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© 2000 Prentice-Hall, Inc.
Statistics for Business and
Economics
Probability
Chapter 3
3-1
Learning Objectives
© 2000 Prentice-Hall, Inc.
1. Define Experiment, Outcome, Event,
Sample Space, & Probability
2. Explain How to Assign Probabilities
3. Use a Contingency Table, Venn
Diagram, or Tree to Find Probabilities
4. Describe & Use Probability Rules
3-2
Thinking Challenge
© 2000 Prentice-Hall, Inc.
What’s the probability
of getting a head on
the toss of a single fair
coin? Use a scale
from 0 (no way) to 1
(sure thing).
So toss a coin twice.
Do it! Did you get one
head & one tail?
What’s it all mean?
3-3
Many Repetitions!*
© 2000 Prentice-Hall, Inc.
Total Heads /
Number of Tosses
1.00
0.75
0.50
0.25
0.00
0
25
50
75
Number of Tosses
3-4
100
125
© 2000 Prentice-Hall, Inc.
Experiments,
Outcomes, & Events
3-5
Experiments & Outcomes
© 2000 Prentice-Hall, Inc.
1. Experiment

Process of Obtaining an Observation,
Outcome or Simple Event
2. Sample Point

Most Basic Outcome of
an Experiment
Sample Space
Depends on
Experimenter!
3. Sample Space (S)

Collection of All Possible Outcomes
3-6
Outcome Examples
© 2000 Prentice-Hall, Inc.
Experiment
Sample Space
Toss a Coin, Note Face
Toss 2 Coins, Note Faces
Select 1 Card, Note Kind
Select 1 Card, Note Color
Play a Football Game
Inspect a Part, Note Quality
Observe Gender
Head, Tail
HH, HT, TH, TT
2, 2, ..., A (52)
Red, Black
Win, Lose, Tie
Defective, OK
Male, Female
3-7
Outcome Properties
© 2000 Prentice-Hall, Inc.
1.
Mutually Exclusive

2.

Experiment: Observe
Gender
2 Outcomes Can Not
Occur at the Same Time
 Both Male & Female in
Same Person
Collectively Exhaustive
1 Outcome in Sample Space Must
Occur
 Male or Female
© 1984-1994 T/Maker Co.
3-8
Events
© 2000 Prentice-Hall, Inc.
1. Any Collection of Sample Points
2. Simple Event

Outcome With 1 Characteristic
3. Compound Event



Collection of Outcomes or Simple Events
2 or More Characteristics
Joint Event Is a Special Case

3-9
2 Events Occurring Simultaneously
Event Examples
© 2000 Prentice-Hall, Inc.
Experiment: Toss 2 Coins. Note Faces.
Event
Sample Space
1 Head & 1 Tail
Heads on 1st Coin
At Least 1 Head
Heads on Both
3 - 10
Outcomes in Event
HH, HT, TH, TT
HT, TH
HH, HT
HH, HT, TH
HH
© 2000 Prentice-Hall, Inc.
Sample Space
3 - 11
Visualizing
Sample Space
© 2000 Prentice-Hall, Inc.
1.
Listing

S = {Head, Tail}
2.
Venn Diagram
3.
Contingency Table
4.
Decision Tree Diagram
3 - 12
Venn Diagram
© 2000 Prentice-Hall, Inc.
Experiment: Toss 2 Coins. Note Faces.
Tail
TH
Outcome
HH
HT
TT
S
S = {HH, HT, TH, TT}
3 - 13
Sample Space
Compound
Event
Contingency Table
© 2000 Prentice-Hall, Inc.
Experiment: Toss 2 Coins. Note Faces.
2
st
Tail
Total
Head
HH
HT
HH, HT
Tail
TH
TT
TH, TT
Total
HH, TH HT, TT
S = {HH, HT, TH, TT}
3 - 14
Coin
Head
1 Coin
Simple
Event
(Head on
1st Coin)
nd
S
Sample Space
Outcome
(Count,
Total %
Shown
Usually)
Tree Diagram
© 2000 Prentice-Hall, Inc.
Experiment: Toss 2 Coins. Note Faces.
H
HH
T
HT
H
Outcome
H
TH
T
TT
T
S = {HH, HT, TH, TT}
3 - 15
Sample Space
© 2000 Prentice-Hall, Inc.
Compound Events
3 - 16
Forming
Compound Events
© 2000 Prentice-Hall, Inc.
1. Intersection



