I mean when you think about Conan, the entire saga is begun on a coincidence where the bad guys kill his tribe / family. They could have hit any other village, or they could have killed him, or he might have been away at the time and not known who did what or been captured. But a string of unlikely events propel him to become Conan.
I completely agree. And I think that this is something that people either miss or overlook quite often without even realising it. Almost all of the greatest stories, films, etc, have spurring moments that kickstart a plot that are, basically, coincidence and chance. Sometimes the events are near unbelievable when you think it through. Often they might seem to be something you just accept, as otherwise there isn't a story, but the simple fact is, these things DO happen.
In fact, they happen absolutely all the time, we just rationalise them, or don't even notice them, as we go about our daily lives, blissfully unaware.
As such, I actually try and make a point of including coincidence and luck quite regularly in my writing, it helps make it seem more natural that sometimes the best plans go wrong, and sometimes the desperate ones go right, because of things that protagonists couldn't possibly predict. Sometimes a last ditch effort works against the odds, other times it's a total failure. Occasionally that one-in-a-million shot DOES work.
I usually make sure the characters themselves are well aware that it's coincidence and/or muse on the fact, and, as others have mentioned, be
very careful about using such luck as the
sole reason the hero wins or survives, if for no other reason than it greatly reduces the importance of the character themselves.