Bernoulli's Theorem

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Theorem

Let the probability of the occurrence of an event be $p$.

Let $n$ independent trials be made, with $k$ successes.


Then:

$\displaystyle \lim_{n \mathop \to \infty} \frac k n = p$


Proof


Also presented as

This result can also be presented in the form:

$\forall \epsilon \in \R_{>0}: \displaystyle \lim_{n \mathop \to \infty} \map \Pr {\size {\frac k n - p} < \epsilon} = 1$


Also known as

This theorem is also popularly known as the Law of Large Numbers.


Source of Name

This entry was named for Jacob Bernoulli.


Sources