Bertrand's Problem

What is the probability that a chord drawn at random on a circle of radius (i.e., circle line picking) has length (or sometimes greater than or equal to the side length of an inscribed equilateral triangle; Solomon 1978, p. 2)? The answer depends on the interpretation of "two points drawn at random," or more specifically on the "natural" measure for the problem.

In the most commonly considered measure, the angles and are picked at random on the circumference of the circle. Without loss of generality, this can be formulated as the probability that the chord length of a single point at random angle measured from the intersection of the positive x-axis along the unit circle. Since the length as a function of (circle line picking) is given by

(1)

solving for gives , so the fraction of the top unit semicircle having chord length greater than 1 is

(2)

However, if a point is instead placed at random on a radius of the circle and a chord drawn perpendicular to it, then

(3)

The latter interpretation is more satisfactory in the sense that the result remains the same for a rotated circle, a slightly smaller circle inscribed in the first, or for a circle of the same size but with its center slightly offset. Jaynes (1983) shows that the interpretation of "random" as a continuous uniform distribution over the radius is the only one possessing all these three invariances.

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