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:
(2) The probability of the certain event (i.e. the event that any event from the considered probability space occurs) equals \(1\).
(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: \[p(A_1\cup \ldots\cup A_n\cup\ldots)=p(A_1)+\ldots +p(A_n)+\ldots\].
Table of Contents
- Axiom: Probability of the Certain Event
- Axiom: Probability as a Non-Negative Number
- 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
- Bosch, Karl: "Elementare Einführung in die Wahrscheinlichkeitsrechnung", vieweg Studium, 1995, 6th Edition