Bertrand’s Paradox Resolution and Its Implications for the Bing–Fisher Problem

Abstract

Bertrand’s paradox is a problem in geometric probability that has resisted resolution for more than one hundred years. Bertrand provided three seemingly reasonable solutions to his problem — hence the paradox. Bertrand’s paradox has also been influential in philosophical debates about frequentist versus Bayesian approaches to statistical inference. In this paper, the paradox is resolved (1) by the clarification of the primary variate upon which the principle of maximum entropy is employed and (2) by imposing constraints, based on a mathematical analysis, on the random process for any subsequent nonlinear transformation to a secondary variable. These steps result in a unique solution to Bertrand’s problem, and this solution differs from the classic answers that Bertrand proposed. It is shown that the solutions proposed by Bertrand and others reflected sampling processes that are not purely random. It is also shown that the same two steps result in the resolution of the Bing–Fisher problem, which has to do with the selection of a consistent prior for Bayesian inference. The resolution of Bertrand’s paradox and the Bing–Fisher problem rebuts philosophical arguments against the Bayesian approach to statistical inference, which were based on those two ostensible problems.

New Ways to Calculate the Probability in the Bertrand Problem

“…The other approach is the mathematical one, as in [8,9,12]. In [8], Chiu and Larson derive the probability distributions of the chord lengths in the three classical solutions.…”

“…In [9], Chechile states a novel method to sample a chord of the circle in a random way. It is based on four steps: first a generation of a random angle, then a random value from a beta distribution, after the construction of an auxiliary circle to define two chords, and finally a random election of one of these chords by flipping a coin.…”

“…2 in [9]). In this paper, we give two methods for calculating the probability in Bertrand's problem.…”

“…None of these approaches seems to carry a disambiguation since there are different results for the probability with procedures that respect the two principles of indifference and invariance (for further analysis of Bertrand's paradox, see [4][5][6][7][8] and the more recent paper of Chechile [9]. They give new solutions to the problem, such as 0.609 or 0.75 in [8] and 1 − √ 3…”