PageI
vd16ljra0.o|!&riq
Mrrh 166- Ilnd Etrn Revlew
4.3,44, 5.1,5.2,M.l-M.3,R l-F.4,G.1,and
L.l, L.2, l,t-1.7,2.1-2.4,3.1-3.4,
ThcfollowiogreviewcorcrsSectiotrs
C.2 andcontainsaboutI Foblem (or I part of a Foblcm) Fr scction. Sime manysectionscontainmultiplc topicg
probldns,this reviewshouldnot be uscdasyour solesouceof practic€prcblemsas
thst canbe addr€$€dh separate
in lccturc,problcmson thc cxamoand
you shrdyfor thc final exam.Youshould,lso revicwall cxrmplesdemon8ratcd
honcwor* ptoblciE from tour lcrtbook, thc Problcmsfrom thc
quizzcayou havct kcn this s€m€stli,thc ouEgcstod
onlinchorneworts,andproblcrnsfrom prcviourwcck-itHrvic{s.
for I ycar.Pastrccor& indicateth$
drivcruaslo$,-risktf thcy dlesccidcnr-frca
l An insumncrcompanycl8ssinca
lhc ocrt ycst, 8nd?8% of thc
98%ol lhe drivarsin thc low-riskcategoryonc ycsr will rcm{in in that catagoay
dd\€$ who a$ not in the low-dskcatcgoryorc ycsr will rmve lo lhc low-riskcstcgorythc ftxt ycar.(F.lr.dtu
dfulfu,b'lr&a{h.d.tb)
(a) If 90% of $c driveNin drc cornmunityo& in thc low-riskcs&gorydrisycar,whotb thc probabilitydroro
drivq cho6€nat randomfrom thc comlnunitywi bc in thc low-riskcatcgorytq,oyaarsftorn now?
L:-lJt)l |t ^ ' rt b { r'
X.
L'.!l'.'ttq<I t)./tlt
I
tc
r
/ t_
L I ,SS ,1xl
I -,".,
I
I
N q r t l , o A .? .*I
LJ
', /o =
L:i;
r.qn1
-'
Xr'T'X, /.rar1
11.412'
O) If this t$nd continu.s,whst pdccnt of tlr€drivcrsin thc comnunitywill bc in $c lolP{irk catcgoryin thc
longrun?
inul r \\ \P(l?r.
t|ud'X-;+tu+edith
vX,u
re
X=[[]
T X,X u,v'r.c
l. 4 ,tt' ]r " ,' '
L'iaa"]e)? ) ',
. q x+
\ .tx\: I
, ) ] I | 'raJ.
1
I.r
- o'oeiI01tt)= o
a'oa'Y-o:1(5=o
q1,s70
*F
\ 9:T,1'J,1,
fi'u, ll1l3.,::;
Ii-
fl
inthcca'k*
2. PedoIm thc ncxt two ro* operations
l'l
0
2
lo 6) a
I oy'--ct
tl\4k4c I
il
a
LDoI u J
JhlJ,
";.u*J'
|
I
al
"t
ll, ili lt'ovq*'"$'
1'or q
-
^l-
t1-t1
lo'+
v^t4
W\.LLL^u
lo
I
1l
03 q n)
Page2
M16turrao|l4kry
wasconductedto oe€\r'ttichof thefotlowingthftc countaies
3. A sunry ol $onrccollcgcstrrdents
thcy hadvisit€d:
Mexico,Csnad4andEngland.It wasfoundthat
Ir-+r"rr.r..,ril
:frf::l#
]|:lff
llT#',JI
**'.""
q{h05c ,u"qfl 44 snrdenrs
hadvisiredcxrrty t of rhcse3 counnies.
wfi. hadVitrt{
f\Ulico.
3?
53
126
l6E
,16
shrdd'rshadvisiicdonly McxicoatrdEngland.
st|rdcntshadvisitedMcxicoandCsnad{.
stud.ntshadvtuiledEngllnd,
studcnbhd visiEd Canadao! Bngland
studcnbhadvisitcdncithcrMcxiconor England.
(a) How manysodcntswcrcsurvcyad?
l ot a4+ ljt 41+ e4+ + + f* l
=Fil
+aa
O) EunAtly Btud.nbsurvcycdhodvisibd Canada8!d Englsnd?
.!il+44-- [,n
r* j' l
(c) Erylolly
sudentssulvcycdhadviritcd Mcxicoor c.n.da butnot borh?
