ELEMENTARY
STATISTICS
Section 3-3
Addition Rule
EIGHTH
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION
MARIO F. TRIOLA
1
Definition
 Compound Event
Any event combining 2 or more simple
events
 Notation
P(A or B) = P (event A occurs or event B
occurs or they both occur)
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
Compound Event
General Rule
When finding the probability that event A
occurs or event B occurs, find the total
number of ways A can occur and the
number of ways B can occur, but find the
total in such a way that no outcome is
counted more than once.
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
Compound Event
Formal Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
where P(A and B) denotes the probability that A and B
both occur at the same time.
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
Compound Event
Formal Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
where P(A and B) denotes the probability that A and B
both occur at the same time.
Intuitive Addition Rule
To find P(A or B), find the sum of the number of ways
event A can occur and the number of ways event B can
occur, adding in such a way that every outcome is
counted only once. P(A or B) is equal to that sum,
divided by the total number of outcomes.
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Definition
Events A and B are
mutually exclusive
if they cannot occur
simultaneously.
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Definition
Events A and B are mutually exclusive if they
cannot occur simultaneously.
Total Area = 1
P(A)
P(B)
P(A and B)
Overlapping Events
Figures 3-5
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
Definition
Events A and B are mutually exclusive if they
cannot occur simultaneously.
Total Area = 1
P(A)
P(B)
Total Area = 1
P(A)
P(B)
P(A and B)
Overlapping Events
Non-overlapping Events
Figures 3-5 and 3-6
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Figure 3-7
Applying the Addition Rule
P(A or B)
Addition Rule
Are
A and B
mutually
exclusive
?
Yes
P(A or B) = P(A) + P(B)
No
P(A or B) = P(A)+ P(B) - P(A and B)
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
Contingency Table
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
Find the probability of randomly selecting a man or a boy.
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Contingency Table
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
Find the probability of randomly selecting a man or a boy.
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Contingency Table
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
Find the probability of randomly selecting a man or a boy.
P(man or boy) = 1692 + 64 = 1756 = 0.790
2223 2223 2223
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
Contingency Table
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
Find the probability of randomly selecting a man or a boy.
P(man or boy) = 1692 + 64 = 1756 = 0.790
2223 2223 2223
* Mutually Exclusive *
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13
Contingency Table
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
Find the probability of randomly selecting a man or
someone who survived.
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
Contingency Table
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
Find the probability of randomly selecting a man or
someone who survived.
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
Contingency Table
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
Find the probability of randomly selecting a man or
someone who survived.
P(man or survivor) = 1692 + 706 - 332 = 2066
2223 2223 2223
2223
= 0.929
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
Contingency Table
Survived
Died
Total
Men
332
1360
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
Totals
706
1517
2223
Find the probability of randomly selecting a man or
someone who survived.
P(man or survivor) = 1692 + 706 - 332 = 2066
2223 2223 2223
2223
= 0.929
* NOT Mutually Exclusive *
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
17
Complementary Events
P(A) and P(A)
are
mutually exclusive
All simple events are either in A or A.
P(A) + P(A) = 1
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
18
Rules of Complementary Events
P(A) + P(A) = 1
P(A)
= 1 - P(A)
P(A) = 1 - P(A)
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
19
Figure 3-8
Venn Diagram for the
Complement of Event A
Total Area = 1
P (A)
P (A) = 1 - P (A)
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
20
Examples
1) A restaurant has 3 pieces of apple pie, 5 pieces of
cherry and 4 pieces of pumpkin pie in its dessert case.
If a customer selects a piece of pie what is the
probability that it is cherry or pumpkin?
Events are mutually exclusive
P(Cherry or Pumpkin)
= P(Cherry) + P(Pumpkin)
= 5/12 + 4/12 = 9/12 = 3/4.
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
21
Examples
1) A single card is selected from a standard deck of cards.
What is the probability that it is a king or club?
Events are not mutually exclusive
P(King or Club) = P(King) + P(Club) – P(King and Club)
= 4/52 + 13/52 - 1/52 = 16/52 = 4/13.
Chapter 3. Section 3-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
22