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Midterm 1 Spring 2001

# Midterm 1 Spring 2001 - Ii'ﬂDTEHJi-i 1 EDLUTIDNS MATH 913...

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Unformatted text preview: Ii'ﬂDTEHJi-i 1 EDLUTIDNS MATH 913, FEBRUARY 9+ illlﬂl thlem 1. {155} In. a certain. wheel, 33% ef the et-mi-eete Jetty in. the smear tmgue. ﬁfth-e et‘rutertte [the Film in. the term-rte. EEK are will. ﬁfthe etreteete 1m lie mat play in the league. 53% m Elem. {at “the: it the prﬂl'ﬂlil'l'ttt'ﬂ that r: mien: medemiy MEI Fem the eeheet it 11 hey? [hi “the: it the prﬂl'ﬂlil'l'ttt'ﬂ that r: mien: medemiy MEI Fem the eeheettee girtwheﬁteyeiatheeemerteegue?‘ [ej Mery'e daughter Mte—mte the nether-it. Khmmhg anthem; etee client her, what it the writ-ital ehe pie-5w in the m termite? Em Althenrgh thie can he H-Dl'IHE't'l HH- a mutihgeuq; table prehlerri1 meet etm'lertu lel plutmhly he more amulet-tattle Iit'ith the tree eehltieu, arlileh ie the clue we‘ll give. We‘ll eteﬂ ett' with a tree. elemilying biLl1t‘lE-tltﬂ. he. te- whether they {lay eeeeer {”Ee-eeer" i :11- :ten‘t [‘Semfj; then He: te whether they are heye ei- girha. The iniernmtieii we have can he retrained-inert L'l'l em ﬂeet the: any SWEET.- 41.3 "'5“ G111 n.5e “'3" E111 Elf L'DIIIHE? “30% at the etiulente play eerieer, then Zﬂ'ﬁ. :leri‘t: eimilm'ly1 that which in. net 14 gh-l he a hey1 and that which in. hat 11 hey it a gh-J: with bﬂlL‘lI commonplace: we can ﬁll in name more at" the tree: any D.lﬂ Soccer ﬁ.3 '3'“ E111 D.5£ Fer ﬁ.iﬂ 9.43 61:1 Finally, wem‘epletetlietreehy multiplying the prehehliitlee etheaediee: 11.111 hf III-'IJIE‘IH I hill “their “'tﬂ elri u.eu:e.tl - e.t: til-T? he millet-Ah I I:-.I-Clt til-2|? 0": Glrl e-te-e.=i - e.e1e New we‘re IT-lti‘lJ" tn hurl-we: the mealtime. {a} The prehehiiity a student in 11. hey is D32 + ELM-1 - ISL-124. That [Deane the prehahiiity the etuﬂerit 'Le a girl in ELB'FE [which we“ need in part (en. [h] The probability a ra.I1.rlI:rI1'.1l:.r eheeen etudent lﬁ agiI] whe- plwe in the eeeeer league hi H.415. That'e HS and Hi. It} IWe're Hill-ting tar PESIG}. the prehah-ility that a ﬁturlertt plays eeeeer given that Hl'ie'e a git]. [“Mﬂry" hint withing to ill: with it: it‘s the gender et' her daughter which he the irepertant datum-J New PIE IGJ _ FiSD-teerahdﬁ'i _ me PM} new ' neeeee. El Problem. 2‘. {EEK} Ate-met emery niﬂri' gete 'I'lu-t-ey Fer. at nee time 01' aneﬂier. Ewe that at Jayme time: 395' efﬂte ehiLItfe-r: in. the [Hui-tee Shttee have Thu-hey Per. Fm Meteeheehigy my rte-reheat M. were deter-1mm test with the fettrmﬁg ehmuetertttim: e ﬁfehﬂntren whe- iemre My Fete. 939E tee! Tim-tithe and 2% tea-t negttiue: e ﬂfehilih'e-Ie wee ulte- th'ilti MIME Thrhey Pun-.11. 19\$ teatmmhirae and .9135 teat negative. H I'I mhtH £1 pitfall! ﬁle-ﬁt the: Malian I11: Tamil-1m. and. tut-II Fmiiilne. when! :'.Ir % Innﬁtthit-ity 1'.th hnﬁnhr. hm ﬁrth-:3; Fem? Set-I'm Agile, we‘lJ tie thle by a tree. Here‘s the pleture: int-t . I.:I.--I.-eI-I.:I:-e-. ~- < 5.: i-M - I.e.-'I.e:-I.eaee 9.31 . I.-r.-:.|:.