Сборник задач по высшей математике 2 том

6.7.14. CKOJIbKO pa3 Ha,Il;O IIo,n;6poCHTb cHMMeTpH'IHYlOMOHeTY, 'IT06bIC BepOHTHOCTblO 0,90 'IacToTar;: IIOHBJIeHHH rep6a OTJIH'IaJIacbOT

~ (BepoHTHoCTH BbIIIa,Il;eHHH rep6a) He 60JIee 'IeMHa 0,01 ?

<) IIycTb IIpoH3Be,n;eHo n HcnbITaHHa (6pocaHHa MOHeTbI). Tor,n;a, COrJIacHo yCJIOBHlO 3a,n;a'IH,'IHCJIOm BbIIIa,Il;eHHa rep6a ,n;OJDKHO y,n;oBJIeTBOpHTb Hepa-

oTKy,n;a 4>0(0,02· vIn) + 4>0(0,02 . vIn) = 0,90, HJIH 4>0(0,02· vIn) = 0,45. IIc':

IIOJIb3YH Ta6JIHII.Y 3Ha'IeHHa<PYHKIJ;HH JIaIIJIaCa, IIOJIY'IaeM0,02· vIn = 1,65. OTclO,n;a CJIe,n;yeT, 'ITOvIn = 82,5, T. e. n = 6807.

TaKHM 06Pa30M, HY:>KHO IIo,n;6poCHTb MOHeTY OKOJIO 7000 pa3.

3aMe'laHUe.3a,Il;a'IYMO:>KHO 6bIJIO pelIIHTb IIpOlIle, HCIIOJIb3YH <P0PMYJIY

«BepOHTHOCTH OTKJIOHeHHH OTHOCHTeJIbHOa 'IaCTOTbIOT IIOCTOHHHOa BepOHTHOCTH B n He3aBHCHMbIX HCIIbITaHHHX He 60JIee 'IeMHa 'IHCJIOc > 0»

Pn (Ir;: - pi < c) = 24>0 (c. ffq).

330

6.7.16. KaKoBa Bep05lTHOCTh TOrO, 'ITOH3 2450 JIaMII, OCBem.aIOm.HX YJIHII,y, K KOHII,y rO,ll;a 6Y,ll;eT ropeTh OT 1500 ,ll;0 1600 JIaMII? C'IHTaTh,'ITO KruK,ll;a5I JIaMIIa 6Y,ll;eT rOpeTh B Te'IeHHerO,ll;a C Bep05lTHOCThIO 0,64.

6.7.17. 3aMe'IeHO,'ITOB Cpe,ll;HeM 80% IIOCruKeHHhlX CeM5IH BCXO>KH. CKOJIhKO HY)KHO IIOCa,!l;HTh CeM5IH, 'IT06hIC Bep05lTHOCThIO 0,90 MO)KHO 6hIJIO 6hI O)KH,ll;aTh, 'ITOHe MeHee 100 IIOCruKeHHhIX CeM5IH B30it-

.ll:YT?

6.7.18. MCIIOJIh3Y5I YCJIOBHe 3a,!l;a'IH6.7.11, HaitTH Bep05lTHOCTh Tom, 'ITO 'IHCJIOIIpOpOCIIIHX CeM5IH 6Y,ll;eT JIe)KaTh B "pOMe)KYTKe [450; 520].

6.7.19. IIpH60p CO,ll;ep)KHT 1000 9JIeMeHTOB, KruK,ll;hIit H3 KOTOPhIX 3a BpeM5I t MO)KeT BhlitTH H3 CTP05l, He3aBHCHMO OT ,ll;pyrHx, C Bep05lTHOCThIO 0,002. KaKoBa Bep05lTHOCTh BhIXO,ll;a H3 CTP051 3a BpeM5I t IIpH60pa,

eCJIH 9TO "POHCXO,ll;HT IIpH OTKa3e XOT5I 6hI O,ll;HOrO H3 9JIeMeHToB? Bep05lTHOCTh IIOIIa,!l;aHH5I B u;eJIh IIpH KruK,ll;OM BhICTpeJIe paBHa

0,005. KaKoBa Bep05lTHOCTh IIOIIa,ll;aHH5I B u;eJIh He MeHee Tpex pa3, eCJIH 'IHCJIOBhICTpeJIO~ paBHO 800?

