Download Document

Statistics for Business and
Economics
Chapter 3
Probability
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-1
Thinking Challenge
• 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?
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-2
Many Repetitions!*
Total Heads
Number of Tosses
1.00
0.75
0.50
0.25
0.00
0
25
50
75
100
125
Number of Tosses
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-3
3.1
Events, Sample Spaces,
and Probability
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-4
Experiments & Sample Spaces
1. Experiment
• Process of observation that leads to a single
outcome that cannot be predicted with certainty
1. Sample point
• Most basic outcome of an
experiment
Sample Space
Depends on
Experimenter!
1. Sample space (S)
• Collection of all sample points
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-5
Visualizing
Sample Space
1.
Listing
S = {Head, Tail}
2.
Venn Diagram
H
T
S
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-6
Sample Space Examples
•
•
•
•
•
•
•
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, Good}
{Male, Female}
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-7
Events
1. Specific collection of sample points
2. Simple Event
• Contains only one sample point
1. Compound Event
• Contains two or more sample points
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-8
Venn Diagram
Experiment: Toss 2 Coins. Note Faces.
Sample Space S = {HH, HT, TH, TT}
TH
Outcome
HH
Compound
Event: At
least one
Tail
HT
TT
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
S
3-9
Event Examples
Experiment: Toss 2 Coins. Note Faces.
Sample Space: HH, HT, TH, TT
•
•
•
•
Event
1 Head & 1 Tail
Head on 1st Coin
At Least 1 Head
Heads on Both
Outcomes in Event
HT, TH
HH, HT
HH, HT, TH
HH
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-10
Probabilities
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-11
What is Probability?
1. Numerical measure of the
likelihood that event will 1
occur
• P(Event)
• P(A)
.5
• Prob(A)
Certain
2. Lies between 0 & 1
3. Sum of sample points is 1
0
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Impossible
3-12
Probability Rules
for Sample Points
Let pi represent the probability of sample point i.
1. All sample point probabilities must lie between 0
and 1 (i.e., 0 ≤ pi ≤ 1).
2. The probabilities of all sample points within a
sample space must sum to 1 (i.e., Σ pi = 1).
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-13
Equally Likely Probability
P(Event) = X / T
• X = Number of outcomes in the
event
• T = Total number of sample points
in Sample Space
• Each of T sample points is equally
likely
— P(sample point) = 1/T
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
© 1984-1994 T/Maker Co.
3-14
Steps for Calculating Probability
1. Define the experiment; describe the process used to
make an observation and the type of observation
that will be recorded
2. List the sample points
3. Assign probabilities to the sample points
4. Determine the collection of sample points contained
in the event of interest
5. Sum the sample points probabilities to get the event
probability
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-15
Combinations Rule
A sample of n elements is to be drawn from a set of N
elements. The, the number of different samples possible
N 
is denoted by   and is equal to
n 
N 
N !
 n  = n ! ( N − n ) !
where the factorial symbol (!) means that
For example, 5 ! = 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1
0! is defined to be 1.
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-16
3.2
Unions and Intersections
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-17
Compound Events
Compound events:
Composition of two or more other events.
Can be formed in two different ways.
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-18
Unions & Intersections
1. Union
•
•
•
Outcomes in either events A or B or both
‘OR’ statement
Denoted by ∪ symbol (i.e., A ∪ B)
2. Intersection
•
•
•
Outcomes in both events A and B
‘AND’ statement
Denoted by ∩ symbol (i.e., A ∩ B)
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-19
Event Union:
Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Sample
Space:
Ace
Black
Event
Black:
2♣,
2♥, 2♦,
S
2♣, ..., A♠
Event Ace:
Event Ace ∪ Black:
A♥, A♦, A♣, A♠
A♥, ..., A♠, 2♣, ..., K♠
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
2♠, ...,
A♠
3-20
Event Union:
Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Color
Simple
Sample Space
(S):
2♥, 2♦,
2♣, ..., A♠
Type
Ace
Non-Ace
Total
Event
Ace ∪ Black:
A♥,..., A♠,  2♣, ..., K♠
Total
Ace & Ace & Ace
Red
Black
Non & Non & NonRed
Black Ace
Red
Black
S
Red
Black
Event
Ace:
A♥,
A♦,
Simple Event Black:
A♣,
2♣, ..., A♠
A♠
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-21
Event Intersection:
Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Sample
Space:
Ace
Black
Event
Black:
2♣,...,A♠
2♥, 2♦,
S
2♣, ..., A♠
Event Ace:
Event Ace ∩ Black:
A♥, A♦, A♣, A♠
A♣, A♠
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-22
Event Intersection:
Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color
& Suit.
