Archive for Borel

Bertrand’s paradox

Posted in Books, Statistics, Travel, University life with tags , , , on March 17, 2011 by xi'an

While progressing through Jaynes’ Probability Theory for my classes next week, I reached the end of Chapter 12 where he proposed a “resolution” of Bertrand’s paradox. Jaynes gives the following description of the paradox:

Bertrand’s problem was stated originally in terms of drawing a straight line `at random’ intersecting a circle (…) we do no violence to the problem if we suppose we are tossing straws onto the circle (…) What is the probability that the chord thus defined [by a random straw] is longer than the side of the inscribed equilateral triangle?

My understanding of the paradox is that it provides a perfect illustration of the lack of meaning of “random” and of the need for a proper definition of the σ-algebra leading to a probabilised space. Different σ-algebras lead to different probabilities, e.g., 1/4, 1/3, 1/2… However, Jaynes considers there is a “correct” answer and endeavours to construct an invariant distribution on the location of the centre of the chord, achieving Borel’s distribution

p(x) = \frac{x}{\sqrt{1-x^2}}

on the chord proportion L/2R. When illustrating this approach, I tried to bring an empirical vision and generated “random” straws by picking both ends at random on the (-10,10)^2 square. Here is a subsample of 10³ such straws interesting with the unit circle. The empirical distribution of the chord proportion is actually quite in agreement with Borel’s distribution, with a probability of being longer than the side of 1/2, as shown below but this does not validate Jaynes’ argument, simply illustrates that I picked the same σ-algebra as his. (Every σ-algebra considered by Bertrand could as well have been used for the simulation.)