Bertrand's Paradox

Dear Alexander,

I have found your site and find it very interesting. However, I have found a comment on your page on Bertrand's Paradox.

You give two different solutions:

First Solution.

Assign a uniform probability distribution to the angles of intersection of the cord on the circumference. Then p=1/3.

Second Solution

Assign a uniform probability distribution to the center of the chord over the area of the circle. p=1/4.

There actually is a third intuitive solution:

Assign a uniform probability distribution to the linear distance between centers of chord and circle midpoint.

E.T.Jaynes has given a very sound argument for this third solution in his paper "The Well-Posed Problem". His own, very careful words about his viewpoint:

If we can find another viewpoint according to which such problems do have definite solutions, and define the conditions under which these solutions are experimentally verifiable, then while it would perhaps be overstating the case to say that this new viewpoint is more "correct" in principle than the conventional one, it will surely be more useful in practice.

So let's have a look at his practical solution. You write:

Thus the "paradox" merely indicates that probability distributions in two dimensions warrant a more careful consideration than the second solution suggested. The notion of uniform randomness is actually less obvious than appeared at the first glance.

Jaynes answers this as follows: randomly distributed lines could be made experimentally, by throwing straws on a circle which is sufficiently small and sufficiently distant such that the "rain of straws" falling down on the circle is random. We are interested in the probability that the chord of those straws intersecting the circle is longer than the side of an inscribed equilateral triangle. Now what properties do we this "rain of straws" expect to have? Jaynes's answer to that, in a nutshell: the "rain of straws" does not "know" onto what target it is thrown. The solution must therefore have three properties:

First there is rotational invariance: the outcome must be the same if we turn the target circle by any angle. This is of course trivially true.

Second there is scaling invariance: the outcome must be the same no matter what size of target circle we use, as long as it is small enough. E.g. if we use two concentric target circles of different sizes at the same time, the same "rain of straws" must produce the same outcome in both circles. This is, however, not the case for the first solution. The probability distribution assumed in the first solution is therefore critically dependent on the circle's size, or, in other words, the random "rain of straws" carries information about the circle's size and is not really as random as we would like it to be.

Third there is translational invariance: the outcome must be independent of the exact location of the target circle. E.g. if we use two target circles of the same size, but with different midpoints, the outcome must be the same as long as the distance between the midpoints is small enough. This is not the case for the second solution: in this case the "rain of straws" carries information about the circle's exact location and is again not really random.

Jaynes has actually proved analytically that solution three is the only possible solution for which the "rain of straws" carries no information at all about the target circle it is thrown on. However, he still does not say that he has "solved" Bertrand's paradox:

While it would perhaps be overstating the case to say that this new viewpoint is more `correct' in principle than the conventional one, it will surely be more useful in practice."

(By the way: Dr. Charles Tyler has really thrown straws, until he had 128 hits, and has clearly confirmed the third solution by measurements).

I hope my letter was interesting for you.

Yours sincerely,

Hanspeter.

Reference

  1. E.T. Jaynes, The well-posed problem, in Foundations of Physics, vol. 3, pp. 477--492, 1973.
--
Hanspeter Schmid
Researcher (Analog IC Design)
Signal Processing Laboratory, ETHZ.
Every halfway intelligent theory has
of course
lots of idiots among its defenders
(P.K.Feyerabend)

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