{ri
s
PageI
,mc oH<ebry
Math 166 - Exam 1 Reyiew
NOTE: For reriews of the other sectionson Etam l, rctrerto thefrst page of WIR #I and #2.
Secrion 1 5 - Rules for hobabilitv
. El€mentary Rules for Probability - For any event E in a samplespace.t, we have
Rule l:
0<P(E)<1
Rule2:
Pl$:
Rule3:
PfO):0
I
. Union Rul€ for Probability - If E and F are any two everts of a samplespace,t, then
P(Er F): P(E)+ P(F) P(ErF).
. Union Rule for Mufualy Exclusive Ev€nts - If E and F are mutually exclusive events,then
P(Eu F) : P(E) + P(F).
. If Et, E2,... ,E^ arcmutuallyexclusive
events,
lhenP(tr U E2U ..UE,):P(E1)+P(E2)+.
.+P(E").
. Complem€nt Rule for Probability - If E is an event of an experiment and E denotesthe event that t does not
occur,lhenP(E') : t - P(E) andP(E) : I - P(E ).
. Odds - Odds are anotherway of representingthe probability of an event and are given as a mtio of the number of
times the event will occur to the numberof times the event will not occur in the long run. Mathematically, the odds
in frvor of an event E are define to be a ratio of P(E) to P(E'): odds in favor of E :
the ratio
A;.
NOTE: we reduce
PG\
!o lowestterms. andthenwe saythatthe oddsin favorof E te ato b or a: b.
6
;.
. Obtaining Probability ft,om Odds - ff the odds ir favor of an eventt occurring are d to ,, then the probability of
rsrlz ):
c occurTrns
a+b
Section1.6- ConditionalProbability
. Conditional Pmbability - the probability of an event occuting given that anothereventhas alrcady occurred.
. We denote'1he probability of the eventA given that the event B has already occuned" by P(A B) .
. Conditional Prob&bility of an Event - If A and B are eventsin an experinent and P(B) I 0, fien the conditional
P(A,n,A).
probabilitythatthe eventA will occurgiventhatthe eventI hasalreadyoccunedis PIA Al :
P(BI
. Independent Events - Two eventsA and B are independentif the outcomeof one does not affect the outcome of
the other
. If A andB areindependent
events,thenP(A a) : P(A) andP(8 A) : P(B).
. Ind€p€ndent Events Theorem - Let A and B be two events with P(A) > 0 and P(8) > 0. Then A and I are
if andonly it P(ArB): P(A)P(B).
independent
. NOTE:To determineif two eventsareindependent,
you mustcomputethe tlree FobabilitiesP(A nt), P(A), and
P(B) separuteLy,and then substitutelhese thfte numbers into the equation P(A n 8) : P(A)P('). ff the equality
holds,thenA and B are independent.If after substitutingyou 6nd that P(A n B) I P(A)P(B), thenA and , are
NC'I independent(i.e., A and B are dependentand somehowafiect each other).
Md rft Fi' 03
P^ge 2
Rdq
ox!,h.
.IndependenceofMoreThanT\yoEvents-ffEt,E2,...,E,areindependentevents,then
P(ErnEzn nE,): P(81)P(E2)P(E")
Seation1.7- Baves'Theorem
. Thereis a hugeformulain the bookfor Bayes'Theorem,but you don't haveto memorizeitl Justmakesureyou
knowtheconditionalprobabilityformulaP(,4ID): ffi,
andknowtro* to usea treediagram(or Venndiagram
probabiliries.
to
find
lhese
in somecases)
1. Consider
thepropositions
p: Bob will havea hamburger
for lunch.
4: Bob will havepizzafor lunch.
r: Fredwill havea hamburger
for lunch.
(a) write thepmpositionr A (p y 4) in wods.
Frad oi\\ h^\)c d, harn.tt.vgcr- k Luntt+,awl ?vb .:\l\ z)I\ar
a-M'abur1f.r 0r l\tu<, tx Lqn*,. (bv+ nsS-rrorl_1.
hl, ,<-
(b) Write the proposition (4V - p) A r in words.
r ht,,o'{t
Mrt ohtrnbu-rgt-ffti. lt^^c(,
bot,r't\ hr.u<prtu- kr lar.ttLt,0
v,4.'<a ta u.ls6rq(/ k*^01
L,a{,.kcd
tuun.
