Bertrand's Paradox (Chord)

A problem showing how different methods of random selection yield different probabilities.

Simulation Control

Selection Method

Probability Analysis

Observed Probability

0.0%

Theoretical: 33.33%

Total Chords0

Longer than Side0

Why the difference?

Choosing two random points on the circumference. The chord is longer only if the points lie in specific arcs (1/3 of circle).

Overview

Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?

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Pro Tips

Switch between methods to see how the definition of 'random' changes the result.

Fun Facts

"Method 1 (Random Endpoints): Gives 1/3."
"Method 2 (Random Radius): Gives 1/2."
"Method 3 (Random Midpoint): Gives 1/4."

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