Chapter 4

ProbabilityStatistika

Statistics for Business and Economics, 6e 2007 Pearson Education, Inc.

Setelah mempelajari bab ini, anda mampu :

Menjelaskan konsep dasar dan definisi probabilitasMenghitung probabilitas bersyaratMenentukan apakah suatu kejadian independen dengan kejadian lainMenggunakan teorema Bayes untuk probabilitas bersyaratTujuan Bab

Random Experiment a process leading to an uncertain outcomeBasic Outcome a possible outcome of a random experiment Sample Space the collection of all possible outcomes of a random experimentEvent any subset of basic outcomes from the sample spaceImportant Terms

A dan B dikatakan kejadian saling asing , jika tidak saling beririsanYaitu, A B himpunan kosongImportant TermsABS

Kejadian E1, E2, Ek dikatakan Collectively Exhaustive jika E1 U E2 U . . . U Ek = S

Complement suatu kejadian A adalah semua kejadian di luar A, dinotasikan Important TermsAS

Probabilitas Peluang suatu kejadian tak pasti akan terjadi (antara 0 dan 1)

Probabilitas0 P(A) 1 Untuk sebarang Kejadian ACertainImpossible.510

1. Untuk suatu kejadian A, berlaku

2. Misalkan A adalah kejadian dalam S, dan Oi adalah kejadian dasar dalam A. Maka

(the notation means that the summation is over all the basic outcomes in A)

3.P(S) = 1Sifat-Sifat Probabilitas

A Probability TableProbabilitas dan probabilitas bersama untuk dua kejadian A dan B diringkas dalam tabel berikut :

BA

Probabilitas bersyarat adalah probabilitas suatu kejadian, bersyarat kejadian lain telah terjadi :Conditional ProbabilityProbabilitas A bersyarat BProbabilitas B bersyarat A

Berapakah probabilitas mobil berAC, mempunyai CD player ?

Kita akan menghitung P(CD | AC)ContohDari survey mobil, 70% menggunakan (AC) dan 40% memupunyai CD player (CD). 20%-nya mempunyai kedua fasilitas di atas.

ContohNo CDCDTotalAC.2.5.7No AC.2.1.3Total.4.61.070% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

ContohNo CDCDTotalAC.2.5.7No AC.2.1.3Total.4.61.0Given AC, we only consider the top row (70% of the cars). Of these, 20% have a CD player. 20% of 70% is 28.57%.

Multiplication rule for two events A and B:

alsoMultiplication Rule

Two events are statistically independent if and only if:

Events A and B are independent when the probability of one event is not affected by the other eventIf A and B are independent, thenStatistical Independence if P(B)>0 if P(A)>0

Statistical Independence ExampleNo CDCDTotalAC.2.5.7No AC.2.1.3Total.4.61.0Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

Are the events AC and CD statistically independent?

Statistical Independence ExampleNo CDCDTotalAC.2.5.7No AC.2.1.3Total.4.61.0P(AC CD) = 0.2P(AC) = 0.7P(CD) = 0.4P(AC)P(CD) = (0.7)(0.4) = 0.28P(AC CD) = 0.2 P(AC)P(CD) = 0.28So the two events are not statistically independent

The probability of a joint event, A B:

Computing a marginal probability:

Where B1, B2, , Bk are k mutually exclusive and collectively exhaustive eventsJoint and Marginal Probabilities

where:Ei = ith event of k mutually exclusive and collectivelyexhaustive eventsA = new event that might impact P(Ei)Teorema Bayes

Perusahaan pengeboran memprediksi peluang sukses 40% untuk sumur baru. Dari data historis, 60% sumur yang sukses, mempunyai test detil, dan 20% sumur yang tidak sukses, juga mempunyai test detil. Berapakah peluang sumur baru akan sukses, jika diketahui telah dilakukan test detil ? Bayes Theorem Example

Let S = successful well U = unsuccessful wellP(S) = .4 , P(U) = .6 (prior probabilities)Define the detailed test event as DConditional probabilities:P(D|S) = .6 P(D|U) = .2Goal is to find P(S|D)Bayes Theorem Example

Bayes Theorem ExampleApply Bayes Theorem:So the revised probability of success (from the original estimate of .4), given that this well has been scheduled for a detailed test, is .667