Medical case histories indicate that different illnesses

QUESTION:

Medical case histories indicate that different illnesses may produce identical symptoms. Suppose that a particular set of symptoms, denoted , occurs only when any one of three illnesses, \(I_{1}\), \(I_{2}\), or \(I_{3}\), occurs. Assume that the simultaneous occurrence of more that one of these illnesses is impossible and that

          \(P\left(I_{1}\right)=.01\),     \(P\left(I_{2}\right)=.005\),     \(P\left(I_{3}\right)=.02\).

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

          \(P\left(H \mid I_{1}\right)=.90\),     \(P\left(H \mid I_{2}\right)=.95\),     \(P\left(H \mid I_{3}\right)=.75\).

Assuming that an ill person exhibits the symptoms, , what is the probability that the person has illness \(I_{1}\)?

Equation Transcription:

Text Transcription:

I_1

I_2

I_3

P(I_1)=.01

P(I_2)=.005

P(I_3)=.02

P(H|I_1)=.90

P(H|I_2)=.95

P(H|I_3)=.75

I_1