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": rather, by definition, given any two points on a circle, we call _chord_ the straight line *segment* connecting the two points. Whence 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.

But 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