1. © 2011 Pearson Education, Inc

2. Statistics for Business and Economics
Chapter 3
Probability
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3. © 2011 Pearson Education, Inc
Contents
Events, Sample Spaces, and Probability
Unions and Intersections
Complementary Events
The Additive Rule and Mutually Exclusive Events
Conditional Probability
The Multiplicative Rule and Independent Events
Random Sampling
Baye’s Rule
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Learning Objectives
Develop probability as a measure of uncertainty
Introduce basic rules for finding probabilities
Use probability as a measure of reliability for an inference
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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?
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Many Repetitions!*
Total Heads Number of Tosses
1.00
0.75
0.50
0.25
0.00 25 50 75
100
125
Number of Tosses
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7. Events, Sample Spaces, and Probability
3.1
Events, Sample Spaces, and Probability
:1, 1, 3
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8. Experiments & Sample Spaces
Process of observation that leads to a single outcome that cannot be predicted with certainty
Sample point
Most basic outcome of an experiment
Sample space (S)
Collection of all possible outcomes
Sample Space Depends on Experimenter!
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9. Sample Space Properties
Mutually Exclusive
2 outcomes can not occur at the same time
Male & Female in same person
Collectively Exhaustive
One outcome in sample space must occur.
Male or Female
Experiment: Observe Gender
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10. Visualizing Sample Space
1. Listing
S = {Head, Tail}
2. Venn Diagram H T S
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Sample Space Examples
Experiment Sample Space
Toss a Coin, Note Face {Head, Tail}
Toss 2 Coins, Note Faces {HH, HT, TH, TT}
Select 1 Card, Note Kind {2♥, 2♠, ..., A♦} (52)
Select 1 Card, Note Color {Red, Black}
Play a Football Game {Win, Lose, Tie}
Inspect a Part, Note Quality {Defective, Good}
Observe Gender {Male, Female}
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Events
Specific collection of sample points
Simple Event
Contains only one sample point
Compound Event
Contains two or more sample points
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Venn Diagram
Experiment: Toss 2 Coins. Note Faces.
Sample Space S = {HH, HT, TH, TT}
Compound Event: At least one Tail
Other compound events could be formed:
Tail on the second toss {HT, TT}
At least 1 Head {HH, HT, TH} TH HT
Outcome HH TT S
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Event Examples
Experiment: Toss 2 Coins. Note Faces.
Sample Space: HH, HT, TH, TT
Event Outcomes in Event
Typically, the last event (Heads on Both) is called a simple event.
1 Head & 1 Tail HT, TH
Head on 1st Coin HH, HT
At Least 1 Head HH, HT, TH
Heads on Both HH
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Probabilities
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What is Probability?
1. Numerical measure of the likelihood that event will cccur
P(Event)
P(A)
Prob(A)
2. Lies between 0 & 1
3. Sum of sample points is 1 1
Certain .5
Impossible
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17. 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.,  pi = 1).
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18. 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
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19. 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
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Combinations Rule
A sample of n elements is to be drawn from a set of N elements. The, the number of different samples possible
is denoted by
and is equal to
where the factorial symbol (!) means that
For example,
0! is defined to be 1.
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21. Unions and Intersections
3.2
Unions and Intersections
:1, 1, 3
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Compound Events
Compound events:
Composition of two or more other events.
Can be formed in two different ways.
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23. Unions & Intersections
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)
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24. Event Union: Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Ace
Black
Event Black: 2, 2,..., A
Sample Space: 2, 2, 2, ..., A S
Event Ace: A, A, A, A
Event Ace  Black: A, ..., A, 2, ..., K
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25. Event Union: Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Color
Simple Event Ace:
A, A, A, A
Sample Space (S): 2, 2, 2, ..., A
Type
Red
Black
Total
Ace
Ace &
Ace &
Ace
Red
Black
Non-Ace
Non &
Non &
Non-
Red
Black
Ace
Total
Red
Black S
Event Ace  Black: A,..., A, 2, ..., K
Simple Event Black: 2, ..., A
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26. Event Intersection: Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Ace
Black
Event Black: 2,...,A
Sample Space: 2, 2, 2, ..., A S
Event Ace: A, A, A, A
Event Ace  Black: A, A
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27. Event Intersection: Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Color
Simple Event Ace:
A, A, A, A
Sample Space (S): 2, 2, 2, ..., A
Type
Red
Black
Total
Ace
Ace &
Ace &
Ace
Red
Black
Non-Ace
Non &
Non &
Non-
Red
Black
Ace
Event Ace  Black: A, A
Total
Red
Black S
Simple Event Black: 2, ..., A
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28. Compound Event Probability
1. Numerical measure of likelihood that compound event will occur
2. Can often use two–way table
Two variables only
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29. Event Probability Using Two–Way Table
Total 1 2 A
P(A  B )
P(A  B )
P(A ) 1 1 1 1 2 1 A
P(A  B )
P(A  B )
P(A ) 2 2 1 2 2 2
Total
P(B )
P(B ) 1 1 2
Joint Probability
Marginal (Simple) Probability
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Two–Way Table Example
Experiment: Draw 1 Card. Note Kind & Color.
Color
Type
Red
Black
Total
Ace
2/52
2/52
4/52
Non-Ace
24/52
24/52
48/52
P(Ace)
Total
26/52
26/52
52/52
P(Red)
P(Ace  Red)
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Thinking Challenge
P(A) =
P(D) =
P(C  B) =
P(A  D) =
P(B  D) =
What’s the Probability?
