Let $\{X_n\}$ be a Markov chain with transition probabilities, as shown in the figure. If $X_0=5$, using first step analysis or whatever method you prefer, evaluate the probability that in an even number of steps, the chain abandons the communication class it begins from.
Attempt. The communication class that contains state $5$ is $\{4,5,6\}$, which is open (leading to classes $\{1,2,3\},~\{7,8\}$). For the first step analysis, if $T=\inf\{k\geq 0:~X_k\notin \{4,5,6\}\}$ the first time our chain leaves the above class, we should have:
$$P(T \in 2\mathbb{N}~|~X_0=5)=\sum_{i\in I}P(E_i)~P(T \in 2\mathbb{N}~|~X_0=5,~E_i)$$ for some proper sets $E_i,~i\in I$ that partition the prob. space. I am having difficulty to find these intermediate sets though.
Thank you in advance!