Bertrand’s paradox

Solving the “hard problem”

Joseph Louis François Bertrand (1822 — 1900) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics.

In his influential book on probability theory, Calcul des probabilités (1889), he introduced a problem challenging the classical interpretation of probability theory, today known as Bertrand’s paradox. In Bertrand’s own words, the problem is the following

We draw at random a chord onto a circle. What is the probability that it is longer than the side of the inscribed equilateral triangle?

Bertrand gave three different answers to his question, yielding three different values for the sought probability. And since these three answers appear to be based, all three, on a valid reasoning, that’s why Bertrand’s problem (and similar problems) was subsequently qualified, by Poincaré, as a paradox.

The first solution

The first proposed answer consists in choosing an arbitrary point on the circle, considering it as one of the vertexes of an inscribed equilateral triangle. This point, describing one of the two points of intersection of the chord with the circle, is kept fixed, whereas the second point is varied (so that the chord moves like a sort of pendulum). One then observes that by considering all possible (second) points on the circle, the chord will rotate of a total angle of 180°, but that only the chords lying within the arc subtended by an angle of 60° at the vertex (see the figure), satisfy the condition of being longer than the side of the inscribed equilateral triangle. Thus, one finds that the probability is:

P = 60° divided by 180° = 1/3.

The second solution

The second proposed solution consists in first choosing an arbitrary direction, and then considering chords which are all parallel to that direction. Then, moving the chords along the circle, one observes that those intersecting its diameter in its central segment, whose length is half the diameter of the circle, satisfy the condition of being longer than the side of the inscribed equilateral triangle. Thus, one finds this time that the probability is:

P = half-diameter divided by diameter = 1/2.

The third solution

The third solution proposed by Bertrand consists in choosing an arbitrary point inside the circle, considering it as the middle point of the chord. Then, moving this point within the entire area of the circle, one observes that all the chords having their middle point within an internal smaller circle, whose radius is one half the radius of the big circle (and which is the incircle of the equilateral triangle), satisfy the condition of being longer than the side of the inscribed equilateral triangle. Being the area of the internal circle one fourth of the area of the big circle, this time the probability is:

P = area-small-circle divided by area-big-circle = 1/4.

According to a recent analysis of philosopher Nicholas Shackel, the current situation is that after more than a century, the paradox remains unresolved, and continues to stand in refutation of the so-called principle of indifference [N. Shackel, “Bertrand’s Paradox and the Principle of Indifference,” Philosophy of Science, 74, April 2007, pp. 150–175]

Even more pessimistically, philosopher Darrell P. Rowbottom recently affirmed that Bertrand’s proposed solutions to his own question, which generate his chord paradox, are all inapplicable, so that there is no solace for the defenders of the principle of indifference, as it emerges that the paradox is much harder to solve than previously anticipated. [D. P. Rowbottom, “Bertrand’s Paradox Revisited: Why Bertrand’s ‘Solutions’ Are All Inapplicable,” Philosophia Mathematica (III) Vol. 21 No. 1, 2012].

Let me remind that the principle of indifference, originally formulated by Jakob Bernoulli as the principle of insufficient reason, and later on by John Maynard Keynes (who strenuously opposed the principle, and devoted an entire chapter of his book in an attempt to refute it), tells us that [Keynes, John Maynard, A Treatise on Probability, London: Macmillan ([1921] 1963)]:

If there is no known reason for predicating of our subject one rather than another of several alternatives, then relatively to such knowledge the assertions of each of these alternatives have an equal probability.

This principle is usually assumed to incorporate a necessary truth about the relation between “possibilities” and “probabilities”:

Possibilities of which we have equal (objective) ignorance have equal probabilities.

And it is generally assumed that its application is sufficient for solving probability problems and finding for them unique solutions!

But this belief is precisely what has been undermined by Bertrand, with his three different “solutions”, all three apparently based on that fundamental principle.

Now, recently, together with Diederik Aerts, we could propose what we think is a convincing solution to this old and important problem, lying at the foundation of probability theory. The solution came about to us as a consequence of a mathematical problem we were able to solve with respect to the measurement problem of quantum theory. In fact, Bertrand’s paradox stood in the way of the solution we searched, so that by solving the latter we also obtained what we think is a convincing solution to the former.

This, however, should not come as a surprise. The intimate connection between fundamental problems of probability theory, like Bertrand’s paradox, and of quantum mechanics, like the measurement problem, is in fact not coincidental. Both disciplines deal with the description of systems subjected to specific experimental actions, according to protocols which incorporate the presence of irreducible fluctuations, so that the outcomes of these actions cannot be predicted in advance with certainty, not even in principle.

In that sense, we can certainly affirm that that the founding fathers of probability theory, without knowing it, were actually quantum physicists ante litteram!

Solving the hard problem of Bertrand’s paradox
by Diederik Aerts and Massimiliano Sassoli de Bianchi

J. Math. Phys. 55, 083503 (2014);

Abstract: Bertrand’s paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace’s principle of insufficient reason. In this article we show that Bertrand’s paradox contains two different problems: an “easy” problem and a “hard” problem. The easy problem can be solved by formulating Bertrand’s question in sufficiently precise terms, so allowing for a non ambiguous modelization of the entity subjected to the randomization. We then show that once the easy problem is settled, also the hard problem becomes solvable, provided Laplace’s principle of insufficient reason is applied not to the outcomes of the experiment, but to the different possible “ways of selecting” an interaction between the entity under investigation and that producing the randomization. This consists in evaluating a huge average over all possible “ways of selecting” an interaction, which we call a universal average. Following a strategy similar to that used in the definition of the Wiener measure, we calculate such universal average and therefore solve the hard problem of Bertrand’s paradox. The link between Bertrand’s problem of probability theory and the measurement problem of quantum mechanics is also briefly discussed.
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