Puzzle of the Week #146 - Tetrahedral Ants

Four ants are positioned at the four corners of a tetrahedron (triangular-based pyramid). At once they all move along one of the edges to another corner, each choosing at random from the three other corners available.

What is the probability that the ants will perform this manoeuvre without any of them having to pass another coming the other way along the same edge or ending up at the same corner as another ant?