1. Statistics for Business and Economics
Probability Chapter 3

2. Learning Objectives
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
As a result of this class, you will be able to ...

3. 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?

4. Many Repetitions!* Total Heads / Number of Tosses Number of Tosses
1.00
0.75
0.50
0.25
0.00 25 50 75
100
125
Number of Tosses

5. Experiments, Outcomes, & Events

6. Experiments & Outcomes
Process of Obtaining an Observation, Outcome or Simple Event
2. Sample Point
Most Basic Outcome of an Experiment
3. Sample Space (S)
Collection of All Possible Outcomes
Sample Space Depends on Experimenter!

7. Outcome 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, OK
Observe Gender Male, Female

8. Outcome Properties 1. Mutually Exclusive 2. Collectively Exhaustive
Experiment: Observe Gender
1. Mutually Exclusive
2 Outcomes Can Not Occur at the Same Time
Both Male & Female in Same Person
2. Collectively Exhaustive
1 Outcome in Sample Space Must Occur
Male or Female
© T/Maker Co.

9. Events 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
2 Events Occurring Simultaneously

10. Event Examples Experiment: Toss 2 Coins. Note Faces.
Event Outcomes in Event
Sample Space HH, HT, TH, TT
1 Head & 1 Tail HT, TH
Heads on 1st Coin HH, HT
At Least 1 Head HH, HT, TH
Heads on Both HH
Typically, the last event (Heads on Both) is called a simple event. Berenson & Levine do not follow this.

11. Sample Space

12. Visualizing Sample Space
1. Listing
S = {Head, Tail}
2. Venn Diagram
3. Contingency Table
4. Decision Tree Diagram

13. S Venn Diagram Tail Experiment: Toss 2 Coins. Note Faces. TH HT HH TT
Compound Event
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
S = {HH, HT, TH, TT}
Sample Space

14. Contingency Table Experiment: Toss 2 Coins. Note Faces. 2 Coin 1 Coinnd 2
Coin st 1
Coin
Head
Tail
Total
To be consistent with the Berenson & Levine text, a simple event is shown. Typically, this is not considered an event since it is not an outcome of the experiment.
Outcome (Count, Total % Shown Usually)
Simple Event (Head on 1st Coin)
Head HH HT
HH, HT
Tail TH TT
TH, TT
Total
HH, TH
HT, TT S
S = {HH, HT, TH, TT}
Sample Space

15. Tree Diagram H HH H T HT H TH T T TT
Experiment: Toss 2 Coins. Note Faces. H HH H T HT
Outcome H TH T T TT
S = {HH, HT, TH, TT}
Sample Space

16. Compound Events

17. Forming Compound Events
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)

18. Event Intersection: Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Black
Event Black: 2B, ..., AB
Sample Space: 2R, 2R, 2B, ..., AB
Ace S
Event Ace: AR, AR, AB, AB
Joint Event (Ace  Black): AB, AB

19. Event Intersection: Contingency Table
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Color
Simple Event Ace:
AR, AR, AB, AB
Sample Space (S): 2R, 2R, 2B, ..., AB
Type
Red
Black
Total
Ace
Ace &
Ace &
Ace
Red
Black
Non-Ace
Non &
Non &
Non-
Red
Black
Ace
Joint Event Ace AND Black: AB, AB
Total
Red
Black S
Simple Event Black: 2B, ..., AB

20. Event Union : Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Black
Event Black: 2B, 2B,..., AB
Sample Space: 2R, 2R, 2B, ..., AB
Ace S
Event Ace: AR, AR, AB, AB
Event (Ace  Black): AR, ..., AB, 2B, ..., KB

21. Event Union : Contingency Table
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Color
Simple Event Ace:
AR, AR, AB, AB
Sample Space (S): 2R, 2R, 2B, ..., AB
Type
Red
Black
Total
Ace
Ace &
Ace &
Ace
Red
Black
Non-Ace
Non &
Non &
Non-
Red
Black
Ace
Joint Event Ace OR Black: AR, ..., AB,2B, ..., KB
Total
Red
Black S
Simple Event Black: 2B, ..., AB

22.  Special Events 1. Null Event 2. Complement of Event
Club & Diamond on 1 Card Draw
2. Complement of Event
For Event A, All Events Not In A: A’
3. Mutually Exclusive Event
Events Do Not Occur Simultaneously
Null Event 

23. Complement of Event Example
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Black
Sample Space: 2R, 2R, 2B, ..., AB S
Event Black: 2B, 2B, ..., AB
Complement of Event Black, Black ’: 2R, 2R, ..., AR, AR

