The classic example of an indeterministic system is a radioactive isotope, e.g. the one that kills Schrödinger's cat.
I get there are arguments against hidden variables in quantum mechanics, but how could they be so sure, back in the twenties, that the strong nuclear forces involved in radioactivity were not governed by hidden variables rather than true randomness?
Einstein was very unhappy about the indeterminism of quantum mechanics regarding even well understood effects like Young's slit experiments, but it seems kind of ideological and brash on behalf of Heisenberg & Co to extend the indeterminism over to phenomena they hadn't even begun to understand, like alpha decay.
Is there a reason for this early self-assuredness in postulating indeterminsm?