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To prevent any confusion, I am not considering the Bayesian interpretation here (though anyone is welcome to include that in an answer if it helps clarification).

The Wikipedia page on the "classical definition of probability" states that

The classical definition of probability was called into question, [and] The frequentist definition of probability became widely accepted as a result of [this] criticism

I did some reading, but I don't quite understand the difference between the classical interpretation and the frequentist interpretation, since (in general terms) they both deal with frequencies.

So what is the essential difference?

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The classical definition considered a finite set of outcomes each of which was considered equally likely. So for example by symmetry you consider the chances of each face of a die as being equally likely. The probability is then one over the number of possible events (so 1/6 for a standard cubic die). The frequentist interpretation used the concept of long-run frequency so could deal with infinite sequences.

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