The Addition Theorem of Probability

The Addition Theorem answers the question: what is the probability that AT LEAST ONE of two events occurs, i.e. $P(A\cup B)$?

General Addition Theorem (for any two events)

The term $P(A\cap B)$ is subtracted because, if $A$ and $B$ overlap, simply adding $P(A)+P(B)$ counts every outcome common to both events TWICE.

Special case — mutually exclusive events

If $A$ and $B$ cannot occur together, $A\cap B=\varnothing$, so $P(A\cap B)=0$, and the formula reduces to the simpler:

For three events, the general formula extends to:

Complement rule (used constantly alongside the addition theorem): since an event $A$ and its complement $A'$ are always mutually exclusive and exhaustive,

$P(A)+P(A')=1 \quad\Rightarrow\quad P(A')=1-P(A)$ …