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, confirming that he does have prostate cancer? equation editorEquation Editor (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? equation editorEquation Editor (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? equation editorEquation Editor (use four decimals)

Explanation / Answer

Since the probability of having cancer is given as 0.16,therefore the probability of not having cancer =1-0.16=0.84

Now,given false postive rate =0.16,therefore test negative and does not have cancer =1-0.16 =0.84

Howver the total probability of not having cancer is 0.84.

Therefore multiplying with 0.84.

P(Test negative and does no have cancer) = 0.84*0.84 =0.7056

P(Test Positive and Donot have cancer) =0.16*0.84 = 0.1344

Similarly given that the positive test is 0.79 for postrate cancer.Thus probability of test positive and having cancer =0.16*0.79 = 0.1264

and P(Cancer and test negative) =0.16*(1-0.79) = 0.0336

The two by two table of the information is shown above.

Now using the conditional probability

a)Probability that the PSA test will yield a positive result is given he has a cancer = 0.79*0.16 = 0.1264

A)Probability that he does not have prostate cancer given the test result is positive = 0.1344(see the above table)

c) Probability that he does not have a postrate cancer given that the test result is negaive =0.7056

Cancer No Cancer Total Test Positive 0.79*0.16 =0.1264 0.16*0.84=0.1344 0.2608 Test Negative 0.21*0.16=0.0336 0.84*0.84 =0.7056 0.7392 0.16 0.84

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