De Moivre–Laplace theorem

The De Moivre–Laplace theorem states that suitably standardized binomial probabilities approach the standard normal distribution as the number of independent trials grows.

Binomial distribution: The probability distribution of the number of successes in a fixed number of independent trials with constant success probability. The theorem approximates this distribution when the trial count grows.
Bernoulli trial: A random experiment with two outcomes and a fixed probability of success. A binomial count sums the outcomes of independent Bernoulli trials.
Abraham de Moivre: A French-born mathematician whose work advanced probability theory and analytic methods in the eighteenth century. His 1733 approximation of binomial probabilities established the theorem's central result.
Statistical inference: The use of data to estimate population properties and assess hypotheses under uncertainty. Normal approximations help justify inference about proportions from large samples.
Exact binomial test: A hypothesis test that calculates probabilities directly from the binomial distribution without a large-sample approximation. It avoids approximation error when sample sizes are small or probabilities are extreme.
Normal distribution: A continuous probability distribution with a symmetric bell-shaped density determined by its mean and variance. It is the limiting shape of standardized binomial probabilities.
Probability mass function: A function assigning probabilities to each possible value of a discrete random variable. The theorem approximates binomial probability masses with values from a normal density.
The Doctrine of Chances: A book by Abraham de Moivre that developed mathematical methods for games of chance and probability. Its later editions presented de Moivre's normal approximation to the binomial distribution.
Confidence interval for a proportion: An interval estimate for an unknown population proportion, calculated from sample data. Large-sample intervals often rely on the binomial proportion's approximate normality.
Poisson limit theorem: A limit result in which binomial distributions with rare successes approach a Poisson distribution. It gives a different binomial limit when success probability shrinks as trial count grows.

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