As a side note, it's not just radioactive decay. Quantum mechanics always gives a probabilistic prediction for the behavior of any system, whether it's a radioactive isotope emitting an alpha particle, the decay of a single muon, a randomly polarized photon passing through a polarized filter, where an electron lands in a diffraction experiment, or anything else. We have no idea whether those processes are fundamentally random, or if there are underlying mechanisms that produce the apparently random behavior we see. It's entirely possible that we may not ever be able to know the answer to that.
Conserve energy. Commute with the Hamiltonian.