Bertrand paradox and angels' sex

A good example to help understand the problems outlined in the previous section is the so-called Bertrand paradox.

Problem:

Given a circle of radius $R$ and a chord drawn randomly on it, what is the probability that the length $L$ of the chord is smaller than $R$?

Solution 1:

Choose ``randomly'' two points on the circumference and draw a chord between them: $\Rightarrow P(L<R)=1/3=0.33$.

Solution 2:

Choose a straight line passing through the centre of the circle; then draw a second line, orthogonal to the first, and which intersects it inside the circle at a ``random'' distance from the center: $\Rightarrow P(L<R)=1-\sqrt{3}/2 = 0.13$.

Solution 3:

Choose ``randomly'' a point inside the circle and draw a straight line orthogonal to the radius that passes through the chosen point $\Rightarrow P(L<R)=1/4 = 0.25$.

Your solution:

$\ldots$ $\ldots$ $\ldots$?

Question:

What is the origin of the paradox?

Answer:

The problem does not specify how to ``randomly'' choose the chord. The three solutions take a uniform distribution: along the circumference; along the the radius; inside the circle. What is uniform in one variable is not uniform in the others!

Question:

Which is the right solution?

In principle you may imagine an infinite number of different solutions. From a physicist's viewpoint any attempt to answer this question is a waste of time. The reason why the paradox has been compared to the Byzantine discussions about the sex of angels is that there are indeed people arguing about it. For example, there is a school of thought which insists that Solution 2 is the right one.

In fact this kind of paradox, together with abuse of the Indifference Principle for problems like ``what is the probability that the sun will rise tomorrow morning'' threw a shadow over Bayesian methods at the end of the last century. The maximum likelihood method, which does not make explicit use of prior distributions, was then seen as a valid solution to the problem. But in reality the ambiguity of the proper metrics on which the initial distribution is uniform has an equivalent in the arbitrariness of the variable used in the likelihood function. In the end, what was criticized when it was stated explicitly in the Bayes formula is accepted passively when it is hidden in the maximum likelihood method.