Bertrand’s Paradox Calculator (random chord: 1/3, 1/2 or 1/4?)

Understanding the Bertrand's paradox calculator — concept and principle

This calculator shows Bertrand's paradox: the probability that a "random" chord of a circle is longer than a given length depends on how the chord is chosen. It gives the answer for the three classic methods, for the side of the inscribed equilateral triangle or a length you enter, plus the average chord length under each method.

How it is calculated

With c = L ÷ (2r), where L is the chord length to beat and r the radius:

• Random endpoints on the circle: P = 1 − (2/π) arcsin c
• Random point on a random radius (the chord's midpoint): P = √(1 − c²)
• Random midpoint anywhere inside the disc: P = 1 − c²

Mean chord lengths: 4r/π (endpoints), πr/2 (radius), 4r/3 (midpoint).

• The default L is the side of the inscribed equilateral triangle, r√3.

The principle

A chord is fixed by its midpoint, or by its two endpoints. "Choose a chord at random" sounds precise, but there are several natural ways to do it, and they spread probability differently. Picking two uniform endpoints makes the central angle uniform. Picking a uniform point along a radius makes the midpoint's distance from the centre uniform. Picking a uniform point in the disc favours midpoints far from the centre, because there is more area there, which gives more short chords. Each method is reasonable; they simply answer different questions. The lesson is that "at random" means nothing until the sampling method is stated.

Worked example

Default inputs: r = 1, chord longer than the triangle side.

1. Chord length used L = 1 × √3 = 1.73205, so c = √3 ÷ 2 = 0.866.
2. Random endpoints: arcsin 0.866 = π/3, so P = 1 − (2/π)(π/3) = 33.33 %.
3. Random radius: P = √(1 − 0.75) = 50 %.
4. Random midpoint: P = 1 − 0.75 = 25 %.
5. Mean chords: 4/π = 1.27324, π/2 = 1.5708, 4/3 = 1.33333.

How to read the result

The three answers, 1/3, 1/2 and 1/4, are all correct for their own method. If you are modelling a physical process, choose the method that matches it: for example, straws dropped at random onto a circle drawn on the floor behave like the random-radius method, which is the one invariant to moving and rescaling the circle.

Common mistakes

• Treating "random" as if it had one obvious meaning.
• Using degrees inside arcsin by hand.
• Entering a chord longer than the diameter.
• Thinking that one of the three answers is the true one in every situation.

Frequently asked questions

What is Bertrand's paradox?

Three reasonable ways of choosing a random chord give three different probabilities, 1/3, 1/2 and 1/4.

Which answer is right?

Each is right for its method; the problem is not fully specified until the method is.

What is the average chord length of a unit circle?

4/π, π/2 or 4/3, depending on the method.

Sources: Encyclopedia of Mathematics, "Bertrand paradox" · J. Bertrand (1889), Calcul des probabilités, Gauthier-Villars