Multiple illnesses produce similar symptoms. If illnesses A, B, and C produce id

Question

Multiple illnesses produce similar symptoms. If illnesses A, B, and C produce identical symptoms designated by U. It is also known that illnesses are mutually exclusive and there are no other illnesses with those symptoms. Suppose the probabilities of contracting those symptoms are p(A)=0.03, p(B)=0.01 and p(C)=.02. The chances of developing the symptoms are p(U/A)=0.85, P(U/B)=0.92 and p(U/C)=0.8. Suppose a person develops symptoms U.

(a) What is the probability that the person is afflicted with B?

(b) What is the probability that the person was healthy?

Explanation / Answer

P(A) = 0.03

P(B) = 0.01

P(C) = 0.02

P(U|A) = 0.85

P(U|B) = 0.92

P(U|C) = 0.8

Suppose a person develops symptoms U

P(U) = P(U|A)*P(A)+P(U|B)*P(B)+P(U|C)*P(C)

P(U) = 0.85*0.03+0.92*0.01+0.8*0.02 = 0.0255+0.0092+0.016 = 0.0507

a) the probability that the person is afflicted with B

P(B|U) = P(U|B)*P(B)/P(U) = 0.92*0.01/0.0507 = 0.1814

b) the probability that the person was healthy

P(healthy) = 1-P(U) = 1-0.0507 = 0.9493

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