Outcomes in Both Events A and B
‘AND’ Statement
 Symbol (i.e., A  B)
2. Union



Outcomes in Either Events A or B or Both
‘OR’ Statement
 Symbol (i.e., A  B)
3 - 17
© 2000 Prentice-Hall, Inc.
Event Intersection:
Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Black
Sample
Space:
2R, 2R,
2B, ..., AB
Ace
Event Ace:
AR, AR, AB, AB
3 - 18
Event
Black:
2B, ...,
AB
S
Joint Event (Ace  Black):
AB, AB
© 2000 Prentice-Hall, Inc.
Event Intersection:
Contingency Table
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Color
Sample
Type
Space (S):
Ace
2R, 2R,
2B, ..., AB
Non-Ace
Joint Event
Ace AND
Black:
AB, AB
3 - 19
Total
Red
Black
Total
Ace & Ace & Ace
Red Black
Non & Non & NonRed Black Ace
Red Black
S
Simple
Event
Ace:
AR,
AR,
AB,
AB
Simple Event Black: 2B, ..., AB
© 2000 Prentice-Hall, Inc.
Event Union :
Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Black
Sample
Space:
2R, 2R,
2B, ..., AB
Ace
Event Ace:
AR, AR, AB, AB
3 - 20
S
Event
Black:
2B,
2B, ...,
AB
Event (Ace  Black):
AR, ..., AB, 2B, ..., KB
© 2000 Prentice-Hall, Inc.
Event Union :
Contingency Table
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Color
Sample
Type
Space (S):
Ace
2R, 2R,
2B, ..., AB
Non-Ace
Joint Event
Ace OR
Total
Black:
AR, ..., AB, 2B, ..., KB
3 - 21
Red
Black
Total
Ace & Ace & Ace
Red Black
Non & Non & NonRed Black Ace
Red Black
S
Simple
Event
Ace:
AR,
AR,
AB,
AB
Simple Event Black:
2B, ..., AB
Special Events
© 2000 Prentice-Hall, Inc.
1.
Null Event

2.
Club & Diamond on
1 Card Draw
Complement of Event

For Event A, All
Events Not In A: A’
3.
Mutually Exclusive
Event

Events Do Not Occur
Simultaneously
3 - 22
Null Event

© 2000 Prentice-Hall, Inc.
Complement of Event
Example
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Black
Sample
Space:
2R, 2R,
2B, ..., AB
Event Black:
2B, 2B, ..., AB
3 - 23
S
Complement of Event Black,
Black ’: 2R, 2R, ..., AR, AR
© 2000 Prentice-Hall, Inc.
Mutually Exclusive Events
Example
Experiment: Draw 1 Card. Note Kind & Suit.
Sample
Space:
2, 2,
2, ..., A


Event Spade:
2, 3, 4, ..., A
3 - 24
S
Outcomes
in Event
Heart:
2, 3,
4, ..., A
Events  & Mutually Exclusive
© 2000 Prentice-Hall, Inc.
Probabilities
3 - 25
What is Probability?
© 2000 Prentice-Hall, Inc.
1.
Numerical
Measure of Likelihood
that Event Will Occur



P(Event)
P(A)
Prob(A)
2.
&1
Lies Between 0
3.
1
Sum of Events is
3 - 26
1
Certain
.5
0
Impossible
© 2000 Prentice-Hall, Inc.
Assigning Event
Probabilities
1.
a priori Classical
Method
2.
Empirical
Classical Method
3.
Subjective
Method
3 - 27
What’s the
probability?
a priori Classical Method
© 2000 Prentice-Hall, Inc.
1.
Prior Knowledge of
Process
2.
Before Experiment
3.
P(Event) = X / T



