I was working on a question and I doubt the solution given is the right one. To simplify, consider a state space $S = \{1,2,3,4,5,6\}$ and an initial distribution $a^{(0)}=(1,0,0,0,0,0)$ with transition matrix P.
The question asks for the probability of reaching state 6 in at most 4 turns if we start from state 1. Their answer is given as the 6th element from $$a^{(4)}=a^{(0)}P^4$$.
I doubt this is the answer as it coincides with $P(X_4 = 6|X_{0}=1)$ which is not what we want. Could anyone help? Thanks!
I have added the actual problem where it looks for the probability of reaching the last state, 8, in at most 4 turns, starting from state 4. I am more interested in the method, but I added the actual question if people want to know it.
$P= \begin{bmatrix} 0 & 1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1/2 & 1/2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1/2 & 0 & 0 & 1/2 & 0 \\ 0 & 0 & 0 & 0 & 1/2 & 0 & 1/2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1/2 & 1/2 & 0 \\ 0 & 0 & 1/2 & 0 & 0 & 0 & 1/2 & 0 \\ 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 1/2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix} $ and $a^{(0)} = \begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{pmatrix}.$