Background: The binomial distribution is fundamental in probability theory and statistics. While approximations such as the Poisson distribution are useful, an exact algebraic simplification can be advantageous. Material and methods: For large sample sizes \ (n\), computational precision issues may arise even when using the exact binomial formula. Results: An alternative exact representation of the binomial PMF using the substitution \ (x = n - k\) and Pochhammer symbol, leading to a form that separates the factorial ratio into a product and highlights the role of the complement probability is derived. The resulting expression isalignP (X = k) = (k+1) ^{ (x) x! (1-p) ^x e^k p}alignConclusion: The simplified exact formulation of the binomial probability mass function offer various computational and other benefits in certain settings.
Ilija Barukčić (Sat,) studied this question.