The binomial distribution models the total count of successes in a fixed number of independent Bernoulli trials.
A Bernoulli trial is an experiment with exactly two mutually exclusive outcomes: success, coded as one, or failure, coded as zero.
The distribution requires two parameters: n, the fixed number of trials, and p, the constant probability of success on each trial.
The binomial formula calculates probability by combining the combination coefficient with success and failure probability powers.
The combination coefficient counts the number of distinct sequences in which k successes can be arranged across n trials.
The expected value of a binomial distribution equals n multiplied by p, representing the average success count over repeated runs.
Its variance equals n multiplied by p multiplied by 1 minus p, reaching its maximum when p is 0.5.
If trials are dependent, such as sampling without replacement from a small population, the hypergeometric distribution must be used instead.
When n is large and p is small, the binomial distribution converges toward a Poisson distribution with parameter lambda equal to n times p.
When n is large and p is moderate, the binomial distribution approximates a continuous normal distribution via the de Moivre-Laplace theorem.
This normal approximation allows analysts to calculate binomial confidence intervals and significance tests rapidly without factorials.
In conversion rate optimisation, binomial models evaluate whether an updated landing page produces genuinely higher conversion rates.
In manufacturing quality inspection, binomial sampling determines whether batch defect levels breach agreed tolerance thresholds.
A common failure mode occurs when trial probabilities vary over time, violating the constant probability assumption.
Analysts must always verify trial independence and constant likelihood before fitting binomial models to transactional data.