Can randomness be defined?
Someone who knows more about the physics than I do would need to answer the latter question, but I believe that certain physical phenomena are supposed to be random in a much stronger sense that just "unintelligible to humans". There is supposed to be nothing other than probabilistic facts about when and how the phenomenon might occur. Even God cannot know in advance when and what is to happen, for there is simply no such fact to be known.
There is a sophisticated mathematical theory of randomness. See the Wikipedia entry on Kolmogorov complexity. It describes a way, in terms of informational complexity, to characterize when a sequence of 0s and 1s is random.