Anonymous

### Question Description

Alice and Bob play the following game:

In each round, Alice first rolls a single standard fair die. Bob then rolls a single standard fair die. If the difference between Bob’s roll and Alice’s roll is at most one, Bob wins the round. Otherwise, Alice wins the round.

(a) What is the probability that Bob wins a single round?

(b) Alice and Bob play until one of them wins three rounds. The first player to three wins is declared the winner of the series. What is the probability that Bob wins the series?

(c) In a single series, what is the expected number of wins for Bob?

I have solved (a) to get 17/36, but I am not sure how to go about solving (b) and (c).

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