Thou shalt not say "at random" in vain: Bertrand's paradox exposed

TL;DR

This paper clarifies that Bertrand's paradoxes are due to careless language use rather than probabilistic inconsistency, highlighting issues in defining geometrical probabilities on infinite sets and distinguishing passive versus active randomness.

Contribution

It demonstrates that Bertrand's paradoxes are linked to language and definition issues in probability, not fundamental inconsistencies, and extends the discussion to Buffon's needle problem.

Findings

Bertrand's paradoxes stem from language misuse, not probability contradictions.

Probabilistic issues arise from defining measures on infinite sets.

Distinction between passive and active randomness impacts interpretation.

Abstract

We review the well known Bertrand paradoxes, and we first maintain that they do not point to any probabilistic inconsistency, but rather to the risks incurred with a careless use of the locution "at random". We claim then that these paradoxes spring up also in the discussion of the celebrated Buffon's needle problem, and that they are essentially related to the definition of (geometrical) probabilities on "uncountably" infinite sets. A few empirical remarks are finally added to underline the difference between "passive" and "active" randomness, and the prospects of any experimental decision