Faculty Mentor(s)
Dr. Dyana Harrelson, Mathematics & Statistics
Document Type
Poster
Event Date
4-17-2026
Abstract
Bertrand’s paradox is a classic problem that highlights how different notions of randomness can lead to different outcomes, even in a simple geometric setting. It concerns the lengths of chords chosen “at random” in a circle. In this talk, we begin by reviewing the three original methods Bertrand proposed for generating random chords, along with several related distributions that have been studied since. We then turn to a geometric application, examining triangles formed by two random chords that share a common endpoint. By joining the remaining endpoints, we obtain a random triangle and compute the probability that it is acute. Using these chord-length distributions, we next give a simplified approach to determining the probability that a triangle formed by three points chosen uniformly at random from the unit disk is acute. We conclude by discussing how Bertrand’s ideas and distributions can be extended to higher-dimensional settings.
Recommended Citation
Repository citation: Moore, Mary; Weeda, Hope; and Cunill Krones, Annika, "Investigations in Bertrand’s Paradox" (2026). 25th Annual A. Paul and Carol C. Schaap Celebration of Undergraduate Research and Creative Activity (2026). Paper 20.
https://digitalcommons.hope.edu/curca_25/20
April 17, 2026. Copyright © 2026 Hope College, Holland, Michigan.
Comments
Hope College Math Department and Global Water Research Institute.