A1 - sample space and events

sample space (

Ω

)

the set of all possible outcomes

discrete: finite outcomes, or countably infinite
continuous: infinite outcomes

mutual exclusivity

if $E_{1} \cap E_{2} = Φ$ , they re mutually exclusive

partition

if $E_{1}, E_{2}, \dots, E_{n}$ such that $E_{i} \cap E_{j} = Φ \forall i \neq j$ and such that $\cup_{i = 1}^{n} E_{i} = F$ , then $E_{1}, E_{2}, \dots, E_{n}$ partition set $F$

ie. if all events $E_{n}$ are mutually exclusive, and every element of F is in at least one of the $E_{i}$
eg: $F = {1, 2, 3, 4, 5, 6}$ , $E_{1} = {1, 2}$ , $E_{2} = {3, 4}$ , $E_{3} = {5, 6}$ , no two sets overlap, and together, they contain every element of $F$

union over a collection of sets

$\cup_{i = 1}^{n} E_{i}$ represents taking the union of all sets, $E_{i}$