Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 2 / 11F / e / Solution

Now the event is to hit $20$ before $1$, starting at $3$. Up to the first of those two hitting times, translate the interval by one unit; its endpoints become $0,19$ and the starting point becomes $2$. The gambler's ruin hitting probability is thereforeStopping at $1$ is essential: using the original absorbing endpoint $0$ would answer a different event.