First return to equal heads and tails with a biased coin

A coin lands heads with probability $p\in(0,1)$. You keep tossing until, for the first time after the start, the number of heads equals the number of tails. Let $f(p,n)$ be the probability this first happens at toss $2n$ ($n\ge1$). Evaluate $f(1/3,5)$ to four decimal places.

About $0.0152$. The first tie at toss $2n$ requires $n$ heads and $n$ tails with one side strictly ahead until the end; by the ballot theorem this is a $\tfrac{1}{2n-1}$ fraction of balanced paths, so $f(p,n)=\tfrac{1}{2n-1}\binom{2n}{n}p^n(1-p)^n$. At $p=\tfrac13$, $n=5$: $\tfrac{252}{9}\left(\tfrac29\right)^5\approx0.0152$.