__Puzzle:__

Three ants are sitting at the three corners of an equilateral triangle. Each ant starts randomly picks a direction and starts to move along the edge of the triangle. What is the probability that none of the ants collide?

__Puzzle Solution:__

So let’s think this through. The ants can only avoid a collision if they all decide to move in the same direction (either clockwise or anti-clockwise). If the ants do not pick the same direction, there will definitely be a collision. Each ant has the option to either move clockwise or anti-clockwise. There is a one in two chance that an ant decides to pick a particular direction. Using simple probability calculations, we can determine the probability of no collision.

P(No collision) = P(All ants go in a clockwise direction) + P( All ants go in an anti-clockwise direction) = 0.5 * 0.5 * 0.5 + 0.5 * 0.5 * 0.5 = **0.25**

A simple way to solve.

P(#collision)= # of way when there is no collision/ total number of way ants can move

= 2/2^3

= 1/4

there is only 2 way when there is no collision when all 3 ants move clockwise or anticlock wise

and there are 3 ants and each can move in 2 direction so there will be 2^3 way they can move.

nice

nice explanations both.

Each ant has 2 possibilities of moving in a direction.. clockwise or anti-clockwise. Now as there are 3 ants .. we have 8 total possibilities of the ants moving in a direction.

Now out of 8 we can have only 2 possibilities where they will not collide.

i.e if all the ants travel clockwise or all the ants travel anti-clockwise.

so the probability is 2/8 = 1/4 = 0.25

thanku for the info.

simple and nice explanation thanx!!

Let the 3 ants x, y, z

anticlockwise- A & clockwise C

{xA,yA,zA}

{xA,yA,zC}

{xA,yC,zC}

{xC,yC,zC}

{xC,yA,zA}

{xC,yC,zA}

{xC,yA,zC}

{xA,yC,zA}

Here in 8 cases only in 2 cases there will be no collision when all the ants will travel in same direction either clockwise or anticlockwise.

i.e

{xA,yA,zA}

{xC,yC,zC}

so 2/8= 1/4 = 0.25

nice

Best explanation. I was confused which are those 8 moves. Thanks for listing those 8 available options.

Three Ants:

0–>clockwise

1–>Anticlockwise

All combinations will be :

A B C

0 0 0 [No collision]

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1 [No collision]

Probability = 2/8=0.25

Question lacks a lot of useful info. Speed of ants should be same. Also, they should start moving simultanesouly, not randomly.

what is probability of colliding???

6/8