A new computerized scan of the liver is used to help identify whether a person has a normal liver (N), benign tumors (B), or liver cancer (C). The computer will automatically classify the output to indicate tumors (T) or no tumors (T). For a normal liver, the computer will indicate that there are NO tumors with probability 0.98. For a benign tumor, the computer will indicate that there are tumors with probability 0.4. For a cancerous liver, the computer will indicate that there are tumors with probability 0.5. If the test is repeated, the scan will observe the liver at a different angle, and the results will be conditionally independent given the patient's true condition (ie., N, B, or C). A healthy patient is one who has no other indications of liver tumors or cancer before having the computerized scan. Such a patient has

probability 0.99 of having a normal liver,

probability 0.009 of having benign liver tumors,

and probability 0.001 of having liver cancer.

(a) What is the probability that a scan of a healthy patient will indicate a tumor?

(b) For a healthy patient, if the computerized scan indicates a tumor, what is the probability that the patient: Has liver cancer? Has benign tumor? Has a healthy liver?

(c) For a healthy patient, if the scan detects a tumor, find the ML and MAP decisions.

(d) If a healthy patient has two computerized liver scans, what is the probability that both indicate tumors?

(e) If a healthy patient has two liver scans and the first one indicated tumors, what is the probability that tumors will also be identified on the second scan?

(f) If a healthy patient has two computerized scans and both indicate tumors, what is the probability that the patient has cancer?

(g) Consider now the case of a single computerized scan, but where other tests have indicated that a patient has a high probability of having liver cancer or benign tumors. Based on these other tests, a doctor believes that the probability that the patient's liver is normal is 0.2. Consider the MAP decision rule if a tumor is detected in the scan. What is the smallest value for the probability that the patient has liver cancer that would result in that MAP decision being that cancer is present?