Table of Contents
The Bertrand Paradox
Proposed by Joseph Bertrand in 1889, this paradox asks: what is the probability that a random chord of a circle is longer than the side of an inscribed equilateral triangle? The paradox is that three seemingly reasonable methods of selecting a "random chord" give three different answers: 1/3, 1/2, and 1/4.
This paradox highlights a fundamental issue in probability: the answer to a probability question depends critically on how "random" is defined. Without a precise specification of the probability measure, the problem is ill-posed.
Three Methods and Their Results
| Method | Description | P(longer) |
|---|---|---|
| Random Endpoints | Choose two random points on circumference | 1/3 |
| Random Radius | Choose random point on a radius | 1/2 |
| Random Midpoint | Choose random point inside circle | 1/4 |
Lesson from the Paradox
- The word "random" is ambiguous without specifying the probability measure.
- Different geometric methods of randomization can yield different probability spaces.
- This is related to the principle of maximum entropy and the invariance principle in probability.
FAQ
Which answer is correct?
All three are correct given their respective definitions of "random." The paradox shows the importance of precisely defining the random process. E.T. Jaynes argued that the answer should be 1/2 based on rotational and scale invariance.
How does this relate to modern probability?
It motivated the development of measure-theoretic probability by Kolmogorov, which requires explicitly defining the probability space before computing probabilities.