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# not happening 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#\}#.

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