Markov Chains & Long-Run Behavior · Medium problem

An absorbing chain has two transient states $A$ and $B$. From $A$ the chain moves to $B$ with probability $1/2$ and is otherwise absorbed; from $B$ it moves to $A$ with probability $1/3$ and is otherwise absorbed. Starting at $A$, what is the expected number of visits to $B$ before absorption? Express your answer as a common fraction.