+ tz r44=@
ro +31
(d) Whatis he probabilitydut r Etrdomly6ehctd studrntfiom thiscurvryh.d b.Grto noncof ficr. lhrc.
counEid? :
r--VT1
J'
I
l- t
I
I
[/ ll
(c) Whatb dtc probabilitythat a randony scloctrdENdcntf,on this surveyhadnot visitcd&Ulad but h.d
b.cn ro at lcarioiE of thcothd two countrics?
lo+ )q + t5
+tl
Cla$ify cacbof thc follo\trirg tyF6 of raddomvuirblcs sscithcr filitc discrrtc,infini& discrc(e,or cot inu,u8.
oodsta&thcpossibtcvrlucsol thcraodomvariabla.
(r) X = ihc numbarofblack cntsbring bodrdadat a kcmcl thatcsnhourcat mo6t45 aaimals.
X- 0 ,1 ,? ,...)4';
(b) y = thc lumbcr of gallod of wat.tuatrlck
NL !L<
Itn;+. d, sLvt:t<
t thai hasa msrifitrm copacityof 5 gdlonr.
Anth,'tutu*
(c) Z = dle lumbcr of timc6 ttly phonerings beforc I answcrit,
7-- O,l,C,..
Q6utz7<l,.r&",;1a-
u-
Pagc3
Md16Ard@I!*rry
5. Aflelia w!nt! !o saveenoughmoneysothat shc*'ill have$3,000to sp€ndon a t ip to Europethatshcis pleqning
to takein 5 yeals.If sl& opcmsn accountpaying6% intattstcompourdcdmonthlywith $400ard makesrnonthly
dcpositsfor 5 ycrrs, wlut is thosizcof themodbly paynctrtthatwil rcachhggg ?
Pfi1r--? ---->
NJ=ta*s
FV -- 30@
I\o-- lP
(/Y'-elY-17
(,,t, -4oo
lo fiodoutif lhcyconsidar
thcmsclvca
to bc! Rcpublicsn,
pcoplcarcrandornly
survcycd
s€vcrat
6. In ancrpcriment,
pcrson's
is
along
with
his
or
h.i
s.i'
rccordcd.
Bach
&spons€'
or rndcpcndctrt.
Dcmocmt,
R- RqoL^lr(ic4+\
(a) Writeanrpgopria&samplc
spsccfor thiscspcrinrcnt.
i - Dp.'vAeVA-+
5--i((,rn),(K,F)1(o,A),tb,F),(t,m),(rf)J ?.f^*n1^**
(b) writ! thcovctrrth.t tlrcF,eonsurvcy.dis femdc
L'-1 ( &,r),to,rl,t r,t{
li1 ^ mhh
g'- s<'n^1"
(c) writc th..\,cntih.t thcpcroon
considcnhimt lf or h.rs.lfto bcanIndcpcdrdt.
survcyod
t^t, (-?- l g { fi \t ( T ,F)J
(d) tu€ thc cvcnefoundin (b) 6nd(c) mutu.lly exclusive?
ca+
Mot \ttt\wa!)^t u&Lulolt( 1i''aa- Ee"\d G
hepq*
UME h^ ei € AG = {rc.,c>\
L+
'+l^!
propositions
tha
7. Conside!
p: Sofircof theprEs€nts
alt wrappcd
gucsb
havc
8riv.d.
All
of
lhc
{:
n Thefoodh mt rsdy.
Writa cymbolicsllythc stdenrcnt,"All of tha gucsbhavcsrrivcd,hot thr food ir not r€adyandnoncof thc
prcscntsarl lvrappcd."
/
\
i t-,'
/\v'/t'w?)
%
^
Writr synbolicatlytllc statcmd{,'EilJrcrthc foodir lcadyor all of thegucatshavaanivEd,hn rct bo6."
^ 'c Y ?
(c) ftitc drcrtatqrEnt (* {v.) n p in Englisb.
(
W aA q.t4.^tqwstsWN a({ |rtl Dr +!^t 6oolit fi 0+ u d! ) bu*
01-r
94+ ptl*fnts are o4oo42d.
. .ar"^L
(d)
WriE dE sratednt (rv pf in English.
-
, . (""Vp) -- 't rA-p
a'^tU* l4 A^u^4#i.l'n"*qil^tyu'xrv1=
"*id
@ri.rnl
hgc 4
@|-b*ry
E. Th€ Capitalfto crrdit arxdconp@y hasfoundthat 23%of its cardholdcnaIElatFin naking tbcir monthly
paymrnts.Findthepmbabilitythatamotlg50rsndornlyselcctcdcardholdc$,at lcastl0 botfcwcrlhan30 arclate
monrh.
in maki4 drcirpsfncnlin a psrocutar
t.
