Medical case histories indicate that different illnesses may

Medical case histories indicate that different illnesses may produce identical symptoms. Suppose that a particular set of symptoms, denoted H , occurs only when any one of three illnesses, I1 , I2 , or I3, occurs. Assume that the simultaneous occurrence of more that one of these illnesses is impossible and that

P(I1) = .01, P(I2) = .005, P(I3) = .02.

The probabilities of developing the set of symptoms H , given each of these illnesses, are known to be

P (H|I1) = .90, P (H|I2) = .95, P (H|I3) = .75

Assuming that an ill person exhibits the symptoms, H, what is the probability that the person has illness I1?

Solution

P(H) = 0.90 * 0.01 + 0.95 * 0.005 + 0.75 * 0.02

P(H) = 0.02875

P( I1 / H) = 0.01 * 0.90 / 0.02875 = 0.3130