Probability Axioms

Given an event $E$ in a Sample Space $S$ which is either finite with $N$ elements or countably infinite with $N=\infty$ elements, then we can write

$$S\equiv \left({\,\bigcup_{i=1}^N E_i}\right),$$

and a quantity $P(E_i)$, called the Probability of event $E_i$, is defined such that

1. $0 \leq P(E_i) < 1$.

2. $P(S) = 1$.

3. Additivity: $P(E_1\cup E_2) = P(E_1)+P(E_2)$, where $E_1$ and $E_2$ are mutually exclusive.

4. Countable additivity: $P\left({\cup_{i=1}^n E_i}\right)= \sum_{i=1}^n P(E_i)$ for $n=1$, 2, ..., $N$ where $E_1$, $E_2$, ...are mutually exclusive (i.e., $E_1\cap E_2 = \emptyset$).

See also Sample Space, Union