Bertrand's paradox: a physical way out along the lines of Buffon's needle throwing experiment

Abstract

Bertrand's paradox (Bertrand 1889 Calcul des Probabilités (Paris: Gauthier-Villars)) can be considered as a cautionary memento, to practitioners and students of probability calculus alike, of the possible ambiguous meaning of the term ‘at random’ when the sample space of events is continuous. It deals with the existence of different possible answers to the following question: what is the probability that a chord, drawn at random in a circle of radius R, is longer than the side of an inscribed equilateral triangle? Physics can help to remove the ambiguity by identifying an actual experiment, whose outcome is obviously unique and prescribes the physical variables to which the term ‘random’ can be correctly applied. In this paper, after briefly describing Bertrand's paradox, we associate it with an experiment, which is basically a variation of the famous Buffon's needle experiment (Buffon 1733 Histoire de l’Acad. Roy. des. Sciences pp 43–5; Buffon 1777 Histoire naturelle, générale et particulièreSupplément 4 46) for estimating the value of π. Its outcome is compared with the analytic predictions of probability calculus, that is with the probability distribution of variables whose uniform distribution can be considered a sound implementation of complete randomness.