Abstract
Bertand’s paradox is a fundamental problem in probability that casts doubt on the applicability of the indifference principle by showing that it may yield contradictory results, depending on the meaning assigned to “randomness”. Jaynes claimed that symmetry requirements (the principle of transformation groups) solve the paradox by selecting a unique solution to the problem. I show that this is not the case and that every variant obtained from the principle of indifference can also be obtained from Jaynes’ principle of transformation groups. This is because the same symmetries can be mathematically implemented in different ways, depending on the procedure of random selection that one uses. I describe a simple experiment that supports a result from symmetry arguments, but the solution is different from Jaynes’. Jaynes’ method is thus best seen as a tool to obtain probability distributions when the principle of indifference is inconvenient, but it cannot resolve ambiguities inherent in the use of that principle and still depends on explicitly defining the selection procedure.
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Notes
One must be historically fair to Bertrand, however. Although it has become standard to seek the probability that a chord is longer than the side of the inscribed triangle, Bertrand’s own version required the probability that it be shorter. This obviously eliminates diameters from the sample space. Furthermore, it also proves that the standard solutions are all valid, since one can calculate the probability that a chord is longer than the side as one minus the probability that is it shorter. When applied to calculating the probability that a chord is shorter than the side, the argument underlying B3 is obviously valid and gives 3/4, which in turns yields indeed 1/4 for the probability that the chord is longer. This shows that the solution is actually sound. Nevertheless, I adopt a different approach here in order to leave no doubt about the matter.
This is because we exclude the diameters, which correspond to \(\beta = 0\), so that \(\beta \) and \(- \beta \) always correspond to different angles.
I use the term “number” informally here, for brevity’s sake. One can easily transfer this to the language of measures.
The only exception to this is the relation between the four angles that we identify as selecting the same chord. These are dependent on the axes chosen. However, this does not influence the independence of the choice of angles and axes in the first place.
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Drory, A. Failure and Uses of Jaynes’ Principle of Transformation Groups. Found Phys 45, 439–460 (2015). https://doi.org/10.1007/s10701-015-9876-7
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DOI: https://doi.org/10.1007/s10701-015-9876-7