Ants moving towards Corners

Puzzle # 11

There are 'n' ants at 'n' corners of a 'n' sided closed regular polygon, they randomly start moving towards another corner that is adjacent to it?

If n = 5, PENTAGON.. i.e., 5 ants positioned at 5 corners are started moving towards other possible corners
If n= 6, HEXAGON.. i.e., 6 ants positioned at 6 corners are started moving towards other possible corners
If n = 8, OCTAGON.. i.e., 8 ants positioned at 8 corners are started moving towards other possible corners
So on and So Forth.

What is the probability that they don't collide?


Solution
PROBABILITY = 1/ 2^{n - 1}

There are 'n' corners in a regular polygon.
Ant placed in 1st corner can go in 2 directions along the closed. Similarly ants placed in any corner can move  in 2 directions.

Total possible directions that ants can move in 'n' sided regular polygon is 2 x 2 x 2 ... n times.
There are only 2 possible solutions where ants cannot collide i.e,
1. CLOCKWISE 
2. ANTICLOCKWISE 

Probability that ants will not collide each other = 2 / 2ⁿ 

                                                                      = 1 / 2^{n - 1}

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