Bertrand's Problem

What is the probability that a chord drawn at random on a circle of radius r (i.e., circle line picking) has length >=r (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 theta_1 and theta_2 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 theta=theta_2-theta_1 measured from the intersection of the positive x-axis along the unit circle. Since the length as a function of theta (circle line picking) is given by


solving for s(theta)=1 gives pi/3, so the fraction of the top unit semicircle having chord length greater than 1 is


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


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.

See also

Chord, Circle Line Picking, Geometric Probability

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Bogomolny, A. "Bertrand's Paradox.", G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 21-23, 1998.Isaac, R. The Pleasures of Probability. New York: Springer-Verlag, 1995.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.Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 11-12, 1984.Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, p. 2, 1978.

Referenced on Wolfram|Alpha

Bertrand's Problem

Cite this as:

Weisstein, Eric W. "Bertrand's Problem." From MathWorld--A Wolfram Web Resource.

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