Bertrand's Paradox Calculator

Bertrand's paradox is a famous probability puzzle posed by Joseph Bertrand in 1889. It asks: what is the probability that a random chord drawn inside a circle is longer than the side of an equilateral triangle inscribed in that circle? The paradox arises because three different but equally valid methods of choosing a 'random' chord each produce a different probability — 1/3, 1/2, and 1/4 — demonstrating that the answer depends entirely on how you define randomness.

The three classic solutions are: (1) Random Endpoints — choose two random points on the circle's circumference; probability ≈ 1/3 (33.33%). (2) Random Radius — choose a random radius and a random point along it as the chord's midpoint; probability = 1/2 (50%). (3) Random Midpoint — choose a random point inside the circle as the chord's midpoint; probability = 1/4 (25%). Each method is mathematically valid, yet each gives a different answer.

The word 'random' is ambiguous without a precise definition of the probability space. Each of the three methods imposes a different uniform distribution over a different geometric parameter — arc length, radial distance, or area. Because the parameter being randomised differs, the resulting probabilities differ. The paradox reveals that probability statements are only well-defined once the sample space and distribution are fully specified.

Fix one endpoint anywhere on the circle. The chord is longer than the triangle's side if and only if the second endpoint falls within the arc of 120° (one-third of 360°) opposite the first point. Since the second endpoint is uniform on the full circle, the probability equals 120°/360° = 1/3 ≈ 33.33%.

The principle of indifference states that if there is no reason to favour one outcome over another, equal probabilities should be assigned to all outcomes. Bertrand's paradox challenges this principle: applied to different parameterisations of the same problem, the principle yields different results. This shows that the principle of indifference is not uniquely applicable without a natural or physically motivated choice of variable.

There is no universally agreed correct solution in pure mathematics — all three are valid given their assumptions. However, in physical experiments (e.g. dropping a straw randomly onto a circle), the random midpoint method (probability 1/4) tends to match empirical results most closely when the straw can land anywhere over the disc. The choice ultimately depends on the physical or conceptual process generating the chord.

Simulation relies on pseudo-random number generation and the law of large numbers. With a finite number of chords, the simulated probability will fluctuate around the theoretical value due to sampling variation. Increasing the number of simulated chords reduces this variance — with 100,000 chords, the estimate typically converges to within a fraction of a percent of the true theoretical probability.

Bertrand's Box paradox is a separate but related problem involving three boxes containing combinations of gold and silver coins. It asks: given that you drew a gold coin, what is the probability the other coin in the same box is also gold? The answer is 2/3, not 1/2 as intuition suggests. While both are Bertrand's paradoxes, the chord problem concerns geometric probability and the definition of randomness, whereas the box problem concerns conditional probability and Bayesian reasoning.