Circular Ant I

We arrange $n$ ants around a circle at equally spaced increments, where $n$ is odd. All ants walk at a common constant speed that allows them to complete $1$ revolution per minute. We run an experiment where we randomly set each ant to face one of the two possible directions (clockwise or counterclockwise) with equal probability, and then have the ants walk for $1$ minute. Whenever two ants bump into each other, they both turn around and start walking in the opposite direction at the same constant speed. At the end of the minute the ants stop and we check their positions. What is the probability that all $n$ ants are in the same positions as they were originally when $n = 9$ ?

Answer format: round to 3 decimal places.