Finite Hitting Time Between Communicating Components

Consider a Markov chain on $\{1, 2, 3, 4, 5\}$ with transition matrix $$P = \begin{pmatrix} \tfrac{1}{2} & \tfrac{1}{2} & 0 & 0 & 0 \\ \tfrac{1}{3} & 0 & \tfrac{2}{3} & 0 & 0 \\ 0 & \tfrac{1}{4} & 0 & \tfrac{1}{2} & \tfrac{1}{4} \\ 0 & 0 & 0 & \tfrac{1}{2} & \tfrac{1}{2} \\ 0 & 0 & 0 & \tfrac{1}{3} & \tfrac{2}{3} \end{pmatrix}.$$

Let $B = \{4, 5\}$ and $T_B = \inf\{n \ge 0 : X_n \in B\}$.

(a) Show that $E[T_B \mid X_0 = i] < \infty$ for all $i \in \{1, 2, 3\}$.

(b) Compute $E[T_B \mid X_0 = 1]$.