A particle moves among n+1 vertices which are situated on the circle in the following manner. At each step it moves one step either in the clockwise direction with the probability p or the counterclockwise direction with the probability q=1-p. Starting at the specified state, call it state 0, let T be the time of first return to state 0. Find out the probability that all states have been visited by time T.
Hint: Condition on the initial transition as well as then employ results from the gambler's ruin problem.