In the statistical analysis of clinical trials, the rule of three states that if no major adverse events occurred in a group of n people, there can be 95% confidence that the chance of major adverse events is less than one in n / 3 (or equivalently, less than 3 in n). This is an approximate result, but is a very good approximation when n > 30.
For example, in a trial of a drug for pain relief in 1500 people, none have a major adverse event. The rule of three says we should have 95% confidence that the rate of adverse events is no more frequent than 1 in 500.
This rule is useful in the interpretation of drug trials, particularly in phase 2 and phase 3, which frequently do not have the statistical power or duration to find the relationship between the intervention and adverse events. They are designed to test the efficacy of a drug, and often the discovery of adverse events is not in the interests of the sponsors.
It should also be noted that this rule applies equally well to any trial done n times. It need not refer to medical or clinical settings. For example, if testing parachutes from the same batch, you test 300 and they all open successfully, the chance of another parachute from the same batch failing to open is likely to be less than 3/300, i.e. less than 1 in 100.
We seek a 95% confidence interval for the probability p of an event occurring, given that it has not been observed to occur in n Bernoulli trials. Denoting the number of events by X, we therefore wish to find the values of the parameter p of a binomial distribution that give Pr(X = 0) ≥ 0.05. The rule can then be derived either from the Poisson approximation to the binomial distribution, or from the formula (1-p)n for the probability of zero events in the binomial distribution by taking logarithms and keeping only the first term of a series expansion of the natural logarithm. In either case, the factor of three arises from –ln(0.05) = ln(20) = 2.9957 ≈ 3.