Consider a random walk $(X_n)$ on the graph below, where we jump from a given vertex to one of its adjacent vertices with equal probability. I want to find:
- the probability that we hit $A$ before hitting $F$, given that $X_0=B$
- the expected first passage time from $B$ to $F$ without hitting $A$.
Thoughts
I thought it might help to modify the state space into $A,X,Y,F$ where $X=\{B,C\}$ and $Y=\{E,D\}$ (I did not clump all four together because getting to $A$ from $B$ is different to getting there from $E$ or $D$).
With some help from the comments, I get the following transition matrix
$$\begin{pmatrix} 0 & 1 & 0 & 0 \\ \frac13 & \frac13 & \frac13 & 0\\ 0 & \frac13 & \frac13 & \frac13\\ 0 & 0 & 1 & 0\end{pmatrix}$$ (states in order $A,X,Y,F$). So we want: $$\mathbb{P}_B(\text{hit}\,A\,\text{before}\,H)=\sum_{k\geqslant 1}\mathbb{P}_B\left(X_k=A\bigcap_{m<k} X_m\neq A,F\right)$$ But I'm stuck on how to evaluate this? It feels like I'm using the wrong method altogether.