Two new Wikipedia articles on little puzzles in geometric probability:
Sylvester's four point problem, https://en.wikipedia.org/wiki/Sylvester%27s_four_point_problem
– if you pick four points randomly in the plane, what is the probability that they form a convex quadrilateral?
Broken stick problem, https://en.wikipedia.org/wiki/Broken_stick_problem – if you break a line segment randomly into three parts, what is the probability that they can form the sides of a triangle?
In both cases the answer depends on how you choose a probability distribution to formalize the problems. And if your answer for the first one is the uniform distribution on the plane, independently for the four points, you're in good company with Sylvester but mistaken: there is no such distribution.
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2026-05-31 06:08:28 UTC
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"content": "Two new Wikipedia articles on little puzzles in geometric probability:\n\nSylvester's four point problem, https://en.wikipedia.org/wiki/Sylvester%27s_four_point_problem\n– if you pick four points randomly in the plane, what is the probability that they form a convex quadrilateral?\n\nBroken stick problem, https://en.wikipedia.org/wiki/Broken_stick_problem – if you break a line segment randomly into three parts, what is the probability that they can form the sides of a triangle?\n\nIn both cases the answer depends on how you choose a probability distribution to formalize the problems. And if your answer for the first one is the uniform distribution on the plane, independently for the four points, you're in good company with Sylvester but mistaken: there is no such distribution.",
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