I'd be careful about throwing around the word "determinism" quite so casually as science has shown that the universe is not deterministic. But this does not suddenly mean that science is redundant as a tool. Rationality and reason certainly do not require determinism, and like science they merely require a probabilistic outcome. Science requires a reasonably high level of probability/confidence of repeatability, but reason and rationality certainly don't.
For example, if I toss a coin three times then reason would suggest it is rational to conclude that it won't land the same way up each time (75% probability that it won't). There's nothing certain that it won't, and in the context of the experiment the outcome of the coin toss is effectively indeterministic (assuming no bias).
Rationality works (or should work) on accepting what is most likely. This is merely probabilistic, not deterministic.
I would say that "by chance" is merely a phrase to express there being no obvious design to the event, or simply to express an occurrence with a perceived low probability. It is a subjective viewpoint, as one person may see the design/cause behind an event, and others may not.
if we understand chance simply as the probability of an event occurring then we would need to prove the world is either deterministic (and then chance simply does not exist) or indeterministic (in which case chance does exist, as outcomes become probabilistic).
Colloquially when we say "by chance" we are simply expressing our lack of knowledge of the actual probability of the event occurring, the lack of any perceived design to the occurrence. But that perception may well be wrong, and what we see as "by chance" might have been more likely than not.
If we ignore the mathematical sense of the word as simply meaning the probability of an event occurring, then in the colloquial sense I think it is more an expression of
unqualified speculation.
