Splitting Probability and Expected Hitting Time with Two Targets

Consider a Markov chain on $\{0, 1, 2, 3, 4\}$ with transition probabilities for interior states: $$p(1,0) = \tfrac{1}{2}, \quad p(1,2) = \tfrac{1}{2}, \quad p(2,1) = \tfrac{1}{3}, \quad p(2,3) = \tfrac{2}{3}, \quad p(3,2) = \tfrac{1}{4}, \quad p(3,4) = \tfrac{3}{4}.$$ States $0$ and $4$ are absorbing. Let $T = \inf\{n \ge 0 : X_n \in \{0, 4\}\}$.

(a) Starting from state $2$, compute $P(X_T = 4 \mid X_0 = 2)$.

(b) Compute $E[T \mid X_0 = 2]$.