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Unformatted text preview: 313 A 813161 N HE N T Qi'm‘L’mbability—The Study of Ruidumn I E'II: ill:13! 4.12 The dermity run'1. J'nt' Illa hLII'ﬂ 1"" uf twu ramlum “Lumbers, For
lIxiI'iLsi ail14. [a]: :11 Irma half of the sampli 1::o11'nwnhatdrug5 aw i112 .un:ium.~s' mus:
.wrlmm problem. {In Lira. Ilam 25% of Ihu: «ampli iwiiavaa Ihm dmga alt' Ihr mm! sarinus
lai'il'nlrlu it} 1119 wampl: pm].‘u_1111'm: is. laclawn. E125 and 0.35. an nplnlrm pal] :Leks an HRH nl' Him adults, "no 1a a1 hamwn tn Jug?"
anpmi [hal tln Impulminn prupnrlinn wlmjug {:i [Humanr} is. p  I111.
Tn l‘hlllllllll‘ :1. m: um. [lu' lll'uimnlunﬁ I'll Ila: 5111HIU WI“! ansWar "Yes." 'I'Il_
.1u1:'.~air lr'J is. a madam Variahh IIMI ina1111r'nxl111aluiy I1IJI'HIH“}' LiiMI'ihLIILil
Wllh '"l‘l'lll .N ' ”IEIIIIEIN:II1II1I'L Iltwlaliranrr ' {].[J'[}U2. I"IIILlI1L'f'nlluwillg I
F1I'liili1lllilllil'H' m: rip . II.II.‘J {h} Pan1 ~_ p *'_' Illa] 4.4 M63 as and Variances of Random Variables Prnl alarm, is the mathennulml language that damHum 11w lungrun regiﬂa ._
hahnvinr ul' madam phannnmna. The pmbabiﬂly dintrlhatinn al' a randu _
variahli' j. an klealitcd rclatirc [I'l'lllilt'llt'jf distribuliun. TIIL' pl'ububililgé hi
Iugramrs aml density curves Ihal pinure pmhnhilily lli‘nll'illllliilllﬁ matml)!
”er wu'h'LI‘ pictm‘ts of dish‘ihmlum nl' dam. In dramHang Llala, Wu mm'ca
I't'mn Hl'arllIH In numerical [lh‘ﬂHlll'ﬂ'h .Im'h a3 [Ileana aml hlllllLllll‘Li duviatiarw'
an WI.“ wlll make ﬂu: awarm' Tumv tn vxpand mar titHt'l'ipllnnﬁ' nll the distri
ImlinlIa ul I'amlum variahlis.. Wr ran ﬁpc‘aI: nf than Ilaan Wllllllllﬂh in a 3:1 .
nl l'hII'IL'I." [ll' IIu standard iiiMalina 41f 1hr: Iundmulfi' ‘r'an'im: number of call.
:1 l1':I'I.':' Luirm'y I'm.mciwa in all I'lnul'. rI H115: acciiun aw Iall“ it'ul'n 1mm: aIJmI
Imw In L'Hlllputu [IIESL‘ iIL'Hi'I'IHIIW 11ll'llHlI1'L'5 anilalmllll1:'LI'I.'.'..'tIL.1,' ulna}: 111i: nwan af .3 raudnm variable The Inca” .1" HF 11 set of ohsat'val Ina i: lheir Ordinm'y swanage. The mean of :
ramlnm variable X is also an awnage uf 1h: possible valueﬂ of I. but with . 4.58 4.59 4.60 4.62 4.63 Chapter 4: Prob ability—The Study of RandomnessI: Exercise 4.44 {pea t 315} gives the distributions oi' the number of people in
households and in families in the United States. sin important difference
is that many households consist oi" one person living alone. whereas a
family must have. at least two members. Some households may contain _
families along. with other people. and so will be larger titan the Family. These '
differences I'I'iE'tltt: it bani to compare the distributions without calculations. '.
