Noisy yes or no message passed down a long chain

A yes/no message is handed to person $P_1$, who passes it to $P_2$, who passes it to $P_3$, and so on along a chain $P_1,\dots,P_n$. Each person, independently of everyone else, relays what they heard faithfully with probability $p$ and relays the opposite with probability $1-p$, where $0<p<1$.

Let $p_n$ be the probability that what $P_n$ passes on matches the message $P_1$ originally received. Find $\lim_{n\to\infty}p_n$, or answer $-1$ if the limit does not exist.

$\tfrac12$. Conditioning on whether $P_n$ heard the right message gives $p_n=p\,p_{n-1}+(1-p)(1-p_{n-1})$, so $p_n-\tfrac12=(2p-1)\left(p_{n-1}-\tfrac12\right)$. Since $|2p-1|<1$, the deviation from $\tfrac12$ shrinks geometrically to zero.