Testing for non-zero probability

There is a thumbrule for the exact question you are asking. If you have N samples with no failures, you can say with a 95% confidence that the failure rate is less than 1 in N/3. So if you have 300 passing values, the worse failure rate you could expect (with a 95% confidence interval) is 1 in 100 or 1%.

You can never say that the percentage is zero. If might be one is a million, or one in a billion, but mathematically, you can't get to zero. Practically, you might be comfortable saying that though. If it takes a thousand years to get one billion samples and your failure rate is lower than one in a billion, then it's probably not going to happen in our lifetimes.