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Problem 10 Dr. Who is treating a terminally ill patient. If no further treatment is given to the patient, he will die in 6 months. There is an experimental single dose drug which can help live the patient longer. However, this drug may have fatal side-effects in certain patients leading to their immediate death after taking the drug. The patient is expected to live for 24 months if this drug works well and does not cause immediate death. The probability that the drug causes immediate death is 0.2. 1. Let U(.'L') denote the patient's utility function where L is the number of months to live. Assuming that U(24) = 1.0 and U(0) = 0, what should the upper bound on the patient's utility for living 6 months be and still have the option of experimental drug be preferred? 2. There is a less risky test procedure that will provide uncertain information that predicts whether or not the patient will have the fatal side effect after taking the drug. When this test is positive, the probability that the drug will work well for the patient is increased. The test has the following characteristics: ' True-positive rate: The probability that the results of this test will be positive if the patient will survive the drug is 0.9. ' False-positive rate: The probability that the results of this test will be positive if the patient will not survive the drug (i.e., the patient will die after taking the drug) is 0.10. What is the patient's probability of surviving the drug if the test is negative? 3. It turns out that the test procedure may have some fatal complications, i.e., the patient may die during the test. Draw a decision tree showing all the options and consequences of Dr. Who's problem. 4. Suppose that the probability of death during the test is 0.05 for the patient. Should Dr. Who advise the patient to have the test prior to deciding on the taking the experimental drug? Take (1(6) : 0.8