A particle moves among n + 1 vertices that are situated on a circle in the following manner: At each step it moves one step either in the clockwise direction with probability p or the counterclockwise direction with probability q 1-p. Starting at a specified state, call it state 0, let T be the time of the first return to state 0.

Find the probability that all states have been visited by time T. Hint: Condition on the initial transition and then use results from the gambler's ruin problem.