Event C D
Total A 4 2 6 B 1 3 5 10
Let students solve first. Allow about 20 minutes for this.
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Solution*
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
Event C D
Total A 4 2 6 B 1 3 5 10
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3.3
Complementary Events
:1, 1, 3
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Complementary Events
Complement of Event A
The event that A does not occur
All events not in A
Denote complement of A by AC S AC A
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Rule of Complements
The sum of the probabilities of complementary events equals 1:
P(A) + P(AC) = 1 S AC A
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36. Complement of Event Example
Experiment: Draw 1 Card. Note Color.
Black
Sample Space: 2, 2, 2, ..., A S
Event Black: 2, 2, ..., A
Complement of Event Black, BlackC: 2, 2, ..., A, A
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37. The Additive Rule and Mutually Exclusive Events
3.4
The Additive Rule and Mutually Exclusive Events
:1, 1, 3
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38. Mutually Exclusive Events
Events do not occur simultaneously
A  B does not contain any sample points 
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39. Mutually Exclusive Events Example
Experiment: Draw 1 Card. Note Kind & Suit. 
Outcomes in Event Heart: 2, 3, 4 , ..., A
Sample Space: 2, 2, 2, ..., A
Mutually Exclusive
What is the intersection of mutually exclusive events?
The null set.  S
Event Spade: 2, 3, 4, ..., A
Events  and are Mutually Exclusive
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Additive Rule
Used to get compound probabilities for union of events
P(A OR B) = P(A  B) = P(A) + P(B) – P(A  B)
For mutually exclusive events: P(A OR B) = P(A  B) = P(A) + P(B)
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Additive Rule Example
Experiment: Draw 1 Card. Note Kind & Color.
Color
Type
Red
Black
Total
Ace 2 4
Non-Ace 24 48 26 52
Try other examples using this table.
P(Ace  Black) =
P(Ace) +
P(Black) –
P(Ace 
Black)
= – =
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Thinking Challenge
Using the additive rule, what is the probability?
P(A  D) =
P(B  C) =
Event C D
Total A 4 2 6 B 1 3 5 10
Let students solve first. Allow about 10 minutes for this.
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Solution*
Using the additive rule, the probabilities are: 1.
P(A  D) = P(A) + P(D) – P(A  D)
= – = 2.
P(B  C) = P(B) + P(C) – P(B  C)
= – =
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44. Conditional Probability
3.5
Conditional Probability
:1, 1, 3
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45. 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(A  B) P(B) P(B)
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46. Conditional Probability Using Venn Diagram
Black ‘Happens’: Eliminates All Other Outcomes
Ace
Black
Black S
(S)
Event (Ace  Black)
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47. Conditional Probability Using Two–Way Table
Experiment: Draw 1 Card. Note Kind & Color.
Color
Type
Red
Black
Total
Ace 2 4
Non-Ace 24 48 26 52
Revised Sample Space
Try other examples using this table.
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Thinking Challenge
Using the table then the formula, what’s the probability?
P(A|D) =
P(C|B) =
Event C D
Total A 4 2 6 B 1 3 5 10
Let students solve first. Allow about 20 minutes for this.
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Solution*
Using the formula, the probabilities are:
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50. The Multiplicative Rule and Independent Events
3.6
The Multiplicative Rule
and Independent Events
:1, 1, 3
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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)
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52. Multiplicative Rule Example
Experiment: Draw 1 Card. Note Kind & Color.
Color
Type
Red
Black
Total
Ace 2 2 4
Try other examples using this table.
Non-Ace 24 24 48
Total 26 26 52
P(Ace  Black) = P(Ace)∙P(Black | Ace)
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53. 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)
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Thinking Challenge
Using the multiplicative rule, what’s the probability?
Event C D
Total A 4 2 6 B 1 3 5 10
P(C  B) =
P(B  D) =
P(A  B) =
Let students solve first. Allow about 10 minutes for this.
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Solution*
Using the multiplicative rule, the probabilities are:
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Tree Diagram
Experiment: Select 2 pens from 20 pens: 14 blue & 6 red. Don’t replace.
Dependent! R
P(R  R)=(6/20)(5/19) =3/38
5/19 R
6/20
14/19 B
P(R  B)=(6/20)(14/19) =21/95 R
6/19
P(B  R)=(14/20)(6/19) =21/95
14/20 B
13/19 B
P(B  B)=(14/20)(13/19) =91/190
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3.7
Random Sampling
:1, 1, 3
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58. Importance of Selection
How a sample is selected from a population is of vital importance in statistical inference because the probability of an observed sample will be used to infer the characteristics of the sampled population.
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Random Sample
If n elements are selected from a population in such a way that every set of n elements in the population has an equal probability of being selected, the n elements are said to be a random sample.
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60. Random Number Generators
Most researchers rely on random number generators to automatically generate the random sample.
Random number generators are available in table form, and they are built into most statistical software packages.
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3.8
Bayes’s Rule
:1, 1, 3
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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
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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?
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Bayes’s Rule Example
Defective
0.02
Factory I
0 .6
0.98
Good
Defective
0.01
0 .4
Factory II
0.99
Good
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Key Ideas
Probability Rules for k Sample Points,
S1, S2, S3, , Sk
≤ P(Si) ≤ 1 2.
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66. © 2011 Pearson Education, Inc
Key Ideas
Random Sample
All possible such samples have equal probability of being selected.
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67. © 2011 Pearson Education, Inc
Key Ideas
Combinations Rule
Counting number of samples of n elements selected from N elements
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68. © 2011 Pearson Education, Inc
Key Ideas
Bayes’s Rule
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