Theorem of Large Numbers for Relative Frequencies Jakob Bernoulli

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:

\[F_n(A)\approx p(A).\]

1.Proof: (related to "Theorem of Large Numbers for Relative Frequencies")

Edit or Add Notation Axiomatic Method

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

1.Proof: (related to "Theorem of Large Numbers for Relative Frequencies")

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