Consider the Markov chain $\left(X_{n}\right)_{n \geq 0}$ on the integers $\mathbb{Z}$ whose non-zero transition probabilities are given by $p_{0,1}=p_{0,-1}=1 / 2$ and

$$\begin{gathered} p_{n, n-1}=1 / 3, \quad p_{n, n+1}=2 / 3, \quad \text { for } n \geq 1 \\ p_{n, n-1}=3 / 4, \quad p_{n, n+1}=1 / 4, \quad \text { for } n \leqslant-1 \end{gathered}$$

(a) Show that, if $X_{0}=1$, then $\left(X_{n}\right)_{n \geq 0}$ hits 0 with probability $1 / 2$.

(b) Suppose now that $X_{0}=0$. Show that, with probability 1 , as $n \rightarrow \infty$ either $X_{n} \rightarrow \infty$ or $X_{n} \rightarrow-\infty$.

(c) In the case $X_{0}=0$ compute $\mathbb{P}\left(X_{n} \rightarrow \infty\right.$ as $\left.n \rightarrow \infty\right)$.