Binomial and normal distributions

If n independent trials are carried out under the same conditions and in each of them event A occurs with probability p (q = 1 − p), this is called a Bernoulli scheme. The probability that A occurs exactly m times is P(n, m) = C(n, m) pᵐ qⁿ⁻ᵐ: there are C(n, m) suitable sequences, each of probability pᵐqⁿ⁻ᵐ. The table of P(n, m) for m = 0, 1, …, n is called the binomial distribution; its entries add up to (p + q)ⁿ = 1. A quantity that takes this or that value as a result of an experiment is a random variable (for example the points on a die, a height, the number of defective items). Quantities shaped by many small random factors (height, measurement error) are normally distributed: the density function is f(x) = (1/(σ√(2π))) · e^(−(x − a)²/(2σ²)); its graph is a bell-shaped curve symmetric about x = a, where a is the centre and σ the standard deviation. About 68 % of values lie within a ± σ and 95 % within a ± 2σ. For large n the binomial distribution approaches the normal distribution.