Nicholas Bernoulli's Theorem

Abstract

Occasioned by a discussion of the variations in the ratio of male to female births in London, Nicholas Bernoulli wanted to test the hypothesis that the probability of the birth of a male is to the probability of the birth of a female as 18 to 17. He therefore needed an approximation to the tail probability of the binomial better than the one provided by James Bernoulli's theorem. He succeeded in finding a much improved version of the theorem and gave his result in a letter of January 23, 1713 to Montmort (1713, pp. 388-394). In the same letter he informed Montmort that the Ars Conjectandi was in the press at Basel. Nicholas Bernoulli's result has been overlooked, perhaps because two otherwise reliable witnesses, namely de Moivre and Todhunter, did not grasp the significance of his result. De Moivre (1730, pp. 96-99) gave a precise account of both theorems with the original examples but without proofs. He did not compare his own result with Nicholas Bernoulli's. Three years later de Moivre (1733; 1738, p. 235; 1756, p. 243) wrote about the results of James and Nicholas Bernoulli that 'what they have done is not so much an Approximation as the determining of very wide limits, within which they demonstrated that the Sum of the Terms was contained'. This is true for James Bernoulli but as we shall see not for Nicholas Bernoulli. Todhunter (1865, p. 131) does not give Nicholas Bernoulli's result. He writes 'His investigation involves a general demonstration of the theorem of his uncle James called Bernoulli's theorem' and furthermore 'The whole investigation bears some resemblance to that of James Bernoulli and may have been suggested by it...'. The last statement is certainly true as also certified by Nicholas Bernoulli in his letter. Recently Sheynin (1970) has noted Nicholas Bernoulli's result without discussing his proof. We shall give a summary and a comparison of the two proofs and show that Nicholas Bernoulli's result is the 'missing link' between the result of James Bernoulli and de Moivre's derivation of the normal distribution as an approximation to the binomial.