Find the mean and standard deviation of both household size and family '
sire. Combine these with your descriptions from Exercise 4.44 to give a
comparison of the two distributions. ir'i’liieh of the two distributions for room counts in Exercises 4.43 and 4.5?
appears more spread out in the probability histograms? Why? Find the _
standard deviation for both distributions. The standard deviation provides '
numerical measure of spread. Example 4.15 gives the distribution of grades {.4 = 4. B = 3. and so on} in
siccounting 210 at North Carolina State University as 0.0? 0.09 0.34 0.32 0.ti Pmbahility Find the. average {that is. ilie mean} grade in this course. You purehase a hot stock for $1000. The stock either gains 30% or loses _
25% each day. each with probability 0.5. its returns on consecutive days a ..
independent of each other. 1if'ou plan to sell the stock other two days. {a} What are the possible values of the stock after. two days. and what is __
the probability for each value? What. is Elie probability that the stock
worth more after two days than the $1000 you paid for it? ibl What is the mean value of the stock after two days? You see that
these two criteria give differtstt answers to the question. "Should I
invest?” it is easier to use lienihrd's Law {Example 4.9. page 202.} to spot suspiciou
patterns when you have ‘t‘et'y many items (for multiple. many invoices lie I:
the some vendor) than when you have only a Few. Explain Why this is true.._ ‘r'ou have two balnIieed. slxsitietl Eliﬂe. The lll'h'l has l. 3. '4. 5.! '5; [“10 l“ 513“}
on Its six Faces. The seeontl die has i. 2. 2. 3. 3. and 4 spots on its anes.
{a} what is the mean number ofspols on the uprﬁiee when you ml] each
these dice?
lib] Write the probability model for the outcomes when you roll liotb dim:
independently. From this. ﬁnd the probability distribution of the sum3.:
the spots on the upEaees of the two dice. ' [all Find the mean number of spots on the two upfaees in two ways: ﬂotil II 1 . 1:.._...,.1'..... 354 Chap!" 1: Pmbabﬂity—Thu Study uf Randmun la the pnmhuhlliq,r nf the neviIll M ul' iii rim: ConsuiiclzmLI will win at] {'35:
um UI tlu inbs? 4.90 In II". .wuing Hf IIIL' iﬂl't‘vinllh rmtim,u:1 rueI115 21 and H imizpundum? Du =I
L‘IIIL'11ln1inI1 Iflnl 11:1ch _vnlu unimm: 4.9] UrnMr il "r’LIm diagram that! iliimlmtm lhE I'uiatinn iwImnn uwnIi al and 3
in Excrriw l ﬂ'n'. Write canIi ul Ilu billowing eweInna in It'lnlh n[ .«L H. A‘,
and ii“ . Imfirzitv 1hr. ewnls can _vnm diagram and Inar ”1i." ininrmnh‘un in
IL'u'I'L'iHvILH‘Jinnaicuinh:11L~11r'nh:lhllit_1..'rifeacli. {u} Cﬂllhlllitlillﬂd wins but]: juhs. (hi (‘nmuliilnled Wins Ihe first _jnI'I lull nm Ihe socmil. [1:1 (_‘tnmuliduted does not win th iinit jnh but does win lhn arcond. {d} Cunm rlinlnlud does nut win ail hur' Ilmb. Chinum: an mini: Alnﬁ'ican wHmin at random. Table 4.] [page 1H} dos.‘I'ilwiu Ihc punpulnriun fmm whirl: m; ilmw. Use Ilic iulurmuliun in Ihat ' “thin; ln lumwr the following quvmiurm. [I1] Whill in Iiir.‘ pl‘ubahiliiy III.iL IIIL' W1 mum chosen h. {:5 warn [JILI rit' older {h} Whui li Ilu' unnditiunnl In'nimhililv thin Ihe wmnnn [humII i1. married.
uh'rn IIInI xllr is {IF ul' mcI 2’ IF} Hummun} “which“11'!!er Illlll'f'il'ililHLlI'I‘IHH" 1'1“} Iu iWL'J' IIHL‘ gmup?
