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On page 9 of Papoulis's book[Probability, Random Variables, and Stochastic Processes], the classical definition of probability is as follows:

The probability of an event equals the ratio of its favorable outcomes to the total number of outcomes provided that all outcomes are equally likely.

Bertrand's paradox is then examined and he concludes:

We have thus found not one but three different solutions for the .. same problem! One might remark that these solutions correspond to three different experiments. This is true but not obvious and. in any case, it demonstrates the ambiguities associated with the classical definition, and the need for a clear specification of the 9utcomes of an experiment and the meaning of the terms "possible" and ''favorable.''

However, what I think is more correct is that the ambiguity in Bertrand's problem has led to such a paradox rather than the ambiguities in the classical definition of probability. In my opinion, ambiguity in the case of Bertrand problem, even by using the axiomatic definition of probability, leads us to such a paradox; That is why I do not understand why we consider this paradox to stem from the ambiguities in the classical definition.

Please guide me in detail and step-by-step to understand how - as Papoulis says - the ambiguities in the classical definition of probability led to such a paradox?

thanks

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    $\begingroup$ The paradox shows various parameterizations have disparate notions of what two "outcomes" we should deem equiprobable (absent further information) look like. In particular, it's unclear which parameters should have a joint uniform distribution. $\endgroup$
    – J.G.
    Commented Jun 14, 2022 at 9:00
  • $\begingroup$ This exact problem has been discussed by many famous probabilists over the past century. Jaynes argues, that in the problem the exact size of the circle and its position are also undetermined and thus must be invariant under small pertubations (changing the size of the circle or shifting its center). If taking this as an assumption the solution is uniquely determined and eliminates otherwise seemingly possible solutions. Details can be found here: Edwin T. Jaynes, The Well-Posed Problem, Foundations of Physics, 3, (1973), pp. 477-493 (bayes.wustl.edu/etj/articles/well.pdf) $\endgroup$
    – Peter Strouvelle
    Commented Jun 14, 2022 at 9:02
  • $\begingroup$ @PeterKeller Good point. Having said that, it's been argued that same reasoning can also be tweaked to give other solutions. $\endgroup$
    – J.G.
    Commented Jun 14, 2022 at 9:37
  • $\begingroup$ that "classical definition of probability" may be a true fact, but it's certainly not a valid definition, being circular (what does "equally likely" mean???) $\endgroup$
    – David C. Ullrich
    Commented Jun 14, 2022 at 11:50

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