Biased ants on a triangle

Question

Three ants sit at the corners of a triangle. Each independently walks clockwise with probability $p$ and counterclockwise with probability $1 - p$ , then sets off at the same speed.

As a function of $p$ , what is the probability that no two ants collide? Which $p$ makes a collision most likely?

Accepted Answer

The ants avoid all collisions only if they all rotate the same way: all clockwise or all counterclockwise. With independent choices, all clockwise has probability p^3 and all counterclockwise has probability (1-p)^3, so P(	ext{no collision}) = p^3 + (1-p)^3. Check p = 	frac12: 	frac18 + 	frac18 = 	frac14, recovering the classic fair-coin answer. Which p makes a collision most likely? A collision is most likely when P(	ext{no collision}) is smallest. Differentiate: \frac{d}{dp}\big[p^3 + (1-p)^3\big] = 3p^2 - 3(1-p)^2 = 3(2p - 1), which is zero at p = 	frac12, and the second derivative …

Biased ants on a triangle

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