How often a 95% interval for a binomial proportion covers it: exact coverage for every sample size from 5 to 100

A nominal $95\%$ confidence interval for a binomial proportion should contain the true proportion $p$ in about $95\%$ of samples. Computed exactly, with nothing simulated, for every sample size and every on a grid of step , the textbook Wald interval covers less than of the time at a fraction of the pairs, and averages . The Wilson score interval falls below at a fraction of them, and the Agresti-Coull interval at . The Clopper-Pearson interval never covers less than , but averages , so it is wider than it needs to be. Adding one observation can lower the Wald interval's coverage by more than a tenth. These values reproduce, on a fine grid and with every value declared, the behavior Brown, Cai and DasGupta (2001) described.