A Canadian male has recently had a Prostate Specific Antigen (PSA) test as to de

Question

A Canadian male has recently had a Prostate Specific Antigen (PSA) test as to determine if he has prostate cancer. The false-positive rate of a PSA test is 16%. If he does have prostate cancer, PSA test will be positive 79% of the time. Because this male is showing symptoms that are consistent with prostate cancer, it is assumed that the chance he has prostate cancer prior to taking the PSA test is 0.16. Part (a) What is the probability that the PSA test will yield a positive result?(use four decimals in your answer) Part (b) If the PSA test gives a positive result, what is the probability that he does not have prostate cancer?(use four decimals) Part (c) Suppose the PSA test result is negative, indicating that he does not have prostate cancer and his symptoms are a result of something else. What is the probability that he does have prostate cancer? (use four decimals)

Explanation / Answer

a)probability that the PSA test will yield a positive result=P( have prostrate and PSA shows positive result+does not have and PSA shows positive result) =0.16*0.79+0.84*0.16=0.2608

b)probability that he does not have prostate cancer given  PSA test gives a positive result

=P( does not have and PSA shows positive result) /P( positive result)=0.84*0.16/0.2608 =0.5153

c)probability that the PSA test will yield a negative result =1-probability that the PSA test will yield a positive result

=1-0.2608 =0.7392

hence probability that he does have prostate cancer given test negative

=P( does have and test negative)/P( test negative)=0.16*(1-0.79)/0.7392=0.0455

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