Consider a Markov chain with $S={1, 2, 3, 4}$ and transition Matrix: $P=\begin{bmatrix} 0 & 1/2 & 1/2 & 0 \\ 0 & 0 & 1/2 & 1/2 \\ 1/2 & 0 & 0 & 1/2 \\ 1/2 & 1/2 & 0 & 0 \\ \end{bmatrix}$
We start in state 3. What is the chance that from this state that we reach state 1 earlier than state 4?
I should first make matrix $P*$, the absorption matrix.
$P^*=\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1/2 & 1/2 \\
1/2 & 0 & 0 & 1/2 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}$
Yet, my answers say we should make this rowvector: $f=\begin{bmatrix} 1\\ x \\ y \\ 0 \\ \end{bmatrix}$ Where $x$ is the chance that we start in state 3 and reach 1 earlier than 4 and $y$ the state that we start in 2 and reach 1 earlier than 4. However, I do not understand why $f(2)=x$ and $f(3)=y$, shouldn't that be reversed?