Bertrand's Paradox Revisited: Calculating, Simulating and Visualizing in Wolfram-Cloud/Mathematica

Published: 25 July 2024
on channel: Thomas Blum

In this video we revisit the problem of Bertrand's Paradox. In a previous video ( • Using Mathematica/Wolfram-Cloud to pl... ) we used Mathematica/Wolfram-Cloud to simulate three versions of generating chords of a unit circle and asking what fraction have a chord length greater than sqrt(3). The methods produced fractions of 1/3, 1/4, and 1/2. A fourth version was added that produced a fraction around 0.6. A possible derivation of that fraction is provided here (1/3 + sqrt(3)/2/pi). In addition we use the opacity of the chords in order to visualize the "chord density." We generalize the "Version 4" simulation to choose the points from a square which contains the circle -- rather than choosing the points from circle itself. As the surrounding square increases in size, the fraction seems to approach the 1/2 from above.

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