I admit that I am not an expert in quantum mechanics, but I do have a friend who is a quantum physicist and I have read up on the subject. So please let me know where I am wrong in what I have said.
OK. First what you said didn't really make much sense, and what sense I could make of it seems to suggest you've got it backwards. So...
We have quantum events with no apparent cause.
The quantum mechanics of a given situation allows us to calculate probabilities, so we have a combination of deterministically defined probabilities and random variation within those probabilities.
But in order to achieve stability at the atomic level, we need the probability of specific quantum events to occur at specific times.
This is where is seems to be backwards. You seem to think we need the probabilities (and some things happening according to them at exact times) to make atoms stable but, in fact, we calculate the structure of atoms using quantum mechanics and it's the solutions to those equations that give the probabilities.
For example, to a first approximation, we model the hydrogen atom as an electron in the electric field of a stationary proton (treating both as point charges) and then set up and solve the Schrödinger equation to find the possible states of the system. It's those solutions that provide us with probability densities. The basic structure of the atom follows from the solutions themselves - that's what defines the various possible 'shells' the election can be in and hence the structure of the atom.
So this would indicate that indeterminate quantum events are not entirely random, but may have a cause which can't be perceived by human investigation.
I really don't know what you're even trying to say here, and what "indeterminate quantum events" you mean. Of course nothing in QM is
entirely random. Again: the theory gives us probabilities in any given situation. The theory itself provides the 'cause' for things to behave in the way they do. There are many strange things about QM but there really is no more mystery about why the laws apply than there is about any other theory.