  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.
