Bertrand’s Paradox Calculator
Enter a circle’s radius and a threshold chord length, pick a random-chord method, and see the probability a randomly drawn chord is longer than the threshold — the famous Bertrand’s paradox.
What this calculator does
Bertrand’s paradox is a famous problem in probability: what is the probability that a randomly drawn chord of a circle is longer than the side of an inscribed equilateral triangle? The surprising answer is that it depends entirely on how you define “random,” giving three different classic answers of 1/3, 1/2, and 1/4. This calculator lets you pick a chord threshold length and a random-chord method to see the resulting probability.
How the calculation works
Three classic methods are supported. The “random endpoints” method picks two uniformly random points on the circle’s circumference and draws the chord between them. The “random point on a radius” method fixes a radius, picks a uniformly random point along it, and draws the chord perpendicular to the radius at that point. The “random midpoint” method picks a point uniformly at random inside the disk and uses it as the chord’s midpoint. Each method leads to a different formula relating chord length to distance from the center, and therefore a different probability.
With the default threshold equal to the side of the inscribed equilateral triangle (radius x square root of 3), the three methods reproduce the classic textbook results of 1/3, 1/2, and 1/4. This tool is for mathematical education and illustrates why the phrase “choose at random” needs a precise definition. It is not a physical measurement tool.