A modified rule of three for the one-sided binomial confidence interval.

Consider the one-sided binomial confidence interval containing the unknown parameter when all trials are successful, and the significance level to be five or one percent. We develop two functions (one for each level) that represent approximations within of the exact lower-bound = . Both the exponential (referred to as a modified rule of three) and the logarithmic function are shown to outperform the standard rule of three ≃ 1 - 3/ over each of their respective ranges, that together encompass all sample sizes ≥ 1. Specifically for the exponential, we find that is a better lower bound when = 0.05 and < 1054 and that is a better bound when = 0.01 and < 209.