1

Pertemuan 03

Peluang Kejadian

Matakuliah : A0392 - Statistik Ekonomi

Tahun : 2006

2

Outline Materi:

• Konsep dasar peluang

• Peluang bebas dan bersyarat

• Peluang total dan kaidah Bayes

3

Basic Business Statistics (9th Edition)

Basic Probability

4

Konsep Dasar Peluang

• Basic Probability Concepts– Sample spaces and events, simple

probability, joint probability

• Conditional Probability– Statistical independence, marginal probability

• Bayes’ Theorem

• Counting Rules

5

Sample Spaces

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

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

6

Kejadian (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

7

Visualizing Events

• Contingency Tables

• Tree Diagrams

Red 2 24 26

Black 2 24 26

Total 4 48 52

Ace Not Ace Total

Full Deck of Cards

Red Cards

Black Cards

Not an Ace

Ace

Ace

Not an Ace

8

Contingency Table

A Deck of 52 Cards

Ace Not anAce

Total

Red

Black

Total

2 24

2 24

26

26

4 48 52

Sample Space

Red Ace

9

Full Deck of Cards

Tree Diagram

Event Possibilities

Red Cards

Black Cards

Ace

Not an Ace

Ace

Not an Ace

10

Probability

• Probability is the Numerical Measure of the Likelihood that an Event Will Occur

• Value is between 0 and 1

• Sum of the Probabilities of All Mutually Exclusive and Collective Exhaustive Events is 1

Certain

Impossible

.5

1

0

11

(There are 2 ways to get one 6 and the other 4)E.g., P( ) = 2/36

Computing Probabilities

• The Probability of an Event E:

• Each of the Outcomes in the Sample Space is Equally Likely to Occur

number of event outcomes( )

total number of possible outcomes in the sample space

P E

X

T

12

Computing Joint Probability

• 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

E.g. (Red Card and Ace)

2 Red Aces 1

52 Total Number of Cards 26

P

13

P(A1 and B2) P(A1)

TotalEvent

Joint Probability Using Contingency Table

P(A2 and B1)

P(A1 and B1)

Event

Total 1

Joint Probability Marginal (Simple) Probability

A1

A2

B1 B2

P(B1) P(B2)

P(A2 and B2) P(A2)

14

Computing Compound Probability

• 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

E.g. (Red Card or Ace)

4 Aces + 26 Red Cards - 2 Red Aces

52 total number of cards28 7

52 13

P

15

P(A1)

P(B2)

P(A1 and B1)

Compound Probability (Addition Rule)

P(A1 or B1 ) = P(A1) + P(B1) - P(A1 and B1)

P(A1 and B2)

TotalEvent

P(A2 and B1)

Event

Total 1

A1

A2

B1 B2

P(B1)

P(A2 and B2) P(A2)

For Mutually Exclusive Events: P(A or B) = P(A) + P(B)

16

Computing Conditional Probability

• The Probability of Event A Given that Event B Has Occurred:

( and )( | )

( )

P A BP A B

P B

E.g.

(Red Card given that it is an Ace)

2 Red Aces 1

4 Aces 2

P

17

Conditional Probability Using Contingency Table

BlackColor

Type Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

Revised Sample Space

(Ace and Red) 2 / 52 2(Ace | Red)

(Red) 26 / 52 26

PP

P

18

Conditional Probability and Statistical Independence

• Conditional Probability:

• Multiplication Rule:

( and )( | )

( )

P A BP A B

P B

( and ) ( | ) ( )

( | ) ( )

P A B P A B P B

P B A P A

19

Conditional Probability and 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

(continued)

( | ) ( )

or ( | ) ( )

or ( and ) ( ) ( )

P A B P A

P B A P B

P A B P A P B

20

Bayes’ Theorem

1 1

||

| |

and

i ii

k k

i

P A B P BP B A

P A B P B P A B P B

P B A

P A

Adding up the parts of A in all the B’s

Same Event

21

Bayes’ Theorem Using Contingency Table

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 borrower who has a college degree will repay the loan?

.50 | .4 | .10P R P C R P C R

| ?P R C

22

Bayes’ Theorem Using Contingency Table

||

| |

.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

(continued)

Repay

Repay

CollegeCollege 1.0.5 .5

.2

.3

.05.45

.25.75

Total

Total

23

Counting Rule 1

• If any one of k different mutually exclusive and collectively exhaustive events can occur on each of the n trials, the number of possible outcomes is equal to kn.– E.g., A six-sided die is rolled 5 times, the

number of possible outcomes is 65 = 7776.

24

Counting Rule 2

• If there are k1 events on the first trial, k2 events on the second trial, …, and kn events on the n th trial, then the number of possible outcomes is (k1)(k2)•••(kn).

– E.g., There are 3 choices of beverages and 2 choices of burgers. The total possible ways to choose a beverage and a burger are (3)(2) = 6.

25

Counting Rule 3

• The number of ways that n objects can be arranged in order is n! = n (n - 1)•••(1).– n! is called n factorial– 0! is 1– E.g., The number of ways that 4 students can

be lined up is 4! = (4)(3)(2)(1)=24.

26

Counting Rule 4: Permutations

• The number of ways of arranging X objects selected from n objects in order is

– The order is important.– E.g., The number of different ways that 5

music chairs can be occupied by 6 children are

!

!

n

n X

! 6!

720! 6 5 !

n

n X

27

Counting Rule 5: Combintations

• The number of ways of selecting X objects out of n objects, irrespective of order, is equal to

– The order is irrelevant.– E.g., The number of ways that 5 children can

be selected from a group of 6 is

!

! !

n

X n X

! 6!

6! ! 5! 6 5 !

n

X n X