You ran a batch of trials and counted the failures — so the failure rate is k ⁄ n, right? Only on average. The true rate could be lower (you got unlucky) or higher (you got lucky), and Clopper–PearsonThe exact confidence interval for a binomial proportion: built by inverting the binomial test, so the stated coverage is guaranteed for every k and n. The Rule of Three is its k=0, 95% upper-bound corner. pins down the range it plausibly sits in. Wider than the textbook ± formula, never wrong at the edges, and unlike the rule of three it works for any k, not just zero.
Observed rate 1.5% (3 of 200 trials) — the true rate is plausibly anywhere from 0.31% to 4.32%, a width of 4.01%.
The dot is the observed rate. Wald is the familiar p̂ ± z·√(p̂q̂ ⁄ n) — clean, but its coverage collapses as k approaches 0 or n (the interval can fall off the edge, or shrink to a point with no failures). Wilson is the practical default — close to Clopper–Pearson but a touch tighter. Clopper–Pearson is the conservative one: a hair wider, but coverage is guaranteed.
Hold the observed rate near 1.5% and watch the width shrink as the run grows. To halve the width you need roughly four times the trials — the √n curse.
| Sample size | k (≈ p̂·N) | 95% interval |
|---|---|---|
| 10 | 0 | 0% – 30.8% width 30.8% |
| 30 | 0 | 0% – 11.6% width 11.6% |
| 100 | 2 | 0.24% – 7.04% width 6.8% |
| 300 | 5 | 0.54% – 3.85% width 3.3% |
| 1,000 | 15 | 0.84% – 2.46% width 1.62% |
| 3,000 | 45 | 1.1% – 2% width 0.91% |