I have a Markov chain with state space $E = \{1,2,3,4,5\}$ and transition matrix below:
$ \begin{bmatrix} 1/2 & 0 & 1/2 & 0 & 0 \\\ 1/3 & 2/3 & 0 & 0 & 0 \\\ 0 & 1/4 & 1/4 & 1/4 & 1/4 \\\ 0 & 0 & 0 & 3/4 & 1/4 \\\ 0 & 0 & 0 & 1/5 & 4/5\ \end{bmatrix} $
How would I find the conditional probabilities of $\mathbb{P}(X_2 = 5 | X_0 =1)$ and $\mathbb{P}(X_3 = 1 | X_0 =1)$?
I am trying to use the formula (or any other formula, if anyone knows of any) $p_{ij}^{(n)} = \mathbb{P}(X_n = j | X_0 =i)$, the probability of going from state $i$ to state $j$ in $n$ steps.
So $\mathbb{P}(X_2 = 5 | X_0 =1) = p_{15}^2$, so I read the entry in $p_{15}$, and get the answer is $0^2$, but the answer in my notes say it is $1/8$?
Also, I get for $\mathbb{P}(X_3 = 1 | X_0 =1) = p_{11}^3 = (\frac{1}{2})^3 = 1/8$, but the answer says it is $1/6$?