Question

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

(b) (7 points) 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) (7 points) In a single series, what is the expected number of wins for Bob?

(d) (6 points) In a single series, how many more games is Alice expected to win than Bob? That is, what is the expected value of the number of wins for Alice minus the number of wins for Bob?

(e) (5 points) In a single series, what is the variance of the expected number of wins for Bob?

Answer 1

Alice and Bob play the following games : in its 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 role is at most one, Bob wins the round. Otherwise Alice wins the round.

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