Aparticle 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.