On the Bertrand paradox
The probability that a random chord in a disc is longer than the side of the inscribed equilateral triangle depends on how randomness is defined. In 1888 Bertrand pointed out different answers for that probability. Five classical models are described and also the following. In a point outside the disc a line is taken in a random uniform direction. A random chord is obtained if the line hits the disc. Conditioned on that event the probability that the chord is longer than the side of the inscribed equilateral triangle is calculated by straightforward trigonometry.
scientific article; zbMATH DE number 1239703