Monday 9 October 2023

A Solution to Bertrand's Paradox

Quoting from Wikipedia, "Bertrand paradox (probability)" [1]
(there is also a nice explanation by 3blue1brown in [2]):
<< "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?" >>

<< The argument is that if the method of random selection is specified, the problem will have a well-defined solution (determined by the principle of indifference). >>

<< Bertrand gave three arguments (each using the principle of indifference), all apparently valid, yet yielding different results. >>

<< The three solutions presented by Bertrand correspond to different selection methods, and in the absence of further information there is no reason to prefer one over another; accordingly, the problem as stated has no unique solution. >>

<< Edwin Jaynes proposed a solution to Bertrand's paradox, based on the principle of "maximum ignorance"—that we should not use any information that is not given in the statement of the problem. >>
No, not "determined by the principle of indifference" (and not even by one of "ignorance", if not about how to read, and write, problem statements, i.e. do not assume more than is given): rather, by definition, given any two points on a circle, we call _chord_ the straight line *segment* connecting the two points. Hence a random distribution of chords *is* a random distribution of pairs of points on the circle. Whence there is one correct answer, which happens to be 1/3.

Indeed, as for a logical analysis, the question is "given an arbitrary circle consider drawing random chords", not "given arbitrary lines in space consider those that are secant to a random circle".

[1] Wikipedia, "Bertrand paradox (probability)"

[2] "Bertrand's Paradox (with 3blue1brown) - Numberphile" on YouTube - first part - second part