Ants on a Triangle


There are three ants on a triangle, one at each corner. at a given moment in time, they all set off for a different corner at random. what is the probability that they don’t collide?

Solution

Consider the triangle ABC. We assume that the ants move towards different corners along the edges of the triangle.

Total no. of movements: 8 A->B, B->C, C->A A->B, B->A, C->A A->B, B->A, C->B A->B, B->C, C->B A->C, B->C, C->A A->C, B->A, C->A A->C, B->A, C->B A->C, B->C, C->B

Non-colliding movements: 2 A->B, B->C, C->A A->C, B->A, C->B

(i.e. the all ants move either in the clockwise or anti-clockwise direction at the same time)

P(not colliding) = 2/8 = 0.25