Multicoin I

Question

$n \geq 2$ people labelled $1-n$ play a game where player $1$ starts with a coin that has probability $0 < p < 1$ of showing heads per flip. The first player to flip a head wins. If player $i$ lands a tails, then they pass the coin to player $i+1$ for their turn to flip. Once the coin reaches player $n$, the coin circles back around to the start for player $1$ to flip again, and the cycle repeats. Find the probability that player $1$ wins when $n = 5$ and $p = 1/3$.