a propos "accuracy in speaking": We were talking (at least, I was) about the physical world, not the mathematical. And there we're talking about experiments - not proofs. And until now, no experiment has falsified the hypothesis that the quantum world acts randomly...
Old chestnut: Einstein was uncomfortable with randomness in quantum mechanics and expressed his discomfort with the phrase, "The Lord God does not play dice." To which Neils Bohr retorted, "Who are you to tell God what to do?"
Again, one cannot say something is random in the system if he does not know what the system "is."