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

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.