The following formulas are also provided to save you the trouble of looking them up elsewhere: . Binomial coefficient (# ) n! = k!(n - k)! . Inclusion-exclusion principle P(AUB) = P(A) + P(B) - P(An B) . Chain rule P(An B) = P(A B) P(B) . Odds P(A) P(A) Odds(A) =- P(A) 1- P(A) . Conditional probability P(AB) = P(An B) P(B) . Bayes' theorem P(E Hi) P(Hi) P(HE) = P(E Ho) P(Ho) + P(E H1) P(H1) + ... + P(EH,) P(H,.) . Information content I(A) = - 10g2 P(A) = 10g2(1/P(A)) . Entropy H(X) = - > P(x;) log2 P(xi) i= 12. [18 points) You are preparing for an oral exam on which you will be asked to solve a single problem. This problem can be one of two types: theory (T) or application (A). Before the exam, you estimate that the probability of a theory problem is 0.6 and the probability of an application problem is 0.4. [You might recall this setup from an entropy problem on the midterm.) (a) In study sessions, you have been able to solve 80% of theory problems and 70% of appli- cation problems. Based on this, what is the probability that you will be able to solve the problem? (b) You have been studying with a friend who has been able to solve 50% of theory problems and 90% of application problems. Your friend takes the exam rst, and they tell you that they were not able to solve the problem. Based on this, what is the (posterior) probability that it was a theory problem? (c) Based on your answer to part (b), what is the [posterior predictive) probability that you will be able to solve the