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: \[p(A_1\cup \ldots\cup A_n\cup\ldots)=p(A_1)+\ldots +p(A_n)+\ldots\].

  1. Axiom: Probability of the Certain Event
  2. Axiom: Probability as a Non-Negative Number
  3. Axiom: Addition of the Probability of Mutually Exclusive Events

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 Einf├╝hrung in die Wahrscheinlichkeitsrechnung", vieweg Studium, 1995, 6th Edition