Wlml IH llu pl'nhahilily Ihnl ”Ii' mum”: 'WL! viimm: 1in mm‘l'inlwmnaﬂ
.Li Irmr Mi _‘wm‘s rrld? {ill ‘L’vl'ilir III.II IIIL li'll‘m.‘ prnlmliiiilirx mm [ouml in {in}, ilﬂ. mnl {1:} satisfy .
lht‘ 1:11I1I1'pll'cutl'ﬂn mic. 4.93 i..‘im:mL~. m: mlull American wrmmn m I'mulnm. Think: 4.1 (I: EHL'I'lhmi the
pnpuhiliun Il'nm which urn: draw. {a} I:"c'lml n :Iw mndirimml lmulmbility Illal line woman chusun I5 IE to 29'
yvmw uid. given that she is nmj‘riudi’ [b] in [ixnmpia 4.3lwefoundIhmmmun'iedlag:1811129} n {1.34%. I114 implriu this sentence: ELIHi is; Illnr pruportion of wmmrn win. are
among those umnrn wiln arr ___ [C] IJHIIJI,I~.'::1:1'1'J1md Piagu ”11:12”, lum'riud}. Wn'h‘ :1 ﬁl'lllL‘Ht'L‘! ul' rim
Julm “INCH In (113' that [[t'ﬁl‘l'HJIL'h lin meaning UE'IIIlH 'i'.Hli.1'llL'1Wl'J
1'4 unilliuunl pl'ﬂhﬂbiljlil‘ﬁ yin“ ll‘i wry diffHunt infm'umliun. 4.94 lirw :Il'r Ii'lr unmn l[in lhnummlnl ul 1‘ill'lll'tl tlcglu'x in II»? “Illli'll El
111:: EIIHT Jim} llt‘illfﬂll‘illi.‘ yum: L‘hihHHiiLI h}; luwil and h}; lili
I'L~L‘‘.IL'I1l."" ————————...._.__________ ILIulILlor'a Mmtvl'k i’mfmasionnl lkummlc Tallui [IlL'Si'
Hm: nl' Ihu dug: _ I"L111IL1L E345 22? 32 IH 922 Mnlr 505 Jﬁl 4i! 2E1 T32 M
inmi I lit) 33H T2 44 mm EXAMPLE 4.3!} Conditional
Pmbllllly EXAMPLE 4.31 Slim is a. professional polter playet: He Mattel. at the dealer. who umpires to deal.  What is the 131mInabilityI thnt the card death to Slim is on non? Then: on 51 enrtls in the eleclt. Because the dark woe omﬁtlly shufﬂed. the next enrtl tlealt L1 tqlll'tll'lt' llltely
In be any nl' Ilte cords llntt Sllm has not met. l'mlr of the 51 cattle are lees. Sn
4 l Hm” 5—1 '13 This euleulutIIm Itssumo: Iltel Slim know: nothing about ttttjtr earth ttlrentlﬂr dealt.
Suppose now that he is looking at 4 cards elrentljr lu hh hood. and the! on: of
them is en nee. He kmm nothing about the other 4H etIrtht except lhttt exactly 3 trees are unumu Illerrt. Sllltﬁ [heritability of litlug dealt ttn not: ell!" whet hr know; 13 How Ptaoel  nee in 4 viaihle curds}  3%  1Itil Mtg thnl there is 1 ate mung I11: 4 emit Slim can are changes the nluhnbilin'
that the next eertl dealt is on we. The new notation PM Hi} is a euntliiioual prohtthlllty. Thet lH. ll gites ' the probability of one event {the next etu'tl tleolt is tut nee] under the emulition
tlmt we ltntne smother event [exactly l of lltt‘: 4 ehilltle tfttl‘tls i5 11“ Me}. You
Ciirl. rend the Intel as. "given the infnrlrtttlltut that." In limtmlﬂe 4.3!] we t‘mtld ﬁnd Inttlmltillllcs because we ill‘l.‘ willluu to use [111 equally llltuljtI prohttl'tilllh" model For n shul'ﬂotl tlet'lt ttl' ont'tle. Here. lﬁ tttt ' example Imam{I on date. 'l'nble 4. l ellhm. the mnrlutl elulus olntlull mutton broken tltmrtt lrynge lump. WI: an.
Interested in the t]rt:tl.tttl:tt'Iltj.I the! n mothnthI t‘ltntien women in the tried. It he tntmnon
Htll'tSE tluIL knowing her Etﬂt'! Ht'ottp 1will Ellllllﬁﬂ llte pmhnhlllty: mﬂnj'yotttttt women
have not Inun‘led. moot middleaged women are ttlarrierl. and nltlerwqrnen me. more
lllt e1].r to he widows. To help us think ettre'hlllygtJ let's deﬁne two events: A = 1]". wontam chosen ht young. ages In to 19
B 2 the women Chosen In married It'l lo 29 .‘lil In M 6.5 nutl twﬂl‘ 'l'olrtl Mﬂt'l'lcd 7.342 43.!“13 3.2?” 59.92“
Meetr married ”.9311 T. 134 ‘I5 I 21.5.55
Widowed 35 1.523 EJHS 11.1944
lllmt'oetl "RH 9.174 I.2II53 l 1. l4 l Total 22.5 [2 62.659 1511639 103.87ll EXAMPLE 4.32 There are [in thousand!) IMHD axlt
women. 22.51.". are aged 13 to 19. choosit
ohnnoe. so the prohnhtltty of choosing a 5M 2 .
