Bertrand `paradox' reloaded (with details on transformations of variables, an introduction to Monte Carlo simulation and an inferential variation of the problem)

TL;DR

This paper clarifies the Bertrand paradox by emphasizing practical questions over abstract principles, providing detailed methods for calculating chord length distributions, and discussing simulation and inferential aspects.

Contribution

It offers a detailed analysis of the Bertrand paradox with explicit calculations, Monte Carlo simulation insights, and an inferential perspective, clarifying misconceptions.

Findings

Explicit formulas for chord length distributions

Monte Carlo simulation techniques for the problem

An inferential approach to the paradox

Abstract

This note is mainly to point out, if needed, that uncertainty about models and their parameters has little to do with a `paradox'. The proposed `solution' is to formulate practical questions instead of seeking refuge into abstract principles. (And, in order to be concrete, some details on how to calculate the probability density functions of the chord lengths are provided, together with some comments on simulations and an appendix on the inferential aspects of the problem.)