Who gets the first head with a one-third coin

Bill and Bob each toss their own coin at the same moments, once per round; each coin shows heads with probability $\tfrac13$ on every toss. What is the probability that Bill's first head arrives on a strictly earlier toss than Bob's first head?

$\tfrac25$. Both waiting times are i.i.d. geometric with $p=\tfrac13$. They tie with probability $\sum_n p^2(1-p)^{2n-2}=\tfrac{p}{2-p}=\tfrac15$, and by symmetry the remaining $\tfrac45$ splits evenly between Bill first and Bob first.