/
Introduction
to Calculus*Probability
1
Name
Goalsfor today
usethe Fundamental
Countinqprinciple
calculatea fnnyta{on with anOwitfroutrepetition
cahulateacombination
1.
to set up 3-stg1,eo
system.by purchqsingsepararecomponents.In
YgY-1l"^qoing
you
find
5
different
receiv6rs,
8 differeitcompactOisc'playerJ,
anO
Y9{,P199_lange
12 differen!sppqker
syste.ms.lf you what.on.e
of eachof thesecompoheits,-howmanyditferentstereo-systems
aie possible?
2.
lf a license.plate
is 5 fetters.folloWgg
bV2 d.igits,
determinehow manydifferent
license.plates
arepossible
if (a)digitshndtdtteisianudripeatedlihoiu)oigl'ts
andletters
cannot-be
repeateb?
3.
Useyourcalculatorto evaluate(a) 10!and (b)O!.
4.
Findthe numberofpermutations.
(a)
5.
sPz
(b)
spg
(c)
16po
Findthe numberof distinguishable
permutations
of the lettersin the word.
(a)
PENctL
(c)
(b)
DAD
toWA
(d)
M|SSOURI
6.
To keepcomputerfiles
secure,manyprograms
requirethe usertoentera
password.The shortestallowablepasswordsare typicallysix letterslong. How
manysix lefterpasswordsare possibleif (a)characterccan be repeatedind (b)
characterscannotbe repeated?
7.
A pafticularclassroomhas24 seats and24students.Assumingthe
- seatsare not
moved,how manydifferentseatinganangements
are possiuie"z-
8'
Apizzl thoqlr!: a special4"t"^ygy canbuya largepizzawithonecheese,one
-''"
vegetable,andone m-eatfor $9.00.You have'achoTce
of7 cfreesl;, i'i----'
vegetables,
and6 meats.Additionally,
younavea choiceof g cruiG
--- and2sauces.
Howmanydifferentvariations
of the iir^special are posiioie?
"Ringing.the
change.s"
!t ? ptgggsswherethebetlsin a towerare rungin all possible
pennutations.Westminstdr
Abbeyhas 10 bellsin its tower.in-no* manyways
can its bellsbe rung?
9'
1 0 . A m u s i c . t o I " r y 9 | ! s t o d i s p l a y . 3 b g y b o a r d s , 2 t r u m p-e t s , a n d 2 g u i t a r s i n i t s s t o r e
r-'rwindow. Howmanydistinguishabre
dis,rty;# tbG'b6t
11. Findthe nurnberof combinations.
(a) scz
(b) scs
(c) 16co
12.
Findthe numberof combinations
of n objectstakenr at a time.
(a)
13.
n=8,r=2
(b)
n=5,r=1
Findthehumberof possible5-cardhandsttratcontainthe cardsspecified.
(a)
5 facecards
(b)
2 acesand3 kings
(c)
4 acesand 1 othercard
1 4 . YourEnglishteacherhas askedyou to select3 novelsfrom a list of 10 booksto be
readas an independent
project.In how manywayscanyou choosewhichbooksto
read?
15.
Yourfriendis havinga partyandhas 15 gamesto choosefrom. Thereis enough
timeto play4 games. In howmanywayscan you choosewhichgamesto play?
16.
Youare buyinga newcar. Thereare7 differentcolorsto choosefromand 10
differenttypesof oplionalequipmentyou can buy. You can chooseonly 1 colorfor
yourcarcnd canaffordonly2 of the options.Howmanycombinations
ale therefor
yourcal?
17.
An amusementparkhas20 differentrides.Youwantto rideat least15 of them.
Howmanydifferentcombinations
of ridescan you go on?
D-ecide
whetherthe.problem
requirescombinations
or permutations
to findthe answer.
Thensolvethe problem.
18.
for 3 drummajorpositions.
fight membersof a schoolmarchingbandareauditioning
In how rnanywayscan studentsbe chosento be drumrirajors?
19.
You musttake6 electiveclassesto meetyourgraduationrequirements
for college.
