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Pertemuan 03Teori Peluang (Probabilitas)

Matakuliah : I0262 – Statistik Probabilitas

Tahun : 2007

Versi : Revisi

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Learning Outcomes

Pada akhir pertemuan ini, diharapkan mahasiswa

akan mampu :

• Mahasiswa akan dapat menjelaskan ruang contoh dan peluang kejadian.

• mahasiswa dapat memberi contoh peluang kejadian bebas, bersyarat dan kaidah Bayes.

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Outline Materi

• Istilah/ notasi dalam peluang

• Diagram Venn dan Operasi Himpunan

• Peluang kejadian

• Kaidah-kaidah peluang

• Peluang bersyarat, kejadian bebas dan kaidah Bayes

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Introduction to Probability

• Experiments, Counting Rules, and

Assigning Probabilities

• Events and Their Probability

• Some Basic Relationships of Probability

• Conditional Probability

• Bayes’ Theorem

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Probability

• Probability is a numerical measure of the likelihood that an event will occur.

• Probability values are always assigned on a scale from 0 to 1.

• A probability near 0 indicates an event is very unlikely to occur.

• A probability near 1 indicates an event is almost certain to occur.

• A probability of 0.5 indicates the occurrence of the event is just as likely as it is unlikely.

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Another useful counting rule enables us to count the

number of experimental outcomes when n objects are to

be selected from a set of N objects.• Number of combinations of N objects taken n

at a time

where N! = N(N - 1)(N - 2) . . . (2)(1)

n! = n(n - 1)( n - 2) . . . (2)(1)

0! = 1

Counting Rule for Combinations

CN

nN

n N nnN

!

!( )!C

N

nN

n N nnN

!

!( )!

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Counting Rule for Permutations

A third useful counting rule enables us to count the

number of experimental outcomes when n objects are to

be selected from a set of N objects where the order of

selection is important.• Number of permutations of N objects taken n at

a time

P nN

nN

N nnN

!!

( )!P n

N

nN

N nnN

!!

( )!

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Complement of an Event

• The complement of event A is defined to be the event consisting of all sample points that are not in A.

• The complement of A is denoted by Ac.• The Venn diagram below illustrates the concept

of a complement.

Event Event AA AAcc

Sample Space SSample Space S

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• The union of events A and B is the event containing all sample points that are in A or B or both.

• The union is denoted by A B• The union of A and B is illustrated below.

Sample Space SSample Space S

Event Event AA Event Event BB

Union of Two Events

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Intersection of Two Events

• The intersection of events A and B is the set of all sample points that are in both A and B.

• The intersection is denoted by A • The intersection of A and B is the area of

overlap in the illustration below.Sample Space SSample Space S

Event Event AA Event Event BB

IntersectionIntersection

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Addition Law

• The addition law provides a way to compute the probability of event A, or B, or both A and B occurring.

• The law is written as:

P(A B) = P(A) + P(B) - P(A B

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Mutually Exclusive Events

• Addition Law for Mutually Exclusive Events

P(A B) = P(A) + P(B)

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Conditional Probability

• The probability of an event given that another event has occurred is called a conditional probability.

• The conditional probability of A given B is denoted by P(A|B).

• A conditional probability is computed as follows:

PPP

( | )( )( )

A BA BB

P

PP

( | )( )( )

A BA BB

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Multiplication Law

• The multiplication law provides a way to compute the probability of an intersection of two events.

• The law is written as:

P(A B) = P(B)P(A|B)

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Independent Events

• Events A and B are independent if P(A|B) = P(A).

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Independent Events

• Multiplication Law for Independent Events

P(A B) = P(A)P(B)

• The multiplication law also can be used as a test to see if two events are independent.

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• Tree Diagram

Contoh Soal: L. S. Clothiers

P(Bc|A1) = .8P(Bc|A1) = .8

P(A1) = .7P(A1) = .7

P(A2) = .3P(A2) = .3

P(B|A2) = .9P(B|A2) = .9

P(Bc|A2) = .1P(Bc|A2) = .1

P(B|A1) = .2P(B|A1) = .2 P(A1 B) = .14P(A1 B) = .14

P(A2 B) = .27P(A2 B) = .27

P(A2 Bc) = .03P(A2 Bc) = .03

P(A1 Bc) = .56P(A1 Bc) = .56

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Bayes’ Theorem

• To find the posterior probability that event Ai will occur given that event B has occurred we apply Bayes’ theorem.

• Bayes’ theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space.

P A BA B A

A B A A B A A B Aii i

n n

( | )( ) ( | )

( ) ( | ) ( ) ( | ) ... ( ) ( | )

P P

P P P P P P1 1 2 2

P A BA B A

A B A A B A A B Aii i

n n

( | )( ) ( | )

( ) ( | ) ( ) ( | ) ... ( ) ( | )

P P

P P P P P P1 1 2 2

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• Selamat Belajar Semoga Sukses.