24. Mutually Exclusive Events Example
Experiment: Draw 1 Card. Note Kind & Suit. 
Outcomes in Event Heart: 2, 3, 4, ..., A
Mutually Exclusive
What is the intersection of mutually exclusive events?
The null set.
Sample Space: 2, 2, 2, ..., A  S
Event Spade: 2, 3, 4, ..., A
Events  & Mutually Exclusive

25. Probabilities

26. What is Probability?
1. Numerical Measure of Likelihood that Event Will Occur
P(Event)
P(A)
Prob(A)
2. Lies Between 0 & 1
3. Sum of Events is 1 1
Certain .5
Impossible

27. Assigning Event Probabilities
What’s the probability?
1. a priori Classical Method
2. Empirical Classical Method
3. Subjective Method

28. a priori Classical Method
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
© T/Maker Co.

29. 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
Of 100 Parts Inspected, Only 2 Defects!

30. Subjective Method 1. Individual Knowledge of Situation
2. Before Experiment
3. Unique Process
Not Repeatable
4. Different Probabilities from Different People
© T/Maker Co.

31. Thinking Challenge
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?
1. Empirical Classical
2. a priori Classical
3. Subjective
Tom in this situation is unique. He probably hasn’t had many other PLUS loans.
An interesting variation is ‘That a randomly selected student will default on their PLUS loans?’ This would be Empirical Classical since old records could be checked.
4. Empirical Classical
One can check old grade records to get frequencies.
Another variation, similar to question 3, is ‘What is the probability that Tom will get an A in this class?’ This is Subjective since this is a unique experience (experiment?) for Tom.
5. Subjective

32. Compound Event Probability
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

33. Event Probability Using Contingency 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

34. Contingency Table Example
Experiment: Draw 1 Card. Note Kind, Color & Suit.
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 AND Red)

35. Thinking Challenge What’s the Probability? P(A) = P(D) = P(C  B) =
P(A  D) =
P(B  D) =
Event
Let students solve first. Allow about 20 minutes for this.
Event C D
Total A 4 2 6 B 1 3 4
Total 5 5 10

36. 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
Event C D
Total A 4 2 6 B 1 3 4
Total 5 5 10

37. Additive Rule

38. 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)

39. Additive Rule Example
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Color
Type
Red
Black
Total
Try other examples using this table.
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52
P(Ace OR B
lack) =
P(Ace) +
P(Black) -
P(Ace 
Black) 4 26 2 28     52 52 52 52

40. Thinking Challenge Using the Additive Rule, What’s the Probability?
P(A  D) =
P(B  C) =
Event
Let students solve first. Allow about 10 minutes for this.
Event C D
Total A 4 2 6 B 1 3 4
Total 5 5 10

41. Solution* Using the Additive Rule, the Probabilities Are: P(A  D) =+
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

42. Conditional Probability

43. 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)

44. Conditional Probability Using Venn Diagram
Black ‘Happens’: Eliminates All Other Outcomes
Black
Black
Ace S
(S)
Event (Ace AND Black)

45. Conditional Probability Using Contingency Table
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Color
Try other examples using this table.
Type
Red
Black
Total
Revised Sample Space
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52

46. Statistical Independence
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)

47. Tree Diagram
Experiment: Select 2 Pens from 20 Pens: 14 Blue & 6 Red. Don’t Replace. R
P(R|R) = 5/19
P(R) = 6/20 R B
P(B|R) = 14/19 R
P(R|B) = 6/19
Dependent! B
P(B) = 14/20 B
P(B|B) = 13/19

48. Thinking Challenge
Using the Table Then the Formula, What’s the Probability?
P(A|D) =
P(C|B) =
Are C & B Independent?
Event
Let students solve first. Allow about 20 minutes for this.
Event C D
Total A 4 2 6 B 1 3 4
Total 5 5 10

49. Solution* Using the Formula, the Probabilities Are: P(A  D) 2 / 10 2=  
P(D) 5 / 10 5
P(C  B) 1 / 10 1
P(C | B) =  
P(B) 4 / 10 4 5 1
P(C) = 
Dependent 10 4

50. Multiplicative Rule

51. Multiplicative Rule
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)

52. Multiplicative Rule Example
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Color
Type
Red
Black
Total
Try other examples using this table.
Ace 2 2 4
Non-Ace 24 24 48
Total 26 26 52

53. Thinking Challenge
Using the Multiplicative Rule, What’s the Probability?
P(C  B) =
P(B  D) =
P(A  B) =
Event
Let students solve first. Allow about 10 minutes for this.
Event C D
Total A 4 2 6 B 1 3 4
Total 5 5 10

54. Solution* Using the Multiplicative Rule, the Probabilities Are: P(C =
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) 

55. Conclusion
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

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End of Chapter
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