PROBLEM 1 (MoRE PROBABILITY RULES FROM SET THEORY) 1. In class we saw the inclusion-exclusion rule that states that for any two events A and B, we have B). Generalize the inclusion-exclusion rule to three sets and Pr(A U B) = Pr(A) + Pr(B)-Pr(A show that, for any three events A, B, C, we have Pr(A U B U C) Pr(A) + Pr(B) + Pr(C) _ Pr(A n B) _ Pr(A n C)-Pr(Bn C) + Pr(A n B n C). 2. Let A and B be two events. The event that exactly one of A and B occurs is the event (AnB)U(An B). Show that Pr( (An B) U (An B)) = Pr(A) + Pr(B) _ 2 Pr(A nB) 3. Let A and B be two events. Show that Pr(AnB) 2 Pr(A) +Pr(B) 4. A partition of the sample space is a collection of disjoint events E1, E2. . . . , En such that -U-1 Ei. For example, the events 1,2,4), 13,6), (5] partition the sample space [1,2,3,4,5,6) Show that for every event A, we have 1 Pr(A) = Pr(ANE). 5. Show that for any events A, B, C, we have Pr(A) = Pr(A n B) + Pr(Anc) + Pr(AnBnc) _ Pr(A n B n C)