Conditional Probability, Independence & Bayes · Superchallenge problem

Ann and Bob each start with $\$2$. They play a sequence of independent rounds; Ann wins each round with probability $\tfrac23$, and the loser of a round pays the winner $\$1$. They stop when one of them has all $\$4$. Given that Ann ends up with all the money, what is the probability that she won the first round? Express your answer as a common fraction.