Color
Sample Space
(S):
2♥, 2♦,
2♣, ..., A♠
Event
Ace ∩ Black:
A♣, A♠
Type
Ace
Non-Ace
Total
Total
Ace & Ace & Ace
Red
Black
Non & Non & NonRed
Black Ace
Red
Black
S
Red
Black
Simple
Event
Ace:
A♥, A♦,
A♣, A♠
Simple Event Black: 2♣, ..., A♠
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-23
Compound Event Probability
1. Numerical measure of likelihood that
compound event will occur
2. Can often use two–way table
• Two variables only
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-24
Event Probability Using
Two–Way Table
Event
Event
B1
B2
Total
A1
P(A 1 ∩ B1) P(A 1 ∩ B2) P(A 1)
A2
P(A 2 ∩ B1) P(A 2 ∩ B2) P(A 2)
P(B 1)
Total
Joint Probability
P(B 2)
1
Marginal (Simple) Probability
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-25
Two–Way Table Example
Experiment: Draw 1 Card. Note Kind &
Color.
Color
Type
Red
Black
Ace
2/52
2/52
Total
4/52
Non-Ace
24/52
24/52
48/52
Total
26/52
26/52
52/52
P(Red)
P(Ace)
P(Ace ∩ Red)
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-26
Thinking Challenge
What’s the Probability?
1.
P(A) =
2.
P(D) =
3.
P(C ∩ B) =
Event
A
4.
P(A ∪ D) =
B
1
3
4
5.
P(B ∩ D) =
Total
5
5
10
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Event
C
D
4
2
Total
6
3-27
Solution*
The Probabilities Are:
1.
P(A) = 6/10
2.
P(D) = 5/10
3.
P(C ∩ B) = 1/10
Event
A
4.
P(A ∪ D) = 9/10
B
1
3
4
5.
P(B ∩ D) = 3/10 Total
5
5
10
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Event
C
D
4
2
Total
6
3-28
3.3
Complementary Events
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-29
Complementary Events
Complement of Event A
• The event that A does not occur
• All events not in A
• Denote complement of A by AC
AC
A
S
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-30
Rule of Complements
The sum of the probabilities of complementary events
equals 1:
P(A) + P(AC) = 1
AC
A
S
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-31
Complement of Event
Example
Experiment: Draw 1 Card. Note Color.
Black
Sample
Space:
2♥, 2♦,
S
2♣, ..., A♠
Event Black:
Complement of Event Black,
2♣, 2♠, ..., A♠
BlackC: 2♥, 2♦, ..., A♥, A♦
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-32
3.4
The Additive Rule and
Mutually Exclusive Events
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-33
Mutually Exclusive Events
Mutually Exclusive Events
• Events do not occur
simultaneously
• A ∩ Β does not contain
any sample points
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
♠♠
3-34
Mutually Exclusive
Events Example
Experiment: Draw 1 Card. Note Kind & Suit.
Sample
Space:
2♥, 2♦,
2♣, ..., A♠
♥
♠
Event Spade:
2♠, 3♠, 4♠, ..., A♠
S
Outcomes
in Event
Heart:
2♥, 3♥, 4♥ ,
..., A♥
Events ♠ and♥ are Mutually Exclusive
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-35
Additive Rule
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)
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-36
Additive Rule Example
Experiment: Draw 1 Card. Note Kind &
Color.
Color
Type
Ace
Red
Black
2
2
Total
4
Non-Ace
24
24
48
Total
26
26
52
P(Ace ∪ Black) = P(Ace) + P(Black) – P(Ace ∩ Black)
4
26 2 28
=
+
–
=
52
52 52 52
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-37
Thinking Challenge
Using the additive rule, what is the probability?
1.
P(A ∪ D) =
2.
P(B ∪ C) =
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-38
Solution*
Using the additive rule, the probabilities are:
1. P(A ∪ D) = P(A) + P(D) – P(A ∩ D)
6
5
2
9
=
+
–
=
10 10 10 10
2. P(B ∪ C) = P(B) + P(C) – P(B ∩ C)
4
5
1
8
=
+
–
=
10 10 10 10
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-39
3.5
Conditional Probability
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-40
Conditional Probability
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)
=
P(A ∩ B)
P(B)
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-41
Conditional Probability Using
Venn Diagram
Ace
Black
S
Black ‘Happens’:
Eliminates All
Other Outcomes
Black
(S)
Event (Ace ∩ Black)
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-42
Conditional Probability Using
Two–Way Table
Experiment: Draw 1 Card. Note Kind &
Color.