A.
.'i..! (.rts " - [a*< i-,a4'"4
X , * fl,ic. e ezsscS
(n,.vn)',
?(rctX < 5o)=bi'm,/{
I c"4r1^1^r
5.j . . lo ,
bino-ncl((Qt'x"1)
4. 1lt'P{
0,1'!Vt"
Frcd,Bob,oid SucaI! all aticnswilh cxtrlrnelyIonglifc cxp.ctancics.Bob andSuo'8conbircd !8. in }!€rs is
onlyhslf tl|srofFrcd's,alldBobhasbccnslivcfof tbtlc tim€ss! m{!y ycsrsasSuc.If thcthftr fricrds' cornbincd
ffiT-ji,XT#iiffi^iT'qva.[-t^'l'rpa]n{* ylzlll
il {;*r.,"9.t\*r:.'.,[ff
t[: i-il3J-l:;Itiy|
L . +t^,-alLtL;luzr)',
I
tt
=5a+
t+
' ' ut+z
x+g+t- so+)
'
lhtrl.
"", t - yl --------iy.r"5rt
' o
'
5' ",1.it
,
ttL, btt'
tt bSlo4tnr.s
\tsltuu4-t
s{/., ''-'4 a'"e
dl:--,
\ ii ,te'b-,t
i-u"'
10. Sl.f,pcopL g.t tnlo an clevalorthat ocwicls 12 noon i-oa ccrtainhrilding. What ir thc probabtlitythat ftcsc 6
Foplc cxi! thc elcaatorvith at lc&st2 peoplc€xitingon tbc rsmcfloof?
ooY
-
td""L
L- er*r"a \{,o-t a.r' -rras{tuto a"{(\ 6'r.r+1
l(L\, t-!(e)
8'
*diff;1lb'"'
tbqtr1"0"7aaa(i+
ec--t''txnt
d^-0.^t
\ft+3 |
r t( ( ,) - 1 _ t lr'! '9'L'1 = l- . llag. \ =V7^ * J
- 6 a t\ ' I
13w$')l'r''t *D '
: ,l-
ll. Constructthetruthtablcfot N (pA - q)Y p.
P
to '-%
T
T
f
F
v
f
Y
F
F
(
p A-
F
T
"" (l A.vt6)
1-
r
\-
Y
I
f
(
T
A.k
I
T
r
T
No{.i&is ;z a-Iau*duSg
Page5
'6ft6tum@lrier,@
12. Bobjust boughrr houscfor $200,000.Hc madca 15%do|ld paymentrnd financedthc balanccwith a loan that
monthly.
compounded
hssaninteEstrateof 5.45%/ycar
(a) ff lhc loanis amonizodfor 25yc.rs, whatis Bob'srDodhlypayrncnt?
Pn? l, --z
$ loa.i,tt
flo-* 5,t15
Fr/= o
0u.t10wt
vl\--e.l\-,t1-,
(b) Aftcr l0 ycaNof polment!,howmuch.quity hosBob.atncdon his housc?
N. WPs
: U*QA. 6l'kl'rr.
€'ruitr
u u
-!&ro\
| . =7:liil-t
-
^AL'I
Nr 6|,ll fttL:
; 215e8- ' 3- l
N-- l1)rro Plnl-=- roat,tt
11"--q4'@=9
fv-t'tooa/?t\--otv
=o
I I L 1 rst,3-7
-\
loSz.az+la* as - l-'loooo
i 141
,uu,l
I 3. At I cortlin univcrsig,65%of drcahdlnts arclib€rd .rts najols, ard 35%st€ rcienccmtjors. Fifty-twopcrccnt
of thc libaralsrfi mqjorsand47%of thc tcianccmajorsa$ fcmslc.
(.) Whattudlc probrbilitythat. nndor y s.lcct€dstudcntat thisunivcrsityis I rci.ncc majorifit is hon'n that
shcisfcmala?
L- U'wtt! l,'t ts w!'r
n
v - 4 |.ak-
L- '>nI L't U vr^Jot
'.K:,ffi
lq _ ln J4
0/t,At\
[r(clF)='wpg\
-- - h'os\(o'"t)
t(o,rt;)0.*'tl
{o,U6Yus})
0.iA14
o(
lDot
(b) Whrt is drcprobabilitythata randony schct d studct iBmslc anda libc.rlort' m.jof?
