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

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

where $0<\alpha<1$ and $0<\beta<1$.

Calculate $\mathbb{P}\left(X_{n}=0 \mid X_{0}=0\right)$ for each $n \geqslant 0$.