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## Theorem: Theorem of Large Numbers for Relative Frequencies (Jakob Bernoulli)

Let the probability of an event $$A$$ occurring in a Bernoulli experiment be $$P:=p(A)$$. We define members of a real sequence $$(F_n)_{n\in\mathbb N}$$ as follows:

• Let $$F_1:=F_1(A)$$ be the relative frequency of $$A$$ occurring, if we repeat the experiment $$n=1$$ times.
• Let $$F_2:=F_2(A)$$ be the relative frequency of $$A$$ occurring, if we repeat the experiment $$n=2$$ times.
• Let $$F_3:=F_3(A)$$ be the relative frequency of $$A$$ occurring, if we repeat the experiment $$n=3$$ times.
• etc.

Then it is almost certain that the sequence members $$F_n$$ will approximate the probability $$P$$ with virtually any accuracy, if $$n$$ is large enough. Formally,

$\lim_{n\to\infty}p(|F_n(A)-P|\le \epsilon)=1$

for arbitrarily small (but fixed) real number $$\epsilon > 0$$. We can also say that the relative frequencies of an event in a Bernoulli experiment (if repeated a large number of times) approximate the probability of that event:

$F_n(A)\approx p(A).$

| | | | | created: 2016-03-30 22:00:27 | modified: 2016-04-04 23:04:58 | by: bookofproofs | references: [856]

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