James McGill, age 68, was recently diagnosed with a particular type of brain tumor, and was referred to Dr. Mitchell Zylber, chief of surgery at the…

James McGill, age 68, was recently diagnosed with a particulartype of brain tumor, and was referred to Dr. Mitchell Zylber, chief of surgery at theuniversity hospital, for further evaluation. This type of tumor is benign in 50% ofcases and is malignant in the other 50% of cases. James McGill’s remaining lifetimewill depend on the type of tumor (benign or malignant) and on the decisionwhether or not to remove the tumor. Table 1.7 shows estimates of James McGill’sremaining lifetime according to the most up-to-date information known about thistype of tumor.Dr. Zylber could perform exploratory surgery prior to the decision whether ornot to remove the tumor, in order to better assess the status of the tumor. Exploratorysurgery is known to indicate a benign tumor 75% of the time, if the tumor is indeedbenign. The surgery is known to indicate a malignant tumor 65% of the time, if thetumor is indeed malignant. Exploratory surgery itself is dangerous: there is a 5%chance that patients with profiles like James McGill’s will not survive such surgerydue to complications from anesthesia, etc.If no exploratory surgery is performed, James McGill must decide whether ornot to have the tumor removed. And if exploratory surgery is performed, James mustdecide whether or not to have the tumor removed based on the results of the exploratory surgery.(a) Draw the decision tree for this problem.(b) What probabilities need to be computed in order to solve the decision tree?(c) After reading Chapter 2, compute the necessary probabilities, and solve for thedecision strategy that maximizes James McGill’s expected lifetime.(d) James McGill’s children are expected to have children of their own within thenext two or three years. Suppose James McGill wants to maximize the probability that he will live to see his grandchildren. How should this affect his decisionstrategy?(e) What ethical questions does this medical problem pose?

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