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 , a chord exceeds the side of an inscribed equilateral triangle exactly when its midpoint distance from the center is below . Uniform midpoint area gives probability , independent uniform endpoints give , and uniform midpoint distance gives . The difference is in the sampling laws, not a contradiction in probability theory.
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The Bertrand paradox is a problem in probability theory that highlights the ambiguities that can arise when dealing with random experiments that seem intuitively straightforward. It was formulated by the French mathematician Joseph Bertrand in the 19th century. The paradox demonstrates that different methods of defining a "random" choice can lead to different probabilities for the same event. The classic version of the Bertrand paradox involves the following situation: 1. **A Circle and a Chord**: Imagine a circle with a diameter.