Binomial and Geometric · Superchallenge problem

A man keeps two matchboxes, one in each pocket, each starting with $5$ matches. Whenever he needs a match he chooses a pocket at random (each with probability $\dfrac12$, independently) and takes a match from that box. Eventually he reaches into a pocket and finds that box empty. What is the probability that at that moment the other box is also empty? Express your answer as a common fraction.