--IDJlr \
AL)=(.r,5y,40
PUl1
(c) At! dracvcntsof bcinga scienccrujor andbcidgfctDalcind€pcndcnt?
Why or why not?
{a3t $(
aft.nalq0?G\
Pr> 1oud1s>)+/o'3txo4?)
(o.as)(o.soas)
L,c)('q1):
Uut i
o .r v #il 0' 115 q
-- u.\ 07Y
Page6
xd6drm@|&drq
14. tf ,{ is a 4 x 4 matsix,I is a 2 x 4 rnatix, C ie a 4 x 5 mstrir, andD k a 5 x 4 matsbl whichof lh. following
arl possiblc?If lhe operationIS pos6iblc,givc dresizcof dlc r€sultin8nratdx.If th€opcrationis NOT possible,
explai! why.
A ;> A*'l
^*Ll
6\ Arc+D
* s1At S.,-rQ.(
-,
f;LlLz
k-c
+-9_.
<:4-€> (€4_
(c) cD - 5A
* 5A
3!.
4aq
4..+
a.dd-watrius-.
Wwvts+
ni^taY W )l,a- 4&rvrL hral',
L T - 5 Ai 5 4 \ +
5t n 7ta ,./
(d)rl(r.)r
Typo:
A'(ctDr
ora,4Ar g,*-@dt
{Y{
t-!=:
r'
[(co)' i5 4\+
4 tr.l
J_f
15. Niko has? *hit , 3 bl..k, .nd 8 bluc shilts.Hc nndon y s.L.1s 6 shir$ to takpwith him o &bolinc,lsEip. lrt
X &prarcDtdlc numbcrof bl6ckshinr rclcctld.
(.) Writc rhcptobdblutydistritotiotrfor x.
U
^l^!-
p {X 'x ) c1l5&)_
ct16,b)
(<
io{
Ct lt'lr)
t
3!
ttg
Ltt,7)cttr4jur,t)atE7)
C(r(u)
f
6
) ot
(b) Howmlny bhct shirc cln hc cxFct to scl.ct?
+ ll#) +al*\ r'rl&\
o(F..)
E(x)=
'
Pagc7
tur66Md@rlr*brlq
16. nle od& in favorofthc ev€ntA are2;3. Theoddsin hvor ofthe cventI are4:3. IfA {trdI srcmutudly erclusir€
----evcn6,fnd theprobabilitythatA or, occurs.
-
eA^a)=o
?(Aug)=?(A\+?r0)-Pl*nb\
= fto+ h2
--%r) =l*,1
17. Sevcralsnack-siradbaSsol M&M's l,crE exarnincdto dclami4c dlc numbc.of M&M's in aachpnckagr,'Illc
--rcsul$ofthc sody arEgivcnin thc tnblcbctow.
X-
Le ---.t nunbcrofpackrgcs| 5
o
4
2
numbcf
ofM&M's I l0 13 ll
fu*-Xinto L1 ----a l=
7
9 15
(a) Wh6tshouldtharandomvarisbloX rggr€scnt,
numbcrof p&kagcsor numborof M&M's?
X---wnrtxa\/Yl&fnb
G)' Gi\rccxchof thc followingfor thisdat! 8ct.Roundto 4 dacinalplacarvhcrl nc€cssrry,
,. prxt=-Zjotslt
ii. variancc= 5tr0l
'- lV,osrb
(Mts. Rrmrcy'scl&ss:Is thir a sampleor a populadon?)
f
iii. modc=--tfrv. fil.(lltn
=
d lY','' 6D qw'lL
,t
__tL
x- 2y+a= - 3
2x+ Y- 22= 2
- U=
- 2.x+ 4y
It. Solvcthc followingsystcmofcquatiomi
(r -L r t-71
It r-tlel
v L 1- 1 - l v )
f I
. i.^"n=si--to,l-:6e
6
*
(vi -vrlvsl
lo{t) -1/f I'tt I
to " o o loJ
d"*t-v,w&r.^s'tt"3-t
l , -2t
vo
-L
'5
x--4t *
'r--QtrLs
^u'"^Ia
''
1 - f a= *
!'L tt
!'+t
,L
75
,t
+--
inftni+e$vrull11ol*+)'rnt.
