## Bertrand's Problem

What is the Probability that a Chord drawn at Random on a Circle of Radius has length ? The answer, it turns out, depends on the interpretation of two points drawn at Random.'' In the usual interpretation that Angles and are picked at Random on the Circumference,

However, if a point is instead placed at Random on a Radius of the Circle and a Chord drawn Perpendicular to it,

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.

References

Bogomolny, A. Bertrand's Paradox.'' http://www.cut-the-knot.com/bertrand.html.

Jaynes, E. T. Papers on Probability, Statistics, and Statistical Physics. Dordrecht, Netherlands: Reidel, 1983.

Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 42-45, 1995.