In this particular case, all we need to do is encode our "state of knowledge" formally into the relevant probabilities, mooting all appeals to intuition.

However this is a "toy problem", in real-world situations I expect that it will not be practical to enumerate all possible outcomes.

I am helping a colleague of mine investigate application of Bayesian inference methods to the question of software testing, and we're seeing much the same difficulty: on an extremely simplified problem we can draw definite conclusions, but we don't yet know how to extend those conclusions to situations the industry would consider relevant.