Complement of an Event

3. Probability: Randomness

Complement of an Event

When observing an experiment we are interested in what is happening (in other words, we are interested in whether an outcome is classified as a certain event). However, we might also be interested in what is NOT happening (in other words, we are interested in whether an outcome is not classified as a certain event). For example, you might want to observe the chances of tossing a coin and NOT getting Heads, or rolling a $6$ -sided die and NOT rolling a $6$ .

Let $A$ be an event, then the set of all possible outcomes of $A$ is called the complement of $A$ .
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Complement of an event

Definition

The complement of an event $A$ is the set of all outcomes in the sample space $\Omega$ that are not classified as $A$ .

Notation

$A^c$

Consider the random experiment of a single coin toss with sample space $\Omega =\{\text{H, T}\}$ .

If event $A$ is defined as $A =$ 'the coin comes up heads' $= \{$ H $\}$ , then what is the complement of $A$ ?

$A^c$ = 'the coin does NOT come up heads' = 'the coin comes up tails' = $\{$ T $\}$ .

New example