Bertrand’s Paradox Revisited: More Lessons about that Ambiguous Word, Random

Document Type: Research Paper

Authors

1 Department of Management Science and Engineering, Stanford University, Stanford, CA 94305 USA

2 Engineering Systems Division and Department of Civil & Environmental Engineering, E40-233, Massachusetts Institute of Technology, Cambridge, MA 02139 USA

Abstract

The Bertrand paradox question is: “Consider a unit-radius circle for which the length of a side of an inscribed equilateral triangle equals 3 . Determine the probability that the length of a ‘random’ chord of a unit-radius circle has length greater than 3 .” Bertrand derived three different ‘correct’ answers, the correctness depending on interpretation of the word, random. Here we employ geometric and probability arguments to extend Bertrand’s analysis in two ways: (1) for his three classic examples, we derive the probability distributions of the chord lengths; and (2) we also derive the distribution of chord lengths for five new plausible interpretations of randomness. This includes connecting (and extending) two random points within the circle to form a random chord, perhaps being a most natural interpretation of random.

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