Bertrand's Problem

Bertrand's problem asks for 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 two chord endpoints are chosen independently and uniformly on the circumference of the circle. Equivalently, one endpoint can be fixed at (1, 0) on the unit circle and the other chosen uniformly on its circumference. The smaller angle between their radii is then uniformly distributed on . The chord length (circle line picking) is

(1)

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

(2)

For the alternative threshold , the angular separation must be at least , giving probability 1/3 instead.

However, if a point is instead chosen uniformly on a radius of the circle and a chord drawn perpendicular to the radius through that point, then its length is at least when the point's distance from the center is at most . Thus,

(3)

For the alternative threshold , the point must instead be within distance of the center, giving probability 1/2.

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.