In this paper we develop some new confidence intervals for the binomial proportion. The Clopper-Pearson interval is interpreted as an outcome of randomised confidence interval theory. The problem of randomised intervals possibly being empty is solved using a new technique involving 'tail functions, with the offshoot being a new class of randomised and Clopper-Pearson intervals. Some of the new intervals are investigated and shown to have attractive frequentist properties. Coverage probabilities and expected widths are compared and guidelines are established for constructing the optimal generalised Clopper-Pearson interval in any given situation.