There's always ____ possibility that he will come.
Actually, scratch my last comment. It isn't entirely accurate in this scenario.
In this scenario I'd say 'the', is the better answer, because he either shows up, or he doesn't.
Whether or not someone shows up is restricted to two outcomes. They either turn up, or they don't. There is nothing else; no middle ground. Being a singular probability between 'true' and 'false', how can 'a' apply?
'a' would imply that this is but one outcome out of many. That makes no sense because there's just the one possibility - there are NOT multiple possibilities whether he shows up or not.