PM] ' 7% The table shows that Iltele the 1.341
pmhuhllttgt tl'trtl we ehootte at women who PH undﬂl I  To ﬁnd the conditional [nuhnhllity thnl
that she ls young. look only M the “18 to
this oolurnn. no the information given on
conditional prtﬂlllb‘ll'll‘F is _ T
HEM]  5: You can tret'liz.r that the conditional prob:
know she is under age 30 it much lower tl
woman. [t is on an; toeonl'use the three prob;
ttt Tobie 4.l end he sure you ttt'ttlerstu
“ﬁtting these three pmlJultlllllett. The
ram! motrlotl Is the. ]Jl‘DLlI.lt.'l of the oral
h. married ”he" that .Ithe in mung. ’l'h PM and Ill 9‘ PM] 1‘!
22.5” " 103.3? 1.342 .. 1D3.E!T The to thinlt your way through th
then. given that she ht young. she is
fundamental multiplication rule of pa Mnlﬂpllention Rule The pmhnhlllty that lltllli of two t‘.'.
found In;
F I: at and ﬂ} Here NEH] is the eontlltionnl pro
motion lhttt at occurs. Slim is still at the poker uthle. In the mo:
month in :1 row. its he ell: II lhe [Able lot
on the table. Slim sees 11 reeds. Of the: Employed 11.135}
35.13?
31.9?5 341.259 4. 1 [)5 . ni t'tilIL'nlioI'l. Her.1.
I' npiain rarel'uliy _
..1 empInyed are net .I.n.un 25 years of a 1 _.
.;I dilate, what is t 
r' r ﬂute? .4 ﬁnale" independ  1 :5
. I .
'I''.'1 ilitinrti .'rr'olJal:iilit_;I.r.'_ ..... watul ]i.‘l'.'i[il‘.l is ,
..m._. that he or she is vi.ilﬁ I Illiit‘l'ﬂ il'nln u
If“ Iiieie i5. Inoliahiii I:
e yen in the next I
lllll lite contract b‘i
 I Hurtinea not fall. it" « ..I'ilittiitilll.’ {Use a tee
4.107 . n at random. It '
. 'I 'xnol answer or I
 n. and that lﬂm '
'."r.'nl and Ema of tilt: ' : ' n It Hale?
 .l anti Eil'fiir iiiapanic.
.‘ II! it: this case we I'
II Iliitiale anlicimllcs _" Ir, anti Sifai: Hi the
. :._'i lnr the rare {whi " .
II litlntei of a rantiontl _
' tthe t'ttl ILi Itlale expect ' _. I .
"i r.
' “'1 N are made 11; wenme" a I.Iii‘1i In the election describeti itI Linwine I1. ”)2, what percent andidme's.
rotes came From black enters? {Write this an a cnnditinnability.) Genetic munseling. f 'muiirimmi pru‘udu'iimn and ﬂames in! inerti fineouitireiini: Iretrit'tt't' wit” "my irrttlr'ur'rlerit.‘ richtits their cussed Fri tinii r'in'irimn. Extenrixert #JI’J‘E ' vi. HJU r'rrrrr'er'rr Hemw? r'emrs'eii‘tgﬁ. Mblnlsnl. People will: aihinihan have i1lle11itanent in tin hair. and em. The gene that [.11 wen“. itiilillikl'ﬂ has hm fonns illicithi. which we denote life a and 11. liarll luIwn il;1.'..:l lmir ui thee. one inherited h‘orn caul ralrnt. a‘t rhlltl iIIiH'l'ii‘ﬁ one oi each 1WCI ﬂiiﬂiuﬁ. independently with prrihahility {15. r'iihittlrirn is a recertei'tSﬂ a person iH alhinn only if the initerlletl iar iii an. [a] EIetil'e parents are. Iiol :iil'JiI'IU htll Hill"! ha}; an albino le'liﬁ
implies. that both of Heth'ﬁ parents have Type An. Wh {11] Which of the lynen mi. H'Iﬂ. JiJi rouiti a ehiid of Beth's}: have?
