According to a 2018 article in Esquire magazine, approximately 70% of males over age 70 will develop cancerous cells in their bowel. Bowel cancer is second only to skin cancer as the most common form of cancer for males in New Zealand. One of the most common tests for the detection of bowel cancer is the bowel-specific antigen (BSA) test. However, this test is known to have a high false-positive rate (tests that come back positive for cancer when no cancer is present). Suppose there is a .04 probability that a male patient has bowel cancer before testing. The probability of a false-positive test is .85, and the probability of a false-negative (no indication of cancer when cancer is actually present) is .30.

A) What is the probability that the male patient has prostate cancer if the BSA test comes back positive?

B) What is the probability that the male patient has prostate cancer if the BSA test comes back negative?

C) For older men, the prior probability of having cancer increases. Suppose that the prior probability of the male patient is .35 rather than .04. What is the probability that the male patient has bowel cancer if the BSA test comes back positive? What is the probability that the male patient has bowel cancer if the BSA test comes back negative?

D) What can you infer about the BSA test from the results of parts (a), (b) and (c)?