Three ants on a triangle
Three ants sit at the three corners of a triangle. Each ant picks one of the two edges at its corner, uniformly at random and independently of the others, and walks along it at the same speed as the others.
Find the probability that no two ants meet.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Each ant goes either clockwise or counterclockwise.
If two ants go in different directions, does some pair have to meet?
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Each ant chooses clockwise or counterclockwise. Nobody meets only if all three choose the same direction.
Step 1. If all three go clockwise, they chase each other around the triangle at the same speed and never meet; likewise if all go counterclockwise.
Step 2. Otherwise some ant goes clockwise and some ant counterclockwise. Going around the triangle, there must be an ant going clockwise whose clockwise neighbour goes counterclockwise; those two walk along the same edge towards each other and meet.
Step 3. There are equally likely choices, and 2 of them avoid any meeting:
In general. With ants on the corners of an -sided polygon, the probability is .