IIoce5lJIH 1000 CeM5IH. Bep05lTHOCTh He IIpopaCTH ,ll;JI5I KruK,ll;OrO ceMeHH paBHa 0,002. HaitTH Bep05lTHOCTh Toro, 'ITO:

a) He IIpopacTeT 10 CeM5IH; 6) Bce ceMeHa IIpOpacTyT.

Ca,!l;OBO,ll;'IeCKHitKooIIepaTHB 3acTpaXOBaJI Ha ro,ll; CBOH ,ll;a'IHhle,ll;0- Ma OT IIO)Kapa. KruK,ll;hlit H3 600 ,ll;OMOBJIa,!l;eJIhu;eB BHec IIO 150 py6JIeit. Bep05lTHOCTh IIO)Kapa (B O,ll;HOM ,ll;OMe) B Te'IeHHerO,ll;a paBHa 0,005, a CTpaXOBa5I CYMMa, BhIIIJIa'IHBaeMa5IIIOCTpa,!l;aBIIIeMY, coCTaBJI5IeT 12000 py6JIeit. KaKoBa Bep05lTHOCTh Tom, 'ITOCTpaxoBa5I KOMIIaHH5I IIOHeceT y6hITOK?

KHHra H3,ll;aHa THPruKOM 10000 9K3eMIIJI5IpoB. Bep05lTHOCTh Tom, 'ITOKHHra 6Y,ll;eT C6POIIIIOpOBaHa HeIIpaBHJIhHO, paBHa 0,0002. HaitTH Bep05lTHOCTh Toro, 'ITOTHpruK CO,ll;ep)KHT MeHee 5 6paKoBaHHhlx KHHr.

CTpeJIOK C,ll;eJIaJI 80 BhICTpeJIOB; BepOHTHOCTh IIOIIa,!l;aHH5I IIpH Ka- )K,ll;OM BhICTpeJIe paBHa 0,7. HaitTH Bep05lTHOCTh Toro, 'ITO:

a)CTpeJIOK IIOIIa,!l;eT 56 pas;

6)'IHCJIOIIOIIa,!l;aHHit 6Y,ll;eT 3aKJIIO'IeHOMe)K.ll:Y 50 H 60.

6.7.25.Bep05lTHOCTh H3rOTOBJIeHH5I ,ll;06pOKa'IeCTBeHHoroH3,ll;eJIH5I paBHa 0,9. HaitTH Bep05lTHOCTh Toro, 'ITOCpe,ll;H B351ThIX HaY,ll;a'IY300 H3- ,ll;eJIHit 95% OKruKeTC5I ,ll;06pOKa'IeCTBeHHhlx.

6.7.26.MCIIOJIh3Y51 YCJIOBHe 3a,!l;a'IH6.7.11, HaitTH Bep05lTHOCTh Tom, 'ITO H3 700 IIOCruKeHHhlX CeM5IH 'IHCJIOIIpOpOCIIIHX 6Y,ll;eT 3aKJIIO'IeHO Me)K.ll:Y 460 H 510.

331

:>Karn.He CMOrJIH O,ll;HOBpeMeHHO n006e,ll;aTh. CKOJIhKO MeCT ,ll;OJI:>KHO ,ll;JIjf 9Toro 6hITh B ero Kaq,e?

6.7.35.B ypHe CO,ll;ep:>KaTCjf 6eJIhle H qepHhle IIIaphI B OTHOIIIeHHH 4 : 3. 1I3

Hee H3BJIeKaeTCjf IIIap, <pHKcHpyeTCjf II.BeT H B03Bparn.aeTCjf B ypHy. qeMY paBHO MHHHMaJIhHOe 'fHCJIOn H3BJIe'feHHii,npH KO'FOPOM C BepOjfTHOCThlO 0,9545 MO:>KHO O:>KH,ll;aTh, 'fTOOTKJIOHeHne OTHaCHTeJIhHOii 'facTOThInOjfBJIeHHjf 6eJIoro IIIapa OT BepOjfTHOCTH ero nOjfBJIeHHjf B O,ll;HOM onhITe He npeBhliliaeT, no MO,!l;yJIIO, BeJIHqHHhI 0,05?

