1. Bayes' Rule Assume the probability of having tuberculosis (TB) is 0.4%. 10% of TB tests result in false negatives and 5% of TB tests result in false positives. Let TB denote the event that one has tuberculosis, and let P denote the event that a TB test is positive. a. Find the probability of having TB if one tests positive for the disease. b. What is the probability that a person has TB if she has 2 positive TB tests in a row? (Assume that each test is statistically independent.) Interpret the result in part (b) compared to that in part (a) c. Suppose the government wants the accuracy of a TB test to be the same as the accuracy of 2 positive TB tests in a row (as in part b). Can the government meet this objective by reducing the occurrence of a false negative? d. Suppose the government wants the accuracy of a TB test to be the same as the accuracy of 2 positive TB tests in a row (as in part b). Can the government meet this objective by reducing the occurrence of a false positive? 2. Projection Bias a. Suppose that John will purchase some amount of coffee tomorrow (at time t+1) to maximize the following utility function: U(C++1|St+1) = 5(C++1 2St+1) - Where C++1 is the amount of coffee purchased at time t+1 and S++1 is how sleepy John is at time t+1. Solve for the optimal amount of coffee that John will purchase tomorrow. b. Suppose John purchases coffee today to be consumed tomorrow to maximize the following utility function: U(Ct+1|St, St+1) = 2(C++1 st (1 )St+1) Where a [0,1]. The amount of coffee purchased for tomorrow depends on how sleepy John is today and how sleepy he is tomorrow (we assume that John forecasts with perfect accuracy). Solve for the optimal amount of coffee that John will purchase today for tomorrow. c. How do we interpret the result in part b if a = 1?