If there are max(SHA-1) hashes, and a 1/max(SHA-1) chance any particular one will hash to itself, does that mean we would expect only a single instance of a hash hashing to itself?
I have no idea how you might go about finding that one expected instance.
You would expect about 1 per unflawed hash function tested. But the law of averages is less powerful when you focus on a specific hash. The chance of having no fixed points in a particular unflawed hash is 1/e, so there is probably a fixed point, and possibly more than one.
I have no idea how you might go about finding that one expected instance.