Chapter 5 of Part IV of Ars conjectandi proves the first limit theorem of probability theory. The intended interpretation of this result is still a matter of controversy, but there is no dispute about what Bernoulli actually proved. He takes for granted a chance set-up on which he can make repeated trials. There is a constant unknown chance p of ‘success’ S on any given trial. When n trials are made a proportion sn of successes is observed. Bernoulli proves what is now called the weak law of large numbers: the probability of an n-fold sequence in which ∣p − sn∣<ε increases to 1 as n grows without bound. Moreover, for any given error ε, he shows how to compute a number n such that the probability of getting sn in the interval [p − ε, P + ε], itself exceeds any given probability 1 − δ. In particular, if (1 −δ) = 0·999, we have a moral certainty that sn will fall in the assigned interval. For example if p is 3/5 then a moral certainty of error less than 1/50 is guaranteed by an n in excess of 25 550.

Bernoulli's proof is chiefly a consequence of his earlier investigation of combinatorics, for it proceeds by summing the middle terms in the binomial expansion. Notice that this result is a theorem of pure probability theory, and holds under any interpretation of the calculus. There is a familiar frequency interpretation of the weak law of large numbers.