This sounds snarky, but I mean it as a genuine question about your revised post: what do you mean by asking about the probability of something that either definitely happens or definitely doesn't? That is to say, either there is a fixed point or there isn't; in either case, I can't see even an informal way to assign any probability other than 0 or 1 to the question.
Without using any knowledge about how they are
computed, & only using the fact that a hash is,
in effect, a result chosen uniformly at random
from a set of results, what is the probability
that ...
So in particular, consider a collection of numbers, and for each one, choose a target uniformly at random from the same collection. So for the set X you have a function f:X->X. What is the probability that there is a fixed point x such that f(x)=x?
This, by the way, is a very well known calculation/result in my area of math.
Colin explained it, but I wanted to apologize for the dissonance. Essentially, I'm phrasing the abstract "pick an element, then pick again" problem in terms of SHA collisions. Just for fun.
