Question

In: Math

Canadian male has recently had a Prostate Specific Antigen (PSA) test as to determine if he...

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 14%. 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 0.17.

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)

Solutions

Expert Solution

P(false-positive) = P(tests positive | don't have cancer) = 0.14

P(tests positive | have cancer) = 0.79

P(has cancer) = 0.17

a) P(tests positive) = P(tests positive | have cancer) * P(have cancer) + P(tests positive | don't have cancer) * P(don't have cancer)

                               = 0.79 * 0.17 + 0.14 * (1 - 0.17)

                               = 0.2505

b) P(don't have cancer | tests positive) = P(tests positive | don't have cancer) * P(don't have cancer) / P(tests positive)

                                                              = 0.14 * (1 - 0.17) / 0.2505

                                                              = 0.4639

c) P(tests negative | have cancer) = 1 - P(tests positive | have cancer) = 1 - 0.79 = 0.21

P(have cancer | tests negative) = P(tests negative | have cancer) * P(have cancer) / P(tests negative)

                                                   = 0.21 * 0.17 / (1 - 0.2505)

                                                   = 0.0476

milcah answered 1 year ago
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