BA 201 Lecture 6 Basic Probability Concepts. Topics Basic Probability Concepts Approaches to...
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Transcript of BA 201 Lecture 6 Basic Probability Concepts. Topics Basic Probability Concepts Approaches to...
BA 201
Lecture 6Basic 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 Sample
Use parameters to summarize features
Use statistics to summarize features
Inference on the population from the sample
p.??
Approaches to Probability A priori classical probability
Based on prior knowledge of the process involved E.g. Analyze the scenarios when tossing a fair coin
Empirical classical probability 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.155
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.156
Special Events Impossible Event
Impossible event E.g. Club & Diamond on 1 card
draw 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
Null Event
p.159
Special Events
Mutually Exclusive Events Two events cannot occur together E.g. A: Queen of Diamond; B: Queen of Club
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
(continued)
p.159
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.
.50 .2 .25P R P C R P C
Let : Had a college degree
: Did not have a college degree
: Repaid the loan
: Did not repay the loan
C
C
R
R
pp. ??-??
Using Contingency Table (Joint Probability Table, Venn Diagram)
R R
0.5 0.5
Attribute A
C
C
0.25
0.75
AttributeB
0.2
1.0
0.050.3 0.45
pp. ??-??
(continued)
Using Contingency Table (Joint Probability Table, Venn Diagram)
Attribute AAttributeB RR
C
C
0.5 0.5
0.25
0.75
0.2
1.0
0.050.3 0.45
Total
Total
Joint probabilitiesMarginal probabilities
pp. ??-??
(continued)
Using Contingency Table (Joint Probability Table, Venn Diagram)
What is the probability that a randomly selected borrower will have a college degree or repay the loan? 0.3 0.2 0.05 0.55P C R What is the probability that a randomly selected borrower will have a college degree and default on the loan?
0.05P C R
pp. ??-??
(continued)
Using Contingency Table (Joint Probability Table, Venn Diagram)
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| 0.1
0.5P C R
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| 0.6
0.75P R C
pp. ??-??
(continued)
Computing Joint Probability:The Multiplication Rule
The Probability of a Joint Event, A and B:
( and ) = ( )
number of outcomes from both A and B
total number of possible outcomes in sample space
|
|
P A B P A B
P A B P B
P B A P A
p.170
Computing Compound Probability: The Addition Rule
Probability of a Compound Event, A or B:
( or ) ( )
number of outcomes from either A or B or both
total number of outcomes in sample space
P A B P A B
P A P B P A B
p.160
Conditional Probability Conditional Probability:
( )( | )
( )
|
P A BP A B
P B
P A BP B A
P A
p.166
Bayes’ Theorem Using Contingency Table50% 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?
.50 | .4 | .10P R P C R P C R
| ?P R C
pp. ??-??
Bayes’ Theorem Using Contingency Table(continued
)
R R
Attribute A
C
C
AttributeB
0.5 0.5
0.4 0.5 0.2 0.1 0.5 0.05
0.3 0.45
0.25
0.75
1.0
0.2| 0.8
0.25P R C
pp. ??-??
Bayes’ Theorem Using the Formula
||
| |
.4 .5 .2 .8
.4 .5 .1 .5 .25
P C R P RP R C
P C R P R P C R P R
p.175
Statistical Independence Events A and B are Independent if
Events A and B are Independent when the Probability of One Event, A, is Not Affected by Another Event, B
( | ) ( )
or ( | ) ( )
or ( and ) ( ) ( )
P A B P A
P B A P B
P A B P A P B
p.169
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