Problem 2.5 (Video 1.5, 1.6, 2.1, 2.2, Quick Calculations) Calculate each of the requested quantities. (a) Let A and B be independent events with P[A] = 1/5 and P[B] = 1/4. Calculate P[An B] and P[A U B]. (b) Let A1, A2, A3 be events that are conditionally independent given B. Additionally, assume that A1, A2, A3 are conditionally independent given Bc. Assume that P[A; | B] = 1/4 and P[A; |B"] = 1/2 for i = 1, 2, 3 and P[B] = 1/3. Calculate P[AinAnA3| B] and P[AinASnA3]. (c) Consider a packet of jellybeans that contains 9 jellybeans, of which 4 are lemon and the remaining 5 are raspberry. You reach in and pull out 3 jellybeans. What is the probability that they are all lemon? What is the probability that they are all raspberry? (d) Let X be a random variable with PMF Px () = 1/6 x= -1, +1 2/3 x=0 . Calculate P[X * 0] and P[X > 0|X * 0]. 0 y