There are three ants on different vertices of a triangle. What is the probability of collision (between any two or all of them) if they start walking on the sides of the triangle?
Assume that each ant randomly picks a direction, with either direction being equally likely to be chosen, and that they walk at the same speed.




Solution: The ants will collide if any of them are moving towards each other. The only way they wont collide is if they are all moving in the same direction.
We can compute this probability and work backwaards from there.

two directions and three ants.

p(clockwise) = (1/2)^3
p(counter clockwise) = (1/2)^3
p(same direction) = (1/2)^3 + (1/2)^3 = (1/4)

the probability of collision is therefore  the probability of the ants NOT moving in the same direction:

p(collision) = 1 - p(same direction) = 1 - 1/4 = 3/4





// they do not collide only if:
// clockwise, clockwise, clockwise or
// anticlockwise, anticlockwise, anticlockwise

// since the probability of not colliding is 2 / (2 * 2 * 2)
// then the probability of colliding is 1 - 2 / (2 ^ 3)

// the generalized solution for non-collision is 1 - 2 / (2 ^ n)