What is the prohahiiitr ol each lrpe'.’ {1:} Beth is not albino. 1Ir'tt'I'ual are the euntiitlnnai probahiI'EEth'S
possibie genetic typex. yieen this Incl? {Ilse the deﬁn'contlitintttti
nt'ﬂbﬂbiiiiﬁl albinism. continued. lieth knows the [imhahiiiliaa ior lilit: MicH
iittm 11:1[1 (chili1h: tilt't'hllls t'iit'l'l ila‘. Hl'u' I1llll'l'il‘ii Bﬂh.'nill'1ilir1. iirll‘llh
Ireltetir trope muhl he rm. {It} 1ic't’hal iii the runlitmnal II1 h.lhiiil1.' that a rlliltl ul 3 liﬂh is non
aihinn ii Beth has» Intr ii.’ What I‘ Iin' L nntiilinnal litV HI 3
nonalbino chihi ii lietii hm trtn' xiii? [hi Beth anti Bahia iil‘ﬁt rhllti ih IIHnalhlnn. What iii the anal
llrﬁbabililj'iliili lirth IHttttttI'I'IL11;t}'re iirt? Cynic ﬁbrosis. Cystic lihnniit ilh a June. tiimn'der that oi‘liLﬁ in dﬂﬂtlt.
It is. inherited but can he inherited only iiholh parents 331's of [in
abnormal gen.2. ltI I939. the CF gene that is abnormal ins of cystic
Iilil'Dﬁis was idenliliui. The pl'oinihililr that .1 randomly CEISDD oi
European ancestry carries. an nhnnmiai IIIT gene is l."25.ni'mbiiite is less in other ethnic gt'tittfh.} The Ci‘lﬂm leat detects. r: not all
itarmi‘ul mutations Iii lite {IF Helm. The test ih positive intf pUUt‘tiﬁ
who are earriem. it ih' [Ignoring human error] 11erer].1t.15i]:tﬂuplt‘ who
:u'e not carriers. .iliniJJ team ptmhh'e. Whnt i51iie nrolJalJl he is a
rai'riel‘l’ Cystic ﬁbrmtiit, conlinnetl. .Iawn LIIMK that he if. a eatyxlir
liln'nsis. Iiizi wile, Julianne. him a Mother wilh ryetic Iihi‘lit‘h Ineam
the I'.n1'nit~:uiiiiit3.r is 23.1 titanhe i‘ttit'lIr'l'it'1'. [I'Jtliianneis a'.i!11Ci1v:.'i1iiti
lthe has with Jason itah nmhtlhillly .I'ai ni' having cystic ﬁll' Hilt'. :ih' not
It L'art‘ior, her children l'ltltt‘lnl have the tii‘u‘t‘ilh't'. Jansen mum have one child. who tlﬂt‘h not haee et'iilit' i'lhl‘nsih'. This infot‘nullltiﬂﬁ ilti‘
Iimhabililj'r' that Julianne iii l'i t'rn'lirl'. Haa I‘itn'ea'e role. to: contiitil'mal 4.97 4.9.0 4. 1'00 . Total In labor
Highest education population force Employed Did not finish high school 212325 12,0?3 111139
High scltooi but no college 51,221 36,055 35.13"."
