PROBLEM 1 (Correlated Risk). Jimmy is applying to college. His grades are good, but he can't tell whether the admissions officers will think his essay is good or bad. He knows that all admissions officers think alike, so that if one thinks his essay is good, they all will think it's good. Other than that, school decisions are independent from one another. At school Q, if his essay is good, he will be admitted with probability 0.8, and if his essay is bad, he will be admitted with probability 0.5. At school R, if his essay is good, he will be admitted with probability 0.6, and if his essay is bad, he will be admitted with probability 0.2. Assuming his essay is a priori equally likely to be good or bad... (a) What is the chance he will be admitted to at least one of Q or R? (b) He is now considering applying to safety schools 51 and 52. At 51, he will be admitted with proba bility 1 if his essay is good and probability 0.8 if his essay is bad. At 52, he can elect not to submit an essay at all and will be admitted with probability 0.9 based on his grades. To maximize the chance of being admitted to at least one school, is it better for him to apply to {(2, R, 51}, or to {Q, R, 52}? (c) Given that he gets into R, what is the chance his essay is good? (d) Given that he gets into both Q and R, what is the chance his essay is good