xd 16A!26
lbgc E
O{rEriq
19. Tbatuc volumcof sodnh 2-litd bod€spsckagodby Acnc, Irrc. ts normallydistribulcrlwith a nra& ol 1.95L
atrda standsddcviationof 0.35L.
|
0(a) Find 6c pmbability tha! a ratrdonly sclcc'tcdbotdc cooiai$ nror! than 2 L of coda"
,L
J,
PU>*)-nnvMclf('4 tdt, ,lt|€,0,75)
rhstislcssthu | '(rtl [s!.
f) 35of s[ htlca csitaina volurnc
(o.zr;.1r2at) --t.t tt I
0-'-\n lJunm
:;-w-
f
A{a\
20. Solvcfor r, t, a, qndl9 h thsfollowingmafix cqurtiql
lx
u1l
lr rl[-r
A,L
'
t i--=
2
(-,t o1
|
|
- tl
LJ\/r
pr-tt-t
xtl
"l
l-
I
f-t +-
-rrw--f ,
It{
L""-
7Vl
[- +
?.\L
----a
Ie-ar -yIr l *
l * -z 5J
,
-8
t1 .l
"' l'
o]*[ o -z]=[s
rJ-?- l
J
-1 210
5 ffi
L:L-)
l0 l
rD/
(-,t ,rl
t/
LE "J
Lx-2t-8 -- -4
(tr)'2t = 4
7'5=t L
x(q-2
,v --| I
21, Horvmany\{ayscrn 4 trd cr.yor8, 6 bluc c.ayous,aad5 ycllow ctayo$ bc !!!g!
lamccolorart idctrtical?
?e(
ffiTistrnt | (l,vxlttltt$flF|
l5l
Lltvl.s!
'-.--'?
0'
(-t ru1
I
-lr
-l
t
p^blr")
b4o,b3
-lt=L
? --l
if cr&yo$ of hc
Page9
b6roder9brlq
22. Blliot santsto tuy Olivia 8 rnoviesfor hcr birdday. Olivia hasgivenElliot a list of 15diffcrrnt moviq!rhatshe
wouldlil(e to have,but Elliot lcff {|e list at honc. rlq-orov wars c€nElliot rsndomlychooscE moviasoff a
shelfthlt has35difidctrt novies (15of whichareooOlivi{l list) sndgc{3!@Z movi€sthatarconthc list?
I Ot- r+\Or<-
(1actr\-7
,.
I
E{ o .\
Qtr,l\Uto,t)+ c(15,8)
l3 s,rc<
23. A nndon vad.blc X hass rl,an of 34 anda standardd.vidtiooof 2.5. Us. Chcbychcv'sin qudity to cstirnstr
P(28.15
<X <39.25).
lrL
ii1€
--ifr
5.e5=\Lrl,6
t , t =b
b1.7t
,24
-6e<
fZ-,1-\
> l- # =
p(^sxs4N'+31.25)
\'.r-l-J
x 0,1 1 i 1 -
andlctA = {a.c..,3,51,8={1,2,4,d},andC = {c,?.3}.w}ich ofqtcfollor24. LEau= {a.b.c,d,.,1.2.3,4.5}.
L- .la,b,4,t,L,+,f5
in8arctruc?
@aaararcaisjoint.frr.,.a
(d){c,?,3}
€ {arc,e,3,5}
F^1s4.
@a,z,sl e nuetut.Sw'.lr;,Nd,a,t",5J
@tu"n*a**rr*
{",e,2\9J",",rO,1
gg. b2-
i5 Uiur-
= {3,5}F..l* ,ry.lC,el\
(D(AnC)"
(cr.cc FAlr4
le!cLr,.,^t si q€c-(r-tin
accountpayitrg
25. Hownuch shouldBobiivcst now a ssvings
l0 yc{s hc has$25,000in th. accoutrt?
to
N-'3u5x
Pmf- o
t\o').7s
({--AsaD
I'P 1
l/\'lh'Ao5
,off.,;,1J,5,jj,r,+s1
2.25%Ecs!compoirdcddaily-solhat at lhc cnd-of
-
Page | 0
6r6tud@ld6r!q
of all customers
at AcrncHardwaFonsnyparticulardry buycault
Ae4'rdingto compadyrEcords,24%
(a) Findtllc probabilitythrt among35randornlys€lecledcllstodlcrs,at most7 buycaulkthatday.
n--3i
"srcura' - lll.-Jv^tLL
?'-.u
{nrq/
rpinawdf(aq.
-- 0. 31^8
^4,'t)
O) How manyofthc 35custonErscstrbeexpectcdto hry c{ulk?
