Bertrand paradox (probability)

The Bertrand paradox, named after Joseph Bertrand (1822-1900), in the stochastic states that probabilities need not be well defined if the underlying probability space and the method that produces the random variable of interest is not clearly defined.

Bertrand's formulation of the problem

We consider a circle and an inscribed equilateral triangle. A chord is selected at random. What is the probability that the chord is longer than a side of the triangle?

Bertrand gave three ways to solve the problem, the all seem valid, but yield different results.

The selection methods can be visualized as follows: A tendon is uniquely defined by its center. Each of the three methods results in a different distribution of midpoints: Methods 1 and 2 result in two different, non- uniformly distributed distributions, method 3 produces a uniform distribution. On the other hand, the tendons from method 2 seem more evenly distributed over the circle than those from the other two methods.

Many of the other possible ways to draw the string, leading to different probabilities.