The correct answer is 'either'.
Because technically both 'a' and 'the' are valid.
A: you use this when you’re not necessarily referring to a specific thing (such as a nail, any nail). It’s called an indefinite article, because you’re not being definite or particular. There are many nails in this big world.
The: you’re talking about a definite item, which is why the is called a definite article. Of course it only makes sense if both you and your listeners know which item you mean. If I commanded you, “Give me the money,” you would rightfully ask, “What money? I don’t owe you any money.”
If you were working at an office you'd be more likely to hear "the possibility of promotion" because typically the succeeding job/role is either predetermined, discussed in advance, or otherwise known. Said individual either gets promoted, or he/she doesn't, therefore the possibility is either there or it is not.
On the flip-side, 'a' would imply that there are multiple avenues of promotion available - perhaps one of many different jobs/roles to choose from. It could also mean merely being offered a promotion has the potential to happen more than once at the same time.
Since 'a' could refer to one of many, said employee could receive 10 offers of promotion all at once ('a' promotion), or just one offer, referred to as 'the' promotion.
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.