A particle moves among n 1 vertices that arc situated on a

A particle moves among n + 1 vertices that arc 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. Stating 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.

Solution