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Pertemuan 03Teori Peluang (Probabilitas)
Matakuliah : I0272 – Statistik ProbabilitasTahun : 2005Versi : 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
CNn
Nn N nn
N
!
!( )!
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Counting Rule for Permutations
A third useful counting rule enables us to count thenumber of experimental outcomes when n objects
are tobe selected from a set of N objects where the
order ofselection is important.• Number of permutations of N objects taken n at
a time
P nNn
NN nn
N
! !( )!
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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 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 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 S
Event Event AA Event Event BB
Intersection
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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:
P PP
( | ) ( )( )
A B A 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(A1) = .7
P(A2) = .3P(B|A2) = .9
P(Bc|A2) = .1
P(B|A1) = .2 P(A1 B) = .14
P(A2 B) = .27
P(A2 Bc) = .03
P(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 B A B AA B A A B A A B Ai
i i
n n( | ) ( ) ( | )
( ) ( | ) ( ) ( | ) ... ( ) ( | )
P P
P P P P P P1 1 2 2
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• Selamat Belajar Semoga Sukses.
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