"But hidden variables aren't the way our universe works—it really is random, right down to its gnarly, subatomic roots. In 1964, the physicist John Bell proved a theorem which showed hidden variable (little clock in the nucleus) theories inconsistent with the foundations of quantum mechanics. In 1982, Alain Aspect and his colleagues performed an experiment to test Bell's theoretical result and discovered, to nobody's surprise, that the predictions of quantum theory were correct: the randomness is inherent—not due to limitations in our ability to make measurements. So, given a Cæsium-137 nucleus, there is no way whatsoever to predict when it will decay. If we have a large number of them, we can be confident half will decay in 30.17 years; but if we have a single atom, pinned in a laser ion trap, all we can say is that is there's even odds it will decay sometime in the next 30.17 years, but as to precisely when we're fundamentally quantum clueless. The only way to know when a given Cæsium-137 nucleus decays is after the fact—by detecting the ejecta. A Cæsium-137 nucleus which has “beat the reaper” by surviving a century, during which time only one in a thousand of its litter-mates haven't taken the plunge and turned into Barium, has precisely the same chance of surviving another hundred years as a newly-minted Cæsium-137, fresh from the reactor core."
Go read the overcomingbias essay on the Many-Worlds interpretation posted above. (Or is it below?)
The short answer is that in the many-worlds interpretation, this too is a deterministic process. The total wavefunction of the whole system (atom + observer) evolves deterministically. What isn't deterministic is "your" subjective view of it, but "you" only view a vanishingly small slice of reality.
Sorry, that's the best three-sentence explanation I can come up with right now, and I admit it's only a shade better than "trust me, I'm a physicist". But trust me, I'm a physicist.
From [http://www.fourmilab.ch/hotbits/how3.html]
"But hidden variables aren't the way our universe works—it really is random, right down to its gnarly, subatomic roots. In 1964, the physicist John Bell proved a theorem which showed hidden variable (little clock in the nucleus) theories inconsistent with the foundations of quantum mechanics. In 1982, Alain Aspect and his colleagues performed an experiment to test Bell's theoretical result and discovered, to nobody's surprise, that the predictions of quantum theory were correct: the randomness is inherent—not due to limitations in our ability to make measurements. So, given a Cæsium-137 nucleus, there is no way whatsoever to predict when it will decay. If we have a large number of them, we can be confident half will decay in 30.17 years; but if we have a single atom, pinned in a laser ion trap, all we can say is that is there's even odds it will decay sometime in the next 30.17 years, but as to precisely when we're fundamentally quantum clueless. The only way to know when a given Cæsium-137 nucleus decays is after the fact—by detecting the ejecta. A Cæsium-137 nucleus which has “beat the reaper” by surviving a century, during which time only one in a thousand of its litter-mates haven't taken the plunge and turned into Barium, has precisely the same chance of surviving another hundred years as a newly-minted Cæsium-137, fresh from the reactor core."