6.7.36. BepOjfTHOCTh nOpaJKeHHjf II.eJIH npH O,ll;HOM BhICTpeJIe paBHa 0,4. BhIJIO npOH3Be,ll;eHO 600 BhICTpeJIOB. HaiiTH:

332

a) rpaHHU;bI, B KOTOPbIX C BepOJlTHOCTblO 0,9948 6Y)J,eT 3aKJIlOqeHO qHCJIO nOna,II,aHHft B u;eJIbj

6) qHCJIO BbICTpeJIOB, KOTopble Ha,II,O npOH3BeCTH no MHmeHH, qTo6bI C BepOJlTHOCTblO 0,9948 O:>KH)J,aTb, qTO OTKJIOHeHHe OTHOCHTeJIbHOft qaCTOTbI OT BepOJlTHOCTH nOna,II,aHHJI npH O)J,HOM BbICTpeJIe 6y- )J,eT MeHbme no MO.rr.yJIIO BeJIHqHHbI 0,05.

KOHTPonbHASI PA60TA

BapillaHT 1

1.12 cTY)J,eHToB CJIyqaftHbIM 06pa30M paCCa:>KHBaIOTCJI Ha 12 nepBbIX MecTax O)J,Horo pJl)J,a napTepa. KaKoBa BepOJlTHOCTb, qTO cTY)J,eHTbI M H H 6y.rr.yT

CI1)J,eTb PJl)J,OM?

2.BaTapeJl, COCTOJlIIJ;aJl H3 10 Opy)J,Hft, Be)J,eT orOHb no 15 Kopa6JIJlM HenpHRTeJIJI. HaftTH BepOJlTHOCTb TOro, qTO Bce OPY)J,HJI CTpeJIJlIOT: a) no O)J,HOft ueJIHj 6) no pa3HbIM u;eJIJlM (BbI60p u;eJIH CJIyqaeH H He 3aBHCHT OT )J,pyrHX).

3.B JlIIJ;HKe HaXO)J,JlTCJI 20 JIaMnOqeK, cpe)J,H KOTOPbIX 3 neperopeBmHe. HaftTI1 BepOJlTHOCTb Toro, qTO 10 JIaMnOqeK, B3J1TbIX HaY)J,aqy H3 JlIIJ;HKa, 6y.rr.yT ropeTb.

4.Ha ATC MoryT nocTynaTb BbI30BbI Tpex THnOB. BepOJlTHOCTH nocTynJIeHHJI BbI30BOB l-ro, 2-ro H 3-ro THna COOTBeTCTBeHHO paBHbI 0,2j 0,3j 0,5. IIocTYIII1JIO TpH BbI30Ba. KaKoBa BepOJlTHOCTb Toro, qTO a) Bce OHH pa3HbIX THnOBj

6) cpe)J,H HHX HeT BbI30Ba 2-ro THna?

5. Ha eJIOqHblft 6a3ap nocTynalOT eJIKH C Tpex JIeCX030B, npHqeM I-ft JIeCX03

rOJIy60ft. KaKoBa BepOJlTHOCTb, qTO OHa nOCTaBJIeHa 2-M JIeCX030M?

6.BepOJlTHOCTb Toro, qTO H3)J,eJIHe He BbI)J,ep:>KHT HcnbITaHHJI, paBHa 0,004. KaKoBa BepOJlTHOCTb TOro, qTO H3 750 npOBepJleMbIX H3)J,eJIHft 60JIee Tpex 113)J,eJIHft He BbI)J,ep:>KaT HcnbITaHHJI?

7.HaftTH BepOJlTHOCTb OTKa3a cxeMbI (pHC. 73), npe)J,nOJIarM, qTO OTKa3bI OT.n:eJIbHbIX 9JIeMeHTOB He3aBHCHMbI. BepOJlTHOCTb OTKa3a Ka:>K)J,oro 9JIeMeHTa paBHa q.

BapillaHT 2

1. 9 TypHCTOB HaY)J,aqy paCCa:>KHBaIOTCJI no 12 BarOHaM 9JIeKTpHqKH. HaftTH BepOJlTHOCTb Toro, qTO Bce OHH OKa:>KYTCJI: a) B O)J,HOM BarOHej 6) BO BTOPOM llarOHej B) B pa3HbIX BaroHax.

