Complement of an Event¶

Definition¶

Let $A$ be an event in a sample space $S$ . The complement of $A$ , denoted $A^c$ , is the event

A^c=\{\omega\in S\colon\omega\notin A\}.

Interpretation¶

The complement $A^c$ is the event " $A$ does not happen". It contains exactly the outcomes of the sample space that are not in $A$ , so every outcome lies in exactly one of $A$ and $A^c$ .

A rectangle for the sample space S with a disc for the event A; the shaded region outside the disc is the complement of A.

The shaded region outside the disc is $A^c$ . It and the disc do not overlap and together fill the rectangle $S$ , so the axioms of probability give $\mathbb{P}(A)+\mathbb{P}(A^c)=\mathbb{P}(S)=1$ $1$ , the rule in the Key Results.

The rule pays off for "at least one" events. In four rolls of a fair die, "no six" happens in $5^4=625$ of the $6^4=1296$ equally likely ordered outcomes, so $\mathbb{P}(\text{at least one six})=1-\tfrac{625}{1296}\approx0.52$ $0.52$ , with no case split over the number of sixes, as in the birthday problem.

A common slip is to negate the wrong word: the complement of "all four rolls show a six" is "at least one roll is not a six", not "no roll shows a six".

Key Results¶

$\mathbb{P}(A^c)=1-\mathbb{P}(A)$ $P (A)$ .
$A\cup A^c=S$ and $A\cap A^c=\varnothing$ $\emptyset$ .
The complement method is useful for events described by 'at least one'.

Examples¶

The complement of 'at least one machine fails' is 'no machine fails'.
The complement of rolling an even number on a die is rolling an odd number.
In the birthday problem, the complement of a shared birthday is that all birthdays are different.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 1

Aliases: complementary event, event complement Keywords: complement rule Symbols: element of, intersection, omega, union

Last updated: 20 October 2026 03:02