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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:
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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
- E.T. Jaynes, The well-posed problem, in Foundations of Physics,
vol. 3, pp. 477--492, 1973.
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