A screeding test which measures the level of a prostate specic antigen (PSA) is

Question

A screeding test which measures the level of a prostate specic antigen (PSA) is a commonly used tool for the detection of prostate cancer. Men with PSA levels greater than 10 (ng/ml) have on average a chance of 67 % of prostate cancer, whereas men whose PSA levels are between 4 and 10 have on average a 25 % chance of having prostate cancer. For those whose PSA levels are below 4 %, the average risk of developing prostate cancer is only 5 %. Suppose that 15 % of men have PSA levels greater than 10, 10 % of men have PSA levels between 4 and 10, and 75 % of men have PSA levels lower than 4.

(a) What is the probability that a randomly chosen man will develop prostate cancer? (b) What is the probability that a randomly chosen man has a PSA level greater than 10, given that he has prostate cancer?

Explanation / Answer

A

probability that a randomly chosen man will develop prostate cancer = 0.15*67 + 0.1*25 + 0.75*4 = 15.55%

B

probability that a randomly chosen man has a PSA level greater than 10, given that he has prostate cancer
= 0.15*67 / [0.15*67 + 0.1*25 + 0.75*4]
= 64.63%

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