Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / M / M5 / a / iii

After the first step to $+1$, the right-hand walk moves away with probability $3/4$ and toward zero with probability $1/4$. Part (ii) therefore gives return probability $1/3$ from that side. After a first step to $-1$, reverse the sign of the coordinate: movement away from zero has probability $2/3$ and toward it probability $1/3$, giving return probability $1/2$. Weight by the probabilities of the first step from zero:There is no holding step at zero, so every positive-time return is included by this decomposition.