Bertrand paradox

Example

Take 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? Three natural ways to choose the chord give three different answers. Choose two independent uniform points on the circle and join them. The probability is $1/3$. Choose a uniform radius, then a uniform point on it, and take the chord perpendicular to the radius through that point. The probability is $1/2$. Choose a uniform point in the disc and take the chord with that point as midpoint. The probability is $1/4$.

Remarks

The 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. Joseph Bertrand gave the example in his book Calcul des probabilités of 1889, as a criticism of the principle of indifference.

Explanations

L1 human

The question sounds complete, but it is not: it does not say how to choose the chord. Each 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.