Medical case histories indicate that different illnesses may produce identical symptoms. Suppose a particular set of symptoms, which we will denote as event H, occurs only when any one of three illnesses-A, B, or C-occurs. (For the sake of simplicity, we will assume that illnesses A, B, and C are mutually exclusive.) Studies show these probabilities of getting the three illnesses:
P(A) = .01
P(B) = .005
P(C) = .02
The probabilities of developing the symptoms H, given a specific illness, are
P(H | A) = .90
P(H | B) = .95
P(H | C) = .75
Assuming that an ill person shows the symptoms H, what is the probability that the person has illness A?