To cut a very long story short, it has been proven that the randomness in quantum mechanics cannot be the result of a lack of information (the same way the randomness in a dice roll can) or missing information about a deeper theory, unless that deeper theory obeys all of the following:
(a) Is non-local, that is allows signals to be sent faster than light and back in time. So that the past, present and future of the universe communicate with each other. So every particle is in contact with its own past and future self in addition to the pasts and futures of every other particle.
(b) Every particle carries an infinite amount of information (essentially the data concerning the communications above)
(c) The world is contextual. That is there is no universal quantities like "Energy", only "Energy in my lab in Singapore given that I ate lunch at 14:05 yesterday", which is a different physical quantity from "Energy in my lab in Singapore given that I ate lunch at 14:06 yesterday".
So either the world is genuinely random (i.e. subatomic events have no cause, no physical mechanism which results in their occurrence, so that let us say there is no answer to the question "Why did that atom decay?") or (a),(b) and (c) above are true.
(a) comes from Bell's theorem.
(b) from Hardy's ontological excess baggage theorem.
(c) from the Kochen-Specker theorem.