The MMPI-2 has been administered to both normal people and people suffering from schizophrenia. Suppose that a particular profile is representative of 95% of people with schizophrenia and only 5% of other people. Suppose further that schizophrenia occurs in only about 1 person out of every 100. Of all those with the schizophrenia-typical profile, about how many actually have schizophrenia?

Question

Quizplus · Accepted Answer

This question involves the use of Bayes' theorem, which requires us to estimate the proportion of true positives (people with schizophrenia who have the schizophrenia-typical profile) and false positives (people without schizophrenia who have the schizophrenia-typical profile). 
Let P(S) be the probability of schizophrenia, P(SP) be the probability of having the schizophrenia-typical profile, P(SP|S) be the probability of having the profile given a diagnosis of schizophrenia, and P(S|SP) be the probability of having schizophrenia given the profile. 
From the problem, we have:
P(SP|S) = 0.95 (95% of people with schizophrenia have the profile)
P(SP|~S) = 0.05 (5% of people without schizophrenia have the profile)
P(S) = 0.01 (schizophrenia occurs in only 1 person out of 100, i.e., 1%)
Using Bayes' theorem, we can calculate:
P(S|SP) = P(SP|S) * P(S) / (P(SP|S) * P(S) + P(SP|~S) * P(~S))
         = 0.95 * 0.01 / (0.95 * 0.01 + 0.05 * 0.99)
         = 0.16
This means that of all those with the schizophrenia-typical profile, only about 16% actually have schizophrenia. Therefore, the best choice is D, far fewer than 50%.

The MMPI-2 Has Been Administered to Both Normal People and People