Puzzle - Three Ants Problem

Puzzle

Three ants are sitting in the corners of the equilateral triangle. All ants start at same time and move at same speed constantly. The speeds of three ants are equal. Each ant picks up a direction randomly at first and moves in that direction. What is the probability that none of the ants will collide with each other ?

Solution

The problem might seem hard at the start. But it is a simple probability problem.

The ants won't collide with each only when all the ants move either in clockwise direction or in anti - clockwise direction.

So, the answer is

P( no ants collide )  = P( all ants move in clockwise ) + P( all ants move in anti - clockwise )

                             =  0. 5 * 0. 5 * 0. 5   + 0. 5 * 0. 5 * 0. 5

                            =  0. 25.

Another way of approaching the problem is,

All possible ways in which ants move is  2 * 2 * 2.

C C C

C C A

C A C

C A A

A C C

A C A

A A C

A A A

Ants won't collide only in two cases either in " C C C" or in "A A A"

So,

P(none of the ants collide ) = 2/8 = 0. 25