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BA 201
Lecture 6
Basic Probability Concepts
Topics

Basic Probability Concepts
Approaches to probability
 Sample spaces
 Events and special events
Using Contingency Table (Joint Probability Table,
Venn Diagram)
 The multiplication rule
 The addition rule
 Conditional probability
 The Bayes’ theorem
Statistical Independence
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Population and Sample
Population
p.??
Sample
Use statistics to
summarize features
Use parameters to
summarize features
Inference on the population from the sample
p.155
Approaches to Probability

A priori classical probability
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Empirical classical probability
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Based on prior knowledge of the process involved
E.g. Analyze the scenarios when tossing a fair coin
Based on observed data
E.g. Record the number of heads and tails in
repeated trials of tossing a coin
Subjective probability


Based on individual’s past experience, personal
opinion and analysis of a particular situation
E.g. Evaluate the status of a coin someone has
offered to use to gamble with
p.156
Sample Spaces
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Collection of All Possible Outcomes
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E.g. All 6 faces of a die:
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E.g. All 52 cards of a bridge deck:
p.156
Events
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Simple Event
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Outcome from a sample space with 1
characteristic
E.g. A Red Card from a deck of cards
Joint Event
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Involves 2 outcomes simultaneously
E.g. An Ace which is also a Red Card from a deck
of cards
p.159
Special Events
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Impossible Event
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Impossible event
E.g. Club & Diamond on 1 card
draw
Null Event

Complement of Event
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For event A, all events not in A
Denoted as A’
E.g. A: Queen of Diamond
A’: All cards in a deck that are not Queen of
Diamond
p.159
Special Events
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Mutually Exclusive Events
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Two events cannot occur together
E.g. A: Queen of Diamond; B: Queen of Club
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(continued)
Events A and B are mutually exclusive
Collectively Exhaustive Events
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One of the events must occur
The set of events covers the whole sample space
E.g. A: All the Aces; B: All the Black Cards; C: All the
Diamonds; D: All the Hearts
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Events A, B, C and D are collectively exhaustive
Events B, C and D are also collectively exhaustive
pp. ??-??
Using Contingency Table (Joint
Probability Table, Venn Diagram)
50% of borrowers repaid their student loans. 20%
of the borrowers were students who had a college
degree and repaid their loans. 25% of the students
earned a college degree.
Let C : Had a college degree
C : Did not have a college degree
R : Repaid the loan
R : Did not repay the loan
P  R   .50
P C
R   .2
P C   .25
pp. ??-??
Using Contingency Table (Joint
Probability Table, Venn Diagram)
(continued)
Attribute
B
Attribute A
R
R
C
0.2
0.05
0.25
C
0.3
0.45
0.75
0.5
0.5
1.0
pp. ??-??
Using Contingency Table (Joint
Probability Table, Venn Diagram)
(continued)
Attribute
B
Attribute A
Total
R
R
C
0.2
0.05
0.25
C
0.3
0.45
0.75
Total
0.5
0.5
1.0
Joint probabilities
Marginal probabilities
pp. ??-??
Using Contingency Table (Joint
Probability Table, Venn Diagram)
(continued)
What is the probability that a randomly selected
borrower will have a college degree or repay the loan?
P C
R   0.3  0.2  0.05  0.55
What is the probability that a randomly selected
borrower will have a college degree and default on the
loan?
P C
R   0.05
pp. ??-??
Using Contingency Table (Joint
Probability Table, Venn Diagram)
(continued)
If you randomly select a borrower and have found out
that he/she has defaulted on the loan, what is the
probability that he/she has a college degree?
0.05
P C | R  
 0.1
0.5
If you randomly select a borrower and have found out
that he/she does not have a college degree, what is the
probability that he/she will default?
0.45
PR |C 
 0.6
0.75
p.170
Computing Joint Probability:
The Multiplication Rule

The Probability of a Joint Event, A and B:
P(A and B ) = P(A B )
number of outcomes from both A and B

total number of possible outcomes in sample space
 P  A | B P  B
 P  B | A P  A 
p.160
Computing Compound
Probability: The Addition Rule

Probability of a Compound Event, A or B:
P( A or B )  P( A B )
number of outcomes from either A or B or both

total number of outcomes in sample space
 P  A  P  B   P  A B 
p.166
Conditional Probability

Conditional Probability:
P( A B )
P( A | B ) 
P( B )
P  A B
P  B | A 
P  A
Bayes’ Theorem
Using Contingency Table
pp. ??-??
50% of borrowers repaid their loans. Out of those
who repaid, 40% had a college degree. 10% of
those who defaulted had a college degree. What is
the probability that a randomly selected borrow who
has a college degree will repay the loan?
P  R   .50
PR | C  ?
P  C | R   .4
P  C | R   .10
Bayes’ Theorem
Using Contingency Table
Attribute
B
C
C
pp. ??-??
(continued)
Attribute A
R
R
0.4  0.5  0.2 0.1 0.5  0.05
0.25
0.3
0.45
0.75
0.5
0.5
1.0
0.2
PR |C 
 0.8
0.25
Bayes’ Theorem Using the
Formula
PR | C 
P C | R  P  R 
P C | R  P  R   P C | R  P  R 
.4 .5 

.2


 .8
.4 .5  .1.5  .25
p.175
p.169
Statistical Independence

Events A and B are Independent if
P( A | B)  P ( A)
or P( B | A)  P( B)
or P( A and B)  P ( A) P ( B )
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Events A and B are Independent when the
Probability of One Event, A, is Not Affected by
Another Event, B
Summary

Introduced Basic Probability Concepts
Approaches to probability
 Sample spaces
 Events and special events
Illustrated Using Contingency Table (Joint Probability
Table, Venn Diagram)
 The multiplication rule
 The addition rule
 Conditional probability
 The Bayes’ theorem
Discussed Statistical Independence