-\.r'BobandFredwitl bothhavea h)inburgerfor lunch.or Bobwillhavepiz,,ator lunch.'
tcr Wrile'tfidproposition
symbolically.
/
\
(PA()r'V
(d) Write theproposition"Bob will not havea hambugeror pizzafor lunch,but Fredwill havea hamburger
for
lunch,"symbolically.
I
- ( pvy ) Ar
2. write a truti Lableforeachot tie lollowing.
( a)
I
r
{
3
- ( - p vq)v(p^ -q)
tg
:t
_f vt
T
FF
T
F
F1
F
T
F T
f{-
F r
(t
{
"Oev,h)
F
T
F
tr
pn^,b
F
{
F
(
(p/l-r.)
-(_gvt) V
l-
T
tr
F
Mdi r# d' 43
otue.
(b)
p)
- r^ (4v -
v-P
k
(
7 T
T
T
TFF
FgT
T
\-
{
F
Itse
rkf
T
fT€
F
r
Page3
Rliey
T
(
F
T
r
F
tT.
'r'
FrT
TTF
F{r
F
F
-1-
F
-r
-l-
Pmbl€ms3,4 6, andE are courtesyofJoe Kahlig.
Trueor False.U : {0, I ,2,3,4,5,6,7,8, 9a)n d A: { 0 , 1 , 2, 3 , 4 , 5 1
r6
r (O
oea
( T) F
T \F )
I1.2.31cA
2CA
=s
nr.q.r
;" ,P ;t ': '*ee
Y@
tot=o
*'A
n({3,4})= 2
n(o): t
T (F ,' 3eAc
.r l;1 0:0
- \'-./
A= I a , b , cj
(a) Listall subsets
ofA.
Lc3,
I",cJ,
fi, 1^3,18,
fu,,v,c\
La,L&,
1b,cJ,
(b) List all ofthe propersubsetsofA.
ili
\4t
at tl' a+u(( 1a,b,4
(c) Giveanexampleof two subsetsof A thataredisjoift. If this is not possible,thenexplainwhy.
|
' 'n
z )
,1t7
t rl
fu\ a^a lD\
a'^!d t5)D\^?1t^u t^3 (\ tbl = FJ
5. Shadethe part of the Venn diagram that is rcpresentedby
(a) (^cu8)n (cuA)
{
^&r14*
it:',if,J"-@ia
Page4
Mri 16 h[ 1M Osedr tu&y
r,(-
(b) (, uc)nAc
l 1 - l,a,J't
.
,,i,r.. -\
'
'' ' '
^
^^
bvc-ttlW)2to
,G;i;;;|,
A"-GW(ut^
E
^.6-
6.U :lo ,t,2,3,4,5,6,7,8,9j,A :{r,3,5, 7 , 9 } , B : { 1 , 2 , 4 , 7 , 8 } , a n d C: { 2 , 4 , 6 , 8 } . Co mp u t e t h e f o l l o w i n s .
(a) (AnB)uC
av--lt,l\
(eno)vc --f t,t,z,+,a,t\
@)AcnB
A,.10,L,4,t,,5J
A'to --{t,,1,lJ
(c)An (BuC)c
g,tc.lto,+,'t,t,6\
(0,t4"
,10,2,t ,4J
ftn (&uc)'-1t,s,n!
7. l.€t u bethe setof all A&M students.Ilt
A : {r € U r ownsanautomobite}
D: {_x€ Ulr livesin a dormoncampus}
F : tr € Ulr is a fteshmanl
(a) Descdbethe s€t(A n D) u 1.' in words.
,I"!zf
altle
qaJ.Ilttll:ht-lu+s vrX,o WlaOvl a""rarl*frrtobiie
\;.^dJh^ 6r.-\.AA.PtL5)
or u.'Loa.^e {\0+fn*Lwn
(b) Usesetnotation(n, U, ") to $.ritethesetof all A&M studentswho arefreshmenliving on campusin a dorm
but do not ownanautomobile,
FAD OAC
Mzr rm Fjr ,4
@Hdb
Page5
bry
8. In a survey of 300 high school senio$:
I2O h^d not rcad Macbeth bfi had rcadAs you Likt It or Romeoand Juliet.
