Complement of an Event
Statistics & ProbabilityThe complement of an event is the set of all outcomes in the sample space that are NOT in the event.
Formula
P(\text{not } A) = 1 - P(A)
Definition
The complement of an event is everything that is NOT that event. If an event is "rolling a 5," its complement is "rolling anything but 5" (1, 2, 3, 4, or 6).
Example
$P(\text{rain tomorrow}) = 0.3$. The complement is $P(\text{no rain}) = 1 - 0.3 = 0.7$. These two events together cover everything that could happen.
Key Insight
The complement rule is a useful shortcut: instead of finding the probability of a complex event directly, find the probability of its complement and subtract from 1.
Definition
The complement of event $A$, written $A'$ or $A^c$, consists of all outcomes in the sample space that are not in $A$. Complement rule: $P(A') = 1 - P(A)$. Together, $A$ and $A'$ are mutually exclusive and exhaustive: they cover all possible outcomes with no overlap.
Example
Drawing at least one ace in $5$ cards is complex to calculate directly. Complement: $P(\text{at least one ace}) = 1 - P(\text{no aces})$. $P(\text{no aces in } 5 \text{ cards}) = \binom{48}{5}/\binom{52}{5} = 1712304/2598960$, approximately $0.659$. So $P(\text{at least one ace}) = 1 - 0.659 = 0.341$.
Key Insight
"At least one" problems are almost always easiest via the complement: $P(\text{at least one}) = 1 - P(\text{none})$.
Definition
The complement $A^c = \Omega \setminus A$ satisfies $A \cup A^c = \Omega$ and $A \cap A^c = \emptyset$. By countable additivity, $P(A) + P(A^c) = P(\Omega) = 1$. De Morgan's laws connect complements to unions and intersections: $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$.
Example
Bonferroni's inequality: $P(\text{at least one } A_i) \le \sum P(A_i)$. The complement gives the union bound: $P(\bigcup A_i) = 1 - P(\bigcap A_i^c)$, useful for bounding probabilities of rare events in high-dimensional probability.
Key Insight
De Morgan's laws are fundamental in computing probabilities of complex events. They allow conversion between union and intersection problems, enabling use of inclusion-exclusion or independence more easily.