333

Puc. 13

2.B aBTonapKe 20 9KCKypCMOHHbIX aBT06ycOB ,1l.ByX MapOK: 12 M 8 COOTBeTCTBeHHO. BepO.HTHOCTb Bble3,1l.a Ha 9KCKypCMIO aBT06ycOB KIDK,1l.ofi MapKH O,1l.Ha M Ta :lKe. KaKoBa BepO.HTHOCTb Toro, qTO nOCJIe Bble3,1l.a Ha 9KCKypCMIO

16aBT06ycOB, B aBTonapKe OCTaJIMCb aBT06ycbI: a) nepBofi MapKM; 6) O,1l.HOit MapKM; B) pa3HbIX MapOK.

3.C BepO.HTHOCTblO 0,4 nOCJIaHHOe co06w;eHMe npMHMMaeTC.H npM o,1l.Hofi nepe,1l.aqe. CKOJIbKO Ha,Il.O C,1l.eJIaTb nepe,1l.aq, qT06bI C BepO.HTHOCTblO He MeHee 0,9 OHa 6bIJIa npMH.HTa XOT.H 6bI O,1l.MH pa3?

4.B o,1l.Hofi Kop06Ke HaXO,1l.MTC.H 4 KpaCHbIX, 5 3eJIeHbIX M 3 qepHbIX KapaH- ,1l.ama, a B ,1l.pyrofi - 3 KpacHbIX M 2 qepHbIX. Ih nepBofi Kop06KM B3.HTbl TpM KapaH,1l.ama, a M3 BTOpofi - ,1l.Ba. KaKoBa BepO.HTHOCTb Toro, qTO Bce BbITaw;eHHble KapaH,1l.amM O,1l.HOrO II;BeTa?

JIaMn. HaY,1l.aqy BbI6MpaeTC.H O,1l.Ha JIaMna. KaKoBa BepO.HTHOCTb Toro, qTO OHa 6paKOBaHHM?

6. IIpoBe,1l.eHo 8 He3aBMCMMbIX McnbITaHMfi, KIDK,1l.0e M3 KOTOPbIX 3aKJIlOqaeTC!I

BO,1l.HOBpeMeHHOM nO,1l.6paCbIBaHMM ,1l.ByX MOHeT. HafiTM BepO.HTHOCTb Toro, qTO

a)B Tpex McnbITaHM.HX M3 BOCbMM nO.HBMTC.H no 2 rep6a;

7. HafiTM BepO.HTHOCTb 6e30TKa3Hofi pa60TbI cxeMbI (pMC. 74), npe,1l.nOJIarM qTO OTKa3bI OT,1l.eJIbHbIX 9JIeMeHTOB He3aBMCMMbI. BepO.HTHOCTb OTKa3a Ka-· :lK,1l.0ro 9JIeMeHTa paBHa q.

8

- -- -t-_-_-_-_-_-_-_1- ---

Puc. 14

334

BaplilaHT 3

1. B ceMH3Ha'lHOM TeJIe<poHHoM HOMepe CTepJIHCh TpH nOCJIe.D:HHe U;H<PPhI. HafiTH BepOjfTHOCTh TOrO, 'ITOCTepJIHCh: a) O.D:HHaKOBhle U;H<PPhI; 6) pa3Hhle lJ,11<pPhI.

2. Ha YCTPOitCTBO nocTynalOT 2 CHrHaJIa, npH'IeMnocTynJIeHHe KaJK.D:OrO CHrHaJIa, B Te'leHHe'1aca,paBHOB03MO)KHO.

YCTPOitCTBO cpa6aThIBaeT, eCJIH pa3HOCTh Me)K,D;y MOMeHTaMH nocTynJIeHl1jf CHrHaJIOB MeHhilie 10 MHHyT. RaitTH BepOjfTHOCTh Toro, 'ITOYCTPOitCTBO cpa6oTaeT.

3.B ypHe HaX0.D:HTCjf 40 IIIapOB. BepOjfTHOCTh Toro, 'ITO2 H3BJIe'leHHhIXIIIapa OKaJKYTCjf 6eJIhIMH, paBHa ;0. CKOJIhKO B ypHe 6eJIhIX IIIapoB?

