Question: Compute probabilities for the following compound events:
A physician has diagnosed a patient as having either hepatitis or liver cancer (but not both). Statistics reveal that hepatitis occurs in the general population twice as frequently as liver cancer. Thus, the physician tentatively concludes that the patient probably has hepatitis. Later the physician conducts a test on the patient that turns out positive. On this test, nine out of ten cases of liver cancer trigger a positive outcome, and one out of six cases of hepatitis triggers a positive outcome. What is the new probability that the patient has liver cancer?