-:.:Ir.1 1.1-1 E 2-” l.!?'l.!lul.ll?II Fem“ thin we aee that _ W _ ﬁ _ . . PinteIH — Pt+i ‘ neee4+neer 'Mm' Duly aheut nﬂ'ﬁ- ehanee. mere then might he eemtertahle tent Hehilri‘e methee. hut net [IS-hill" He hurl. El Problem at {1593’} A meme efem-j-ie tepteyed My faith-1g epe-ireffmlr d-iee. {in the ﬁrst rett, e telnet of eeuen e1- ElE‘ll'E'il- m Wielding- e. mli of E, .':'I' or 113 twee-e ineieniiy. Anything etee reemm meme We}- Him! it the pﬁlhrItI-ﬂity: {a} ﬂfwhmhgtheewmhteﬁrﬂmtt {h} Greenﬁehtegnmemeiheﬁmtmtl Sell-4m- {a} The gene hwenhy rellhige Tel-11. 11 ml he teller] hitelet'mya. 5.61 er 13.5. while 1' can he relJeﬂ in ash: maybe 1. E at 2,5- er 3. -l car 4.3 or 5,2 he ii, ]. There are thua eight diethiet euteemea which 11:1th in en outright Iit'ih1 and enrh hen-i pmhehllity lﬂﬁ'. thua the gautmhillty of winning an the Eﬂil lull in 2,135 = 2.19 a: 0.222. [h} A2eah herelletlih enlyeneway: ].l. Athreeeahherelletl in ten:- wage-1:112 er 2,1- And a. tweltveﬁm he rolled in only one way, a. 6.5- 11th there are exactly inur tlietdnet mltmmm Th1L‘l‘I Mint in an entrlght lane on the Eret roll. end] ntwhieh hae plinhmhillt}.r 1.135: there-fate the [rehal'iillty ef1eeihg an the Era: roll inefee-1feme.111. At thin point it eeeme m‘lil‘lwl'litﬁ te [tIEIziEI'It- H. tuIJ mtalyelﬁ efthe game at crepe. {'t'eu didn‘t have tn the this an the ﬁne]. tet'tIItheini This int-elven it further amdyatie ef the ether 2.13 of the pent-dhle eaeee. where the Ere-t tell in e -l. 5-. El. 3. 5-}. er 10- Fe: eitnmple. Hippeee the tint DUI] in II- E: the queetlen thm heeeme-i. ‘What‘e the pmhehiJity that we'll rel] H-tilll heft-Ire we tell e T." [11 theory. the me eeulrl gen e11 I'etever [With neither e nix net Hm ever being; relied}: M e [.Irneticnl mutter. rate :a' the ether will eventually he rolled. Nething; elee eeuntn: eiﬂyaeixeeaeetlen- Aetherenreﬁuemyeelreﬂiugaei; Muriel-e may»: at milling a eeeeu. there :u-e enly eleven [tremble euteemee Ia.Ih.ie1-i will lend tn 3. In:eat-eluahzm1 all equiv-ﬂy likely. alert in live at then: we‘ll til-hi the game. Se- il em 'peint‘ b: :de we have :1 Wu ehniiee ef wihnhig the game. Shiiileu-ly. it elu- ‘pelht‘ in. live, we have n dfii} prehehllity at winning the game {the lam way-H wie- eeaIJd tell a live, out at the ten arm-e ween-um rell alive he aeeuenj. The eemplete nnalye'e.‘ hi how we can win the game in; theretere: I A I}? [Ir-:Ittal'iillt3.r {It winning an the ﬁrﬁt roll. I A efee I l.."l2 prehehiJity at rolling a lpenint' of leer. Thin ili inhaled hy aﬂfil I [I3 [a'ehal'iihty of telling :1. font bejere railing in seven: IJIII.‘ plelmhihty el? wihnhlg thin my lli therefere If]: ill: H3 = [135. I .i'l. lfﬂﬂ = 1.15 probability et I'UlliJIE n ‘peint' at ﬁve. Thin is Followed by a Ilﬂﬂ = 2ft]- pml'rehility et' telling the Willi:- hetiere reJIiu; Hm: [reliability el' Winning this war. 1.191! 215 = EHEL I it. 5.136 pmhehiJity el' rellhIE a peiut eleiit. I'elleweri ha.- a hill EI'I'IJh-HJ'Jilit}.r ef telling the point hefere telling Reeven: mhnhility at winning thie way in Willi x 5:111 = ﬂfm. i A We; :a-elmlilllty et' mung a paint ef eight. fellemeti by n em Whillty of telling the paint hefere telling HHEWI'L: prehehility at winning thin my. Mill} 1'! 