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

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.