%0 Journal Article
%T Bertrand’s Paradox Revisited: More Lessons about that Ambiguous Word, Random
%J Journal of Industrial and Systems Engineering
%I Iranian Institute of Industrial Engineering
%Z 1735-8272
%A Chiu, Samuel S.
%A Larson, Richard C.
%D 2009
%\ 04/01/2009
%V 3
%N 1
%P 1-26
%! Bertrand’s Paradox Revisited: More Lessons about that Ambiguous Word, Random
%K Bertrand paradox
%K geometrical probability
%K Randomness
%K Mathematical Modeling
%R
%X 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.
%U http://www.jise.ir/article_3997_6a52197644296c9501c5be1f63955e8a.pdf