I believe in luck in the sense that, as Pasteur said, "Chance favours the prepared mind." Unless one is describing the past (e.g. "That was a lucky throw"), pure luck is entirely random by definition, and favours only those who are good at mitigating the bad and enhancing the good.
Coincidences are also entirely random. The reason they occur so often is because, although each specific coincidence is an unlikely event, there is an absurd number of possible ones.
Good luck/Bad luck. It is completely separated from statistics, it is simply chance.
Good luck charms and such are bull but luck as an intangible idea certainly exists.
I would say that luck as an idea does not exist. Luck is something subjective and relative to someone (the lucky one). As I see it, probability exists and luck is just a consequence of our subjective perception of it.