De Moivre’s Normal Approximation to the Binomial, 1733, and Its Generalization

Abstract

Abraham de Moivre (1667–1754) was of a French Protestant family; from 1684 he studied mathematics in Paris. The persecution of the French Protestants caused him at the age of 21 to seek asylum in England. For the rest of his life he lived in London, earning his livelihood as a private tutor of mathematics and later also as a consultant to gamblers and insurance brokers. He became a prominent mathematician and a Fellow of the Royal Society in 1697, but he never got a university appointment as he had hoped. He wrote three outstanding books: Miscellanea Analytica (1730), containing papers on mathematics and probability theory; The Doctrine of Chances: or, A Method of Calculating the Probability of Events in Play (1718, 1738, 1756); and Annuities upon Lives (1725, 1743, 1750, 1752), each new edition being an enlarged version of the previous one. His Doctrine contained new solutions to old problems and an astounding number of new results; it was the best textbook on probability theory until Laplace [159]. Here we only discuss his two proofs of Bernoulli’s law of large numbers and his two approximations to the binomial.