Bernoulli's Theorem

HISTORY

Bernoulli's theorem, sometimes called the “weak law of large numbers,” was first described by Bernoulli, Jakob (1713) in his work Ars Conjectandi, which was published (with the help of his nephew Nikolaus) seven years after his death.

MATHEMATICAL ASPECTS

If S represents the number of successes obtained during n Bernoulli trials, and if p is the probability of success, then we have:

$$ \lim_{n\rightarrow \infty} P\left(\left|\frac{S}{n}-p\right| \geq \varepsilon \right) = 0\:, $$

or

$$ \lim_{n\rightarrow \infty}P\left(\left| \frac{S}{n}-p\right|<\varepsilon \right) =1\:, $$

where ${ \varepsilon > 0 }$ and arbitrarily small.

In an equivalent manner, we can write:

$$...