Conditional Probability, Independence & Bayes · Challenge problem

Ann and Bea take turns shooting at a target, Ann first, and the first to hit it wins. Ann hits on each of her shots with probability $\tfrac13$ and Bea on each of hers with probability $\tfrac12$, independently. Given that Bea wins, what is the probability that she won with her very first shot? Express your answer as a common fraction.