Bertrand Paradox

The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Bertrand introduced it in his work Calcul des probabilités(1889)^[1] as an example to show that probabilities may not be well defined if the mechanism or method that produces the random variable is not clearly defined. The link below has to be read to understand the paradox.

The Bertrand paradox goes 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 Paradox - Wiki

In his 2007 paper, "Bertrand’s Paradox and the Principle of Indifference",^[7] Nicholas Shackel affirms that after more than a century the paradox remains unresolved, and continues to stand in refutation of the principle of indifference. Also, in his 2013 paper, "Bertrand’s paradox revisited: Why Bertrand’s ‘solutions’ are all inapplicable",^[8] Darrell P. Rowbottom shows that Bertrand’s proposed solutions are all inapplicable to his own question, so that the paradox would be much harder to solve than previously anticipated.

The paradox has been investigated for more than a century. I offer the following simple solution. Drop a circle randomly on a plane containing a line - or alternatively drop a line on a plane containing a circle. If the circle contacts the line a chord is created and its length tested. If the circle does not contact the line no chord is created and this trial is discarded.

It is easy to see that when the circle contacts the line, it can fall anywhere on a diameter of the circle perpendicular to the line with equal probability. In the figure below the line extends to infinity in both directions.

It is known that the side of an inscribed equilateral triangle bisects the the radius of the circle when the radius is drawn perpendicular to the side of the inscribed equilateral triangle. Therefore 1/2 the random events described above (all positions on the diameter above are equally likely) will result in a chord less than the length of the side of an inscribed equilateral triangle i.e. exactly one half the diameter crossings result in a chord less than the length of the side of an inscribed equilateral triangle. In the figure above chords that intersect the highlighted red "half diameter" will be longer than the side of an inscribed equilateral triangle.

This method of selecting the chord is truly random and abides by the principle of "maximum ignorance" (as well as scale and translation invariance).

Q.E.D.