Bertrand's paradox on a monitor

TL;DR

This paper examines Bertrand's paradox using discrete geometry and probability, approximating the circle with grid boxes to compute the probability of certain distances, providing a numerical approximation.

Contribution

It introduces a discrete geometric approach to analyze Bertrand's paradox, offering a novel approximation method for the probability calculation.

Findings

The probability of pairs separated by more than √3 approaches approximately 0.33273.

The method bridges classical probability with discrete geometric approximation.

Provides a new perspective on classical paradox through computational approximation.

Abstract

We investigate Bertrand's probabilistic paradox through the lens of discrete geometry and old-fashioned but reliable discrete probability. We approximate the plane unit circle with $1/n$ times $1/n$ boxes and count the pairs of boxes separated by distance more than $\sqrt{3}$. For $n\to\infty$ the proportion of such pairs goes to $$ \frac{1+\sqrt{3}}{8}-\frac{\pi(2-\sqrt{3})}{96}=0.33273\dots\;. $$