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#.
#\phantom{0}#
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#:
\[A^c = \text{'the coin does NOT come up heads'} = \text{'the coin comes up tails'} = \{\text{T}\}\]
Or visit omptest.org if jou are taking an OMPT exam.