Thereare 12classestlTatTou-dre
interesrd-inl ln howma-nyWayScan'youSeECt
yourelectiveclasses?
2A.
Yourschoolyearbook
hasan editor-inchief
andan assistanteditor{n-chief.Thestaff
of the yearbookhas 15 students.In howmanywayscan studentsbe chosenfor
these2 positions?
Introduction
to Calculus*Probability
2
Name
Goalsfor today
calculateprobabilities-theoretical
and experimental
calculatethe probabilities
of compound'events
calculatethe probabilities
of independent
anddependentevents
1.
2.
P(A) = .2 and P(B) = .6. Which event is more likely to occur? Explain.
A jar contains2 red marbles,s blue marbles,and 1 greenmarble. Findthe
probabilityof randomlydrawingthe giventype of m?rUle.
(a)
3.
a red marble
(b)
Youhavea.11.equ.qlly
likelychancgof choosingany integerfrom 1 through20. Find
the probabilitybf thiegivenevent.
(a)
an odd numberis chosen
(c)
4.
(b)
a perfectsquareis chosen
a muttipleof 3 is chosen
A cardis drawnrandomlyfiom a standardS2-carddeck. Findthe probabilityof
drawingthe givencard.
(a)
a redcard
(b)
(c)
5'
a blueor a greenmarble
a cardotherthan10
afacecard(a king,queen,orjack)
The resultsof rollinga.six-sided
die 120timesare shown.Use tabletofindthe
experimentalprobabilityof eachevent.'Also-findne n6br"tiJJthe
p'totd6iri-iv.'
d*=
do the probabilities
corirpare?
Roll
1
2
3
4
5
6
# of times
15
18
20
17
24
26
(a)
rollinga 6
(b)
(c)
6.
rollingan oddnumber
rollinga numbergreaterthan
2
ptbabifity of winningthe lotteryaccordingto
-'- givenrutes.Assume
liEJ!:
r -- the
numbersareselectedat randbm
(a)
You-mustcoffectlyselect6 out of 51 numbers.Theorderof the numbersis
not important.
(b)
YoumustcgffectlVselect3 numbers,eachfrom0 to 9. The orderof
the
numbersis important.
7.
8.
Findthe indicatedprobability.StatewhetherAand B are mutuallyexctusive.
(a)
P(A)= 0.a; P(B)- .35; P(Aor B) = .5; P(A and B) = ---
(b)
P(A)= 13117;P(B)=
(c)
P(A) = 5%; P(B) - 29%; P(A or B) = -;
P(A) = e.34
(b)
P(A)= g/+
A card is randomlydrawnfroma standardS2-carddeck. Findthe probabilitypfthe
givenevent.
(a)
a queenanda heart
(b)
(c)
10.
P(A and B) = oo/o
FindP(A').
(a)
9.
; P(Aor Bl = 14t17;p(A and B) = 6117
a heartora diamond
a five and a six
Two six-sideddice are rolled.,Findthe'probability
of the givenevent.
(a)
the sumis not3
(b)
(c)
the sumis neither3 norT
thesumis greaterthan2
1 1 " An employqeo! a larggnational,cgmpany
is promoledto manasementand will be
th91eG-;dCt"piiuluiritv.otn";:q$H."lgg$l'".^T_"gppdye€iqlgtgthaf
movedro.
Gotorado,and a 507oprobability,
ofibeing'rnovedto
RgtTg
!?en-v.eL
L'allas,Texas. What
is the probabilitythat the e'rnployee
ivitlUem6veOto Dal6s or
Denver?
12.
35 lightgreenparakeets(14 femalesand 21 males)and 44 skv
L p"t storecontains-
(28femat6s
ino l6mates).Youianodty-;d;"
i-i 6r1",i
!l1g,ql3k".gF
parakeets.
\Athatis,theprobability
thatit ida femaleor a si<yntuepaiafeetZ
13.
