Three distinct methods of randomly selecting a chord yield different probabilities—demonstrating the importance of precisely defining randomness.
Last updated: April 2026
Choose two random points on the circle and connect them
More trials = more accurate simulation
Simulation Trials
10,000
Theoretical P
33.3%
Simulated P
33.0%
Difference: 0.37%
| Method | How It Works | P(Chord > Side) | Intuition |
|---|---|---|---|
| Random Endpoints | Two random points on circumference | 1/3 (33%) | Angular separation 60°–120° |
| Random Radius | Random radius, random point on it | 1/2 (50%) | Midpoint closer than r/2 |
| Random Midpoint | Random point inside circle (midpoint) | 1/4 (25%) | Midpoint in inner circle (r/2) |
The paradox: same question, three different "randomness" definitions, three different valid answers. This shows probability requires precise specification of the random process.
Bertrand's Paradox is a classical problem in probability theory that shows how the outcome of a probability calculation depends critically on the method used to generate random events. Published by Joseph Bertrand in 1889, it illustrates that "randomness" must be precisely defined.
The Problem: A random chord is drawn in a circle. What's the probability that this chord is longer than the side of an inscribed equilateral triangle?
Setup: You have a circle with an inscribed equilateral triangle. The triangle's side equals radius × √3 ≈ 1.732r.
Method 1 Process: Pick two random points on the circle's edge and connect them with a chord. Does this chord exceed 1.732r?
Analysis: The angle between two random points on a circle is uniformly distributed from 0° to 180°. The chord is longer than the triangle's side when the angle is between 60° and 120°. That's 60° out of 180° = 1/3.
Key insight: If you instead used Method 2 (random radius) or Method 3 (random midpoint), you'd get different answers—but each is mathematically correct for its method. The paradox reveals that "randomly select a chord" is ambiguous.
Why do the three methods give different answers?
Each method defines 'random' differently, producing different probability distributions of chords. There's no single correct answer—only different valid interpretations.
Which method is 'correct'?
None is universally correct. The paradox shows that probability requires careful specification of how randomness is generated. Different applications may justify different methods.
What's the equilateral triangle reference?
For an equilateral triangle inscribed in a circle, the side length equals the radius times √3 ≈ 1.732. Chords longer than this are the 'success' condition.
How is this paradox resolved?
Modern probability theory recognizes that 'uniform randomness' must be precisely defined. Different measures (probability measures) give different valid results.
Is simulation or theory more accurate?
With sufficient trials, simulation converges to the theoretical value. Both are correct once the method is specified. More trials = better agreement.
What's the practical significance?
It demonstrates the importance of precisely specifying probability models in real applications—ambiguity can lead to drastically different predictions.
Can I visualize the three methods geometrically?
Yes! Method 1 samples arcs, Method 2 samples radial distances, Method 3 samples disk area. Each creates different chord distributions.
Who discovered this paradox?
Joseph Bertrand presented it in his 1889 textbook. It influenced development of modern measure theory and probability foundations.
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