Three ants are arranged on vertices of a triangle, one ant to a vertex. At some moment, all the ants begin crawiling along the sides of the triangel. Each one crawls along one of the two sides that connect to the vertex it is sitting on, with an equal probability of picking either.

Assuming that all the ants move with an equal speed, and that they keep crawling forever in the same direction along the triangle, what are the odds that no two will collide?

Unless I've missed something obvious then...

P(no collision) = P(all 3 go in same direction)

...because all 3 going in the same direction is the only condition under which they won't collide

= P(all 3 go clockwise) + P(all 3 go anticlockwise)

Assuming that each ant is independent and (as stated in the problem) there is a 0.5 probability that an ant will go in either of the 2 directions;

P(clockwise) = P(anticlockwise) = 0.5*0.5*0.5 = 0.125

and

P(no collision) = 0.125 + 0.125 = 0.25

In the general case, for n ants on any closed polygon, you would have;

P(no collision) = 2 * 0.5^n
= 0.5^(n - 1)

Posted by riz on 2002-11-21 15:13:50