Mr. X, aged 85, was recently diagnosed with a particular type of brain tumor and was referred to Dr. Y, chief of surgery at the nearby university hospital, for further evaluation. This type of tumor is benign in 60% of cases and is malignant in the other 40% of cases. Mr. X’s remaining lifetime will depend on the type of tumor (benign or malignant) and on the decision whether or not to remove the tumor. The table below shows the estimates of Mr. X’s remaining lifetime according to the most up-to-date information known about this type of tumor. Tumor Leave tumor Remove tumor Benign 8 5 Malignant 1 5 Dr. Y could perform exploratory surgery before deciding whether or not to remove the tumor to assess the status of the tumor better. Exploratory surgery indicates a benign tumor 75% of the time if the tumor is indeed benign. The surgery is known to indicate a malignant tumor 65% of the time if the tumor is indeed malignant. Exploratory surgery itself is dangerous: there is a 5% chance that patients with profiles like Mr. X’s will not survive such a surgery due to complications from anesthesia, etc. If no exploratory surgery is performed, Mr. X must decide whether or not to have the tumor removed. And if exploratory surgery is performed, Mr. X must decide whether or not to remove the tumor based on the exploratory surgery results. a) Draw the decision tree for this problem, and find the optimal strategy. (Hint: You must use both prior and posterior probabilities in the decision tree.) State explicitly the optimal policy and its expected value. b) Mr. X’s children are expected to have their own children within the next two or three years. Suppose Mr. X wants to maximize the probability that he will live to see his grandchildren. How should this affect his decision strategy? c) What ethical questions does this medical problem pose?
Mr. X, aged 85, was recently diagnosed with a particular type of brain tumor and was referred to Dr. Y, chief of surgery at the nearby university hospital, for further evaluation. This type of tumor is benign in 60% of cases and is malignant in the other 40% of cases. Mr. X’s remaining lifetime will depend on the type of tumor (benign or malignant) and on the decision whether or not to remove the tumor. The table below shows the estimates of Mr. X’s remaining lifetime according to the most up-to-date information known about this type of tumor. Tumor Leave tumor Remove tumor Benign 8 5 Malignant 1 5 Dr. Y could perform exploratory surgery before deciding whether or not to remove the tumor to assess the status of the tumor better. Exploratory surgery indicates a benign tumor 75% of the time if the tumor is indeed benign. The surgery is known to indicate a malignant tumor 65% of the time if the tumor is indeed malignant. Exploratory surgery itself is dangerous: there is a 5% chance that patients with profiles like Mr. X’s will not survive such a surgery due to complications from anesthesia, etc. If no exploratory surgery is performed, Mr. X must decide whether or not to have the tumor removed. And if exploratory surgery is performed, Mr. X must decide whether or not to remove the tumor based on the exploratory surgery results. a) Draw the decision tree for this problem, and find the optimal strategy. (Hint: You must use both prior and posterior probabilities in the decision tree.) State explicitly the optimal policy and its expected value. b) Mr. X’s children are expected to have their own children within the next two or three years. Suppose Mr. X wants to maximize the probability that he will live to see his grandchildren. How should this affect his decision strategy? c) What ethical questions does this medical problem pose?
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] Mr. X, aged 85, was recently diagnosed with a particular type of brain tumor and was referred to Dr. Y, chief of surgery at the nearby university hospital, for further evaluation. This type of tumor is benign in 60% of cases and is malignant in the other 40% of cases. Mr. X’s remaining lifetime will depend on the type of tumor (benign or malignant) and on the decision whether or not to remove the tumor. The table below shows the estimates of Mr. X’s remaining lifetime according to the most up-to-date information known about this type of tumor.
Tumor Leave tumor Remove tumor
Benign 8 5
Malignant 1 5
Dr. Y could perform exploratory surgery before deciding whether or not to remove the tumor to assess the status of the tumor better. Exploratory surgery indicates a benign tumor 75% of the time if the tumor is indeed benign. The surgery is known to indicate a malignant tumor 65% of the time if the tumor is indeed malignant. Exploratory surgery itself is dangerous: there is a 5% chance that patients with profiles like Mr. X’s will not survive such a surgery due to complications from anesthesia, etc.
If no exploratory surgery is performed, Mr. X must decide whether or not to have the tumor removed. And if exploratory surgery is performed, Mr. X must decide whether or not to remove the tumor based on the exploratory surgery results.
a) Draw the decision tree for this problem, and find the optimal strategy. (Hint: You must use both prior and posterior probabilities in the decision tree.) State explicitly the optimal policy and its expected value.
b) Mr. X’s children are expected to have their own children within the next two or three years. Suppose Mr. X wants to maximize the probability that he will live to see his grandchildren. How should this affect his decision strategy?
c) What ethical questions does this medical problem pose?
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