# Proof

(related to Corollary: Probability of Laplace Experiments)

According to the axiom probability of certain event we have that $p(\Omega)=p(\{\omega_1,\ldots,\omega_n\})=1.$ By definition of Laplace experiments, all elementary events $$\omega_i$$ are mutually exclusive, we can add their probabilities, resulting in $p(\Omega)=p(\{\omega_1\})+\ldots+p(\{\omega_n\})=1.$ Because we have a Laplace experiment by hypothesis, it follows that $p(\Omega)=p(\{\omega_1\})+\ldots+p(\{\omega_n\})=\underbrace{p + \ldots + p}_{n\text{ times}}=1,$ and therefore $p(\omega_i)=\frac 1n~~~~~~~~~~~~(i=1,\ldots n).$

It follows for the finite subset $$A\subseteq\Omega$$ that we have to sum the probability $$\frac 1n$$ the number of times, the event $$\omega_i$$ belongs to the event $$A$$. Again, because all events are mutualy exclusive, we get

$p(A)=\sum_{\omega_i\in A}\frac 1n=\frac{|A|}{|\Omega|}.$

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### References

#### Bibliography

1. Bosch, Karl: "Elementare Einführung in die Wahrscheinlichkeitsrechnung", vieweg Studium, 1995, 6th Edition