# 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

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 1.6$: The Brachistochrone. Fermat and the Bernoullis - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Bernoulli's theorem** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.20$: The Bernoulli Brothers - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Bernoulli's theorem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Bernoulli's theorem** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Bernoulli's Theorem**