### Chapter 3. Probability: Relationships between Events

### 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#

The sample space of this experiment is:

\[\Omega =\{\text{H, T}\}\]

The outcome belonging to event #A# is:

\[A=\text{'the coin comes up Heads'}=\{\text{H}\}\]

The complement of #A# is the set of all outcomes in #\Omega# that are not part of #A#:

\[A^c = \text{'the coin does NOT come up heads'} = \text{'the coin comes up tails'} = \{\text{T}\}\]

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