An event is any subset of the [[Sample space]].

Example: Given
$$S = \{1,2,3,4,5,6\}$$

then
$$A = \{2,4\}$$
is an event.

Can be visualized with [[Venn diagram]]s.

Events happen with a [[Probability]].

## Special events

Some special events are the empty event ($\phi$) and the [[Sample space]] itself ($S$).

# Operations

Just like there are normal operations (+, \*, ..) on numbers there exist
operations on events. They work much like events on normal [[Set]]s.

## Intersection

$$A \cap B$$

## Union

$$A \cup B$$

## Complement

$$A'$$

# Disjoint

Two events are disjoint if they don't "overlap" in any way.

# Independent

Two events are independent if $P(A \cap B) = P(A) \cdot P(B)$.

Intuition: $A$ does not affect $B$ and vice versa.

$(A, B)$ independent $\Leftrightarrow$ $(A', B)$ independent.

For example, given a fair dice roll and

$$A = \{2\}, \ B = \{2, 3\}, \ C = S$$

then $A$ and $B$ are dependent but $A$ and $C$ are independent (since the
outcome of $A$ doesn't affect the outcome of $C$ since $P(C) = 1$.

In general, $A_1, ..., A_n$ are independent if...