  We consider the classic problem of interval estimation of a proportion $p$
based on binomial sampling. The "exact" Clopper-Pearson confidence interval for
$p$ is known to be unnecessarily conservative. We propose coverage-adjustments
of the Clopper-Pearson interval using prior and posterior distributions of $p$.
The adjusted intervals have improved coverage and are often shorter than
competing intervals found in the literature. Using new heatmap-type plots for
comparing confidence intervals, we find that the coverage-adjusted intervals
are particularly suitable for $p$ close to 0 or 1.
