Axioms of Probability

Let us define a probability function, $P$, with

• Domain: all possible events.
• Range: $\mathbb{R}$

$P$ satisfies the following:

1. $P(S)=1;P(\varnothing )=0$
2. $A_1$, $A_2$, … are disjoint events then $P({\cup }_{i=1}^{\infty })={\sum }_{i=1}^{\infty }P(A_i)$

title: Subsets 
Suppose we have events $A$ and $B$, if $A \subseteq B$ then $P(A) \leq P(B)$.

title: Complements
$P(A^C) = 1 - P(A)$ where $A^C$ is every event that is not $A$.