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Alice and Bob play the following game: in each round, Alicefirst rolls a single standard fair die. Bob then rolls a singlestandard fair die. If the difference between Bob’s roll and Alice'sroll is at most one, Bob wins the round. Otherwise, Alice wins theround.
(a) (5 points) What is the probability that Bob wins a singleround?
(b) (7 points) Alice and Bob play until one of them wins threerounds. The first player to three wins is declared the winner ofthe series. What is the probability that Bob wins the series?
(c) (7 points) In a single series, what is the expected numberof wins for Bob?
(d) (6 points) In a single series, how many more games is Aliceexpected to win than Bob? That is, what is the expected value ofthe number of wins for Alice minus the number of wins for Bob?
(e) (5 points) In a single series, what is the variance of theexpected number of wins for Bob?
