A Tutorial on How to Efficiently Calculate and Format Tables of the Binomial Distribution

Abstract. Owing to the importance of the binomial distribution, tablesfor its probability mass function (PMF) and its cumulative distributionfunction (CDF) have been extensively published. This paper presents atutorial on a novel technique for constructing and recording thesetables. The tables are to be compiled with a high precision and aminimal round-off error through the use of highly efficient iterativealgorithms based on binary recursive relations for the PMF and theCDF. The computational cost for each additional table entry is merely asingle multiplication plus two additions. This remarkably low cost isdue to the fact that each entry is computed in terms of exactly twopreceding entries. Certain symmetries and limiting values are noted,thereby allowing a considerable reduction in the number of listedentries. Table entries are given in compact forms of fixed-point orfloating-point format, with an objective of minimizing the numericaluncertainty that arises when representing an entry in a limited fieldwidth, i.e., when using a limited number of digit or character positions.This kind of novel format is crucial for accurate evaluation ofprobabilities associated with ultra-high reliability systems or with veryrare events.1