P(k) = \binom{n}{k} p^k (1-p)^{n-k}

Fri, 01 May 2026 03:56:36 +0000

# Understanding the Binomial Distribution: The Probability Mass Function P(k) = \binom{n}{k} p^k (1-p)^{n-k} The binomial distribution is a cornerstone of probability theory and statistics, widely applied in fields ranging from genetics and business analytics to machine learning and quality control. At its heart lies the probability mass function (PMF) for a binomial random variable: \[
P(k) = \binom{n}{k} p^k (1 - p)^{n - k}
\] This elegant formula calculates the probability of obtaining exactly \( k \) successes in \( n \) independent trials, where each trial has two outcomes—commonly termed "success" (with probability \( p \)) and "failure" (with probability \( 1 - p \)). In this article, we’ll break down the components of this equation, explore its significance, and highlight practical applications where it shines.