 Solution

After a first step to $+1$, the walk moves away from zero with probability $3/4$ and towards it with probability $1/4$, so the previous result gives return probability $1/3$. After a first step to $-1$, reflect the line: movement away from zero has probability $2/3$ and towards zero $1/3$, giving return probability $1/2$. Weighting by the first-step probabilities gives