_Bab 5 Peluang.ppt

8/15/2019 _Bab 5 Peluang.ppt

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PELUANG

Bab 5Peluang


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Peluang:Konsep


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Suatu fenomena atau percobaan

dikatakan acak ( random ) jika hasil

(outcome) dari masing –masing

kejadian dalam fenomena tersebuttidak pasti. Tetapi untuk jumlah

ulangan yang besar, distribusi dari hasil

bersifat teratur

Randomness and probability

Peluang ( probability ) dari suatu hasil atau outcome pada fenomena

acak dapat didefinisikan sebagai proporsi banyaknya hasil dari suatu

kejadian tertentu pada sejumlah deretan ulangan yang besar.


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Pelantunan Koinasil dari setiap pelantunan koin bersifat acak. Tetapi hasil dari

keseluruhan pelantunan koin yang diulang berkali!kali (menuju tak hingga

kali) itu dapat diprediksi, asalkan ulangan dari pelantunan bersifat bebas

atau independent satu sama lain (hasil suatu pelantunan tidak

dipengaruhi hasil pelantunan sebelumnya).

asil dari setiap pelantunan koin bersifat acak. Tetapi hasil dari

keseluruhan pelantunan koin yang diulang berkali!kali (menuju tak hingga

kali) itu dapat diprediksi, asalkan ulangan dari pelantunan bersifat bebas

atau independent satu sama lain (hasil suatu pelantunan tidak

dipengaruhi hasil pelantunan sebelumnya).

First series of tossesSecond series

The probability of headsis ".# $ the proportion oftimes you get heads inmany repeated trials.


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Model Peluan Probability models menggambarkan secaramatematis hasil atau outcome dari suatu proses acak%odel peluang terdiri atas:

&) S ! "uan Sampel #Sample Space$% This is a set, or list, of all

possible outcomes of a random process. 'n e&ent is a subset ofthe sample space.

) Peluan untuk masing!masing kejadian yang mungkin dalamruang sampel

Model Peluang

Example: Probability Model for a Coin Toss

S $ ead, Tail*

Probability of heads $ ".#

Probability of tails $ ".#


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Probability Concepts

Mutually E'clusi&e E&ents+f & occurs, then ( cannot occur

& and ( ha-e no common elements

lack/ards

0ed/ards

' card cannot belack and 0ed at

the same time.

&


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Probability, Experiment, Outcome, Event:Defned

P01 ' +2+T3 ' -alue bet4een5ero and one, inclusi-e, describingthe relati-e possibility (chance or

likelihood) an e-ent 4ill occur.

'n e6periment is a processthat leads to the occurrence

of one and only one of se-eralpossible obser-ations.

'n outcome is the particularresult of an e6periment. 'n e-ent is the collection ofone or more outcomes of an

e6periment.

#!7


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Mutually Exclusive Events andCollectively Exhaustive Events

-ents are mutually e'clusi&e if the occurrence of any one e-ent means that none of the others can occur at the same time.-ents are collecti&ely e'(austi&e if at least one of the e-ents must oc cur 4hen an e6periment is conducted.

The sum of all collecti-ely e6hausti-e and mutually e6clusi-e e-ents is &." (or &""8)

-ents are independent if the occurrence of one e-ent does not affect the occurr ence of another.

collecti-ely e6hausti-eand mutually e6clusi-e

e-ents


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Classical and Empirical Probability

/onsider an e6periment of rolling a si6!sided die. 9hat is the probability of thee-ent an e-en number of spots appearface up;<

The possible outcomes are:

There are three fa-orable; outcomes (at4o, a four, and a si6) in the collection ofsi6 e=ually likely possible outcomes.

The empirical approach to probability is based on 4hatis called the la4 of large numbers. The key to

establishing probabilities empirically is that more

obser-ations 4ill pro-ide a more accurate estimate ofthe probability.

>'%P2 :1n ?ebruary &, ""@, the Space Shuttle /olumbia

e6ploded. This 4as the second disaster in & @ spacemissions for A'S'. 1n the basis of this information,

4hat is the probability that a future mission issuccessfully completed<

98.0123121

flightsof numberTotalflightssuccessfulof Number

flightsuccessfulaof yProbabilit

==

=


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u ect ve ro a ty "Example

+f there is little or no past e6perience or information on 4hich to base a probability,it may be arri-ed at subjecti-ely.

+llustrations of subjecti-e probability are:&. stimating the likelihood the Ae4 ngland Patriots 4ill play in the Super o4l ne6t year.

. stimating the likelihood you 4ill be married before the age of @".@. stimating the likelihood the B.S. budget deficit 4ill be reduced by half in the ne6t &"

years.

21@

#!&"


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ummary o ypes oProbability

21@

#!&&


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Rules o# %ddition

"ules of AdditionCika kejadian A dan B kejadian salingterpisah (mutually e6clusi-e), maka

P#A atau B$ ! P#A$ ) P#B$

Cika kejadian A dan B kejadian tidak

saling terpisah (mutually e6clusi-e),makaP#A atau B$ ! P#A$ ) P#B$ * P#A dan B$


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Rules o# %ddition>'%P2 :

'n automatic Sha4 machine fills plastic bags 4ith a mi6ture of beans,broccoli, and other -egetables. %ost of the bags contain the correct4eight, but because of the -ariation in the si5e of the beans and other-egetables, a package might be under4eight or o-er4eight. ' check ofD,""" packages filled in the past month re-ealed:

9hat is the probability that a particular package 4ill be either

under4eight or o-er4eight


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$he Complement Rule

The complement rule is used todetermine the probability of an e-entoccurring by subtracting the probabilityof the e-ent not occurring from &.