4.BepOjfTHOCTh nOTepH nHChMa B nO'lTOBOM OT.D:eJIeHHH paBHa 0,03, a TeJIerpaMMhI - 0,01. OTnpaBJIeHO .D:Ba nHChMa H O.D:Ha TeJIerpaMMa. KaKoBa BepOjfTHOCTh Toro, 'ITO.D:Ofi.D:eT: a) TOJIhKO TeJIerpaMMa; 6) XOTjf 6hI O.D:HO H3 OTnpaBJIeHHit?

5. B nYHKTe npoKaTa HMeeTCjf 8 HOBhIX H 10 nO.D:ep)KaHHhIX (T. e. XOTjf 6hI pa3 HCnOJIh30BaHHhlx) aBTOM06HJIefi. 3 MaIIIHHhI B3jfJIH HaY.D:a'lY B npOKaT 11 cnYCTjf HeKOTopoe BpeMjf BepHYJIH. IIocJIe 9Toro BHOBh HaY.D:a'lYB3jfJIH B npOKaT .D:Ba aBTOM06HJIjf. KaKoBa BepOjfTHOCTh Toro, 'ITOo6a aBTOM06HJIjf HOBhle?

6. BepOjfTHOCTh nOna.D:aHHjf B u;eJIh npH KaJK.D:OM BhICTpeJIe paBHa 0,8. RaitTH BepOjfTHOCTh Toro, 'ITOnpH 5 BhICTpeJIaX u;eJIh 6Y.D:eT nOpaJKeHa: a) 2 pa3a;

6) He MeHee 2 pa3; B) He 6Y.D:eT nOpaJKeHa HH pa3Y.

7. RaitTH BepOjfTHOCTh OTKa3a cxeMhI (pHC. 75), npe.D:noJIar~, 'ITOOTKa3hI OT.D:eJIhHhIX 9JIeMeHTOB He3aBHCHMhI. BepOjfTHOCTH qi OTKa30B 9JIeMeHTOB co-

OTBeTCTBeHHO 0,1; 0,2; 0,05; 0,2; 0,1.

Puc. 75

BaplilaHT 4

1. )J;Ba npHjfTeJIjf B H C peIIIHJIH, 'ITO3a 6HJIeTaMH B KHHO nOfi.D:eT TOT, Y I<oro BhIlla,D;eT MeHhIIIee '1HCJIOO'IKOBnpH 6pocaHHH HrpaJIhHoit KOCTH. Ka- I<OBa BepOjfTHOCTh Toro, 'ITO3a 6HJIeTaMH nOfi.D:eT: a) C; 6) npoHrpaBiliHit; 8) BhmrpaBiliHit?

335

2. B .Hru;HKe 50 ro,n;HbIX H 16 ,n;eq,eKTHblx ,n;eTaJIeit. C60pru;HK Hay,n;aqy ,n;OCTaeT

8 ,n;eTaJIeit. HaitTH BepO.HTHOCTb TOro, qTO cpe,n;H HHX: a) HeT ,n;eq,eKTHblx; 6) 3 ,n;eq,eKTHblx.

3. BepO.HTHOCTb Toro, qTO B pe3YJIbTaTe 5 He3aBHCHMbIX onbITOB c06blTHe A

(npe,n;nOJIaraeTC.H, qTO OHa o,n;Ha H Ta }J{e BO Bcex onbITax) npoH30it,n;eT XOT.H 6bI O,Il;HH pa3, paBHa 0,99757. Onpe,n;eJIHTb BepO.HTHOCTb nO.HBJIeHH.H C06bITH.H npH O,n;HOM onbITe.

4. B MacTepcKoit TpH CTaHKa. OHH Tpe6ylOT HaJIa,rr.KH B TeqeHHe CMeHbI C BepO.HTHOCT.HMH 0,05; 0,1; 0,3 COOTBeTCTBeHHO. KaKoBa BepO.HTHOCTb Toro, qTO B TeqeHHe CMeHbI nOTpe6yeTC.H HaJIa,rr.HTb: a) Bce CTaHKH; 6) TOJIbKO O,n;HH CTaHOK.

