Bertrand's paradox: a physical solution

TL;DR

This paper resolves Bertrand's paradox by defining a physical experiment involving throwing a straw of finite length onto a circle, resulting in a unique probability that depends on the ratio of straw length to circle radius.

Contribution

It introduces a physical, operational approach to Bertrand's paradox, providing a definitive probability solution based on a tangible experimental setup.

Findings

Probability depends on the ratio L/R of straw length to circle radius.

Provides a unique, physically grounded solution to the paradox.

Clarifies the interpretation of randomness in geometric probability.

Abstract

We present a conclusive answer to Bertrand's paradox, a long standing open issue in the basic physical interpretation of probability. The paradox deals with the existence of mutually inconsistent results when looking for the probability that a chord, drawn at random in a circle, is longer than the side of an inscribed equilateral triangle. We obtain a unique solution by substituting chord drawing with the throwing of a straw of finite length L on a circle of radius R, thus providing a satisfactory operative definition of the associated experiment. The obtained probability turns out to be a function of the ratio L/R, as intuitively expected.