Basic Probabilities

Probability of an Event occurring

$A$ - event we want to know the probability of
$S$ - the sample space, or all possible outcomes
$n(A)$ - number of outcomes that result in A
$n(S)$ - total number of outcomes possible

Here's a visualization using a Venn diagram:

Complementary Event

If the whole Sample Space is equal to 1 (the probability of any event occurring), then we can write all non $A$ events as follows:

Multiple Events

Intersecting Events

Given events $A$ and $B$, they are considered intersecting when the two outcomes can occur simultaneously, and one outcome does not limit the other from being possible.

$A\cap B$ is the notation for the intersection of events $A$ and $B$.

In a Venn diagram, this would be region $A\cap B$:

$A\cup B$ is the notation for the union of events $A$ and $B$.

In a Venn diagram, this would be the entirety of the colored region:

If $A\cup B=1$, then $A$ and $B$ are said to be exhaustive. Between them, they make up the whole of $S$. They exhaust all possibilities.

To find the probability of $A\cup B$, use this equation:

Mutually Exclusive Events

Given events $A$ and $B$, they are considered mutually exclusive when they don't share any outcomes:

Conditional Probability

The probability of event $A$ given event $B$ has already occurred:

Visualized in a Venn diagram:

Visualized with a probability tree:

Law of Total Probability

If you have $n$ mutually exclusive and exhaustive events, $A$ through to $A_n$, and events $B$ through to $B_n$, then

Bayes Theorem

If you have $n$ mutually exclusive and exhaustive events, $A$ through to $A_n$, and $B$ is another event, then

my favorite theorem. Use this to prove a test that is 99% accurate is only correct half of the time.

Independence

Two events are independent if the occurrence of one does not affect the probability of occurrence of the other.
If two events $A$ and $B$ are independent, then

Also

Sources: 1