Random Ants

The Problem

Three ants are placed at the three vertices of an equilateral triangle. Each ant picks a direction (clockwise or counterclockwise) uniformly at random and starts walking along the perimeter at unit speed.

1. What is the probability that no ants collide?
2. Generalize: $n$ ants on a regular $n$-gon, each picking a direction uniformly at random. What is $\mathbb{P}(\text{no collision})$?
3. If the ants walk at different speeds (each ant's speed chosen uniformly from {1, 2, \ldots, k\}), how does the answer change?