# probability

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1 Pertemuan 03 Teori Peluang (Probabilitas) Matakuliah: I0272 Statistik Probabilitas Tahun: 2005 Versi: Revisi 2 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswaakan mampu : Mahasiswa akan dapat menjelaskan ruang contoh dan peluang kejadian. mahasiswa dapat memberi contoh peluang kejadian bebas, bersyarat dan kaidah Bayes. 3 Outline Materi Istilah/ notasi dalam peluang Diagram Venn dan Operasi Himpunan Peluang kejadian Kaidah-kaidah peluang Peluang bersyarat, kejadian bebas dan kaidah Bayes 4 Introduction to Probability Experiments, Counting Rules, andAssigning Probabilities Events and Their Probability Some Basic Relationships of Probability Conditional Probability Bayes Theorem 5 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. 6 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 whereN! = N(N - 1)(N - 2) . . . (2)(1) n! =n(n - 1)( n - 2) . . . (2)(1) 0! = 1 Counting Rule for Combinations CNnNn N nnN= |\

|.| =!!( )!7 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 nNnNN nnN=|\

|.| =!!( )!8 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 A Ac Sample Space S 9 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 AB. The union of A and B is illustrated below. Sample Space S Event AEvent B Union of Two Events 10 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 B. The intersection of A and B is the area of overlap in the illustration below. Sample Space S Event AEvent B Intersection 11 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(AB) = P(A) + P(B) - P(A B) 12 Mutually Exclusive Events Addition Law for Mutually Exclusive Events P(AB) = P(A) + P(B) 13 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=14 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) 15 Independent Events Events A and B are independent if P(A|B) = P(A). 16 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. 17 Tree Diagram Contoh Soal:L. S. Clothiers P(Bc|A1) = .8 P(A1) = .7 P(A2) = .3 P(B|A2) = .9 P(Bc|A2) = .1 P(B|A1) = .2P(A1 B)= .14 P(A2 B)= .27 P(A2 Bc) = .03 P(A1 Bc) = .56 18 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. PABA B AA B A A B A A B Aii in n( | )( ) ( | )( ) ( | ) ( ) ( | ) ... ( ) ( | )=+ + +P PP P P P P P1 1 2 219 Selamat Belajar Semoga Sukses.