Are there any pure coincidences? Are there any totally meaningless happenings?
cf. Soph. Trachiniae 1.1-3
Here’s a brief argument that there are not.
I think you can show at least that (a) one can never know whether a given coincidence is meaningless, and that (b) since fact a is itself meaningful, it follows that every coincidence is at least ‘formally’ meaningful — meaningful as a reminder of fact a.
Here goes:
Lessons are meaningful. Some coincidences are lessons. Some lessons take time to realize. Some lessons take a lifetime to realize. Then we ask: "how do we know that this coincidence isn't a lesson that takes a lifetime to realize?"
Insofar as we are still living (and we are always still living!), we can't know the answer to this question. As such, we can always conclude for everything that (following Vergil) "perhaps one day it will help to remember even these things." (A. 1.203). Every coincidence (including every misfortune!) is at least potentially meaningful, even supremely meaningful.
Above, I called this kind of meaning ‘formal.’ That’s because it holds true regardless of the ‘content’ of any particular experience. No matter what you’re going through, that experience always might be a lesson.
Hence, any experience can validly remind you of Vergil's postulate (that everything is potentially meaningful). And hence, perhaps surprisingly, every experience becomes not just potentially, but actually meaningful. That is: meaningful as a reminder of the fact that everything is potentially meaningful.
How meaningful.