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Lecture 7

Lecture 7

9 Pages
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Department
Quantitative Methods
Course Code
QMS 102
Professor
Clare Chua

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10/19/2010
1
Probability
Topicscoveredtoday:
1. BasicProbabilitytheory(Chapter4)
2. DiscreteProbabilityDistribution
(Chapter5,section5.1)
TypesofProbability
Discrete Continuous
Numerical
(Random
variable)
Categorical
Data
Chapters 1,2,3
Binomial
Hypergeometric
Poisson
Probability
Distribution
Normal
Distrbution
Uniform
Distribution
Exponential
DIstribution
Probability
Distribution
Chapters
4,5, 6
Probability
to cope
with uncertainty
Definitions
Probability: Ameasureofthelikelihoodthatanevent
willhappen.(Youcanalsorefertotheprobabilityofanevent
thatmayhavealreadyoccurred,butonlyifyoudon'tknow
whetherornottheeventoccurred.)
Experiment:Somethingdonetoobtaindata.Itmay
involveaphysicalactiontomakesomethingtakeplaceormay
j
ustinvolveobservin
g
somethin
g
ha
pp
en.
jg g pp
Outcome: Theresultofonetrialoftheexperiment.
SampleSpace: Thelistofallpossibleoutcomes.
Event:Acollectionofoneormoreoutcomes.
Example1:
Tos sacoin:(Interestedinresults,thatis,heads/tails.)
a)once.Experiment:Tosscoinonetime
SampleSpace:{HT}
Event:e.g.Getaheadr{H}
b)twice.(ortoss2coins) Experiment:Tosscointwice.
Sam
p
leS
p
ace:
{
HHHTTHTT
}
p p { }
Event:e.g.Getatleastoneheadr{HHHTTH}
c)threetimes.(ortoss3coins)Experiment:Tosscoin3times
SampleSpace:{HHHHHTHTHTHHTTHTHTHTTTTT}
Event:e.g.Getoneheadr{HTTTHTTTH}
Example2:
Rolladie: (Interestedintheresultofeachdie.)
a)once.Experiment:Rolldieonce
SampleSpace:{123456}
Event:e.g.Getanevennumberr{246}
b)twice.(orroll2dice)Experiment:Rolldietwice.
SampleSpace:{1r11r21r31r41r51r6
2r12r22r32r42r52r6
3r13r23r33r43r53r6
4r14r24r34r44r54r6
5r15r25r35r45r55r6
6r16r26r36r46r56r6}
Event:e.g.Getatotalof5r{1r42r33r24r1}
Probability
a)Symbol:P(event)=#
b)Answer:Decimalorfractionbetween0and1.(Decimalpreferred)
Chance isexpressedasapercent.
e.g.Tos s acoin.P(head)=0.5
Thereisa50%chanceofgettingahead.
c)IfP(event)=0thenitisnotpossible fortheeventtooccur.
IfP(event)=1thentheeventisacertaintyi.e.guaranteedtooccur.
d)Thesumofalltheprobabilitiesinthesamplespaceis1
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10/19/2010
2
Probability
Not possibleCertainty
0.0 0.10.2 0.30.4 0.50.6 0.70.8 0.91.0
Prob. our sun will
disappear this
year
Prob. the
Leafs will win
the cup
Prob. of
getting a
head on a
coin toss
Prob. of an
increase in
taxes this
year
Prob. of snow in
Toronto this
winter
1. Allprobabilitiesarebetween0and1inclusive
2. Theprobabilityofaneventwhichcannotoccuris0.
3. Theprobabilityofaneventwhichoccurwith
certaintyis1.
4. Theprobabilityofaneventnotoccurringisone
minustheprobabilityofitoccurring.
P(E')=1rP(E)
5.Youcanalsorepresentprobabilityasadecimalorasa
percent.
ApproachestoProbability
A) Subjective
B) Experimental(orrelativefrequency)
C) Theoretical(orClassical)
A)Subjective:
Personaljudgmentandexpressesdegreeof
belief.
l
Examp
l
e:
P(Snowtoday)=0.01Basedonstudyof
weatherconditions.Notcalculatedthrough
theuseofprobabilityformula.
B)Experimental(orrelativefrequency):
Definition:Thechancesofsomeeventoccurredandweeitherconductrepeated
testingorobserveevent.Theprobabilityistheratioofthenumberoftimesan
eventoccurredtothenumberoftimestested.
Example1:Toss athumbtack.Findtheprobabilitythatthethumbtackwilllandonits
headwiththepointstickingup.
Theonlywaytodeterminethisprobabilityistotossthethumbtackandcountthe
numberoftimesitlandswithitspointstickingup.
Wecanonlygettheapproximateanswertotherequiredprobability.
Theexactcorrectanswercanonlybedeterminedbyrepeatingtheexperimentan
infinitenumberoftimes.
Dependingontheaccuracyrequired,wecandeterminethenumberoftimesthe
experimentmustberepeated.
Example2: Inthepastfewmonths,39%ofthetrucksthathavebeeninspectedata
