### Section_04_02 - it

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SECTION 4.2
RULE OF COMPLEMENTS
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Objectives
1.
2.
3.
Compute probabilities by using the General Addition
Rule
Compute probabilities by using the Addition Rule for
Mutually Exclusive Events
Compute probabilities by using the Rule of Complements
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Objective 1
Compute probabilities by using the General
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Events in the Form A or B
A compound event is an event that is formed by combining
two or more events.
One type of compound event is of the form A or B.
The event A or B occurs whenever A occurs, B occurs, or A
and B both occur.
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Example
Consider the following table which presents the result of a survey
that would provide government support for higher education.
Each person was also asked whether they voted in the last
election.
Favor
Oppose
Undecided
Likely to vote
372
262
87
Not likely to vote
151
103
25
The event that a person is likely to vote or favors the law is an
event in the form A or B.
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To compute probabilities of the form P(A or B), we use the
For any two events A and B,
P(A or B) = P(A) + P(B) – P(A and B)
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Example
Find the probability that a randomly selected person is likely to vote or favors
the law.
Favor
Oppose
Undecided
Likely to vote
372
262
87
Not likely to vote
151
103
25
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Solution
Favor
Oppose
Undecided
Likely to vote
372
262
87
Not likely to vote
151
103
25
Using the General Addition Rule, we have
P(Likely to vote OR Favors the law)
= P(Likely to vote) + P(Favors the law) – P(Likely to vote AND Favors the law)
Now, there are 372 + 262 + 87 = 721 people who are likely to vote so
P(Likely to vote) = 721/1000 = 0.721.
There are 372 + 151 = 523 people who favor the law so P(Favor the law) =
523/1000 = 0.523.
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Solution
Favor
Oppose
Undecided
Likely to vote
372
262
87
Not likely to vote
151
103
25
The number of people who are both likely to vote and who favor the law is
372. Therefore, P(Likely to vote AND Favors the law) = 372/1000 = 0.372.
P(Likely to vote OR Favors the law)
= P(Likely to vote) + P(Favors the law) – P(Likely to vote AND Favors the law)
= 0.721 + 0.523 – 0.372 = 0.872
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Objective 2
Compute probabilities by using the Addition Rule
for Mutually Exclusive Events
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Mutually Exclusive Events
Two events are said to be mutually exclusive if it is impossible
for both events to occur.
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Example

A die is rolled. Event A is that the die comes up 3, and event B is that
the die comes up an even number.
These events are mutually exclusive since the die cannot both come up 3
and come up an even number.

A fair coin is tossed twice. Event A is that one of the tosses is heads,
and Event B is that one of the tosses is tails.
These events are not mutually exclusive since, if the two tosses are HT or
TH, then both events occur.
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The Addition Rule for Mutually Exclusive Events
If events A and B are mutually exclusive, then P(A and B) =
If A and B are mutually exclusive events, then
P(A or B) = P(A) + P(B)
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Example
In the 2008 Olympic Games, a total of 11,028 athletes participated.
Of these, 596 represented the United States, 332 represented
Canada, and 85 represented Mexico. What is the probability that an
Olympic athlete chosen at random represents the U.S. or Canada?
Solution:
These events are mutually exclusive, because it is impossible to compete
for both the U.S. and Canada. So,
= 596/11,028 + 332/11,028
= 928/11,028 = 0.08415
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Objective 3
Compute probabilities by using the Rule of
Complements
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Complements
If there is a 60% chance of rain today, then there is a 40%
chance that it will not rain. The events “Rain” and “No rain” are
complements. The complement of an event says that the event
does not occur.
If A is any event, the complement of A is the event that A does
not occur. The complement of A is denoted Ac.
The Rule of Complements states that P(Ac) = 1 – P(A).
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Example
According to the Wall Street Journal, 42% of cars sold in May 2008 were
small cars. What is the probability that a randomly chosen car sold in May
2008 is not a small car?
Solution:
The events of choosing a small car and not choosing a small car are
complements.
P(Not a small car) = 1 – P(Small car) = 1 – 0.42 = 0.58.
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Do You Know…
•
•
•
How to compute probabilities using the General Addition
Rule?
How to compute probabilities of mutually exclusive
events?
How to use the Complement Rule to find probabilities?
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