What is the expected value of this imperfect information/ Computer Science
(This is based on a problem written by John Toczek.) Joe is sick. He may have a cold, the flu, or a bacterial infection. The probability of a cold is 0.5, the probability of flu is 0.2, and the probability of a bacterial infection is 0.3. If his illness is a cold, Joe will miss five days of work, which will cost him $500 (lost income). If his illness is the flu, Joe will miss 8 days of work, which will cost him $800 (lost income). If his illness is a bacterial infection (and he does not treat it), Joe will miss 12 days of work, which will cost him $1200 (lost income). Joe (who is not sure what is wrong with him) has to decide whether or not to go to the doctor. The cost to go to the doctor will be $200. If his illness is a cold or the flu, he will still miss work (the number of days missed and the lost income will not be reduced). However, if he has a bacterial infection, the doctor will prescribe antibiotics, which will cost $50, but he will miss only 2 days of work, which costs Joe $200 in lost income. Before deciding to go to the doctor, Joe can take his temperature, which may be “normal” or “high.” Joe knows that, if he has a cold, then the temperature will certainly be “normal”; if he has the flu, the temperature will certainly be “high”; if he has a bacterial infection, the temperature will certainly be “high.” Thus, taking his temperature is imperfect information about his illness. If it were “high,” the probability of flu is 0.4, and the probability of a bacterial infection is 0.6. What is the expected value of this imperfect information (taking his temperature)?
