Binomial Distribution

The binomial distribution gives the probability of getting exactly k successes in n independent trials, when each trial has the same probability p of success. It is one of the most commonly used distributions in statistics and appears throughout genetics and epidemiology.

The probability of k successes is P(k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ, where C(n, k) is the number of ways to choose which k trials succeed. The mean is np and the variance is np(1 − p). When n is large, the distribution approaches the bell-shaped normal curve, and when n is large and p is small, it approaches the Poisson distribution. Each individual trial, such as one coin toss, is called a Bernoulli trial.

In Mendelian genetics, each child of two carriers has a 1 in 4 chance of inheriting a recessive condition, so the chance that exactly 2 of 4 children are affected is C(4,2) · (1/4)² · (3/4)² ≈ 0.21. The count of copies of an allele in a sample drawn from a population, as in Hardy-Weinberg calculations and in models of genetic drift, also follows this pattern. Gregor Mendel's counts of peas can be compared with expected ratios in this way.

In epidemiology it models the number of infected people in a group, and in clinical trials it describes the number of patients who respond to a treatment. It underlies the confidence intervals for proportions, and the fluctuation patterns analyzed in the Luria-Delbrück experiment built on related probability ideas. It is a basic tool of biostatistics.

• Statistics — the wider field
• Principles of Mendelian Genetics — a setting where it applies
• Biostatistics — where it is used in life sciences