Like the title states, what is the exact chance that a Pokemon from a horde battle will have its hidden ability?
I don't think that you can find an official source for that chance, but I can give you a pretty good alternative. I've found this question on Pokemon Database which is what you're looking for.
Since we can't check the actual chance, a guy tried an experiment testing 200 Whismur hordes, which gives a total of 1,000 individual Whismur. That's a pretty good sample.
What were my results? Well, 952 Whismur had the ability Soundproof and 48 had Rattled. Since I am only interested in the Hidden Ability for this, then 48/1000 = 0.048 or 4.8%
So from this experiment I got the chance of a Pokémon in a Horde with its Hidden Ability is 4.8%. Since GameFreak seems to like round numbers, I suggest the actual answer is probably 5% then.
That's not an official answer, but we can easily assume that the chance of finding a Pokemon from a horde with its Hidden Ability is 5%.