In QM Born's rule is a postulate. It's unlikely that one can proof it somehow from the other postulates (see Weinberg, Lectures on Quantum Mechanics, Cambridge University Press 2012). So far I don't know of any evidence that there is anywhere the classical determinism left. I guess, if one can find a deterministic theory, then it will be even less comprehensible than quantum theory, because given the fact that Bell's inequality is violated as predicted by QT one must give up locality, and this will be a big challenge to be made compatible with the relativistic space-time structure and causality.
Concerning radioactive decay, I've no clue, how it could be understandable within classical mechanics beyond a purely statistical ("random walk") rule: The decay probabilities are given and then implemented in terms of a rate equation, in the most simple case leading just to radioactive decay of A to B+X (like one of the three usual decay mechanisms of radioactivity, called ##\alpha##, ##\beta##, and ##\gamma##, because it was just not understood what's really going on).
With quantum (field) theory it's easy to describe as interactions causing transitions from one state to another, and thus the only microscopic mechanism to "explain" the radioactive decays known today. With "explain" I mean to finally trace it back to the interactions that we take as "fundamental" today,i.e., those described by the Standard Model of Elementary particles; the three decay forms correspond to the strong interaction (cluster formation within nuclei "preforming" ##\alpha## particles, i.e., ##\text{He}^4## nuclei within the nucleus which then tunnel through the potential barrier a la Gamav), the weak interaction (##\beta## decay of one quark flavor to another quark flavor and leptons like ##\text{n} \rightarrow e^-+\bar{\nu}_e + \text{p}##, i.e., the decay of a down quark to an up-quark and the leptons within the neutron, and the electromagnetic interaction, which is nothing else than an electromagnetic transition of an excited nuclei leading to the decay to a less excited nucleus (maybe even to its ground state) and a photon. This is all described in terms of quantum field theory by taking the unstable particles/nuclei as resonances and calculating perturbatively their width, i.e., lifetime which figures into the decay rates to be put into the phenomenological rate equations.
Note that this is an approximation, which is strictly speaking contradicting basic principles of quantum field theory, namely the unitarity of the S-matrix, according to which there cannot be any strictly exponential decay law (see the textbook of Sakurai, 2nd edition). In the energy domain that's the statement that the spectral function of the unstable state cannot be a strict Lorentzian.