6l h^d rcAdAs you Like lt b^tt not Romeoond Juliet.
15 h^d rcAAMacbeth al:,dAs you Like It.
14h^d rcAAAs YouLike It andRomeoand Juliet.
t had rcaAMacbeth andRomeoa.ndJuliet.
5 had readMocbeth andRomeoand Juliet t'/ut not As you Like It.
40 had read only Macbeth.
A= As you Like It.
Ie|M = Macbeth, R = Romeoand Juliet,
^nd
(a) Fill in a Venn diagam illustrating the aboveinformation.
l1-o-5-D-t o
(b) How many studentsread exactly one of thesebooks?
4ut too,.5n=16l
(c) How many studentsdid i$ rcadRomeoa.ndluliet?
4o{-il rs-o+ ao =F4
(d) How many srudentsreadMacbelh or As you LiAe lt and also rcadRoneo and Juliet'!
5++f t0
=lo
(e)computen(MU(AtrnA)):
\\+ 50 f 40 k 4 | 5'nA
Rt- (,u,",r.'(
(cAA
ft - allo,d'eJ
- l,e
f ,!'4 ,c )
^ ^-
1,
V\ U( KAA) - b, "-, f 3, c
h6
Page6
G6 tun'Oklkridry
9. Findr(A n B) ifn(a) :8, r(a) :9, andn(AuB): 14.
n(AuB)--n(A)+ n( 6) - ^t A \g)
t4 -- t +1 l3 = n/Ano)l ^ (A^ a\
10. A bag containsI pennies,a nickel. and two dimes. Two coins are selectedat random{iom the bag and the monetary
valueof the coins(in cents)is recoded.
(a) wlat is thesamplespaceofthis experiment?
1.
a
a^ '- l& t t ot lltlt , ln7
'J
(P -/
?\ .r"
P\ ,'
dd. ',."
O) Write the eventt lhat the monetary value of the coins is lessthan I I cents.
1^ ,n
Y'-=Ld'\!J
(c) wlite theeventf thatthenickel is drawn.
t4
€ =1 ln . 15J 1
(d) Are theeventst andF mutuallyexclusive?Supportyout answer.
Sinus
E Otr,lt"\ *Q, Eand € a,u"vw+{^}lua%.
J(
-!4-$.U.14\
(e) Wdte the event G that the value of the coins is more than 25 cents.
/-Z 7
I--
4
!) '-1
7
\
-
-
l
r-t\
v-)
I
I 1. If p is a tru€ statementand q is a false statement,find the truth value of each of the following statements:
tat - n !-q\
T
(b)pv-s
IVT
..
@
( c ) - ( a ^ -P )
^ {r'nr)
= - Ltr\
hvu
br6Fd@3oq.rbrry
PageT
12. La S: {a,b,c,4e, f} be the samplespaceof ao experimenrwith the following Fobability distribution:
ouko"
b - ,!
l!
,t- .t9 *
n
I-
!- :
*o"o*oToo1
* m(;; ,ao# e- [ -7o'
r*t t - {o.c.e\.s - {c.././}.andc
t
of$e ex""..;;
[b.dI beevents
a 4 z -lo?
r * ^ - *1
J,n;t$ ;,Xi
(a) Fill in tbe missingprobabilitiesin theFobability distributionabove.
(b) Is this a uniformsamplespace?Wlry or why not?
,
srr^e rt %iljll !'k I"J.
1Jo.g*rporrra
3
^
{cl Fiodeachot lhefollowing:
i. p(A)= ?(a>+?G)
lp(O--
2
1
,
f;l
=l 6
|
,io
-qo {o
ii P(c)-_ftL)+ pU)
iii. P(rc)
'k'k &
= l - ? ( b )= l -
2! - e --EI
40
-
50
L-t-J
iv. P(An B)
A^blq ((nno=
k
4q
lu
v.P(Arl--+oI +or 4 e + iu'
I,
q"
,J9
lln
\ut--lac,gdg!
(d) Are theeventsA andC mutuallyexclusive?Why or why not?
tt.
Anc=?c]+V A rvirt",.,Lvurl\.e4a!.u;
\
Page8
M4r 6 a n r @ o H & b 4
13. One card is drawn at mndom from a standarddeck of 52 playing cards. What is the probability that the card is
is aV.*b
(a)acrub?