X = No. of Event Outcomes
T = Total Outcomes in Sample Space
Each of T Outcomes Is Equally Likely
 P(Outcome) = 1/T
3 - 28
© 1984-1994 T/Maker Co.
© 2000 Prentice-Hall, Inc.
Empirical Classical
Method
1.
Actual Data
Collected
2.
After Experiment
3.
P(Event) = X / T


Repeat Experiment
T Times
Event Observed X
Times
4.
Also Called Relative
Frequency
Method
3 - 29
Of 100 Parts
Inspected, Only
2 Defects!
Subjective Method
© 2000 Prentice-Hall, Inc.
1.
Individual
Knowledge of Situation
2.
Before Experiment
3.
Unique Process

Not Repeatable
4.
Different
Probabilities from
Different People
3 - 30
© 1984-1994 T/Maker Co.
Thinking Challenge
© 2000 Prentice-Hall, Inc.
Which Method Should Be Used to Find the
Probability ...
1. That a Box of 24 Bolts Will Be Defective?
2. That a Toss of a Coin Will Be a Tail?
3. That Tom Will Default on His PLUS Loan?
4. That a Student Will Earn an A in This Class?
5. That a New Store on Rte. 1 Will Succeed?
3 - 31
Compound Event
Probability
© 2000 Prentice-Hall, Inc.
1. Numerical Measure of Likelihood that
Compound Event Will Occur
2. Can Often Use Contingency Table

2 Variables Only
3. Formula Methods



Additive Rule
Conditional Probability Formula
Multiplicative Rule
3 - 32
Event Probability Using
Contingency Table
© 2000 Prentice-Hall, Inc.
Event
Event
B1
B2
Total
A1
P(A1  B1) P(A1  B2) P(A1)
A2
P(A2  B1) P(A2  B2) P(A2)
Total
Joint Probability
3 - 33
P(B1)
P(B2)
1
Marginal (Simple) Probability
© 2000 Prentice-Hall, Inc.
Contingency Table
Example
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Color
Type
Ace
Black
Total
2/52
2/52
4/52
Non-Ace 24/52
24/52 48/52
26/52
26/52 52/52
Total
P(Red)
3 - 34
Red
P(Ace)
P(Ace AND Red)
Thinking Challenge
© 2000 Prentice-Hall, Inc.
What’s the Probability?
P(A) =
P(D) =
P(C  B) =
P(A  D) =
P(B  D) =
3 - 35
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Solution*
© 2000 Prentice-Hall, Inc.
The Probabilities Are:
P(A) = 6/10
P(D) = 5/10
P(C  B) = 1/10
P(A  D) = 9/10
P(B  D) = 3/10
3 - 36
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
© 2000 Prentice-Hall, Inc.
Additive Rule
3 - 37
Additive Rule
© 2000 Prentice-Hall, Inc.
1. Used to Get Compound Probabilities
for Union of Events
2. P(A OR B) = P(A  B)
= P(A) + P(B) - P(A  B)
3. For Mutually Exclusive Events:
P(A OR B) = P(A  B) = P(A) + P(B)
3 - 38
Additive Rule Example
© 2000 Prentice-Hall, Inc.
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Color
Red
Black
2
2
Total
4
Non-Ace
24
24
48
Total
26
26
52
Type
Ace
P(Ace OR Black) = P(Ace) + P(Black) - P(Ace  Black)
4
26
2
28




52 52 52 52
3 - 39
Thinking Challenge
© 2000 Prentice-Hall, Inc.
Using the Additive Rule, What’s the
Probability?
P(A  D) =
P(B  C) =
3 - 40
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Solution*
© 2000 Prentice-Hall, Inc.
Using the Additive Rule, the Probabilities
Are:
P(A  D) = P(A) + P(D) - P(A  D)
6
5
2
9




