Medium

Probability

Consider $n$ people $P_1,\dots, P_n$. $P_1$ receives binary information "yes" or "no" and will pass this information along to $P_2$. More generally, $P_i$ will pass information along to $P_{i+1}$. However, $P_i$ will transfer the information that they hear with probability $p$ and transfer the opposite information with probability $1-p$, where $0 < p < 1$, independent of all other people. Let $A_i$ be the event that $P_i$ transfers the original information (i.e. what $P_1$ received) to $P_{i+1}$, and let the probability of this be $p_i$. Compute $\displaystyle \lim_{n \rightarrow \infty} p_n$. If this limit does not exist, answer $-1$.

Notes

Hint