Stand on a large clock, say on the 1. Now flip a coin and move ahead one hour if the coin turns up heads, and back one hour otherwise. Keep repeating the process until you’ve stood on all 12 numbers. How long, on average, will this random walk take? If you generalize to clocks with p positions, how does the expected time vary with p?
Here’s a little Python code to experiment with the problem.
from random import random p = 12 def time_to_cover_circle(): circle = [0 for i in range(p)] count, pos = 0, 0 while True: count += 1 pos += 1 if random() > 0.5 else -1 pos %= p circle[pos] = 1 if min(circle) == 1: return count