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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[856] **Bosch, Karl**: “Elementare Einführung in die Wahrscheinlichkeitsrechnung”, vieweg Studium, 1995, 6. Auflage