P ( A) E P (F A) $ &or P ( A) $ & ! P (F A).

>'%P2 'n automatic Sha4 machine fills plastic bags 4ith a mi6ture ofbeans, broccoli, and other -egetables. %ost of the bags containthe correct 4eight, but because of the -ariation in the si5e of the

beans and other -egetables, a package might be under4eightor o-er4eight. Bse the complement rule to sho4 the probability

of a satisfactory bag is .G""

P ( ) $ & ! P (F ) $ & – P(' or /)

$ & – HP(') E P(/)I$ & – H." # E ."7#I

$ & ! .&" $ .G"

21D


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$he &eneral Rule o# %ddition and 'oint Probability

The Jenn Kiagram sho4s the result of asur-ey of "" tourists 4ho -isited ?lorida

during the year. The sur-ey re-ealed that & "4ent to Kisney 9orld, &"" 4ent to usch

Lardens and M" -isited both.

9hat is the probability a selected person-isited either Kisney 9orld or usch

Lardens<

P(%isney or B&s' ) = P(%isney) + P(B&s' ) P(bot %isney and B&s' ) = $! *! + $ *! , *!

= ., + ." .-

C1+AT P01 ' +2+T3 ' probability that measuresthe likelihood t4o or more e-ents 4ill happen

concurrently.


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pecial and &eneral Rules o#Multiplication

The special rule of multiplication re=uires that t4o e-ents A and B are independent .T4o e-ents A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other.This rule is 4ritten: P#A and B$ ! P#A$P#B$

E+AMPLEA sur&ey by t(e American Automobile association #AAA$ re&ealed ,- percent of its members made airline reser&at ions last year. /0o members are selected at random. Since t(e number of AAAmembers is &ery lar e1 0e can assume t(at R1 and R2 are independent. 2(at is t(e probability both made airline reser&ations last year3

Sol&tion:The probability the first member made an airline reser-ation last year is .M", 4ritten as P ( &) $ .M"The probability that the second member selected made a reser-ation is also .M", so P ( ) $ .M".Since the number of ''' members is -ery large, you may assume that & and are independent.

P #R 4 and R $ ! P #R 4$P #R $ ! #.,-$#.,-$ ! .6,


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pecial and &eneral Rules o#Multiplication

The general rule of multiplication is used to find the joint probability that t4o independent e-ents 4ill occur.

>'%P2 ' golfer has & golf shirts in his closet. Suppose G of these shirts are 4hite and the othersblue. e gets dressed in the dark, so he just grabs a shirt and puts it on. e plays golf t4o

days in a ro4 and does not do laundry.9hat is the likelihood both shirts selected are 4hite<

The e-ent that the first shirt selected is 4hite is / &. The probability is P (/ &) $ GN& The e-ent that the second shirt ( / )selected is also 4hite. The conditional probability that the

second shirt selected is 4hite, gi-en that the first shirt selected is also 4hite, isP (/ O/ &) $ N&&.

To determine the probability of 4hite shirts being selected 4e use formula: P(AB) = P(A) P(B0A)P (/ & and / ) $ P (/ &)P (/ O/ &) $ (GN& )( N&&) $ ".##


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Contingency $ables

A 78N/9NGEN7: /ABLE is a table used to classify sample obser-ations according to t4o or more identifiablecharacteristic

>'%P2 :.g. ' sur-ey of " adults classifiedeach as to gender and the number of

mo-ies attended last month. achrespondent is classified according tot4o criteriaQthe number of mo-ies

attended and gender .

-ent A& happens if a randomly selectede6ecuti-e 4ill remain 4ith the companydespite an e=ual or slightly better offerfrom another company. Since there are

& " e6ecuti-es out of the "" in thesur-ey 4ho 4ould remain 4ith the

companyP ( A&) $ & "N "", or .M".

-ent BD happens if a randomly selectede6ecuti-e has more than &" years of

ser-ice 4ith the company. Thus, P( DO '&)

is the conditional probability that ane6ecuti-e 4ith more than &" years ofser-ice 4ould remain 4ith the company.1f the & " e6ecuti-es 4ho 4ould remain

7# ha-e more than &" years of ser-ice, soP( DO '&) $ 7#N& ".


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$ree Diagrams

' tree dia ram is useful for portrayingconditional and joint probabilities. +t isparticularly useful for analy5ing businessdecisions in-ol-ing se-eral stages. ' tree dia ram is a graph that is helpfulin organi5ing calculations that in-ol-e

se-eral stages. ach segment in the treeis one stage of the problem. The branchesof a tree diagram are 4eighted byprobabilities.


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9ndependent and ;ependent E&ents

+ndependent: 1ccurrence of one does notinfluence the probability ofoccurrence of the other

Kependent: 1ccurrence of one affects theprobability of the other

Probability Concepts

( d d ) d


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+ndependent -ents

& $ heads on one flip of fair coin

$ heads on second flip of same coin

0esult of second flip does not depend on the result ofthe first flip.

Kependent -ents

& $ rain forecasted on the ne4s

$ take umbrella to 4ork

Probability of the second e-ent is affected by theoccurrence of the first e-ent

(ndependent vs) DependentEvents


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The trials are independent

only 4hen you put the coin

back each time. +t is called

T4o e-ents are independent if the probability that one e-ent occurs

on any gi-en trial of an e6periment is not affected or changed by the

occurrence of the other e-ent.

9hen are trials not independent<

+magine that these coins 4ere spread out so that half 4ereheads up and half 4ere tails up. /lose your eyes and pick

one. The probability of it being heads is ".#. o4e-er, if you

donRt put it back in the pile, the probability of picking up

another coin and ha-ing it be heads is no4 less than ".#.

_Bab 5 Peluang.ppt

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