Experimental tests of Bertrand's question and the Duhem-Quine problem

TL;DR

This paper experimentally investigates Bertrand's question on random chords in a circle, confirming theoretical predictions and illustrating the Duhem-Quine problem about the conditional nature of hypothesis testing.

Contribution

It provides an experimental validation of Bertrand's question and highlights the influence of auxiliary assumptions on hypothesis testing in probability.

Findings

Probability depends on the ratio of circle diameter to straw length

Laplace's principle of indifference is only approached in the limit of infinite straw length

Systematic discrepancies occur when the ratio approaches 1

Abstract

In this paper we report on an experimental test of Bertrand's question on the probability to find a random chord drawn inside a unit-radius circle with length greater than $\sqrt{3}$. In an experiment performed by tossing straws onto a circle, we confirm theoretical predictions that the answer depends on the ratio of the circle diameter, $2R$, to the straw length, $L$, and that the special case corresponding to Laplace's principle of indifference is only obtained in the experimentally unattainable limit of infinite straw length, $\tilde{d}=2R/L\rightarrow 0$. In addition, we observe a systematic discrepancy in the limit, $\tilde{d}=2R/L\rightarrow 1$, where a large number of events are rejected. We conclude that the experimental test of Bertrand's paradox provides a good illustration of the Duhem-Quine problem---that hypothesis testing is always conditional on a bundle of real auxiliary assumptions.