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 . In an experiment performed by tossing straws onto a circle, we confirm a theoretical prediction that the answer depends on the ratio of the circle diameter, 2R, to the straw length, L, and that the special case, which follows from rotational and translation invariance using integral geometry, is only obtained in the experimentally unattainable limit of infinite straw length, . In addition, we observe a systematic discrepancy in the limit, , 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.