Bertrand paradox (probability)

Joseph Bertrand introduced it in his work Calcul des probabilités (1889)[1] as an example to show that the principle of indifference may not produce definite, well-defined results for probabilities if it is applied uncritically when the domain of possibilities is infinite.

This issue can be avoided by "regularizing" the problem so as to exclude diameters, without affecting the resulting probabilities.

[4] The problem's classical solution (presented, for example, in Bertrand's own work) depends on the method by which a chord is chosen "at random".

[3] The argument is that if the method of random selection is specified, the problem will have a well-defined solution (determined by the principle of indifference).

"Method 2" is the only solution that fulfills the transformation invariants that are present in certain physical systems—such as in statistical mechanics and gas physics—in the specific case of Jaynes's proposed experiment of throwing straws from a distance onto a small circle.

In order to arrive at the solution of "method 3", one could cover the circle with molasses and mark the first point that a fly lands on as the midpoint of the chord.

[7] Several observers have designed experiments in order to obtain the different solutions and verified the results empirically.