This is, of course, a very important problem. One (extreme) point of view is that any form of classical (=commutative) randomness reflects "only" human uncertainty and does not have an "objective" physical meaning.
(Further answers to this question and more discussion are welcome on the posting entitled "Randomness in nature" on my blog "Combinatorics and More". Here is a link to a subsequent post with further discussion.) Some related material can be found in the site of the conference "The Probable and the Improbable: The Meaning and Role of Probability in Physics".