Transcript BA 201
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
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
Empirical classical probability
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
Collection of All Possible Outcomes
E.g. All 6 faces of a die:
E.g. All 52 cards of a bridge deck:
p.156
Events
Simple Event
Outcome from a sample space with 1
characteristic
E.g. A Red Card from a deck of cards
Joint Event
Involves 2 outcomes simultaneously
E.g. An Ace which is also a Red Card from a deck
of cards
p.159
Special Events
Impossible Event
Impossible event
E.g. Club & Diamond on 1 card
draw
Null Event
Complement of Event
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
Mutually Exclusive Events
Two events cannot occur together
E.g. A: Queen of Diamond; B: Queen of Club
(continued)
Events A and B are mutually exclusive
Collectively Exhaustive Events
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
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
PR |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
PR | 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
PR |C
0.8
0.25
Bayes’ Theorem Using the
Formula
PR | 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 )
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