[shonenknifefan1]

Jaynes has a good chapter on the topic of the distinction between randomness and ignorance within the Bayesian framework [1] in "Probability Theory: The Logic of Science".

The distinction between the two is that in the context of ignorance your belief state is very unstable and susceptible to change with the injection of new information. On the other hand, in the context of a "random" event like a coin flip, your belief state is very stable and unsusceptible to change with the injection of new information.

Whereas you may initially apply the same probability to both a coin flip and the outcome of a wrestling match, those probabilities will evolve in very different ways as you acquire new information such as inspecting the coin/combatants. The probability of the outcome of the coin flip is unlikely to change at all, whereas the probability of the outcome of the wrestling match could change substantially.

This can be modeled by assigning the probability of an outcome to a continuous rate parameter, and assigning a probability distribution to that rate parameter. The entropy of that probability distribution can describe how ignorant your belief state is, and hence how susceptible it is to change.

[1] http://www-biba.inrialpes.fr/Jaynes/cc18i.pdf