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
 The complement of an event consists of all the simple events (outcomes) that are not in the event .
Let us refer back to the experiment of throwing one die. As you know, the sample space of a fair die is . If we define the event as
Then, , which includes all the simple events of the set that are odd. Thus, the complement of is the set of simple events that will not occur in . So will include all the elements that are not odd in the sample space of the set :
The Venn diagram is shown below.
This leads us to say that the event and its complement 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 .
The Complementary Rule
The sum of the probabilities of an event and its complement must equal .
As you will see in the following examples below, it is sometimes easier to calculate the probability of the complement of an event rather than the event itself. Then the probability of the event, , is calculated using the relationship:
Example:
If you know that the probability of getting the flu this winter is , what is the probability that you will not get the flu?
Solution:
First, ask the question, what is the probability of the simple event? It is
The complement is
Example:
Two coins are tossed simultaneously. Here is an event:
What is the complement of and how would you calculate the probability of by using the complementary relationship?
Solution:
Since the event is observing all simple events , then the complement of is defined as the event that occurs when does not occur, namely, all the events that do not have heads, namely,
We can draw a simple Venn diagram that shows and in the toss of two coins.
The second part of the problem is to calculate the probability of using the complementary relationship. Recall that . So by calculating , we can easily calculate by subtracting it from .
and
Obviously, we could have gotten the same result if we had calculated the probability of the event of occurring directly. The next example, however, will show you that sometimes it is easier to calculate the complementary relationship to find the answer that we are seeking.
Example:
Here is a new kind of problem. Consider the experiment of tossing a coin ten times. What is the probability that we will observe at least one head?
Solution:
Before we begin, we can write the event as
What are the simple events of this experiment? As you can imagine, there are many simple events and it would take a very long time to list them. One simple event may look like this: another etc. Is there a way to calculate the number of simple events for this experiment? The answer is yes but we will learn how to do this later in the chapter. For the time being, let us just accept that there are simple events in this experiment.
To calculate the probability, each time we toss the coin, the chance is the same for heads and tails to occur. We can therefore say that each simple event, among events, is equally likely to occur. So
We are being asked to calculate the probability that we will observe at least one head. You may find it difficult to calculate since the heads will most likely occur very frequently during consecutive tosses. However, if we calculate the complement of , i.e., the probability that no heads will be observed, our answer may become a little easier. The complement is easy, it contains only one simple event:
Since this is the only event that no heads appear and since all simple events are equally likely, then
Now, because then
That is a very high percentage chance of observing at least one head in ten tosses of a coin.
Lesson Summary
 The complement of an event consists of all the simple events (outcomes) that are not in the event .
 The Complementary Rule states that the sum of the probabilities of an event and its complement must equal , or for an event
Review Questions
 A fair coin is tossed three times. Two events are defined as follows:
 List the sample space for tossing a coin three times
 List the outcomes of
 List the outcomes of
 List the outcomes of the events
 Find
 The Venn diagram below shows an experiment with five simple events. The two events and are shown. The probabilities of the simple events are: Find and
Review Answers

 all:
 same as same as