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Section 5-3
Binomial Probability
Distribution
BINOMIAL PROBABILITY
DISTRTIBUTION
Abinomialprobabilitydistribution resultsfroma
procedurethatmeetsallthefollowingrequirements:
1. Theprocedurehasafixednumberoftrials.
2. Thetrialsmustbeindependent.(Theoutcome
ofanyindividualtrialdoesn’taffectthe
probabilitiesintheothertrials.)
3. Eachtrialmusthavealloutcomesclassifiedinto
twocategories.Thesecategoriesareusually
“success”or“failure”.
4. Theprobabilityofasuccessremainsthesamein
alltrials.
5% GUIDELINE
TreatingDependentEventsasIndependent:
5%GuidelineforCumbersomeCalculations
Whencalculationswithsamplingarevery
cumbersomeandthesamplesizeisnotmore
than5%ofthesizeofthepopulation,treatthe
selectionsasbeingindependent (evenifthey
areactuallydependent).
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NOTATION FOR BINOMIAL
PROBABILITY DISTRIBUTIONS
“S”and“F”(successandfailure)denotethe
twopossiblecategoriesofalloutcomes;p will
denotetheprobabilityof“S”andq willdenote
theprobabilityof“F”.Thatis,
1
(
probabilityofsuccess)
(
probabilityoffailure)
NOTATION (CONCLUDED)
denotesthefixednumberoftrials
denotesthenumberofsuccessesin trials,
so canbeanynumberbetween0and ,
inclusive.
denotestheprobabilityofsuccess inone of
the trials.
denotestheprobabilityoffailure inone of
the trials.
denotestheprobabilityofgettingexactly
successesamongthe trials.
EXAMPLES
Identifysuccess,failure,n,p,andq forthequestions
below.Wewillanswerthequestionslater.
1. Iftheprobabilityis0.70thatastudentwith
veryhighgradeswillgetintolawschool,what
istheprobabilitythatthreeoffivestudents
withveryhighgradeswillgetintolawschool?
2. Theprobabilityis0.60thatapersonshopping
inacertainmarketwillspend$25ormore.
Findtheprobabilitythatamongeightpersons
shoppingatthismarket,atleast6willspend
$25ormore.
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IMPORTANT CAUTIONS
• Besurethat and bothrefertothesame
categorybeingcalledasuccess.
• Whensamplingwithoutreplacement,the
eventscanbetreatedasiftheywere
independentifthesamplesizeisnomore
than5%ofthepopulationsize.Thatis,
0.05 .(Thisisthe5%Guideline.)
THREE METHODS FOR FINDING
BINOMIAL PROBABILITIES
1. UsingtheBinomialProbabilityFormula.
2. UsingTableA‐1locatedinAppendixAon
pages583.
3. UsingtheTI‐83/84calculator.
METHOD 1: THE BINOMIAL
PROBABILITY FORMULA
!
! !
where
·
·
for
0, 1, 2, . . . ,
numberoftrials
numberofsuccessesin trials
probabilityofsuccessinanyonetrial
probabilityoffailureinanyonetrial
(
1
)
Thisformulacanalsobewrittenas
·
·
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EXAMPLE
Iftheprobabilityis0.70thatastudentwith
veryhighgradeswillgetintolawschool,what
istheprobabilitythatthreeoffivestudents
withveryhighgradeswillgetintolawschool?
UsetheBinomialProbabilityFormula.
METHOD 2: USING TABLE A-1
IN APPENDIX A
PartofTableA‐1isshownbelow.Withn =4andp =
0.2inthebinomialdistribution,theprobabilitiesof0,
1,2,3,and4successesare0.410,0.410,0.154,0.026,
and0.002respectively.
EXAMPLE
Theprobabilityis0.60thataperson
shoppinginacertainmarketwillspend$25
ormore.Findtheprobabilitythatamong
eightpersonsshoppingatthismarket,at
least6willspend$25ormore.UseTableA‐
1.
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METHOD 3: USING THE TI-83/84
CALCULATOR
1. Press2nd VARS (togetDISTR)
2. Selectoption0:binompdf( ontheTI‐83.Use
A:binompdf( ontheTI‐84.
3. Completetheentryofbinompdf(n,p,x) with
specificvaluesforn,p,and .
4. PressENTER, andtheresultwillbethe
probabilityofgetting successesin trials;that
is,
.
NOTE:Formoreinformationon“binompdf(”seethe
UsingTechnologyboxonpage217.
TI-84 WITH NEW OPERATION
SYSTEM
IfusingtheTI‐84withthenewestoperating
system,completethescreenasfollows
Entervaluefor here.
Entervaluefor here.
Entervaluefor here.
HighlightPASTEandpressENTERtocopythe
commandtothehomescreen.PressENTERagainto
executethecommand.
EXAMPLE
Useyourcalculatortocomputetheprobability
of3successesin10trialsiftheprobabilityof
successis0.4.
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MATHEMATICAL
TRANSLATIONS OF ENGLISH
PHRASES
Phrase
MathSymbol
“atleast”or“nolessthan”
≥
“morethan”or“greaterthan”
>
“fewerthan”or“lessthan”
<
“nomorethan”or“atmost”
≤
“exactly”
=
CUMULATIVE BINOMIAL PROBABILITIES
USING THE TI-83/84 CALCULATOR
Thecumulativebinomialprobability isthesumofthe
individualprobabilitieslessthanorequalto .Thatis,
0
1
2
⋯
Tofindthecumulativebinomialprobabilityonthe
calculator:
1. Press2nd VARS (togetDIST)
2. SelectoptionA:binomcdf( ontheTI‐83.Use
B:binomcdf( ontheTI‐84.
3. Completetheentryofbinomcdf(n,p,x) withspecific
valuesfor , ,and .
4. PressENTER,andtheresultwillbetheprobabilityof
gettingatmost successesin trials,thatis,
.
CUMULATIVE BINOMIAL PROBABILITIES
USING THE TI-84 WITH NEW OS
Press2nd VARS (togetDIST)andselectB:binomcdf(.
Entervaluefor here.
Entervaluefor here.
Entervaluefor here.
HighlightPASTEandpressENTERtocopythe
commandtothehomescreen.PressENTERagainto
executethecommand.
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EXAMPLE
AccordingtoNielsonMediaResearch,88%of
allUnitedStateshouseholdsin2014hadHD
television.
(a) Inarandomsampleof15households,what
istheprobabilitythatexactly10hadHD?
(b) Inarandomsampleof15households,what
istheprobabilitythatatleast13hadHD?
(c) Inarandomsampleof15households,what
istheprobabilitythatfewerthan13had
HD?
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