Three Ants on a Triangle and the Collision Probability

Three ants on a triangle collide with probability three quarters, since avoiding a collision requires all three to walk the same rotational direction.

Three ants placed on the three corners of a triangle, each independently choosing to walk along one of the two edges leading away from its corner at random, collide with probability three quarters. The only way to avoid a collision is for all three ants to walk around the triangle in the same rotational direction, an event with probability one quarter.

Place one ant on each of the three vertices of a triangle. Each ant independently picks one of the two edges touching its own vertex, with equal probability one half for each, and walks along that edge to the adjacent vertex, with all three ants moving at the same speed and starting at the same instant. Two ants collide if they meet somewhere along an edge, which happens whenever two ants choose to walk toward each other along the same edge. The question asks for the probability that at least one collision occurs somewhere on the triangle.

Label the vertices 1, 2 and 3 arranged clockwise. Each ant effectively makes a binary choice: walk clockwise, meaning toward the next vertex in clockwise order, or walk counterclockwise, meaning toward the previous vertex. With three ants each making one of two independent choices, there are 2 times 2 times 2, or 8, equally likely combinations of directions overall.

A collision occurs on a given edge whenever the two ants at its endpoints walk toward each other along that edge. If every ant walks clockwise, all three glide around the triangle in the same rotational sense and no two ants ever share an edge heading toward each other, so there is no collision. The identical argument applies if every ant walks counterclockwise. Any other combination of directions necessarily has at least one pair of ants walking toward each other along a shared edge.

Out of the 8 equally likely combinations of individual directions, exactly 2 result in no collision: all three ants clockwise, or all three ants counterclockwise. Every one of the remaining 6 combinations mixes clockwise and counterclockwise choices among the three ants, and any such mix forces at least two ants onto a shared edge walking toward each other, guaranteeing a collision somewhere on the triangle.