10 10 10 10
P(B  C) = P(B) + P(C) - P(B  C)
4
5
1
8




10 10 10 10
3 - 41
© 2000 Prentice-Hall, Inc.
Conditional Probability
3 - 42
Conditional Probability
© 2000 Prentice-Hall, Inc.
1. Event Probability Given that Another
Event Occurred
2. Revise Original Sample Space to
Account for New Information

Eliminates Certain Outcomes
3. P(A | B) = P(A and B)
P(B)
3 - 43
Conditional Probability
Using Venn Diagram
© 2000 Prentice-Hall, Inc.
Black
Ace
S
Event (Ace AND Black)
3 - 44
Black ‘Happens’:
Eliminates All
Other Outcomes
Black
(S)
© 2000 Prentice-Hall, Inc.
Conditional Probability
Using Contingency Table
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Color
Red
Black
Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Type
P(Ace | Black) =
3 - 45
P(Ace AND Black)
P(Black)

2 / 52
26 / 52
Revised
Sample
Space

2
26
Statistical Independence
© 2000 Prentice-Hall, Inc.
1.
Event Occurrence
Does Not Affect Probability
of Another Event

Toss 1 Coin Twice
2.
Causality Not Implied
3.
Tests For


P(A | B) = P(A)
P(A and B) = P(A)*P(B)
3 - 46
Tree Diagram
© 2000 Prentice-Hall, Inc.
Experiment: Select 2 Pens from 20 Pens:
14 Blue & 6 Red. Don’t Replace.
P(R) = 6/20
Dependent!
P(B) = 14/20
3 - 47
P(R|R) = 5/19
R
P(B|R) = 14/19
P(R|B) = 6/19
B
R
P(B|B) = 13/19
B
R
B
Thinking Challenge
© 2000 Prentice-Hall, Inc.
Using the Table Then the Formula, What’s
the Probability?
P(A|D) =
P(C|B) =
Are C & B
Independent?
3 - 48
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Solution*
© 2000 Prentice-Hall, Inc.
Using the Formula, the Probabilities Are:
P(A  D) 2 / 10 2
P(A | D) =


P(D)
5 / 10 5
P(C  B) 1 / 10 1
P(C | B) =


P(B)
4 / 10 4
5 1
P(C) =

10 4
3 - 49
Dependent
© 2000 Prentice-Hall, Inc.
Multiplicative Rule
3 - 50
Multiplicative Rule
© 2000 Prentice-Hall, Inc.
1. Used to Get Compound Probabilities
for Intersection of Events

Called Joint Events
2. P(A and B) = P(A  B)
= P(A)*P(B|A)
= P(B)*P(A|B)
3. For Independent Events:
P(A and B) = P(A  B) = P(A)*P(B)
3 - 51
Multiplicative Rule
Example
© 2000 Prentice-Hall, Inc.
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Color
Red
Black
2
2
Total
4
Non-Ace
24
24
48
Total
26
26
52
Type
Ace
P(Ace AND Black) = P(Ace)  P(Black | Ace)
 4  2
 2 
       
 52   4 
 52 
3 - 52
Thinking Challenge
© 2000 Prentice-Hall, Inc.
Using the Multiplicative Rule, What’s the
Probability?
P(C  B) =
Event
C
D
4
2
P(B  D) =
Event
A
P(A  B) =
B
1
3
4
Total
5
5
10
3 - 53
Total
6
Solution*
© 2000 Prentice-Hall, Inc.
Using the Multiplicative Rule, the
Probabilities Are:
P(C  B) = P(C)  P(B|C) = 5/10 * 1/5 = 1/10
P(B  D) = P(B)  P(D|B) = 4/10 * 3/4 = 3/10
P(A  B) = P(A)  P(B|A)  0
3 - 54
Conclusion
© 2000 Prentice-Hall, Inc.
1. Defined Experiment, Outcome, Event,
Sample Space, & Probability
2. Explained How to Assign Probabilities
3. Used a Contingency Table, Venn
Diagram, or Tree to Find Probabilities
4. Described & Used Probability Rules
3 - 55
End of Chapter
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