5. B nepBoit ypHe 3 6eJIbIX H 7 qepHbIX mapOB, BO BTOPOit 5 6eJIbIX H 2 qepHblx.

113 nepBoit ypHbI nepeJIo}J{HJIH BO BTOPYIO TpH mapa, 3aTeM H3 BTOPOit ypHbI H3BJIeqeH O,n;HH map. KaKoBa BepO.HTHOCTb Toro, qTO OH 6eJIblit?

6. ITo KaHaJIY CB.H3H nepe,n;aIOTC.H 7 co06ru;eHHit, Ka}J{,n;oe H3 KOTOPbIX, He3aBHCHMO OT ,n;pyrHx, MO}J{eT 6bITb HCKa}J{eHO C BepO.HTHOCTblO 0,15. HaitTH BepO.HTHOCTb Toro, qTO 6y,n;eT npaBHJIbHO npHH.HTO He MeHee ,n;ByX co06ru;eHHit.

7. HaitTH BepO.HTHOCTb 6e30TKa3Hoit pa60TbI cxeMbI (pHC. 76), CqHTM qTO OTKa3bI OT,n;eJIbHbIX 9JIeMeHTOB He3aBHCHMbI. BepO.HTHOCTb OTKa3a 9JIeMeHTa C HOMepOM i

Puc. 76

BapMaHT 5

1. B .Hru;HKe JIe}J{aT 9 Ky6HKOB C HOMepaMH OT 1 ,n;o 9. IToCJIe,n;oBaTeJIbHO H3BJIeKaIOTC.H TpH Ky6HKa. HaitTH BepO.HTHOCTb Toro, qTO nO.HB.HTC.H Ky6HKH: a) C HOMepaMH 2, 5, 9; 6) C HOMepaMH 5, 2, 9; B) C HOMepaMH 4, 5, 4.

2. 52 HrpaJIbHble KapTbI pa3,n;aIOTC.H 4 HrpoKaM. HaitTH BepO.HTHOCTb Toro, qTO: a) Bce Ty3bI 6Y)J;yT Y o,n;Horo HrpOKa; 6) Ka}J{,n;blit HrpOK nOJIyqHJI O,n;HH TY3.

3. TPH CTpeJIKa ,n;eJIalOT no O,n;HOMY BbICTpeJIY B n;eJIb. BepO.HTHOCTH nona- ,n;aHHit B n;eJIb COOTBeTCTBeHHO paBHbI 0,6; 0,85; 0,7. KaKoBa BepO.HTHOCTb nona,rr.aHH.H B n;eJIb: a) TOJIbKO BToporo CTpeJIKa; 6) XOT.H 6bI o,n;Horo CTpeJIKa?

336

4. B MeiliKe CMeIIIaHbI HHTH, Cpe)l.H KOTOPbIX 30% KpaCHbIX, 60% CHHHX, a OCTMbHbIe 6eJIbIe. KaKOBa BepOHTHOCTb Toro, qTO TpH BblHYTbIe HaY)l.aqy HIiTIi 6y)l.yT O)l.HOrO IIBeTa?

5.Ha CKJIa)l, C opY)I{HeM COBepIIIaIOT HMeT qeTbIpe CaMOJIeTa. BepoHTHoCTb nOpaJKeHHH CaMOJIeTa CHcTeMoit llBO paBHa 0,8. llpH IIpopbIBe k CaMOJIeTOB aTaKyeMbIit o6-beKT 6Y)l.eT YHHqTO}l{eH C BepOHTHOCTbIO Pk. HaitTH BepOHTHOCTb YHHqTO}l{eHHH CKJIa)l,a.

6.HaitTH BepoHTHocTb Toro, qTO B cepHH H3 9 IIo)l.6paCbIBaHHit HrpMbHoit KOCTH 5 OqKOB BbIIIa)l,eT MeHee Tpex pa3.

7.BepoHTHoCTb Toro, qTO Hayra)l. B3HToe H3)l.eJIHe OKaJKeTCH "pHrO)l.HbIM 6e3 )l.OBO)l.KH, paBHa 0,97. KOHTpOJIep "pOBepHeT 400 H3)l.eJIHit. ECJIH Cpe)l.H HHX OKaJKeTCH 16 HJIH 60JIee HY}l{)l.aIOIIIHXCH B )l.OBO)l.Ke, BCH IIapTHH B03BpaIIIaeTCH Ha )l.Opa60TKY. HaitTH BepOHTHOCTb Toro, qTO IIapTHH H3)l.eJIHit 6Y)l.eT IIpHHHTa.