C-.uftt ,l,ttt* .11^s-c4rd
Pru>-{$
fiZ-l
- ls al
(b) a face card?
il,etavdts altce rcrA
F-arttr,t'LLat
fttr)=ffi =
(c) a club or a face card?
- P{o^F)_P(Lu F) : elu)+PlF\
1 - l_- 4 1
- vs>
t,z. - 5)
_
5n
L5a I
(d) neither a club nor a face card?
= |-g(c ur=)
f( c-"nr')'. p((cur).)
-t-U
-6
14.LetE andF betwoevents
ofanexperiment
s*,Inu'",u, 3Jrr.
",FPo4.
(a) AreE andF mutually
exclusive
events?
Pf E n tr)
fttJab $
^t
?Lev")--P(€)+P(tr)-f{Entr)
- - \ A s + \ , 1 - ? te n r>
A , \ pt
D,g''e(e^F)
(b) FindP(t'uF
andp(t ur) :0.62.
:i*crP(E0O)0, i| i1 @ssible
'&r lLP:+ +t"rOeuent+ *O acour a.,
1l^aaa*qn//.j'. 'ffure{o.rtL a-J
).
F a"l
P[L.uF.),Pl€.nO")
tot
vral lv,a-U.l.ral&Luautt .
U
=l- fllFAF); | _a,lz5
,iO,{tr
(c) Find P(EiF
).
p(rur"):
(d)Find
0,K+ 0 ,L1-l 0,\V
Y-v'.
Y"'o'b
f -U Fc-a' b, c
'aTq
0A-o.tz
\z
0,45'a
-0,?-7
l'0.la - 0.13
M
16 an ,@3 oHc{Ei
Page9
R2ruy
15. A surveywasconductedin which 1,000studentsat RandomUniversitywere askedhow manyhoursthey are
currentlytaking.Theresultsarc givenin thelablebelow Usethetableto answerthe following questionsabouta
randomly selectedstudentwho participated in this survey
F- freshl'u,,r
y or less
Classification
Freshman
l0
Sophomore
l5
Junior
25
Senior
!50
Total
100
fl- 6o0rh-osto.{os
J - Ju{idv't
N - 5{rr,6v5
l0 to 13 14to 17 18 or more Total
130
140
5
285
90
55
20
180
105
80
35
245
ilo
I{J
45
290
435
360
105
1000
(a) What is the probability that the stualentis a freshmanflre or sheis registeredfor 18 or mote hours?
lL.rt'-
?(r\u)=
5'
k.4t',l\^
a' Ptr*| t+r rt\<kre 4 ftk lg on
",.ryf iur{,(j.
'
l, d J
t,'1. ' . ' . , -if ir-J
J
(b) Wlat is theprobability
thata@!fs
' , t l\ , > , n t , o
'
ttv,s
registered
for 14-l?hours?
E'-{-'"s
0/
|\ n
\ _ ll rf '!./\ \ -- A a O < ,
r Qg st' r t' t h t t4 t' tivs
x.n \ {'l5
(c)whatistheprobability
thar
thestudenr
isajun;"ri:,:
li .l"['j"_,iltr"Hx,:
i?: ,:i."t
o / - < 1,.
=
f t J IAU bi=- 6 r +W r1p6'16-ffi-a
l-rA
4e rycc
16. A studenthas two exams in one day. The Fobabiliry that he passesthe first exam is 0.9, and the probability that
he pass€sthe secondexam,is 0.85. If the probability that the studentpassesat least one of the two exams is 0.97.
p"'iir"nt?
*gQ"i9)*o
"u"ot(6
fr- p"cgs lsvu*'
q-pds.".t
O,-/ t t7t"a-
(eiL
Y(AnDt?(A\ftb>
o,1si to4)(o'st)
((Avl)'t qt
p([oh\. p(A)+plb)-P(AAb)
no.frrvl".qrr'd.lrLt
0,4t'04r):16- wA n b )
cl'L i (lnsl
M,h re FdrM3 oHdkr
Page10
Rqe
l'7. I Et A, B, and C be thle€ independent
eventsof an experimenrwith P(A) : 0.4, P(A) : 0.75, and P(C) : 0.3.
