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I present a critical review of techniques for estimating confidence intervalson binomial population proportions inferred from success counts insmall-to-intermediate samples. Population proportions arise frequently asquantities of interest in astronomical research; for instance, in studiesaiming to constrain the bar fraction, AGN fraction, SMBH fraction, mergerfraction, or red sequence fraction from counts of galaxies exhibiting distinctmorphological features or stellar populations. However, two of the mostwidely-used techniques for estimating binomial confidence intervals--the'normal approximation' and the Clopper & Pearson approach--are liable tomisrepresent the degree of statistical uncertainty present under samplingconditions routinely encountered in astronomical surveys, leading to anineffective use of the experimental data (and, worse, an inefficient use of theresources expended in obtaining that data). Hence, I provide here an overviewof the fundamentals of binomial statistics with two principal aims: (i) toreveal the ease with which (Bayesian) binomial confidence intervals with moresatisfactory behaviour may be estimated from the quantiles of the betadistribution using modern mathematical software packages (e.g. R, matlab,mathematica, IDL, python); and (ii) to demonstrate convincingly the major flawsof both the 'normal approximation' and the Clopper & Pearson approach for errorestimation.

On the Estimation of Confidence Intervals for Binomial Population Proportions in Astronomy: The Simplicity and Superiority of the Bayesian Approach