3-3
The Addition Rules for Probability
Objective
Find the probability
of compound events,
using the addition
rules.
Two events are mutually exclusive events if they cannot occur at the same time
(i.e., they have no outcomes in common).
In another situation, the events of getting a 4 and getting a 6 when a single card is
drawn from a deck are mutually exclusive events, since a single card cannot be both a 4
and a 6. On the other hand, the events of getting a 4 and getting a heart on a single draw
are not mutually exclusive, since you can select the 4 of hearts when drawing a single
card from an ordinary deck.
Try This! 4–16
1. Determine whether these events are mutually exclusive.
a.
b.
c.
d.
e.
f.
g.
Roll a die: Get an even number, and get a number less than 3.
Roll a die: Get a prime number (2, 3, 5), and get an odd number.
Roll a die: Get a number greater than 3, and get a number less than 3.
Select a student in your class: The student has blond hair, and the student has blue eyes.
Select a student in your college: The student is a sophomore, and the student is a business major.
Select any course: It is a calculus course, and it is an English course.
Select a registered voter: The voter is a Republican, and the voter is a Democrat.
Try This! 4–15
Rolling a Die
Determine which events are mutually exclusive and which are not, when a single die is
rolled.
a.
b.
c.
d.
Getting an odd number and getting an even number
Getting a 3 and getting an odd number
Getting an odd number and getting a number less than 4
Getting a number greater than 4 and getting a number less than 4
The probability of two or more events can be determined by the addition rules. The first addition rule is used when the events
are mutually exclusive.
Addition Rule 1
When two events A and B are mutually exclusive, the probability that A or B will occur is
P(A or B) ⫽ P(A) ⫹ P(B)
Try This! 4–17
Try This! 4–18
Selecting a Doughnut
A box contains 3 glazed doughnuts, 4 jelly doughnuts, and 5 chocolate doughnuts. If a
person selects a doughnut at random, find the probability that it is either a glazed
doughnut or a chocolate doughnut.
Political Affiliation at a Rally
At a political rally, there are 20 Republicans, 13 Democrats, and 6 Independents. If a
person is selected at random, find the probability that he or she is either a Democrat or
an Independent.
Example 4–19
Selecting a Day of the Week
A day of the week is selected at random. Find the probability that it is a weekend day.
When two events are not mutually exclusive, we must subtract one of the two probabilities of the outcomes that are common to both events, since they have been counted
twice. This technique is illustrated in the next example.
Example 4–20
Drawing a Card
A single card is drawn at random from an ordinary deck of cards. Find the probability
that it is either an ace or a black card.
Solution
Since there are 4 aces and 26 black cards (13 spades and 13 clubs), 2 of the aces are
black cards, namely, the ace of spades and the ace of clubs. Hence the probabilities of
the two outcomes must be subtracted since they have been counted twice.
2
28
7
P(ace or black card) ⫽ P(ace) ⫹ P(black card) ⫺ P(black aces) ⫽ 524 ⫹ 26
52 ⫺ 52 ⫽ 52 ⫽ 13
When events are not mutually exclusive, addition rule 2 can be used to find the probability of the events.
Addition Rule 2
If A and B are not mutually exclusive, then
P(A or B) ⫽ P(A) ⫹ P(B) ⫺ P(A and B)
Note: This rule can also be used when the events are mutually exclusive, since
P(A and B) will always equal 0. However, it is important to make a distinction between
the two situations.
Example 4–21
Selecting a Medical Staff Person
In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are females.
If a staff person is selected, find the probability that the subject is a nurse or a male.
Example 4–22
Driving While Intoxicated
On New Year’s Eve, the probability of a person driving while intoxicated is 0.32, the
probability of a person having a driving accident is 0.09, and the probability of a person
having a driving accident while intoxicated is 0.06. What is the probability of a person
driving while intoxicated or having a driving accident?
In summary, then, when the two events are mutually exclusive, use addition rule 1.
When the events are not mutually exclusive, use addition rule 2.
The probability rules can be extended to three or more events. For three mutually
exclusive events A, B, and C,
P(A or B or C ) ⫽ P(A) ⫹ P(B) ⫹ P(C)
For three events that are not mutually exclusive,
P(A or B or C ) ⫽ P(A) ⫹ P(B) ⫹ P(C) ⫺ P(A and B) ⫺ P(A and C )
⫺ P(B and C ) ⫹ P(A and B and C )
The figure below shows a Venn diagram that represents two mutually exclusive events A
and B. In this case, P(A or B) ⫽ P(A) ⫹ P(B), since these events are mutually exclusive
Figure 4–5
P (A and B )
Venn Diagrams for the
Addition Rules
P (A )
P (S ) = 1
(a) Mutually exclusive events
P (A or B ) = P (A ) + P (B )
P (B )
P (A )
P (B )
P (S ) = 1
(b) Nonmutually exclusive events
P (A or B ) = P (A ) + P (B ) – P (A and B )
Note: Venn diagrams were developed by mathematician John Venn (1834–1923) and are used in set
theory and symbolic logic. They have been adapted to probability theory also. In set theory,
the symbol 傼 represents the union of two sets, and A 傼 B corresponds to A or B.
The symbol 傽 represents the intersection of two sets, and A 傽 B corresponds to A and B.
Venn diagrams show only a general picture of the probability rules and do not portray all situations,
such as P(A) ⫽ 0, accurately.