E(xl--ng=55xoa4=
27. (ScqionO.| ) Roy qndClaficcplay a gam. in which ftcy cach6ip a coin at thc aamctimc. If both coinslsnd
he.ds,RoypaysClstic. $2. If bolhcoinslandtail6,thcnClrtriccpeysRoy $3, If Roy SctrhcadrandClartc€gets
tails,Clrricc paysRoy$5, Odtc isc,RoypaysClaricr $6.
optimalsEatcgic8
for eachplay.r.
(^ ^ \ - ') ' tr ' k l r L l '
il$'6^
itur'
uvo&^,
fffra^'- t5 -l '
(I% bUn'r,l
Ln*"fiAitt1 ?-v^tI a*-AAN iq'J^su-tA
2E. Johnbonow.d$378'fiomBob,whoch{rgcdhim simplcintcrcstat a ratr of 3.?%pory.lr. Aicr sornctim., tohr
Fpaid Bobwith $401.31,Ho', riady monthspals€dbctwccndrctim.0utJohn borowcdthc moncysndtbctimc
Bob?
hclcpaid
A--?rt
lrtot:tl-nll -- t1g *D'061'*t
=fuOrn
t y,u 'kn,ryE
ve-
3w
quaItcily,by whstFrccnt will
29. U Am i6 cardnginicrastir ! s{vingsaccount!t a mtc of 1l.2%/ycarcomgoundcd
ha! &countgrowby in onaycsrif sh€mak s no additionddaporit$?
(htsisoquvoJe.*
awwu.a{
n elVI\ frv {Lc
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analystsordieda groupof cmbuyctsin a nteliopolitanar€a(all with goss alnlral incofirefof$l rnifiion
or mor€)andfoundlhat of thosewhocutendy ownaneconomycar,25%will tuy a mjd-sizedsedanard 5% wiU
buy a luiury s€danfor thcir rcxt car. Of thosewho cun€ntlyo*n a mid-sizeds€dan,40%will hoy an.conomy
car andl5% will buy a lurury sedanfor theirrcxt car. All of tho6€who c|'rcndy o*n a luxury s€drnwill never
buyaly olhertypcof caragain.
(4,
W.irc thc transition matrix for this Markov proc$s. If there are any absoding statcs,list thcm first in the
matrix.
, ry*j
L- l:-
L -t.ut'.o*"1rat
fl-rr,itl,4dda*t'l-w1r/,1?d'a-
ln
Ll I I o.u t0 ,t(
V 0 0.1 0 ,+
Nt''{r
l 0
I i1 a/v\absnbi U Sldlt '
J
0.45o.
(b) If this Eansitionrnatrixis foi a rcgularMarkovprocass,6nd thc sleady-statdistribution,If this transition
rnatdxis for anabsorbingMartov proccss,find thelfunitinSmatrix.
L€
L
I
e--{o'o;ol;f ad
A1rye)-l,pte,a
$ = 1u.r o,41
bF 0.lry1
b^:\11
=€
Mfi?hr\
F
nl 0
(c) Wtst is thc pmbabilityth6t somcon who initially owncda ndd-siz€ds€danwill cwntu.lly own a luxury
scdln?
f v uUa l' i>
l'
garne.
I | . (ScctionC.2) td A = | -j" 'r? tc rtrcpayof nratrixfor ! two-pcrson.
z€rD-sum
L -J- UIJ no{ +,
6pjcr^,n4
-ttl
(a) Finddteoptima.lftixcd straacgrcs
for c.ch player.
e ,]
? =Lp'
(n' --
Alu
c, +4'b'c
t+5
- - 4 + tr - il tS
e*d'e'a1
rptinae.cna'r,*ca(
?,."'n
d.a.1.l,t'5
- !)t7
a,ogt=l - h ,
it P--L th +l'
q
d-b
b-l
n t*,-1
, 'l
,4
=
a Le"l %,=;A:8. - 4F1,:2-rt
6 '"qri,l;{+-it
r 5 t!'. 1 LJZqaJ
lx41a. plnacz s optj /v{"t- w**-eA Snal+tL
(o, tlrnou|c cxlrcclpopayo[ rc urerowprayerrr oourprayersuscurer opumarrmxaosrarcgrc6,
=r llb+El
E=fAQ
L-,i]l:il,].3""JWFT
krd.r.,tv*rt{rd I
,-
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bc
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4
e+d'b'c
4+b-a-+J
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