There is a screening test for prostate cancer that looks at the level of PSA (prostate specific antigen) in the blood. There are a number of reasons besides prostate cancer that a man can have elevated PSA levels. In addition, many types of prostate cancer develop so slowly that they are never a problem. Unfortunately, there is currently no test to distinguish the different types and using the test is controversial because it is hard to quantify the accuracy rates and the harm done by false positives. For this problem we’ll call a positive test a true positive if it catches a dangerous type of prostate cancer. We’ll assume the following numbers: Rate of prostate cancer among men over 50 = 0.0005; True positive rate for the test = 0.9; False positive rate for the test = 0.01. Let T be the event a man has a positive test and let D be the event a man has a dangerous type of the disease. Find P(D|T) and P(D|T̅) (Hint: Draw a tree diagram).
There is a screening test for prostate cancer that looks at the level of PSA (prostate specific antigen) in the blood. There are a number of reasons besides prostate cancer that a man can have elevated PSA levels. In addition, many types of prostate cancer develop so slowly that they are never a problem. Unfortunately, there is currently no test to distinguish the different types and using the test is controversial because it is hard to quantify the accuracy rates and the harm done by false positives. For this problem we’ll call a positive test a true positive if it catches a dangerous type of prostate cancer. We’ll assume the following numbers: Rate of prostate cancer among men over 50 = 0.0005; True positive rate for the test = 0.9; False positive rate for the test = 0.01.
Let T be the
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