Assume you are interested in buying a used vehicle C_1. You are also considering of taking it to a qualified mechanic and then decide whether to buy it or not. The cost of taking it to the mechanic is $100. C_1 can be in good shape (quality q^+) or bad one (quality q^-). The mechanic might help to indicate what shape the vehicle is in. C_1 costs $3,000 to buy and its market value is $4,000 if in good shape; if not, $1, 400 in repairs will be needed to make it in good shape. Your estimate is that C_1 has a 70% chance of being in good shape. Assume that the utility function depends linearly on the vehicle's monetary value. a. Calculate the expected net gain from buying C_1, given no test. b. We also have the following information about whether the vehicle will pass the mechanic's test: P(pass(C_1)|q^+ (C_1)) = 0.8 P(pass(C_1)|q^- (C_1)) = 0.35 Use Bayes' theorem to calculate the probability that the car will pass/fail the test and hence the probability that it is in good/bad shape given what the mechanic will tell you. c. What is the best decision given either a pass or a fail? What is the expected utility in each case? d. What is the value of optimal information for the mechanic's test? Will you take C_1 to the mechanic or not