Probability & Stochastic Proc Q2

[7 marks] An ant moves around the following clock: The ant starts on the number 12 and, every minute, it moves one number clockwise with probability 1/2 or one number counter-clockwise with probability 1/2, inde- pendently of any previous move. The ant stops when it has visited every number around the clock. a) [3 marks] Let Tk the number of moves needed by the ant for reaching number k for the rst time (set To = T12 = 0). Show that 1119 10 ' (Hint: think of the gambler's ruin problem.) P(Tk_1