Calculateeach of the following.
@)P(Anf)-.'PfA) plec)
(.'.',az lxlaywda*)
(0,,1)
I r-urr)
ic
ptA,)+p{c)- ftA,nc1
?(qcyc)=
(c) P(Blc)
+ 0,1-(t-0,+)(e.31
=]F?-i.t
'(r-u..r)
l(b\c)= {lb]l
=EA
[iru r*ur-.p)
18. The personnelmanagerat a certain companyclaims that she appmvesqualified applicantsfor a cedain job 85% of
the time; shercjects an unqualified person 80% of the time. If 70% of all applicantsfor this job are qualified, find
each ofthe following.
(B iu,dn tr'derydbrb.c"-wn.dridq)
(a) Draw a t ee diagram (with probabilities and notation on all branches)representingthe aboveinformation.
afPt'rot v
Q- n'rt,{.,r,,1
+ ;"
ff - opp;cn^
trbpr*r
*Os+@*
P(A-iq=iDE_
kL
** * *i'/-r
rbr whatistheprobabil,o
1;{oort"l*t
-t-------
(o,r)(o
tt) + (o.e)(o,t-)=l---uqf
Pa)=
J
(c) Whatis theprobabilitythatanapplicantis apnrovedfor thejo(i{re or sheis unqualified?
P(At0")
-tg!l
rmk.,)
(d) Whatis theprobabilitythat anapplicantis unqualified{i he or sheis approvedfor thejob?
?g!DP/
, ! e n. lR)
t./"
p{A)
tt - n 09lb
l?\
-"--
!
v
M'r r6F!'4soHrebq
Page 11
19. A ?th gade classwas selectedand the following information was collected about the 30 students.
4 studentshaveonly glasses.
haveonly braces12 snrdents
6 studentshaveglassesandbraces.
t@'".
Det€rmine whether the event that the st!!d9!Lba!gla!!e! and the event that the studenthasbracesare independent.
youranswer
Justify
G- zkuAp aj
|,t.. *as:>f,t
g-
"ta{a(a\
&ft-,
.2 ,
e(Gl'6)?,flcifiP{
f i =-",,'
-to 'lZ6X6)
G 6^l B a^t rnltpt'tdl*4
'-/
20. The oddsthat il_udl not snowin CollegeStationnext winterare 17to 2. Wlat is theprobabilitythat it will snow
dds
11 to ? 'd4 a"
t t. ,. , 11 no +
11 t-t'tq'
in College Station next winter?
7\a-o,i"or e.o"-i!3 +i,wt
,t
P{.;lltoru) -
,t
til. :,.,
'-\t-|..j.
21. An experiment is as follows: One marble is drawn from a bag containing 3 red, 7 blue, and I green marble. The
color of the marble selectedis observed.If the marble is red, then a fah coin is flipped and the side landing up is
observed.If the mmble is green,then a secondmarble is drawn ftom lhe bag and the color is observed.
(a) what is thesample
space
for thisexperiment? Q-td
Vnlblt,
P:ffi^Yi!h'"
K-l
p (a,n!
3.f{x,*1,LR,i)
b)(G/R)r
nr,t
marbler( bluel
Or WnulRU. pro#Ultlrylhatthe6rstmarblersgreenandthesecond
/\,
r
.- ,1\
V(qru,nl*
nt ;p(Vtur
anAH^ a* ) = p(qw
4ilil3,*n'u..
-/
+\l+\
-E\
(c)wrire
*. *ol"l"!?J
lheevenr
rhar
ar,* .-t.
"I "1"."'.1oii"i
Y'--7,b,
''^
')
Md 16 Fd[M
OHdio
Page12
hq
22. A bagcontains5 pennies,3nickels,and 7 dimes.A pursecontains4 nickelsand 6 dimes.A coin is drawnfrom
the bag and dansfened to the purse, but if a nickel is s€lectedfrom the bag, then all 3 of the nickels that were in
the bag are transferredto the purse. A coin is then drawn from the purse. The type of coin drawn from each of the
bagandpurseis recorded.Wlat is the probabililyrhct
tro,
p, ,
1rt" _y k*n
(a) thetransfen€d
coir wasadimeif a nickelwasselected
fromthepurse?
