Suppose I have a connected, finite, graph $G = (V,E)$, and I have some vertex set $U$ such that the subgraph of $G$ induced by the vertices $Y = V \setminus U$ is still connected.
Now, suppose I have a random walk $X_n$ on $G$, that follows the transition probabilities induced by the edge weights of $G$. Let $X_0 = u \in Y$. I want to know how I can go about calculating $P_u(X_1, \dots, X_k \in Y)$.
I keep running into something like $\prod_{i=1}^{k-1}P_u(X_{i+1} \in W | X_{i}\in W)$, which I don't know how to simplify either.
To simplify the problem as much as possible, you can combine all vertices in $U$ into a single vertex $x$. Each edge from $v \in Y$ to a vertex in $U$ becomes an edge from $v$ to $x$; if you get parallel edges this way, you should either keep them both, or else combine them by adding the weights.
Next, turn the random walk into an asymmetric Markov chain by making $x$ an absorbing state. This will make it easy to tell when we've left $Y$: if we do, we go to $x$, and then we stay in $x$ forever.
Now, the complementary probability (the probability that you get from your starting vertex $u$ to $x$ within $k$ steps) can be computed by taking powers of the transition matrix of this Markov chain, as usual.