Mutually exclusive events
Two events are mutually exclusive if they cannot happen at the same time — they have no common outcome. For example, a single die cannot show both a $2$ and a $5$ on one roll. For mutually exclusive events:
Key formula
$P(A \cap B) = 0, \qquad P(A \cup B) = P(A) + P(B)$
Non-mutually exclusive events
Events are non-mutually exclusive when they can occur together, so they share outcomes. Then we must subtract the overlap once to avoid double counting. The general addition rule is:
Key formula
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Example
Roll a die. Let $A =$ “even” $\{2,4,6\}$ and $B =$ “greater than $3$” $\{4,5,6\}$. These overlap at $\{4,6\}$, so $P(A \cup B) = \frac{3}{6} + \frac{3}{6} - \frac{2}{6} = \frac{4}{6} = \frac{2}{3}$.
Example
Roll a die. “$2$” and “$5$” are mutually exclusive, so $P(2 \text{ or } 5) = \frac{1}{6} + \frac{1}{6} = \frac{1}{3}$.
Remember
Always check for overlap first. If $P(A \cap B) = 0$ the events are mutually exclusive and the subtraction term disappears.