Marching ants collision probability

[Guest]

Suppose there are ants at each vertex of a triangle and they all simultaneously crawl along a side of the triangle to the next vertex. The probability that no two ants will encounter one another is 2/8, since the only two cases in which no encounter occurs is when all the ants go left, i.e., clockwise -- LLL -- or all go right, i.e., counterclockwise -- RRR. In the six other cases -- RRL, RLR, RLL, LLR, LRL, and LRR -- an encounter occurs.

Now suppose that, analogously, there is an ant at each vertex of a polyhedron and that the ants all simultaneously move along one edge of the polyhedron to the next vertex, each ant choosing its path randomly. For each of the following polyhedra, what is the probability that no two ants will encounter one another, either en route or at the next vertex? Express your answer reduced to lowest common denominators, e.g., 2/8 must be reduced to 1/4.

A tetrahedron
A cube.
A regular octahedron
A regular dodecahedron