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I think that's an ill-posed question without knowing (or making assumptions I guess) how the distribution was chosen isn't it?

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I'm inclined to agree with you here. The solutions suggested here seem to depend on heuristics that depend on the type of distribution.

A tangential question: real numbers are uncountably infinite. Are probability distributions over the real numbers likewise uncountably infinite, or do they form a higher infinity?

First, a point of terminology: uncountably infinite refers to any infinite cardinality other than countably infinite, so if the number of probability distributions is at least the cardinality of the real numbers, then it's already uncountable.

As for what uncountable cardinality they form, it's the same as the real numbers [0]. Roughly, a probability distribution is determined by the countable collection of real numbers P(X <= q) with q rational. That means the cardinality is no more than R^Q, which is isomorphic to R. (the cardinality is at least R due to, say, uniform distributions on [0, x] for x real).

[0] https://math.stackexchange.com/questions/3698864/why-does-th...

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