Winning with any probability using a fair coin

You only have a fair coin. Describe a game, played with that coin, that you win with probability exactly $p$, where $p\in(0,1)$ is any given number.

Compare random binary digits with the binary expansion of $p$. Write $p=0.p_1p_2p_3\ldots$ in base $2$ and treat flip $i$ as a digit $s_i$ (heads $=1$). Stop at the first $i$ where $s_i\ne p_i$; you win if $s_i<p_i$. The flips form a uniform number $U$, you win exactly when $U<p$, and the game takes $2$ flips on average.