BaplllaHT 6

1. B Kpyr BIIHcaH KBa)l.paT. HaitTH BepOHTHOCTb Toro, qTO CJIyqaitHM TOqKa, 6poIIIeHHM B Kpyr, He IIOIIa)l,eT B KBa)l.paT.

2. B IIBeTOqHOM JIapbKe "pO)l.aIOTCH 8 aCIIaparYCOB H 5 repaHHit. KaKoBa BepOHTHOCTb Toro, qTO Cpe)l.H 5 "pO)l.aHHbIX pacTeHHit: a) 2 aCIIaparycaj 6) Bce repaHH?

3.B HIIIHKe 6 6eJIbIX H 30 qepHbIX IIIapOB. KaKoBa BepOHTHOCTb Toro, qTO H3 )),ByX BblHYTbIX mapOB O)l.HH 6eJIbIit, a )l.pyroit qepHbIit?

4.BepoHTHoCTb )),03BOHHTbCH C IIepBoit IIOIIbITKH B CIIpaBOqHOe 6IOPO BOK3aJIa paBHa 0,4. KaKoBa BepOHTHOCTb Toro, qTO: a) y)l.aCTCH )l.03BOHHTbCH IIpH BTOPOM 3BOHKej 6) "pH)l.eTCH 3BOHHTb He 60JIee Tpex pa3?

5.BaTapeH H3 Tpex OPY)l.Hit "pOH3BeJIa 3MII, "pHqeM )l.Ba CHapMa IIOIIMH B IIeJIb. HaitTH BepOHTHOCTb Toro, qTO TpeTbe opY)l.He IIOIIMO, eCJIH BepOHTHOCTH IIOIIa)l,aHHH B IIeJIb 1-M, 2-M H 3-M Opy)l.HHMH COOTBeTCTBeHHO paBHbI 0,5; 0,3j 0,4.

6.Coo6IIIeHHe CO)l.ep}l{HT 500 CHMBOJIOB. BepoHTHoCTb HCKaJKeHHH CHMBOJIa rrpH IIepe)l.aqe IIOCTOHHHa H paBHa p. ECJIH XOTH 6bI O)l.HH CHMBOJI HCKaJKeH, TO coo6IIIeHHe 6Y)l.eT IIpHHHTO HeBepHo. llpH KaKHX 3HaqeHHHX P BepoHTHoCTb Toro, qTO coo6IIIeHHe 6Y)l.eT YCIIeiliHo IIepe)l.aHO, OKaJKeTCH paBHoit 0,95?

7.HrpMbHM KOCTb 6pocaeTcH 16 pa3. HaitTH HaHBepoHTHeitillee qHCJIO mo rrOHBJIeHHit qHCJIa OqKOB, KpaTHoro TpeM. HaitTH BepOHTHOCTb P16(mO)?

337

§ 8. AlI1CKPETHblE cnYYAVlHblE BEnll1YlI1Hbl

nOHRTllle cIlY"IaMHOM Belllll"lIllHbl. 'IIYHK....IIIR pacnpeJ\elleHIIIR

IIOHHTHe CJIY'Iati:Hoti:BeJIH'IHHbI- O,n;HO 1'13 BalKHeti:llHx B TeOpHH BepOHTHQcTeti:. IIPH 3TOM rro,n; cJ/,y"{ail:HoiJ. aeJ/,u"{U'HoiJ. rrOHHMaIOT BeJIH'IHHY,KOTOPax B pe-

3YJIbTaTe OrrblTa CO CJIY'Iati:HbIMHCXO,n;OM rrpHHHMaeT TO HJIH HHOe '1HCJIOBOe3Ha'leHHe, rrpH'IeM3apaHee HeH3BeCTHO, KaKOe HMeHHO.

CJIY'Iati:HbleBeJIH'IHHbI(KpaTKO: C. B.) 0603Ha'laIOT60JIbllHMH JIaTHHCKHMH 6y-

KBaMH a rrpHHHMaeMble HMH 3Ha'leHHHMaJIblMH 6YKBaMH Xl,

YI, Y2, ...

