It's not entirely out of the question. There have been many previous cases of animals that were not sighted for years eventually showing up again.
The ivory-billed woodpecker _normally_ requires hardwood forests (see Wikipedia), but the problem with assuming that the animal is gone because the hardwood forests were largely cleared is that we can't be sure that the bird could not possibly adapt to another environment, at least for a while (e.g. the population growth might be negative, but it might take a while for the last animals to disappear).
So given that the habitat of the woodpecker is not a blasted wasteland and is not entirely urbanized, I'd be surprised if one could come up with any firm probabilities aside from, "Likely extinct, but don't bet too much on it."