Binomial Distribution
The binomial distribution models the number of successes across a fixed number of independent trials, each with the same probability of success — the number of heads in 20 coin flips, the number of defective units in a batch of 100, the number of conversions out of 500 website visitors. It's defined by exactly two parameters: n, the number of trials, and p, the probability of success on each individual trial, and those two numbers alone fully determine the distribution's entire shape, mean, and spread.
The binomial distribution rests on a specific set of assumptions that have to actually hold for it to validly model a given real-world process, and violating any of them — without anyone necessarily noticing, since the calculation will still run and produce a number either way — is a common, quiet source of statistical error. The number of trials must be fixed in advance, each trial must have exactly two possible outcomes (success or failure), the probability of success must stay constant across every trial, and the trials must be independent of one another, meaning one trial's outcome doesn't affect the probability of any other. This last assumption is the one most often silently violated in practice: sampling without replacement from a finite population technically breaks independence (each draw changes the remaining population's composition, however slightly), though the binomial approximation remains reasonably accurate as long as the sample is small relative to the total population — a common, generally safe simplification rather than a mathematically rigorous requirement.
A/B testing is one of the most common real-world applications of the binomial distribution: each visitor to a test variant either converts or doesn't (a binary outcome), the total number of visitors is fixed once the test concludes, and under the null hypothesis that the true conversion rate is identical between variants, the number of conversions observed follows a binomial distribution — which is the statistical basis used to calculate whether an observed difference in conversion rates between two variants is large enough to be considered statistically significant rather than attributable to the ordinary random variation a binomial process produces even when there's no real underlying difference at all.
As n grows large, the binomial distribution increasingly approximates a normal distribution (a specific instance of the central limit theorem at work, since a binomial outcome is itself the sum of n independent binary trials), a convenient property that historically allowed statisticians to use simpler normal-distribution-based formulas as an approximation for binomial calculations before modern computing made computing exact binomial probabilities for large n computationally trivial — the normal approximation remains useful today primarily as an intuition-building tool and in contexts where a quick, closed-form estimate is preferred over an exact computation.
The binomial distribution's mean is simply n times p (the number of trials times the success probability), and its variance is n times p times (1 minus p) — both derivable directly from the sum of n independent, identically distributed binary trials, a mathematical relationship that makes the binomial distribution one of the more approachable named distributions to reason about from first principles, since its core behavior can be built up directly from the much simpler case of a single trial repeated n times.
Read the binomial distribution as the standard model for counting successes across a known, fixed number of independent yes/no trials: its validity rests entirely on four specific, checkable assumptions — fixed trial count, binary outcomes, constant success probability, and independence — that are easy to violate without any obvious signal that something's wrong, its central role in A/B testing is a direct consequence of conversions being exactly the kind of binary, fixed-trial-count process it was built to model, and its convergence toward a normal distribution as n grows is a concrete, specific instance of the central limit theorem operating on one of the simplest possible underlying random processes.