= Bertrand paradox
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= Bertrand's paradox
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Different precise <measure>[measures] on random chords give different <probability>[probabilities]; the phrase “random chord” does not select a unique <probability distribution>. In a circle of radius $r$, a chord exceeds the side of an inscribed equilateral triangle exactly when its midpoint distance from the center is below $r/2$. Uniform midpoint area gives <probability> $1/4$, <independent> uniform endpoints give $1/3$, and uniform midpoint distance gives $1/2$. The difference is in the sampling laws, not a contradiction in <probability theory>.
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