Luigia Caputo†, Aniello Buonocore‡
Abstract: The “Bertrand’s paradox” arises from a geometric probability problem for which J. Bertrand provides three solutions with different values of the probability sought; from here the “paradox” arises, a term that several authors attribute to J. Poincare. The interest in this issue has lasted for over a hundred years and even today it is possible to find articles in which Bertrand’s paradox, in one way or another, is take into consideration. Indeed, it as well as in the philosophical debate concerning the opposition between the frequentist and Bayesian approaches to Inferential Statistics, it has also used to highlight (hypothetical) logical inconsistencies of the “principle of indifference” in problems in which the total number of cases is not countable.
The Principle of Indifference, whose origin is, often, attributed to P. S. Laplace, is a fundamental concept in the field of probability but also in decision-making under uncertainty. It offers a rational starting point for assigning probabilities in the absence of any information. However, it is essential to be aware of the importance of updating our beliefs as added information about the issue becomes available.
In this article, in order to resolve Bertrand’s paradox, we highlight the role, often overlooked, of the random experiment (and also of the random device for its implementation) that generates the support of a probability space. Afterwards, from the problem itself it is possible to trace the class of generating events and a pre-measure on this class to complete a probability space consistent with the problem. This way of proceeding is illustrated for each of the three solutions identified by Bertrand and for another recently proposed solution. We conclude that all solutions are equally valid because, once the appropriate probability space has specified, the only correct solution will emerge in a logical and formal way.
Keywords: historical definitions and paradoxs, geometric probability, combinatorial probability.
MSC: 60-00, 60-03, 60C05.
DOI 10.56082/annalsarscimath.2025.3.369
† luigia.caputo@unina.it, Universita` di Napoli Federico II–Dipartimento di Matem- atica e Applicazioni “Renato Caccioppoli”
‡ aniello.buonocore@unina.it, Universita` di Napoli Federico II–Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application, Volume 17 no 3, 2025
