Nine Bumper Bots on a Loop

Nine numbered bumper bots sit at nine equally spaced marks on a circular track. At the bell, each bot independently picks clockwise or counterclockwise (fair coin) and drives at a constant speed of exactly one lap per minute. Whenever two bots meet head-on, both instantly reverse direction at full speed. What is the probability that after exactly one minute, every bot is standing on its own starting mark? What does the answer become with ten bots?