And then the nuclei return to the ground state. That process is probabilistic and measured in half-lives. The key point is that the decay back to ground state happens at a very precise rate that is not influenced by effectively anything, and can be measured accurately. Thus, a clock.

> That process is probabilistic and measured in half-lives

> The decay back to ground state happens at a very precise rate that is not influenced by effectively anything

That sounds contradictory to me.

I suppose it could: the term "probabilistic" applies to the quantum probability of any one metastable isomer (excited nucleus) decaying to ground state. In application you measure large numbers of decays, and in great numbers the decay curve is extremely precise.