Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain with state space $\{1,2\}$ and transition matrix

$$P=\left(\begin{array}{cc} 1-\alpha & \alpha \\ \beta & 1-\beta \end{array}\right)$$

where $\alpha, \beta \in(0,1]$. Compute $\mathbb{P}\left(X_{n}=1 \mid X_{0}=1\right)$. Find the value of $\mathbb{P}\left(X_{n}=1 \mid X_{0}=2\right)$.