Ants and Triangle

There are 3 ants sitting on three corners of a triangle. All ants randomly pick a direction and start moving along the edge of the triangle. What is the probability that any two ants collide?

Solution to metapuzzle: show

For the metapuzzle, you should assume

"All ants randomly pick" means "each ant randomly and independently picks"
"A direction" means "one of the two edges meeting at that corner"
When an ant reaches the next corner, it stops. It does not continue along the next edge as a real ant might
When two ants reach each other on the same edge, they collide. They don't just walk past each other as real ants would, with one of the underside of the edge.

So:

There are 3 ants sitting on three corners of a triangle. Each ant randomly and independently picks one of the edges that meeet at its corner, and walks along it, stopping when it reaches another corner. What is the probability that any two ants meet?

Solution to puzzle: show

¾. The first ant goes some way around the triangle. If the other two both go the same way there are no meetings; otherwise there is one.

Ants and Triangle

Problem statement

Solution to metapuzzle: show

Solution to puzzle: show