Zero failures isn't zero risk: the rule of three for evals

The rule of three for evals says zero failures in N runs is a count, not a rate. With 0 failures in N independent runs, the exact 95% upper bound on the true failure rate is 1 - 0.05^(1/N) , which 3/N approximates. After 100 clean runs you still cannot rule out a 2.95% rate, about 1 in 34. Here is the reading that bites you. Your eval harness runs the agent 100 times, prints "0 failures," and the tile goes green. Someone screenshots it into the launch thread. The unspoken translation is "the failure rate is zero." It is not what the data says. I wrote a small script to make the gap concrete, so I ran a real gate over 200 deterministic agent runs first, counted honestly, and got the dashboard everyone trusts: gate: spend<=budget over 200 deterministic agent runs observed failures: 0 (distinct scenarios: 200) naive point rate : 0.00% binomial SE: 0.00 pp naive 95% CI : [0.00%, 0.00%] <- zero width: false certainty Look at the standard error. For a zero count the binomial SE is sqrt(0*1/200) , which is exactly 0, so the naive 95% interval collapses to [0.00%, 0.00%] . A zero-width confidence interval. The math is telling you it is completely certain, from 200 samples, that the true rate is precisely zero. That is obviously wrong, and it is the exact shape of every "all green" board I have ever trusted too much. TL;DR "0 failures in N runs" is an observed count, not a rate. The naive binomial SE of a zero count is 0, which is why a green board looks like proof and isn't. The honest number is the one-sided upper bound. With 0 failures in N runs, the 95% upper confidence limit on the true failure rate is 1 - 0.05^(1/N) . The rule of three, 3/N , approximates it and rounds the risk slightly up. At N=30 the bound is 9.50% (about 1 in 11). At N=100 it is 2.95% (1 in 34). At N=1000 it is 0.30% (1 in 334). Zero failures in 30 runs is compatible with a 1-in-11 true failure rate. To rule out a 0.1% rate at 95% you need about 2995 clean runs, not 50. "We ran it fifty times" and "