The Bertrand “paradox” is the famous observation that asking for the probability that a “random chord” on a circle spans more than 120° is meaningless until you specify exactly how you choose the chord. The Wikipedia article describes three methods that give P = 1/3, 1/2 and 1/4.
So … having played around with a lot of calculations for what happens when you pick two points inside a set and draw a line through them, I couldn’t resist asking what happens if you choose a “random chord” that way.
The answer is:
P(θ > 120°) = 1/3 + (3√3)/(4π) ≈ 0.74683
https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)
Greg Egan
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2025-09-04 05:53:52 UTC
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2025-09-04 05:53:52 UTCEvent JSON
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