Color
Type
Red
Black
Ace
2
2
Total
4
Non-Ace
24
24
48
Total
26
26
52
P(Ace
Black) 2 / 52
P(Ace | Black) =
=
P(Black)
26 / 52
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Revised
Sample
Space
2
=
26
3-43
Thinking Challenge
Using the table then the formula, what’s the
probability?
1.
2.
P(A|D) =
P(C|B) =
Event
A
Event
C
D
4
2
Total
6
B
1
3
4
Total
5
5
10
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-44
Solution*
Using the formula, the probabilities are:
P (A ∩ B
P (A D ) =
P (D )
)=
P (C ∩ B
P (C B ) =
P (B )
)=
2
5 = 2
5
5
10
1
10 = 1
4
4
10
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-45
3.6
The Multiplicative Rule
and Independent Events
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-46
Multiplicative Rule
1. Used to get compound probabilities for
intersection of 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)
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-47
Multiplicative Rule Example
Experiment: Draw 1 Card. Note Kind &
Color.
Color
Type
Ace
Red
Black
2
2
Total
4
Non-Ace
24
24
48
Total
26
26
52
P(Ace ∩ Black) = P(Ace)∙P(Black | Ace)
骣4 2 骣 2
= 琪
=
琪
桫52 4 桫 52
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-48
Statistical Independence
1. Event occurrence does not affect probability of
another event
• Toss 1 coin twice
2. Causality not implied
3. Tests for independence
• P(A | B) = P(A)
• P(B | A) = P(B)
• P(A ∩ B) = P(A) × P(B)
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-49
Thinking Challenge
Using the multiplicative rule, what’s the
probability?
Event
C
D
4
2
1.
P(C ∩ B) =
2.
P(B ∩ D) =
Event
A
3.
P(A ∩ B) =
B
1
3
4
Total
5
5
10
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
Total
6
3-50
Solution*
Using the multiplicative rule, the probabilities
are:
1
1
P (C ∩ B ) = P (C ) ⋅ P (B C ) =
⋅ =
10 5 10
5
3
6
P (B ∩ D ) = P (B ) ⋅ P (D B ) =
⋅ =
10 5
25
4
P (A ∩ B ) = P (A ) ⋅ P (B A ) = 0
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-51
Tree Diagram
Experiment: Select 2 pens from 20 pens: 14
blue & 6 red. Don’t replace.
Dependent!
5/19
6/20
R
14/19
14/20
B
6/19
13/19
R
P(R ∩ R)=(6/20)(5/19) =3/38
B
R
P(R ∩ B)=(6/20)(14/19) =21/95
B
P(B ∩ B)=(14/20)(13/19) =91/190
P(B ∩ R)=(14/20)(6/19) =21/95
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-52
3.7
Bayes’s Rule
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-53
Bayes’s Rule
Given k mutually exclusive and exhaustive
events B1, B1, . . . Bk , such that
P(B1) + P(B2) + … + P(Bk) = 1,
and an observed event A, then
P(Bi | A) =
P(Bi ∩ A)
P( A)
P(Bi )P( A | Bi )
=
P( B1 )P( A | B1 ) + P(B2 )P( A | B2 ) + ... + P(Bk )P( A | Bk )
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-54
Bayes’s Rule Example
A company manufactures MP3 players at two
factories. Factory I produces 60% of the MP3
players and Factory II produces 40%. Two
percent of the MP3 players produced at Factory
I are defective, while 1% of Factory II’s are
defective. An MP3 player is selected at random
and found to be defective. What is the
probability it came from Factory I?
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-55
Bayes’s Rule Example
0 .6
0 .4
Factory
I
Factory
II
0.02
Defective
0.98
Good
0.01
Defective
0.99
Good
P(I | D) =
P(I )P(D | I )
0.6 ⋅ 0.02
=
= 0.75
P(I )P(D | I ) + P(II )P(D | II ) 0.6 ⋅ 0.02 + 0.4 ⋅ 0.01
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-56
Key Ideas
Probability Rules for k Sample Points,
S1, S2, S3, . . . , Sk
1.
0 ≤ P(Si) ≤ 1
2.
∑ P (S ) = 1
i
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-57
Key Ideas
Random Sample
All possible such samples have equal
probability of being selected.
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-58
Key Ideas
Combinations Rule
Counting number of samples of n elements
selected from N elements
Copyright © 2014, 2011, and 2008 Pearson Education, Inc.
3-59