The Rule of Three (R3) states that 3/"n" is an approximate 95% upper limit for the binomial parameter, when there are no events in "n" trials. This rule is based on the one-sided Clopper-Pearson exact limit, but it is shown that none of the other popular frequentist methods lead to it. It can be seen as a special case of a Bayesian R3, but it is shown that among common choices for a non-informative prior, only the Bayes-Laplace and Zellner priors conform with it. R3 has also incorrectly been extended to 3 being a "reasonable" upper limit for the number of events in a future experiment of the same (large) size, when, instead, it applies to the binomial mean. In Bayesian estimation, such a limit should follow from the posterior predictive distribution. This method seems to give more natural results than-though when based on the Bayes-Laplace prior technically converges with-the method of prediction limits, which indicates between 87.5% and 93.75% confidence for this extended R3. These results shed light on R3 in general, suggest an extended Rule of Four for a number of events, provide a unique comparison of Bayesian and frequentist limits, and support the choice of the Bayes-Laplace prior among non-informative contenders. Copyright (c) 2009 The Authors. Journal compilation (c) 2009 International Statistical Institute.
