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

The latter interpretation is more satisfactory in the sense that the result remains the same for a rotated circle, a slightly smaller circleinscribed 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.