Generalizing Bertrand's Paradox
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Dan McClung
College:
The Dorothy and George Hennings College of Science, Mathematics, and Technology
Major:
Mathematical Sciences
Faculty Research Advisor(s):
Raymond Viglione
Abstract:
Bertrand's Paradox begins with a circle with a radius of one with an equilateral triangle inscribed inside. The problem is to determine the probability that a randomly selected chord is longer than one of the sides of the equilateral triangle. Despite seeming like a simple geometry problem, the process by which a chord is randomly selected leads to seemingly contradictory results. It is expected that regardless of the method used, the probability of this occurring should not change, since the diagram of the circle and inscribed triangle do not change. However, three common methods used to solve lead to unequal probability values, which has resulted in the problem receiving much interest from mathematicians. This project is intended to see if the paradox still exists when the equilateral triangle is not fixed. It is desired to see if the paradox exists for an equilateral rectangle, pentagon, hexagon, or a polygon with any number of sides greater than three. Each of the methods used for the original interpretation of the problem is applied to the problem, just without fixing the number of sides of the polygon at three. As a result, equations are derived for each individual method where the probability can be determined by substituting any integer for the number of sides in. This allows a generalized form of Bertand's Paradox so that more meaningful interpretations can be examined.