---
tag: 0T0JWJ3M47
kind: example
name: Bertrand paradox
aliases: [bertrands-paradox]
msc: [60D05]
instance_of: [0924BCD257]
uses: [0AKNPYQ04R]
sources:
  - url: https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)
    locator: Bertrand's formulation of the problem
same_as:
  wikidata: Q828552
  wikipedia: Bertrand paradox (probability)
---

## Example

Take a circle and an equilateral triangle inscribed in it. Choose a chord of the circle "at random". What is the probability that the chord is longer than a side of the triangle?
Three natural ways to choose the chord give three different answers.
Choose two independent uniform points on the circle and join them. The probability is $1/3$.
Choose a uniform radius, then a uniform point on it, and take the chord perpendicular to the radius through that point. The probability is $1/2$.
Choose a uniform point in the disc and take the chord with that point as midpoint. The probability is $1/4$.

## Remarks

The paradox shows that "at random" has no meaning until one fixes a probability measure. The three methods give three different distributions on the set of chords.
Joseph Bertrand gave the example in his book Calcul des probabilités of 1889, as a criticism of the principle of indifference.

## Explanations

### L1 human

The question sounds complete, but it is not: it does not say how to choose the chord.
Each method is fair in its own sense, and each one gives a different answer. In probability you must always say exactly how the random choice is made.