@
-
n,
f)
Dr
- o
' ,..t
't
,,
"
|
',
U-|
f, . ycxr,' t'r\". {}tt.t;t
Ny . ;.ri"t)
D1 . .,1, nt
*
oi3**'"
dPlu'1t'to)'
'W
,nl, 'i--i ?t ,' Q::z
4N
_n *=:@l1o.+
S"<|f,y)' r U,
'fr
_
_?o.
r,\
Nr- t\i Ciel
ffi*,
n rrn\-+
'
j-
vz )
,E
l' D
( 1. \ ( t \
'- - -==:;?4rL1--(7.)(fr)r(L*)(1)
.,ri1,
N!
s.j=
-. ,,.--,
?
'4).->-.-o
tD, Artr\?)'-ifll:ry rt-
- ' fi;;;.-_=--;
''-21Dr)_
fr- "2,
;f,a D r lD
44"\:+
:
)B
tk{
-A*O
o) bothcoinsarepennies?
--
ffi t
IZJ.^.tt"1
t'
LEZ)
f1p,nP1)'
&)(!'\=6
(c) the coin drawn ftom the bag was a nickel or the coin drawn ftom the purse was a dime?
p(t't,ulr;.
D))
P1N)r p(D,\ - P1tu,n
-"
-i;,,\
'-lB
= g +[,{")(+;)r{il(it)r(aXpl
-(?)(?")
(d) thecoin drawnfrom thenurseis a Wnnf if)he coin drawnfrom thebagwasa nickel?
P(l,"lN).g
(ilutU A.{ r1t
i'^'L pu roe)
Pr,!r'r,ftr,r.
23- Two cards are drawn at random fiom a standarddeck of 52 playing cards. What are the odds that the secondcard
drawn is a khg given that the first card drawn was a queen?
lroht
h ; t ; 4 a=
rJ=a
4
)l
tl
ilds 5-
\-t,
,
.l
-4 1
rl-1
Uj:t-,
^
'' ' '
Page13
M'h 16 Fdr03 oudrd rMq
24. Thre€ rnarblesare drawn one at a time without rcplacementfrom a bag containing I I red and 9 blue marbles.
(a) Draw a treediagramwith probabilitiesonall bmnchesfor this €xperiment. Ki
(,.
Lhc
( ti. Wu wtb i , t1a;tt,
i' - 1,7,1
*...i:
61
- f?d w.a rLl( t$
#?i
n
!-
-z--=
rE
is ltue? $
O) Whatis theprobabilirythatthethird marblesetectea
p&it=P,)P)( ) r (),,)(ft
)(&) +(*,)(:) (*) . i*,Xft
Xr -l
(c) Wlat is the probability that th€ third marble selectedis blue if it is known lhat at least one of the first two
marbles\€lecledwas Ied?
r'ol
?U>z-Aa'u'
'r dJtt-1t
(N r?d ,^tt
't
'
Q/A-l.,,00,.{
1d,0.)\) I(,JI{
\t''!
i'"e
'
Ybr
'L'c'it
^1}
'}dta")')
L-r(*)(3<>
'"d't*"t
,[)GlrkX%)(a)(h)
"
=']# =l*\
25. A buildhg on campushasthree vending machhis:'t'ivb Coke ifihines xnd a $ack machine-Basedon the model
of the machines,the nrst Coke machinehas a 12% chanceof breaking down in a particular we€k, and the second
Coke machinehas a 4% chanceof breaking down in a particular we€k. The snackmachine has a 10% chanceof
breaking down in a particular week. Assuming independence,find the probability that exactly one machinebreaks
down
i"t t+ (iltc {Aa(n,toiluyS
A
'Jrr,
r0r,r*,
L lvr "!
U\2 ,'
fr
r,
" a.v\a
r'
.'
ni
t\
J.d-lor\
' '
^t t LW [ h, u
lr*) - F(t,
n cr' rln')+ p(c:Ac,
nr'.r")
+P10lnc,'n
\ Wtnlrs
(1turll1
r
Cirn rvt,rii!'i-i,'40ffrwlit
.-1
-'.^V
fie
a^)tt{tt
(q12)(r
-o,ut
lo
ri|-u,\+ (r-olr)0s[t-u,)* { t- o,rQ[r-o,o
-
-:--,--------'7
Ul\sH-'-