# Definition: Probability and its Axioms

A probability is a function of the probability space to real numbers (i.e. $$p:\Omega\mapsto \mathbb R$$), fulfilling the following axioms:

(1) For all $$A\subseteq \Omega$$, the probability of any event is a non-negative real number: $p(A)\ge 0.$

(2) The probability of the certain event (i.e. the event that any event from the considered probability space occurs) equals $$1$$. $p(\Omega)=1$

(3) The probability of the union event (i.e. the event that any individual event occurs) of countably many events $$A_1,A_2,\ldots\in\Omega$$, which are mutually exclusive, equals the sum of their individual probabilities: $p(A_1\cup \ldots\cup A_n\cup\ldots)=p(A_1)+\ldots +p(A_n)+\ldots$.

Definitions: 1 2 3 4 5
Proofs: 6 7 8 9 10 11 12
Propositions: 13 14 15 16
Theorems: 17

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### References

#### Bibliography

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