5-f11 = 25'1355. {Title lit the name an the rtelmlillty et' meJthig a paint. efeht. [hm yen Etrllﬂlﬂ why‘f'i e A 1,135 = H2 {Helmhillty el" rellhig a print of nine. hallowed by a 41“] = 215 peetmhility el rolling the [mint heleee e elE'll'Et'l: [Helmhillty at winning th'hl way, Will o ."l. SISE = 1f12 protmhiljty ol rolling a. point oi ten, Followed by a ﬁfﬂl = ”'3 [imbuldlity of rolling the point before uneven: [Ifﬂhﬂhill‘t-F of winning thin “my, 1..I'.‘Elﬁ. Time the protmhildty ofonr whining the game in 1' 2 ‘25 25 ‘2 l 2-H §+Th+ﬁ+ﬁ+ﬁ+i+ﬁ_ﬁ‘ Erl' ahont. 0.49719719- Jnet e. nhade under a half—hut what do yon acpeet [mm LFIH Vegan? El Pmblem d. {1'55} d col-runny when home of stream. ltd-mm .50 In a Mr; I'm! the entrant number ofeerenn per hm: Tam-lien, with pmhﬂhﬂi—lﬁ diatrihuﬂion MfoLIn-m: —----IEI {rt} Firm! thr- omootm'l 1min: and thndnlﬂ rlovintioo for the monitor of mean in r1 hat. [1]} him-11g tlune hence-t wlth Eﬂor moreeere'nm, ﬁnd the etpettedmhie ofthe nun-rather arm. {Be mint-[F The proheha'iily dimtnm'en hm moire-1'! To ﬁnd the new yoh-Lﬁihh'ee nun-mute the pmhihih'tp the! A: =.1r_. Elven that .31? EW-! em {a} The oedenleﬂon 1n 1|eg hnt brief: IL=iEK|Jﬂ5+oﬂKE|35+5 Inﬂﬁh+52xﬂ1ﬂ=ﬁﬂl We eoan even do title: by inmeetlon; the men.“ he elem‘ljl hotne- where mind 5|], no it we unhtrnet 5|] from all the when we ﬁnd ;¢=hﬂ+[—2]Hﬂ-Uo+{—l}xﬂ.2h+ﬂxlil.\$h+l xﬂ.2h+2:tﬂ.lﬂ ={d}.11 where we can telte Mhmttnge oi the Feet. that the {—1} :1 [L25 tmleneenthe] Kﬂjﬁendit‘nemytoomnfnneﬂnﬂﬁh. There‘n noenny wear to do the ntnndnrd deﬂation: the1.I'n.l'i;°|nv|:ue1 a" in nonapmted hy- r:rE = Huh—50.1]? x |Z|J3|~fl+lf:l.'lai|—.fnﬂ.1]1 :24: I125 + {ﬁll — 50-1}? :1: I135 +{El—Elllj" x nee+ {52— non“ x 13.10 = 1.09. The: r! = 1.31.09: mum. [h} Ari-Long [hone home with 5:} or more Renew, ﬁnd the expected wane ef the number efnerewn. Many ntndente rentrieted their httention Lo jmt e pert of the above tehle, and computed the —--Eil —--EEI new valneolnm 1: I My: l].3.'_'r+.5-1:-c l].iﬁ+€|.2:-c RID-3.5.45. But this: onn't hr. right-how om the hem-noted 1min: of n quantity Width mm onlyr tnlco 'l'lti'llﬂ" ol'iiﬂ or mom ho er: low 41.! 35?! The [nation in tho. this Jon-t tnhlr. d'nrm't roprr-ron-E ﬁl- [rmhII-Eliﬁ'ly iﬁlrlﬁ- hm'im—tho HUT! of tho probabilities in only 111'. not 1. We hour. to divide oi! the enm'ee I'i'tr 0.1 Thin. injmtlﬁed 113', eg.1 PHI -513] end {.1.’ 21-511]: HIEW? _ FtK=5IJi _ Pt}: 2 on} _ n35 _ it?" Notiontlmtwohm'olmdtholuotthntl{x=W}Hnd[XEﬁﬂJ in: tho sumo hr: X = 50. The oon'oot oonrlita'onnl Enﬂml'ility thI-o :irt thoroForo and Ilo'l-r _:-IEEI__ Prolmhility Home 111.1135? PIX =5ﬁl-1' 2501= we can compute ft == 51] :r: [L5 +5] :r: DEERE +52 n: ISM-11351" = [HIE-129. But HI'I eerier way to do the mme thing quJd h-e tooomgnlhe the oontrihltion to- n from the original table. ﬁﬂnﬂﬂﬁ+ﬁlnﬂ25+ﬁ£nﬂlﬂ=ﬂﬂlﬁ nod I'L'iViIll: 'L’r. l'nr IL?- Notiee that en anew-e1- efM.