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

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Sample Spaces

Collection of All Possible Outcomes E.g. All 6 faces of a die:

E.g. All 52 cards of a bridge deck:

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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

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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

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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)

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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

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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

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(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

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(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

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(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

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(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

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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

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Conditional Probability Conditional Probability:

( )( | )

( )

|

P A BP A B

P B

P A BP B A

P A

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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

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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

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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

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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

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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