449.Whisper Down the Lane

$n$ people stand in a line. $P_1$ receives a binary message ("yes" or "no") and passes it to $P_2$, who passes it to $P_3$, and so on. Each person independently transmits the message they heard with probability $p$ and the opposite message with probability $1-p$, where $0 < p < 1$. Let $p_n$ be the probability that $P_n$ passes along the original message. Find $\lim_{n \to \infty} p_n$.