non-interactive proofs set an acceptable failure level. Because of this, some people call them "arguments".
Generally, the chance of failure is set to something like 1 in 2^256. Which is low enough to be totally trust-worthy.
When talking about this chance of failure, you have to take into account that an attacker can try to repeat as often as he wants. If you presume he has 2^64 attempts (which is unreasonably high) then his chance of finding a correct proof would be 1 in 2^192. Which is still truly negligible.
This wiki page: https://en.wikipedia.org/wiki/Fiat%E2%80%93Shamir_heuristic
Has a nice explanation but I figure you've seen that already.
In short the idea is:
- Set an acceptable chance of failure (like 1 in 2^256)
- Trust the output of a hash function to be random (random oracle model)
- Use the hash function to derive the "challenges" normally made a by an interactive verifier based on the commitment
Because the challenge(s) are chosen according to the random oracle (hash function) they are random. So it is impossible for the attacker to pick his commitment based on the challenge.
