Explore Bertrand's paradox of random chords with three methods, inscribed polygon comparison, probability visualization, and Monte Carlo simulation.
Bertrand's paradox, posed by Joseph Bertrand in 1889, asks a deceptively simple question: if you draw a random chord of a circle, what is the probability that it is longer than the side of an inscribed equilateral triangle? The paradox is that the answer depends on how you define "random chord" — three natural methods yield three different probabilities: 1/3, 1/2, and 1/4.
Method 1 (Random Endpoints) picks two points uniformly on the circumference and connects them, giving P = 1/3. Method 2 (Random Radial Point) picks a random radius and a random distance along it, then draws the perpendicular chord, giving P = 1/2. Method 3 (Random Midpoint) picks a random point inside the disk as the chord's midpoint, giving P = 1/4.
This calculator lets you compare all three methods analytically and through Monte Carlo simulation. You can also generalize from equilateral triangles to any regular inscribed polygon. The paradox demonstrates that "random" is not self-defining — the choice of probability measure changes the answer, a foundational issue in geometric probability and measure theory.
Bertrand's paradox is a cornerstone of probability education that highlights the importance of precise problem formulation. This calculator makes all three methods tangible through side-by-side comparison, visual bars, and Monte Carlo simulation that students can run repeatedly.
It's invaluable for courses on probability, measure theory, and the philosophy of randomness. The generalization to arbitrary inscribed polygons extends the learning beyond the classic triangle case, helping build deeper geometric intuition.
For an inscribed equilateral triangle in a circle of radius r: Side length = 2r·sin(π/3) = r√3 P(chord > side): • Method 1 (endpoints): 1/3 • Method 2 (radial point): 1/2 • Method 3 (midpoint in disk): 1/4 General n-gon: • Method 2: cos(π/n) • Method 3: cos²(π/n)
Result: P(chord > side) = 1/3 ≈ 33.33%
With two points chosen uniformly on the circumference, one-third of random chords exceed the triangle's side length √3 ≈ 1.732.
Joseph Bertrand introduced this paradox in his 1889 textbook "Calcul des probabilités" to argue that classical geometric probability was ill-defined without specifying a probability measure. His challenge undermined the principle of indifference — the assumption that without specific information, all outcomes should be treated as equally likely — because "equally likely" depends on how you parameterize the space of chords.
The paradox remained philosophically unresolved for decades. In 1973, physicist E.T. Jaynes proposed a resolution based on the principle of maximum ignorance (maximum entropy), arguing that the correct answer should be invariant under scale and positional transformations. His analysis favored Method 2 (P = 1/2), but this interpretation is not universally accepted.
**Method 1 (Random Endpoints):** Choose two points uniformly on the circle's circumference. Fix one point; the other must fall in the arc subtended by the triangle's opposite side (1/3 of the circumference) for the chord to exceed the side length. P = 1 - 2/3 = 1/3.
**Method 2 (Random Radial Point):** Choose a random radius and a random point uniformly along it. The chord perpendicular to the radius at that point is longer than the triangle side when the point is closer to the center than r/2. P = 1/2.
**Method 3 (Random Midpoint):** Choose a random point uniformly in the disk. The chord with this midpoint is longer than the triangle side when the midpoint falls inside a concentric circle of radius r/2, which has area πr²/4. P = (πr²/4)/(πr²) = 1/4.
Bertrand's paradox fundamentally shaped the development of measure-theoretic probability in the 20th century. Kolmogorov's axiomatization (1933) resolved the issue by requiring every probability model to explicitly specify a σ-algebra and probability measure. The paradox remains a powerful teaching tool for illustrating why mathematical rigor matters in probability and why intuitive appeals to symmetry or fairness can be misleading.
Each method defines "random chord" using a different probability distribution. Random endpoints use uniform distribution on the circumference, random radial uses uniform distance along a radius, and random midpoint uses uniform distribution over the disk area. These are distinct probability spaces.
None is uniquely correct — that's the paradox. The answer depends on the physical process generating random chords. If you physically throw straws onto a circle, the midpoint method (P = 1/4) most closely matches experiments, but the "right" method depends on context.
It demonstrates that specifying a probability problem requires defining the sample space and probability measure precisely. "Choose at random" is ambiguous when applied to continuous geometric objects — you must state the method of selection.
As the number of sides increases, the inscribed polygon's side length decreases, making it easier for random chords to exceed it. In the limit of infinite sides, the polygon approaches the circle, and almost all chords are "longer" than the vanishingly small side.
Jaynes argued in 1973 that the Method 2 answer (1/2) is uniquely correct because it's the only solution invariant under translations and rotations — it satisfies the maximum ignorance principle. However, this is debated.
With 10,000 trials, the simulation typically agrees with the theoretical value to within 1-2 percentage points. Increasing to 100,000 trials reduces the standard error by about √10 ≈ 3.16×.