Less than bachelor‘s degree 45,4?1 33y331 31.9?5
College graduate 4T3?! SEEM 36,259 Find the unemployment rate for people with each level of education. How does die unemployment rate change with education? Explain carefully why your results show that level oi: education anti being employed are not independent. {a} ‘Nhat is ﬁre probability that a randomly chosen person 25 years of age or older is in the labor force? lb} If you know that the person chosen is a college graduate. what is the conditional probability that he or she is in the labor force? {e} Are the events "in the labor force” and "college graduate" independent? How do you know?1 You know that a person is employed. What is the conditional pmhahility that he or she is a college graduate? You know that a second person is
a college graduate. 1iiihat is the conditional pinbability that he or she is
employed? Functional Robotics Cor'porm ion buys electrical eonttollets hunt a Japanese supplier. ri‘he company’s treasttrcr thinks that there is probability 0.4 that the dollar will [all in value against the Japanese yen in the next month. The pmbability that the supplier will demand that the contract he
renegotiateti is 0.3 if the dollar falls. and 0.2 if the dollar does not fall. itihat
is the probability that the supplier will demand renegotiation? (Use a tree  diagram to organize the information glared]l it telemarketing company calls telephone numbers chosen at random. It
ﬁnds. that T000 of'calls are not completed {the patty does not answer or refuses to talk], that 20% result in tallting to a woman. and that 10% result
in talking to a man. After that point.‘30% of the women and 20% of the men actually buy something. What percent of calls result in a sale? The voters in a large city are 40% white. 40% blackJ anti 20% Hispanic.
[lIiSpanies tnay be of any race in ofﬁcial statistics, but in this case we
are speaking of political blocs.) it lilaclt mayoral candidate anticipates
attracting 30% of the white vote. 90% of the black vote. and 50% of' the Hispanic vote. Draw a tree diagram with probabilities for the race (white.
biaclt, or Hispanic) and vote {tor or against the candidate]I of a randomly
chosen voter. What percent oi the overall rote does the candidate expect to get? In the sEtting of Exercise 4.i01. what pertera of sales are made to women? (Write this as a conditional probability.) 4.104 In the election described in Exercise 4:102. wht
votes come from black voters? [Write this as a t Genetic counseling. Conditional probabilities.
for counseling people who may have genetic doth
children. Exercises 4. idd to 4. .l 00 concern genetr'. 4.105 albinism. People with albinism have little pign‘ and eyes. The gene that governs albinism has tv
which we denote by e and A. Each person has a
inherited from each permit. A child inherits one
independently with probability 0.5. albinism is :
is albino only if the inherited pair is no. {a} Bed1’s parents are not albino but she has an
implies that both of Bed1’s parents have ty'p {h} Which of the types on. Art, AA could a child
What is the probability of each type? {c} Beth is not albino. What are the conditional possible genetic typesr giyen this fact? ("Lise
probability.) 4.1013 albinism, continued. Beth knows the probabili
from part (cl of the previous exercise. She ntan'i
genetic type must be an. {a} 1i'i'ltat is the conditional probabiliiy that a el' albino if Beth has Lypeaio'i What is the cond
nonalbino child if Beth has type AA? Eli} Beth and Bob’s first child is nonalbino. Wht
probability that Beth is a carrier. type do? 4.1021r Cystic ﬁbrosis. Cystic ﬁbrosis is a lung disorder It is inherited but can be inherited only if both p:
abnormal gene. In 1909; the CF gene that is abnc
ﬁbrosis was identiﬁed. The probability that a ran:
European ancestry carries an abnormal CF gene is less in other ethnic grousz The CF20rn test de
harmful mutations of the CF gene. The test is pos
who are carriers. It is (ignoring human error] netr are not carriers. Jason tests positive. What is the]
carrier?I 4.i03 Cystic fibrosis. continued. .lason knows that he :
fibrosis. His wife. Julianne, has a brother with eye
the pmhability is 2i} that site is a earlier. if Julian
sltc has with Jason has probability 1i4 of having c
a carrierr her children cannot have the disease. Ja:
one child. who does not have cystic ﬁbrosis. This .
probability that Julianne is a carrier. Lise Bayes's t mm no... owmﬁ mm... m... or...
. m....._.........o...m 2.2.... in .n. 35.3.... 3 .......m 1.2%.... 5.... En... mm... m...—
UE. 553.2... on. 1......5... ....n 353... 9...... .3... m. m .03. 3%.. E... .9... 5...... ﬂag... c... :............A. 32.... 3:9... E... E . _ __
um... .39.... 5.... w... 93.3%.. _.....E._..... ._.n .55... 45.... ._..
33...? 5.... m3. 5.. 2...... 0.. n¢.....m.......n.. .3... man... .
:5 Egg...— m:.. :5 .5...” .: m... ..............n. 33:3. H. . _ . .
.a u... .m 2.5 523.2... 3.5:... Q... m... 32......“ .532..."
an... “.55.. n... .2. n......3.ﬁ.. .9... .m .. 3.352.. .c F... u . H.533. 1.2.1.315......
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