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1.1 The Axioms of Probability

Denote probability by $\operatorname{\mathsf{P}}\left({}\right)$ , and the probability of an event $A$ in the sample space $\Omega$ by $\operatorname{\mathsf{P}}\left({A}\right)$ . Then the axioms of probability state

Axiom 1 (positivity): $\operatorname{\mathsf{P}}\left({A}\right)\geq 0$ for all $A\subseteq\Omega$ .
Axiom 2 (finiteness): $\operatorname{\mathsf{P}}\left({\Omega}\right)=1$ .
Axiom 3 (additivity): $\operatorname{\mathsf{P}}\left({A\cup B}\right)=\operatorname{\mathsf{P}}\left% ({A}\right)+\operatorname{\mathsf{P}}\left({B}\right)$ for any $A,B\subseteq\Omega$ if $A\cap B=\emptyset$ .