Is radioactive decay deterministic? Suppose you know at time $t$ that there is some atomic nucleus that radioactively decays.
If you were to magically roll back the universe to the exact same state and let it continue as per usual without any intervention, when we get to time $t$ again, would inspection of the same atomic nucleus have the decay happen at that exact time again?
Decay appears to be unpredictable and random, but does this mean it is not deterministic? I am interested in if we know whether or not radioactive decay is deterministic or not.
 A: You can’t know that at a certain time there will be a radioactive decay. You can say that the decay will occur with a certain probability, but never with a certainty.
Even if the decay happened at a certain time $t$, if you could hypothetically roll back the universe$^1$ and then let it continue, there is no guarantee that the same decay will happen at the exact same time. This is exactly what we mean when we say that quantum mechanical processes (like radioactive decay) are not deterministic$^2$.
At the quantum level, all processes have an inherent uncertainty and there is no way around this. It is an intrinsic property of nature.
$^1$ For the same reasons, this would be impossible even in principle. If you could do such a thing, this would imply that the universe is deterministic, and for the reasons discussed, it certainly is not.
$^2$ Note that determinism is considered  to depend on the interpretation of quantum mechanics. I more or less refer to the Copenhagen Interpretation (perhaps the most common amongst physicists) whenever talking about QM though I have never given this deeper consideration.
But one of the cornerstones of quantum mechanics, the Heisenberg uncertainty principle (HUP), supports of the notion of the non-deterministic nature of the universe, and this principle is valid in both Copenhagen and other interpretations (although the HUP may itself be interpreted differently in these other interpretations).
A: Radioactive decay is deterministic only for large populations of radioactive nuclei, for which an average time to decay can be measured and a half-life thereby calculated with statistical tools.
As you go to progressively smaller populations, the statistical tools eventually become inapplicable and the behavior of a single radioactive nucleus becomes nondeterministic i.e., there is no way in principle to predict exactly when it will decay.
A: The question is ultimately not about the particle decay, but about being able to predict the evolution of the Universe from its very beginning. This raises a host of questions:

*

*Whether this is computationally possible (i.e., whether we can access, store and manipulate the information about all the degrees-of-freedom of the Universe... while being a part of this very Universe). If this is indeed a limitation, we end up with having to use statistical physics, where the events are random due to our lack of knowledge about all the degrees-of-freedom.

*Whether the evolution of the Universe is tryly deterministic or whether, e.g., spontaneous symmetry breaking at different levels is truly random (see, e.g., the Anderson's famous More is different).

*Finally, the way the question is formulated assumes the frequentist interpretation of probability - as a frequency of an event in an (infinite) ensemble of trials. Bayesian view is less common in physics, but from philosophical point the primacy of either of them is far from settled.

A: It’s not entirely clear what you mean by “magically roll back time.”  If you include enough magic you can get any outcome you want.
Let’s consider a specific, realizable example: suppose that you have a molecule made of two tritium atoms, one of which will decay before the other.
Before the decay, the two tritium nuclei in the molecule are indistinguishable from each other.  That means all the arguments which create the difference between orthohydrogen and parahydrogen would also apply to pure tritium, and a hypothetical sample of pure tritium would have different macroscopic low-temperature thermal properties depending on how long it had been cold and whether it had flowed over any cold magnetic catalyst.
Indistinguishability means that we can swap the two nuclei with zero observable effects.  If I give you a molecule of diatomic tritium, and one of the nuclei decays, you can’t say “the decay happened to the atom on the left.” That’s just not a degree of freedom that simple molecules have.  Indistinguishability is a fundamental assumption of quantum statistics, supported by mountains of evidence from many surprising directions.  A particle which is “about to decay” is really, fundamentally, exactly the same as a particle which is not going to decay for many half-lives.
A: This is really a question about whether you believe the outcome, the atomic nucleus decays at a given instant of time, can be determined exactly (Einstein) or the best you can do is to assign a probability of the event happening (Bohr and Heisenberg).
As far as I know the best we can do is to use Quantum Mechanics to assign a probability to the event (the decay of an atomic nucleus at an instant of time) happening.
Einstein thought that there must be a hidden layer of reality below the quantum level, and that if we could find this hidden layer, we could get rid of the probabilistic laws of quantum mechanics and not just predict what might happen next but, using the deterministic laws, predict what will happen.
So, in the case of radioactive decay, are we able to determine exactly what will happen to an atomic nucleus?
It appears at present, that the answer to this question is, "no", as we have not yet found the deterministic laws which are required to do this.
A: "Rolling back the universe" is a frequent thought experiment in philosophy; especially in discussions of Time, Free Will, Determinism or the like. Interestingly, whether you believe that everything will happen exactly the same when you roll back is often orthogonal to those other questions in some sense.
Unfortunately, obviously we do not know the real, physical answer. Nobody knows for sure whether the universe is "transactional" in this manner and can be rolled back to a previous snapshot, much less whether atom decay and all the other quantum phenomena will repeat exactly; and we obviously are not able to do it practically.
Since we have no way to falsify the statement by experiment or otherwise, there is no scientific way whatsoever to give a meaningful answer except "we don't know"; the question must stay firmly in the realm of thought experiments or belief systems.
