Download PROBABILITY EVENTS - Gordon State College

2/3/2017
Sections 4-1 and 4-2
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and
Basic Concepts of Probability
RARE EVENT RULE FOR
INFERENTIAL STATISTICS
If,underagivenassumption(suchasalottery
beingfair),theprobabilityofaparticular
observedevent(suchasfiveconsecutivelottery
wins)isextremelysmall,weconcludethatthe
assumptionisprobablynotcorrect.
Statisticiansusetherareeventrulefor
inferentialstatistics.
PROBABILITY
Probability isthemeasureofthelikelihood
thatagiveneventwilloccur.
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EVENTS
• Anevent isanycollectionofresultsor
outcomesofaprocedure.
• Asimpleevent isanoutcomeoreventthat
cannotbefurtherbrokendownintosimpler
components.
• Thesamplespace foraprocedureconsists
ofallpossiblesimple events.Thatis,the
samplespaceconsistsofalloutcomesthat
cannotbebrokendownanyfurther.
PROBABILITY
Probability isameasureofthelikelihoodthat
agiveneventwilloccur.
NOTATION:
• P denotesaprobability.
• A,B,andC denotespecificevents.
•
denotestheprobabilityofeventA
occurring.
RULE 1: RELATIVE FREQUENCY
APPROXIMATION OF PROBABILITY
Conduct(orobserve)aprocedurealargenumber
oftimes,andcountthenumberoftimesthat
eventA actuallyoccurs.Basedontheseactual
isapproximated asfollows:
results,
numberoftimes occurred
numberoftimesprocedurewasrepeated
ThisruleusestheLawofLargeNumbers.
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THE LAW OF LARGE NUMBERS
Asaprocedureisrepeatedagainandagain,the
relativefrequencyprobability(fromRule1)ofan
eventtendstoapproachtheactualprobability.
CAUTION:Thelawoflargenumbersappliesto
behavioroveralargenumberoftrials,anditdoes
notapplytooneoutcome.Don’tmakethefoolish
mistakeoflosingalargesumofmoneyby
incorrectlythinkingthatastringoflosses
increasesthechancesofawinonthenextbet.
EXAMPLE
Afairdiewastossed563times.Thenumber
“4”occurred96times.Ifyoutossafairdie,
whatdoyouestimatetheprobabilityisfor
tossinga“4”?
RULE 2: CLASSICAL
APPROACH TO PROBABILITY
Assumethatagivenprocedurehas different
simpleeventsandthateachofthosesimple
eventshasanequalchanceofoccurring.If
eventA canoccurin ofthose ways,then
numberofways canoccur
numberofdifferentsimpleevents
CAUTION:Whenusingtheclassicalapproach,
alwaysverifythattheoutcomesareequally
likely.
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EXAMPLE
Findtheprobabilityofgettinga“7”whena
pairofdiceisrolled.
RULE 3: SUBJECTIVE
PROBABILITIES
,theprobabilityofeventA,isestimated by
usingknowledgeoftherelevantcircumstances.
PROBABILITY LIMITS
— Likely
• Theprobabilityofan
impossibleeventis0.
• Theprobabilityofaneventhat
iscertaintooccuris1.
•0
1 foranyeventA
— Unlikely
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COMPLEMENTARY EVENTS
ThecomplementofeventA,denotedby ̅,
consistsofalloutcomesinwhicheventA does
not occur.
EXAMPLE
Whatistheprobabilityofnot rollinga“7”
whenapairofdiceisrolled?
ROUNDING OFF PROBABILITIES
Whenexpressingthevalueofaprobability,
eithergivetheexact fractionordecimalor
roundofffinaldecimalresultstothree
significantdigits.
Suggestion: Whentheprobabilityisnota
simplefractionsuchas2/3or5/9,expressit
asadecimalsothatthenumbercanbebetter
understood.
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UNLIKELY AND UNUSUAL
EVENTS
Aneventisunlikely ifitsprobabilityisvery
small,suchas0.05orless.Aneventhasan
unusuallylownumber ofoutcomesofa
particulartypeoranunusuallyhighnumber
ofthoseoutcomesifthatnumberisfarfrom
whatwetypicallyexpect.
SUMMARY:
• Unlikely:smallprobability
• Unusual:extremeresult
ODDS
• Theactualoddsagainst eventA occurringarethe
̅ ⁄
ratio
,usuallyexpressedintheformof
a:b (or“a tob”),wherea andb areintegershaving
nocommonfactors.
• Theactualoddsinfavor ofeventA areratio
̅ ,whichisthereciprocaloftheactual
/
oddsagainstthatevent.IftheoddsagainstA are
a:b,thentheoddsinfavorofA areb:a.
• Thepayoffodds againsteventA representtheratio
ofthenetprofit(ifyouwin)totheamountbet:
payoffoddsagainstA =(netprofit):(amountbet)
EXAMPLE
TheAmericanStatisticalAssociationdecided
toinvestsomeofitsmemberrevenueby
buyingaracehorsenamedMean.Meanis
enteredinaraceinwhichtheactual
probabilityofwinningis3/17.
(a) FindtheactualoddsagainstMean
winning.
(b) Ifthepayoffoddsarelistedas4:1,how
muchprofitdoyoumakeifyoubet$5
andMeanwins.
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