4-tfe in'aforest!s qotgrowingproperfy.A botanistdetermines
thatthereis an
85{" qr.gpapilrtv
the_tree
nasa di#ase or i'sGing damigeoov ilisects,-^'+s%
probability
it hasa disease,anda soy"pro$biifi irE-6;fig a'd;g;ir bt insects.
probability
is-the
that.thetreebotlitraJi d6ei"i ano"iJ#indA;i"ged uy
Wh"t
insects?
14.
EventsA andB areindependent.
Findtheindicated
probability.
(a)
P(A)= 0.3; P(B)= o.9i P(A and B) =
(b)
P(A)=
; p(B) - 0.3; p(A and B) = 0.06
15.
16.
17.
EventsA and B are dependent.Findthe indicatedprobabirity.
(a)
P(A)= 0.1;P(BIA)= 0.8i P(Aand B) =
(b)
P(A)=
; P(BIA)= O.5; P{A and B) = 0.25
Threeftiendsaretaking_an
Englishclassthathasa summerreadinglisl Eachstudent
is requiredtg rgadonebookfromthe list,whichcontains3 biograihies,10classics,
and5 hlstoricalnovels.
(a)
thatthe firstfriendchoosesa biography,the second
flno !t,9 probability
friendchoosesa ciassic,andthe thirdfriendchoosesi nihciri,cal
novel.
(b)
Findthe probability
thatthethreefriendseachchoosea differentclassic.
Findtheprobability
of drawingthe givencardsfroma standardS2-carddeck(a)with
replacement
and (b)withoutreplacement.
(a)
a heart,thena diamond
(c)
(b)
a 2, thena facecard
an ace,hen a 2, thena 3
18.
Youareplayinga gamethatinvolvesdrawingthreenumbersfroma hat. Thereare
25 piecesof papernumbered1 to 25 in the liat. Eachnumberis replacedafterit is
drawn.Whatisthe probability
thateachnumberis greaterthan 2Otir lessthan4?
19.
Theownerof a one-manlawnmowingbusinessownsthreeoldand unreliableriding
mowe.rs.As longas oneof the threelswgrkingne can staypioOu&ive.-fi6mpastexperience,
one of the mowersis unusablel{percent of the time,one 6 perc6ntof
the time,andone 20o/"of the time. Findthe probability
that allthreemowdrs.are
unusableon a givenday.
20.
Thetable,basedon a GallupPoll,showsthe numberof votersfin 1000's)bv party
gffiliation
whowereexpectedto votefor BillClintonand BobDoiein the 1i)96'
Presideritial
election.
Democrat
21.
Republican
Independent
Clinton
31,379
3,340
12,685
Dole
2,092
29,396
8,721
(a)
Findthe probabilitythat a randomlyselectedpersonvotedfor Clinton.
(b)
Findthe probabilitythat a randomlyselectedDemocratvotedfor Clinton.
fn the UniledStatesduring1993-1994schoolyear,39.6o/o
of allmaleteachersand
26.1Y"of all femaleteachershadtwentyye.arsbrmoreof full{imeteicfring
Tlat ygar694,000males-aird1,867,000femateswereteiCfiers.
.e.Tpe.np!9e.
t!"
thata randomlychosentedc*rerin the UnitedStatesthatyear
pr9p3UjtttV
.Yf|i:
was a fematewlthtwentyyearsor moreof full-timeteachingexperience?
\
Pre-Calculus
ProbabilityReviewQuestions
l.
How manytwo-symbolcodewordscanbe formed if the first symbolis a letter and
secondsymbolis a digit?
2.
In how manywayscan5 peoplebe linedup?
3.
How manydifferent words(real or imaginary)canbe formedusingall the letters
in the word CALCULUS?
4.
In how.manyways cana committeeof 4 sfudentsbe formedfrom a pool of 7
students?
5.
You arebuying a flower anangement.The florist has12 typesof flowersand6
typesof vases.If you canaffordexactly3 typesof flowersandcanaffordonly I
type of vase,how many different arrangements
canyou buy?
6.
Fourhigh schoolfriendswill all be attendingthe sameuniversitynextyear.
Therene 14 dormitorieson campus.Find the probabilitythat at least2 friends
Will be in the samedorm?
l