Prostate cancer is the most common form of cancer found in men. Many physicians routinely perform a PSA test, particularly for men older than 50. PSA is a protein produced only by the prostate gland and thus is easy to detect. Higher than normal reading of PSA level is potentially indicative of cancer. However, studies have shown that the test is not very accurate. In fact, the probability of having a high PSA level given that the man does not have cancer (false positive) is 13.5%. If the man does have cancer, the probability that the test gives a normal PSA level (false negative) is 30%. Suppose that you are a manager in a private health insurance company and you want to decide on whether your health insurance plans for elderly men should cover the cost of routine PSA tests. A key factor for your decision making is the cancer detection rate conditional on the PSA test result.

Assume that a study of prostate cancer incidence shows that the probability an elderly man develops prostate cancer is 10%.

Answer the following questions:

a) Calculate the probability that an elderly man has prostate cancer given that his PSA test result is positive. [8 marks]

b) Calculate the probability that an elderly man actually does not have prostate cancer given that his PSA test result is positive.