MCrrOJIb3YH TeopeTHKO-MHO:lKeCTBeHHYIO TpaKTOBKY, MO:lKHO ,n;aTb 60JIee CTporoe orrpe,n;eJIeHHe:

~CJ/,y"{aiJ.'H(J,JI, aeJ/,u,,{U'Ha X eCTb '1HCJIOBaHcl>YHKIJ;HH, orrpe,n;eJIeHHaH Ha rrpOCTpaH-

CTBe 3JIeMeHTapHblX C06bITHti: n. $

TaKHM 06pa30M, C. B. X KalK,n;OMY 3JIeMeHTapHOMY C06b1THIO W CTaBHT B COOTBeTCTBHe ,n;eti:cTBHTeJIbHOe '1HCJIOX (w), T. e. X = X (w), wEn.

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'1HHe,He,n;OCTaTO'lHO3HaTb, KaKHe 3Ha'leHHHOHa rrpHHHMaeT - BalKHO eI:u;e 3HaTb,

HaCKOJIbKO '1aCTOOHH rrpHHHMaIOTCH 3Toti: BeJIH'IHHOti:«<BbIIIa,n;aIOT») B pe3YJIbTaTe HcrrbITaHHti:. ,1J.JIH 3TOti: lleJIH HCrrOJIb3YIOT rrOHHTHe 3aKOHa pacrrpe,n;eJIeHHH.

JII060e rrpaBHJIO (Ta6JIHlla, cl>YHKllHH, rpacl>HK), rr03BOJIHIOI:u;ee HaxO,n;HTb BePOHTHOCTH rrpOH3BOJIbHblX C06bITHti: A ~ n (B '1aCTHOCTH, BepOHTHOCTH Toro, 'ITO ,n;aHHaH C. B. rrpHMeT KOHKpeTHoe 3Ha'leHHeHJIH rrorra,n;eT B 3a,n;aHHblti: HHTepBaJI),

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~CPY'H'lCtJ,ueiJ.pacnpeiJeAe'H'UR, c. a. X Ha3bIBaeTCH cl>YHKllHH Fx(x) (KOpOTKO F(x)),

4I>yHKllHH pacrrpe,n;eJIeHHH 06JIa,n;aeT CJIe.n;yIOI:u;HMH CBoti:cTBaMH:

1.0 ~ F(x) ~ 1;

2.F(x) - Hey6blBaIOI:u;aH cl>YHKllHH, T.e. F(X2) ~ F(xI), eCJIH X2 > Xl;

3.F(-oo) = 0, F(+oo) = 1;

4. F(x) HerrpepblBHa CJIeBa B JII06oti: TO'lKeX, T. e. F(x - 0) = F(x), X E lR;

5. Pia ~ X < b} = F(b) - F(a).

~ECJIH MHO:lKeCTBO B03MO:lKHbIX 3Ha'leHHti:c. B. X KOHe'lHORJIH C'IeTHO(3TO 3Ha-

'1HT, 'ITOero 3JIeMeHTbi MorYT 6blTb rrepeHYMepOBaHbl HaTYPaJIbHblMH '1HCJIaMH),

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338

Jlblit rrpOMe:lKYTOK Ha 'IHCJIOBOit OCH, TO TaKruI CJIY'IaitHaHBe.nH'IHHa.Ha3b1BaeTCH

rpa<pe .

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Puc. 77

<lJYHKJlHH pacrrpe.n;e.neHHH .n;. c. B. HMeeT BH.n;

F(x):::: E Pi,

Xi<X

r.n;e CYMMHpOBaHHe Be.n;eTCH rro BceM HH.n;eKcaM i, )l;JIH KOTOPblX Xi < x.

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rrpHHHMaIOw;eit 3Ha'leHHHYj C BepoHTHoCTHMH qj :::: P{Y :::: Yj}, j :::: 1,2, ... ,m Ha3bIBaeTCH.n;. C. B., rrpHHHMaIOW;ruI Bce 3Ha'leHHHBH.n;a Xi+Yj (COOTBeTCTBeHHO, Xi-Yj

HJIH Xi 'Yj) C BepOHTHOCTHMH Pij :::: p{{X:::: X;}· {Y:::: Yj}}:::: P{X:::: Xi, Y:::: Yj}.

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339