First, note that there can only ever be at most three rounds, because after those three rounds, someone will have won twice.

I'm going to write these possibilities as "winner for each round" (and N for not played)

AAN

ABA

ABB

BAA

BAB

BBN

Which of these possibilities would have A winning the game? What is the probability of each of those possibilities? What would the probability therefore be of A winning the game?