Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?
Obviously the probability depends on how you choose the random chord. Suppose you require the following additional constraints for choosing how the chords must be chosen:
Assume that chords are laid at random onto a circle with a diameter of 2, for example by throwing straws onto it from far away. Now another circle with a smaller diameter (e.g., 1.1) is laid into the larger circle. Then the distribution of the chords on that smaller circle needs to be the same as on the larger circle. If the smaller circle is moved around within the larger circle, the probability must not change either.
Edwin James showed in his 1973 paper that these requirements can be met if and only if we choose the chord as follows: Choose a radius of the circle, choose a point on the radius and construct the chord through this point and perpendicular to the radius.