John and Bill play a guessing game. In each round, the probability that John guesses the answer correctly is p_A and the probability that Bill guesses correctly is p_B. Suppose that Jogn and Bill guess independently of each other and also independently from round to round. They keep playing until at least one of them guesses correctly.

(a)Find the probability that the game ends exactly after n rounds?

(b)Find the probability that the game ends with John guessing the answer (i.e. A guesses correctly before

Bill guesses correctly or both guess correct at the same time.)