“ ﬁlial-hut better heme than one oi \$5.:15; the 't'ﬂi'llﬂt of A: are M154} or more, no- the mean nhonlﬂ him he at Jennt m. I took {tit}r on you and didn‘t ibil-E you to compute the rot-tied atmtdard deﬂation. In cane you‘na inteteetoﬂ. it‘ti |].'F]'FEILB. El Problem 5.. {EEK} A mmtirle mﬂomﬁug eon-war.- hot it trim-Mae which pm m defeetiue M‘Mﬁ‘ eat of eatery four. Stun-ma: a buy of ten. rim-rites mqﬂﬂhﬂ'ﬂi by tutti! meme ia examined. {it} What :'.Ir % Whit-ital that nun: of the, marble: i1 Hermit-Ire? {h} III-"hat i1 HIP. pﬁltlrIHity that menth- two of tin: martin! rlrr. drﬁm- Err-117'“ {It} III-"hat in tin: probability that two or more, of HIP. mutt-«Fol rut tie- JI'eeta'trefl‘ Sate-Pim- Thirl- it: a binomial distribittion pronn. Each marHo has a ﬁxed ;|'Irobttl'tilitztr oi being Matti-ate: “out: out of ﬂour" is it [my [lint-a: erb'tuh meant-t that plﬂ'ttthility pol? hing deﬁnetcivc it p = III-25. Tbm: the WW an: {a} {muoemutnittlﬂ = [n.i'ﬁjll] = net-153135. "an. don‘t need to know anything about binomial prohaleitiot todothia: if tbeprob— ahilltg.r oi a marble being good in 13.15, then the probability that ten othe-tutiee marblee are good it 0351'] {atmtuuh'ug tbe eveuta are independent]. [bi {ﬁtﬂﬂﬂhmjﬂ = .15 it hm II! 0.1m113 = 0.231.563. [t'a not good enough to write (I: {omﬂomﬂ baton-to while I can ho ;|'trl:tt3|r MIDI: you know What [0.2513 and WEE-fl" Hm. I hat-I1: Imol' mgr of Imo'l'iog that you know what {3"} ih.'t!rt'.l1l mutt write the it {or Elhiel get the 11.231565 watt]. to} The 1'1roi‘tabilit3.r that two or more marblee are detective in the sum of the probabilitiaa that: enmity two are defective. plwi: malty three are defective. phti: malty four are defective. phui: etc. eto. It‘ti eager to omnmlte the aoewer RH - Pﬁemotly [I are defective] - Pﬁexaotly ] it dElEmLiﬁl'E]. eapeoially aiuoe: we‘ve alrede oomputeﬂ one oftbeae quantitieti in item {a}. til} we need the probability that emotiy one it detectiw: {lf'jtoamltamtﬂ = 0.13mi Thlta the meet- to part (of in 1 — ﬂiﬁﬁﬂ [35 — ELIET'I'I‘I! = ﬂfia'n'ﬂft. Problem E. {15%} Tana m ore dart: ham .1 dealt of omit. {it} “beat is the prehﬁit'ily ﬂmt both end—Ir arr. mm? {'1} Find the [Probability that neithor rant 5': an m. {o} Fin-Ii H1: [ﬁninﬁittty that lit lore-It on: of Hit orrmh ix rm not. tit-imitate at} This it E x 1%: %, about [LOU-152.183- {It} Thie it! a it ﬂ—l = it]! eastﬂttﬂ- tie} a: taut one “team that either Hwy one, or emetiy talo, are aeea. We have to compute the probabilitim minutely {but Eor- tunatetj', one of them in all'veadg.r done in part [a]:|. Hoar can we draw metly one ante1 and one non-m? Drawh'ug them in. that order lute 1'.-|"ottabillt1.r e e _ h 52 x 51 _ 221' but they eait alt-to he til-aunt in the order: hoihaoe, then ace. I'I'Lh'ﬂ'l haa probabilltjl e t 1 _ h 52 t] — 221' TIue-i tbo [Itlhal'ril'tty oi flowing mouth on: no: in \$1 Which when added to it?! the probability oi drawing two aeea1 yielda ﬁ'i 2: [Ll-lEIBEl aa the anawer to thitt part. El ...
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Midterm 1 Spring 2001 - Ii'ﬂDTEHJi-i 1 EDLUTIDNS MATH 913...

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