The probability thet a noncancerous man will have an elevatedPSA level is approx

Question

The probability thet a noncancerous man will have an elevatedPSA level is approximately .135, increasing to approximately .268if the man does have cancer. If, on the basis of other factors, aphysician is 70% certain that a male has prostate cancer, what isthe conditional probability that he has cancer given that a) the test indicated an elevated PSA level? b) the test did not indicate an elevated PSA level? The probability thet a noncancerous man will have an elevatedPSA level is approximately .135, increasing to approximately .268if the man does have cancer. If, on the basis of other factors, aphysician is 70% certain that a male has prostate cancer, what isthe conditional probability that he has cancer given that a) the test indicated an elevated PSA level? b) the test did not indicate an elevated PSA level?

Explanation / Answer

Let C= has cancer; C'=does not have cancer;  E=test shows elevate PSA; E'=PSA not elevated P(E/C')=.135; P(E/C)=.268; P(C)=.7; P(C')=1-.7=.3 P(E)=P(E/C)P(C)+P(E/C')P(C')=.268(.7)+.135(.3)=.228 a)P(C/E)=P(E/C)P(C)/P(E)=.268(.7)/.228=.823 b)P(C/E')=P(E'/C)P(C)/P(E') Now    P(E')=1-P(E)=.772 .andP(E'/C)=1-P(E/C)=.732 P(C/E')=(.732)(.7)/.772=.664

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