Ants on a pentagon, with a twist

Question

Five ants sit at the corners of a regular pentagon. Each picks one of its two edges uniformly at random and walks at the same speed.

What is the probability that no two collide? And given that no collision occurred, what is the probability they were all walking clockwise?

Accepted Answer

No collision. No two ants collide only if all five walk the same way around the pentagon, all clockwise or all counterclockwise. With 2^5 = 32 equally likely configurations and exactly 2 collision-free ones, P(	ext{no collision}) = \frac{2}{32} = \frac{1}{16}. Conditional probability. Given no collision, only two configurations remain, all-clockwise and all-counterclockwise, and they are equally likely (each had probability 	frac{1}{32} unconditionally). So P(	ext{all clockwise} \mid 	ext{no collision}) = \frac{P(	ext{all clockwise})}{P(	ext{no collision})} = \frac{1/32}{2/32} = \frac{1…

Ants on a pentagon, with a twist

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