The level of PSA (prostate-specific antigen) in the blood is an indicator of pro

Question

The level of PSA (prostate-specific antigen) in the blood is an indicator of prostate cancer. However, it is possible to have elevated PSA levels and not have prostate cancer. Let T be the event that the test is positive and be the event that a man has prostate cancer. A test is a "true positive test" if it is positive and the man has cancer. You are given the following information: The probability that a man has cancer is 10^-5. True positive rate for the test = 0.9. False positive rate lord the test = 0.01. Find and P(C|T^c). Comment on the results.

Explanation / Answer

probabilty of cancer =P(C)=0.00001

P(T|C)=0.9 ; P(T|Cc) =0.01

hence probabilty of test positve =P(T)=P(C)*P(T|C)+P(Cc)*P(T|Cc)=0.00001*0.9+0.99999*0.01=0.010009

and test negative =P(Tc)=1-0.010009=0.989911

hence P(C|T)=P(C)*P(T|C)/P(T)=0.00001*0.9/0.010009=0.000899

and P(C|Tc)=P(C)*P(Tc|C)/P(Tc) =0.00001*0.1/0.989911=1.01*10-6

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