= Bertrand paradox
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= Bertrand's paradox
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Different precise measures on random chords give different 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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