Question

N ants sitting at the corners of a polygon. Each ant randomly picks a direction and start to move along the edge of the polygon. What is the probability that none of the ants collides.

Answer 1

Answer:

$\frac{1}{2^{N-1}}$

Step-by-step explanation:

Each ant has only two option to move, either in the clockwise direction or in the anticlockwise direction.

The ants will not collide if all the ants are either moving in the clockwise direction or all the N ants are either moving in the anticlockwise direction

The probability of one ant to move either in the clockwise or in the anticlockwise direction is 1/2 = 0.5

Ants moving are independent events

Thus the probability that the ants will not collide

= Probability that all the ants move in the clockwise direction + Probability that all the ants move in the anticlockwise direction

$=\frac{1}{2}\times\frac{1}{2}\times \frac{1}{2}\times ......\text{upto N}+\frac{1}{2}\times\frac{1}{2}\times \frac{1}{2}\times ......\text{upto N}$

$=(\frac{1}{2})^N+(\frac{1}{2})^N$

$=2\times \frac{1}{2^N}$

$=\frac{1}{2^{N-1}}$