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  "body": "\n## Example\n\nTake a circle and an equilateral triangle inscribed in it. Choose a chord of the circle \"at random\". What is the probability that the chord is longer than a side of the triangle?\nThree natural ways to choose the chord give three different answers.\nChoose two independent uniform points on the circle and join them. The probability is $1/3$.\nChoose a uniform radius, then a uniform point on it, and take the chord perpendicular to the radius through that point. The probability is $1/2$.\nChoose a uniform point in the disc and take the chord with that point as midpoint. The probability is $1/4$.\n\n## Remarks\n\nThe paradox shows that \"at random\" has no meaning until one fixes a probability measure. The three methods give three different distributions on the set of chords.\nJoseph Bertrand gave the example in his book Calcul des probabilités of 1889, as a criticism of the principle of indifference.\n\n## Explanations\n\n### L1 human\n\nThe question sounds complete, but it is not: it does not say how to choose the chord.\nEach method is fair in its own sense, and each one gives a different answer. In probability you must always say exactly how the random choice is made.\n",
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