3.3. The Complement of an Event http://www.ck12.org
3.3 The Complement of an Event
Learning Objectives
- Know the definition of the complement of an event.
- Using the complement of an event to calculate the probability of an event.
- Understanding the complementary rule.
Definition
ThecomplementA′of an eventAconsists of all the simple events (outcomes) that arenotin the eventA.Let us refer back to the experiment of throwing one die. As you know, the sample space of a fair die isS=
{ 1 , 2 , 3 , 4 , 5 , 6 }. If we define the eventAas
A:{observe an odd number}Then,A={ 1 , 3 , 5 }, which includes all the simple events of the setSthat are odd. Thus, thecomplementofAis the
set of simple events that will not occur inA. SoA′will include all the elements that are not odd in the sample space
of the setS:
A′={ 2 , 4 , 6 }.
The Venn diagram is shown below.
This leads us to say that the eventAand its complementA′are the sum of all the possible outcomes of the sample
space of the experiment. Therefore, the probabilities of an event and its complement must sum to 1.