It seems to me that in the paradox, although the Axiom of Choice plays a role, the bigger issue is how probability behaves in non-measurable sets. His intuitio

It seems to me that in the paradox, although the Axiom of Choice plays a role, the bigger issue is how probability behaves in non-measurable sets. His intuition about how the laws of probability should work is not correct in this scenario. Conceptually, it's not so different from the garden variety probability 101 issues taught in measurable sets - one can make a countably infinite number of picks from a segment in the real line, and the probability of picking any given number from that segment is still zero, despite an 'infinite' number of picks from a finite-measure segment.