually a user of this drug is 0.75?
A) 0.965
B) 0.973
C) 0.978
D) 0.983
E) 0.985

Question

Quizplus · Accepted Answer

Let P(U) be the probability that the selected person is a user, and let P(+) be the probability of a positive test result. We want to find the specificity, denoted by S, such that P(U | +) = 0.75. By Bayes’ theorem,
P(U | +) = P(+ | U) P(U) / P(+)
where
P(+ | U) = 0.99, P(+ | U^c) = 1 – specificity = 0.05, 
P(U) = 0.05, P(U^c) = 0.95
P(+) = P(+ | U) P(U) + P(+ | U^c) P(U^c)
Plugging in the given values,
P(+) = (0.99)(0.05) + (0.05)(0.95) = 0.059
Now, we solve for S:
0.75 = (0.99)(0.05) / 0.059Solve for S, we get S = 0.983. 
Therefore, the answer is D.

A Random Person Is Selected from a Large Population in Which