Surgery: A patient is very sick and will die in 6 months if he goes untreated. The only available treatment is risky surgery. The patient is expected to live for 12 months if the surgery is successful, but the probability that the surgery will fail and the patient will die immediately is 0.3.
a. Draw a decision tree for this decision problem.
b. Let v(x) be the patient's payoff function, where x is the number of months until death. Assuming that v(12) = 1and v(0) = 0, what is the lowest payoff the patient can have for living 3 months so that having surgery is a best response?
For the rest of the problem, assume that v(3) = 0.8.
c. A test is available that will provide some information that predicts whether or not surgery will be successful. A positive test implies an increased likelihood that the patient will survive the surgery as follows: True-positive rate: The probability that the results of this test will be positive if surgery is to be successful is 0.90. False-positive rate: The probability that the results of this test will be positive if the patient will not survive the operation is 0.10. What is the probability of a successful surgery if the test is positive?
d. Assuming that the patient has the test done, at no cost, and the result is positive, should surgery be performed?
e. It turns out that the test may have some fatal complications; that is, the patient may die during the test. Draw a decision tree for this revised problem.
f. If the probability of death during the test is 0.005, should the patient opt to have the test prior to deciding on the operation?