Parent: probability-theory
See also: random-variable

Event $E$

Source: Wiki

$$E = \left{ \omega_i \right} \subseteq \Omega$$

Group of outcomes
Every event $E$ is assigned a probability of it happening
Example event: $E = \left{ \omega \in \Omega \mid X(\omega) \leq x \right}$
“Set of all outcomes $\omega$ which satisfy the condition $X(\omega) \leq x$” $$P(E) = P(X \leq x) = p_E$$

Example

Experiment: flip a coin twice $$\Omega = \left{ (H, H), (H, T), (T, H), (T, T) \right}$$

Event $E_1$: $H$ occurs in either flip $$E_1 = \left\lbrace (H, H), (H, T), (T, H) \right\rbrace \subset \Omega$$

Event $E_2$: same result twice $$E_2 = \left{ (H, H), (T, T) \right} \subset \Omega$$

Source: blitzstein-hwang

Notation	Description
$A \subseteq B$	$A$ implies $B$
$A~\cap~B=\emptyset$	mutually exclusive
$\bigcup_i^n A_i = \Omega,~A_i~\cap~A_j=\emptyset$ for $i \neq j$	partitions of $\Omega$
$(A~\cap~B^c) \cup (A^c~\cap~B)$	$A$ or $B$ but not both