Fair 50-50 outcome from a coin of unknown bias

A coin comes up heads with some unknown probability $p$ strictly between $0$ and $1$. Using only this coin, how can you produce an outcome that is exactly 50-50?

Toss twice: HT means one outcome, TH the other, and on HH or TT start again. Both HT and TH have probability $p(1-p)$, so they are equally likely (von Neumann's trick). Each round succeeds with probability $2p(1-p)$, so the expected number of tosses is $\tfrac{1}{p(1-p)}$.