Suppose we send a sequence of $n$ bits $X=\{x_1,...,x_n\}$ through a channel. Each $x_k=1$ with probability $p$, and $x_k=0$ w.p. $1-p$. The channel flips one input bit with probability $q$. If $Y=\{y_1,...,y_n\}$ is the output of the channel, then
How would you define the conditional probability $P(Y|X)=P(y_1,...,y_n|x_1,...,x_n)$ in terms of $p$ and $q$?
I'm pretty sure there is a simple form but I can't see it.