A Random Walk with Absorbing Walls

A robot plays a shooting game: a made shot scores $+1$, a miss scores $-1$. The game ends when the cumulative score $X$ reaches $2$ (a win) or $-2$ (a loss); otherwise play continues indefinitely. Each shot is made with probability $\alpha$ (where $\dfrac{1}{2} \leqslant \alpha < 1$), independently.

1. Find the probability that the robot wins after exactly $4$ shots. 2. Let $A_n$ be the event that after $n$ shots the game has not yet ended.

1. If $\alpha = \dfrac{1}{2}$, find $P(A_2 \mid A_1)$ and $P(A_{2k+2} \mid A_{2k+1})$ for $k \in \mathbb{N}^*$.
2. If $P(A_{2k+2} \mid A_{2k}) = \dfrac{4}{9}$ for all $k \in \mathbb{N}^*$, find the expected value of $X$ when the game ends.