Bertrand paradox (probability)

The Bertrand paradox is a problem within the classical interpretation of probability theory. It poses a geometrical probability question where the answer appears indeterminate, depending on how the problem is approached. It highlights the necessity of precisely defining the "random" selection method used to solve a probability problem.

The problem, first introduced by Joseph Bertrand in 1889, can be stated as follows:

Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?

Bertrand offered three different, seemingly valid, methods for randomly choosing a chord, each leading to a different probability:

Method 1: Random Endpoints

Choose two points uniformly at random on the circumference of the circle and draw the chord connecting them. Imagine the triangle fixed in place. If both points lie on the arc between two vertices of the triangle (one third of the circumference), then the chord will be shorter than a side of the triangle. Otherwise, it will be longer. The probability that both points land in the same arc is 1/3. Therefore, the probability that the chord is longer than a side of the triangle is 1 - 1/3 = 2/3.

Method 2: Random Radius

Choose a radius of the circle at random and then choose a point on that radius at random. Construct a chord through that point perpendicular to the radius. The chord will be longer than a side of the inscribed equilateral triangle if the chosen point on the radius is closer to the center of the circle than the midpoint of the radius. This occurs half the time. Therefore, the probability that the chord is longer than a side of the triangle is 1/2.

Method 3: Random Midpoint

Choose a point at random inside the circle and construct a chord whose midpoint is that point. The chord will be longer than a side of the inscribed equilateral triangle if the midpoint is inside a circle concentric with the original circle, with radius equal to half the radius of the original circle. The area of the smaller circle is 1/4 the area of the larger circle. Therefore, the probability that the chord is longer than a side of the triangle is 1/4.

The Paradox and its Resolution

The paradox arises because the phrase "at random" is ambiguous. Each method implicitly defines a different probability distribution over the set of all possible chords. There is no single, universally "correct" way to choose a chord at random unless the selection process is explicitly defined.

The resolution to the paradox lies in recognizing that without a clearly defined method for selecting a random chord (i.e., specifying a probability distribution), the question is ill-posed. The different methods demonstrate how different interpretations of "random" can lead to different, yet logically consistent, answers.