# Ant and Triangle Problem 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?

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Somewhat more simply stated: pick an ant.  Each of the other two ants must decide to go the same direction which is 0.5 * 0.5 = 0.25.

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.

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

There are two possibilities if all the ants move in clockwise or counterclockwise direction, they wont collide.
The total no. of case that can happen is 2*2*2 where each ant can choose either of the directions.
So the probability is 2/8=0.25

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