Proposition: No zero-error conclusion from finitely many successes

Fix a circuit input and suppose independent attempted runs have failure probability $\epsilon$. For every finite $M$ and every $0<\epsilon <1$, the probability of observing no failures in $M$ runs is

Therefore an observation of finitely many successes cannot alone distinguish exact zero failure from every positive failure probability.

Proof. Independence multiplies the $M$ success probabilities. The resulting product is strictly positive.