1. Question 1 A partition of the sample space 51 is a collection of disjoint events E1, E2, . . . , En such thatn=ur=1e =E1UE2UH-UEW (i) Show that for any event A, we have n. PM) = 213mm E.) 5:1 {ii} Use part (i) to show that for any events A, B,and C we have PM) = PMB) +P(A c) +P(A 38(1ch P(ABnC) 2. Question 2 Seven people are in an elevator which stops at ten oors. (a) In how many ways can they get off the elevator '? (a) 71\" (bl 1'37 (C) (10% (1-?) (b) In how many ways can they get off the elevator if no two people get o' at the same oor? (a) 71\" (bl 1'37 (6) (10b (17") (c) Now consider the situation when no two people get off at the same oor. Before the elevator begins to travel, each of them pushes a button for his or her oor. In how many ways can the elevator buttons be lighted? (a) 7'\" (b) 107 (c) (10): (17") (give explanation for all your answer} 3. Question 3 Alice and Bob are taking a probability course. The course has only three grades: A, B, and C. The probability that Alice gets a B is 0.3. The probability that Bob gets a B is 0.4. The probability that neither gets an A but at least one gets a B is Ill. What is the probability that at least one gets a B but neither gets a C? 4. Question 4 A computing center has 3 processors that receive 11. jobs, with the jobs assigned to the processors purely at random so that all of the 3\" possible assignments are equally likely. Find the probability that exactly one processor has no jobs