Bertrand’s Paradox Calculator
Compare three random-chord models for a chord longer than the side of an inscribed equilateral triangle.
Formula
Uniform endpoints: 1/3; uniform radius distance: 1/2; uniform midpoint area: 1/4.
Probability of a longer chord ≈ 100÷3
How to use
- Choose how a random chord is sampled.
- The length threshold is √3 times the circle radius.
- Read the probability for that sampling rule.
Examples
- Uniform midpoint area gives (R/2)²/R² = 1/4 = 25%.
Frequently asked questions
Why are the answers different?
Each sampling rule assigns a different probability distribution to chords.
Does the circle radius matter?
No. Scaling preserves the relative geometry and these probabilities.
Is this a simulation?
No. These are exact geometric probabilities before display rounding.
Sources
Updated: