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A-02 - 313 A 813161 N HE N T Qi'm‘L’mbability—The...

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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 “Lumber-s, For l-Ixi-I'i-Lsi- ail-14. [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! sari-nus lai'il'nlrlu it} 1119 wampl: pm].‘u_1111'm: is.- lac-lawn. 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 [Human-r} is. p - I111. Tn l‘hlllllllll‘ :1. m: um.- [lu' lll'uimnlunfi I'll Ila: 5111H|I|U WI“! ans-War "Yes." 'I'Il_ .-1u1:'.~air lr'J is. a madam Variahh- IIMI ina1111r'nxl111aluiy I1IJI'HIH“}' LiiMI'ihLIIL-il Wllh '"l‘l'lll .N ' ”-IEIIIIEIN|:II1I|I1I'L| Iltwlaliranrr '- {].[J'[}U2. I"IIILlI|1L'f'nlluwillg I F1I'liili1lllilllil'H' m:- rip .- II.II.‘-J {h} Pan-1 ~_ p *-'_-' Illa] 4.4 M63 as and Variances of Random Variables Prnl alarm,- is the mathennulml language that dam-Hum 11w lung-run 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 mat-ml)! ”er wu'h'L-I‘ pictm‘ts of dish‘ihmlum nl' dam. In dram-Hang Llala, Wu mm'ca I't'mn Hl'arllIH In numerical [lh‘flHlll'fl'h .I-m'h a3 [Ileana aml hlllllLllll‘Li duviatiarw' an WI.“ wlll make flu: awarm' Tum-v tn vxpand mar tit-Ht'l'ipllnnfi' nll the distri ImlinlI-a ul I'amlum variahli-s.. Wr ran fipc‘a-I: nf than Ila-an Wllllllllflh in a 3:1 .- nl l'h-II'IL'I." [ll' IIu- standard iii-Malina 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 anilalmllll|1:'LI'I.'-.-'.-.'t|IL-.1,' ulna}: 111i: nwan af .3 raudnm variable The Inca” .1" HF 11 set of ohsat'val Ina- i:- lheir Ordinm'y swan-age. The mean of : ramlnm variable X is also an awn-age 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 Randomness-I: 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'tl-tt: 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 sic-counting 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. slx-sitietl 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 sum-3.:- the spots on the up-Eaees of the two dice. ' [all Find the mean number of spots on the two up-faees in two ways: flotil- II 1 .- 1:-.-._...,.1'..-... 354 Chap!" 1: Pmbabflity—Thu Study uf Randmun la the pnmhuhlliq,r nf the nevi-Ill M ul' iii rim: Consuiiclzm-LI will win at] {'35: um UI tlu- inbs? 4.90 In II".- .wuing Hf IIIL' ifll't‘vinllh rmt-im-,u:1- rue-I115 21 and H imiz-pundum? Du =I L‘IIIL'11ln1inI1 Iflnl 11:1ch _vnlu uni-mm: 4.9] Urn-Mr il "r’L-Im diagram that! iliimlmtm lhE I'uiatinn iwIm-nn uwnI-i al and 3 in Excrriw -l fl'n'. Write can-Ii ul Ilu- billowing ewe-Inna in It'lnlh n[ .«L H. A‘, and ii“ . Imfirzitv 1hr.- ewnls can _vnm- diagram and Ina-r ”1i." ininrmnh‘un in IL'u'I'L'iHvILH‘Jinnaicuinh:1|1L~11r'nh:lhllit_1..'rifeacli. {u} Cflllhlllitlillfld wins but]: juhs. (hi (‘nmuliilnled Wins Ihe first _jnI'I lull nm Ihe soc-mil. [1:1 (_‘tnmuliduted does not win th- iin-it jnh but does win lhn arcond. {d} Cunm rlinlnlud does nut win ail hur' Ilmb. Chin-um: an mini: Alnfi'ican w-Hmin at random. Table 4.] [page 1-H} dos-.‘I'ilwiu Ihc- pun-pulnriun 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 l-i Ilu' unnditiunnl In'nimhililv thin Ihe wmnnn [hum-II i1. married. uh'rn IIInI xllr is {IF ul' m-c-I 2’ IF} Hum-mun} “which“11'!!er Illlll'f'il'ililHLlI'I‘IHH" 1'1“} |I|u| iWL'J' IIHL‘ gmup? Wlml IH llu- pl'nhahilily Ihnl ”Ii' mum”: 'WL! vii-mm:- 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'lhm-i 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: ELI-Hi is; Illnr prup-ortion 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 Jul-m “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 diff-Hunt 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‘hihHHii-LI h}; luwil and h}; lili- I'L~L‘||‘.I|L'I1l."" ——-———-———...._.__________ ILIulIL-lor'a Mm-tvl'k i’mfmasionnl lkummlc Tallui [IlL'Si' Hm: nl' Ihu dug: _- I"L-111IL1L- 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 euleulutI-Im Its-sumo: 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 4-H etI-rtht except lhttt exactly 3 trees are unumu Illerrt. Sllltfi [heritability of lit-lug dealt ttn not: ell!" whet hr know; 13 How Ptaoel | nee in 4 viaihle curds} - 3% - 1-Itil 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 git-es ' 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 ell-hm. the mnrlutl elulus olntlull mutton broken tltmrtt lrynge lump. WI: an.- Interested in the t]rt:tl.tttl:tt'Iltj.I the! n moth-nthI t‘ltntien women in the tried. It he t-ntmnon Htll'tSE tluIL knowing her Etflt'! Ht'ottp 1will Ellllllfifl llte pmhnhlllty: mflnj'yotttttt women have not Inun‘led. moot middle-aged 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“ Meet-r married ”.9311 T. 134 ‘I5 I 21.5.55 Widowed 3-5 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- empIny-ed 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; wen-me" a -I.Iii‘1i In the election describe-ti itI Lin-wine I1. ”)2, what percent andidme's. rote-s came From black enters? {Write this an a cnnditinnability.) Genetic munseling. f 'muiirimmi pru-‘udu'iim-n- and flames in! inert-i fineouitireiini: Iretrit'tt't' wit” "my irrttlr'ur'rlerit.‘ rich-tits their cussed Fri tin-ii- r'in'irimn. Extenrixert #JI’J‘E |'|| vi. HJU r'rrrrr'er'rr Hem-w? r'emrs'eii‘tgfi. Mblnlsnl. People will: aihinihan have i|1lle11itanent in tin hair. and em. The gene that [.11 wen“. itiilillikl'fl has hm fonns illicit-hi. which we denote life a and 11. liarll lu-Iwn 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't-Sfl a person iH alhinn only if the initerlletl |ia|r iii an. [a] EI-etil'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 type-x. yieen this Incl? {Ilse the defin'contlitintttti nt'flbflbiiiifil albinism. continued. lieth knows the [imhahiiiliaa ior lilit: Mic-H iittm 11:1[1 (chili-1h:- tilt't'hllls t'iit'l'l ila‘. Hl'u' I1llll'l'il‘i-i Bflh.'nill'1ilir1. iirll‘llh Ireltetir trope muhl he rm. {It} 1ic't’hal iii the run-litmnal |II1 -h.lhiiil1.' that a rlliltl ul 3- liflh is non- aihinn ii Beth has» Int-r -i-i.’ What I‘- Iin' L nntiilinnal |litV HI 3 non-albino chihi ii lietii hm trtn' xiii? [hi Beth anti Bahia iil‘fit rhllti ih IIHn-alhlnn. 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'Dfi-is 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 lea-t detects. r: not all itarmi‘ul mutations Iii lite {IF Helm. The test ih positive int-f 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 LII-MK that he if. a eatyxlir liln'nsis. Iiizi wile, Julianne. him a Mother wilh rye-tic Iihi‘lit‘h Ineam the I'.n1'nit~:|uiiiiit3.r is 23.1 titan-he i‘ttit'l-Ir'l'it'1'. [I'Jtliianneis a'.i!11Ci1v:.'i1-iiti 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. [l-IiSpanies 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 pert-era 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} Bed-1’s parents are not albino but she has an implies that both of Bed-1’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} 1|i'i'ltat is the conditional probabiliiy that a el' albino if Beth has Lypeaio'i What is the cond non-albino child if Beth has type AA? Eli} Beth and Bob’s first child is non-albino. 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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