Let $\mathcal{X}=(X_n:n\in\mathbb{N}_0)$ denote a Markov chain with state space $E=\{1,\dots,5\}$ and transition matrix
$P=\pmatrix{1/2&0&1/2&0&0\\1/3&2/3&0&0&0\\0&1/4&1/4&1/4&1/4\\0&0&0&3/4&1/4\\0&0&0&1/5&4/5}$
Compute the probabilities $\mathbb{P}(X_2=5|X_0=1)$ and $\mathbb{P}(X_3=1|X_0=1)$.
Given an initial distribution $\pi=(1/2,0,0,1/2,0)$, compute $\mathbb{P}(X_2=4)$.
I've got the transient states as $1,2,3$. And the recurrent states as $4,5$, and the communication classes I think are $\{1,2,3\}$ and $\{4,5\}$.
1) To calculate $\mathbb{P}(X_2 = 5|X_0 = 1)$, is it just finding $P^2_{(1,5)}$? Which equals $1/8$?
2) For $\mathbb{P}(X_3 = 1|X_0=1)$, I tried finding $P^3_{(1,1)}$ which I got $1/24$. Is that correct?
3) For finding $\mathbb{P}(X_2=4)$, do I just take $π(P^2)$?