particularinspectionstationhavefailedthesafetyinspection.Whatisthe
probabilitythatthenexttruckinspectedwillfailthesafetyinspection?
Theoretical(orClassical)
Definition:
Probabilityisalikelihoodthataneventwillhappen.
Thetheoreticalprobabilityofaneventcanbecalculatedas:
Ifalltheoutcomesinthesamplespaceareequallylikely,then
space
sample
the
in
outcomes
of
event thein outcomes ofnumber
)( eventP
Example:
Ifyoutossafaircoin,whatistheprobabilitythatatailwillshowup?
Solution:
Whenyoutossacointhereareonly2possibleoutcomes:HeadorTail
Sotheoptionsfortossingatailare1outof2.
space
sample
the
in
outcomes
of
Continued…Theoretical(orClassical)
Example:
Abagcontains100marbles.Thereare30green
marblesand70yellowmarbles.Findthe
probabilityofpickingagreenmarble.
30
bag
the
in
marbles
red
of
Number
3.0
100
30
)( bagtheinmarblesofNumberTotal
bag
the
in
marbles
red
of
Number
marblegreenp
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10/19/2010
3
DiscreteProbabilityDistribution
ProbabilityDistributionForADiscrete
RandomVariable
Aprobabilitydistributionforadiscreterandomvariableisa
mutuallyexclusivelistingofallthepossiblenumerical
outcomesalongwiththeprobabilityofoccurrenceofeach
outcome.Probabilityof
Distribution
X=Number
oftails P(X)
01/8
13/8
23/8
31/8
Probability
Probabilityof
occurrence
Probabilityof
Distribution
p(E) =X
T
P(X)
1/8
X=Number
oftails
T
X=#ofwaysinwhich
theeventoccurs
T=total#ofoutcomes
Example:Tossacoinandguess
Whetheritwillbeaheadoratail
0
1
2
3
1/8
3/8
1/8
3/8
Tossafaircoin3times
P(Head)=1/2
E=Event
Objective
1. IdentifytheProbabilityofDistribution
2. Com
p
utetheMeanandStandard
p
Deviation
RandomVariable
WhatisRandomVariable?
WhatisRandomVariable?
Definition:
Arandomvariableisavariablethattakeson
differentvaluesasaresultoftheoutcomesof
arandomex
p
eriment.
p
Notation:X=RandomVariable
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Description
10192010 TypesofProbability Data Chapters 1,2,3 Numerical Categorical (Random variable) Probability Discrete Continuous Distribution Distribution Topicscoveredtoday: 1. BasicProbabilitytheory(Chapter4) Chapters Binomial Normal Distrbution 2. DiscreteProbabilityDistribution 4,5, 6 (Chapter5,section5.1) Probability Hypergeometric Distribution to cope with uncertainty Poisson Exponential DIstribution Definitions Example1: Probability: Ameasureofthelikelihoodthatanevent Tossacoin:(Interestedinresults,thatis,headstails.) willhappen.(Youcanalsorefertotheprobabilityofanevent a)once. Experiment:Toss coinonetime thatmayhavealreadyoccurred,butonlyifyoudontknow whetherornottheeventoccurred.) SampleSpace:{HT} Event:e.g.GetaheadH {H} Experiment: Somethingdonetoobtaindata.Itmay involveaphysicalactiontomakesomethingtakeplaceormay b)twice.(ortoss2coins) Experiment:Tosscointwice. jjustinvolveobservigsomething ghappen. Sam plepace:{{HHHTTHTT}} Event:e.g.GetatleastoneheadH {HHHTTH} Outcome: Theresultofonetrialoftheexperiment. SampleSpace: Thelistofallpossibleoutcomes. c)threetimes.(ortoss3coins)Experiment:Tosscoin3times SampleSpace:{HHHHHTHTHTHHTTHTHTHTTTTT} Event: Acollectionofoneormoreoutcomes. Event:e.g.GetoneheadH {HTTTHTTTH} Example2: Probability Rolladie: (Interestedintheresultofeachdie.) a)Symbol:P(event)=# a)once.Experiment:Rolldieonce b)Answer:Decimalorfractionbetween0and1.(Decimalpreferred) Chance isexpressedasapercent. SampleSpace:{123456} Event:e.g.GetanevennumberH {246} e.g.Tossacoin.P(head)=0.5 Thereisa50%chanceofgettingahead. b)twice.(orroll2dice)Experiment:Rolldietwice. SampleSpace:{1H11H21H31H4H51H6 c)IfP(event)=0thenitisnotpossibleventtooccur. 2H12H22H32H42H52H6 IfP(event)=1thentheeventisacertainty i.e.guaranteedtooccur. 3H13H23H33H43H53H6 d)Thesumofalltheprobabilitiesinthesamplespaceis1 4H14H24H34H44H54H6 5H15H25H35H45H55H6 6H16H2636H46H56H6} Event:e.g.Getatotalof5H {1H42H33H24H1} 1 www.notesolution.com
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