Probability of absorption in Markov chain Given the probability matrix $P$ with states $s_1...s_5$ where $s_1$ and $s_5$ are absorbing
$$
P =
\left[
\begin{matrix}
1 & 0.7 & 0 & 0 & 0 \\
0 & 0 & 0.5 & 0 & 0 \\
0 & 0.3 & 0 & 0.65 & 0 \\
0 & 0 & 0.5 & 0 & 0 \\
0 & 0 & 0 & 0.35 & 1
\end{matrix}
\right]
$$
I am trying to find the probability of the process starting at $s_2...s_4$ being absorbed by one of the absorbing states given infinite steps. Since $s_1$ and $s_5$ are absorbing states, after infinite steps the process will end up in either of the states so the sum of the two probabilities must equal one, but how can we find the distribution of the two states?

Let $p_i$ be the probability that the process is eventually absorbed by $s_1$ after starting at $s_i$. Then $p_1=1$, $p_5=0$ and
\begin{align}
p_2&=0.7p_1+0.3p_3\;,\\
p_3&=0.5p_2+0.5p_4\;,\\
p_4&=0.65p_3+0.35p_5\;.
\end{align}
This system of three linear equations in three unknowns has a unique solution.
