Сборник задач по высшей математике 2 том

6.4.33. nHTH3HaqHOe qHCJIO 06pa30BaHO npH nOMO:W;H nepecTaHoBKH n;HcPpl

2, 2, 2, 5, 5. Bce pa3Me:w;eHHH paBHOB03MO:>KHbI. HaiiTH BepoHT-' HOCTb Toro, qTO 06e nHTepKH CTOHT PH,II;OM npH YCJIOBHH, qTO noJIyqeHHOe qHCJIO qeTHoe.

6.4.34. .Il,Ba,II;n;aTb nHTb 9K3aMeHan;HOHHbIX 6HJIeTOB cO,II;ep:>KaT no ,II;Ba BO-

npoca, KOTopble He nOBTopHIOTCH. 8K3aMeHYIO:W;Hil:cH MO:>KeT OTBeTHTb TOJIbKO Ha 45 BonpOCOB. HaiiTH BepOHTHOCTb Toro, qTO 9K3aMeH 6Y,II;eT C,II;aH, eCJIH ,II;JIH 9Toro ,II;OCTaTOqHO OTBeTHTb Ha ,II;Ba Bonpoca H3 O,II;HOrO 6HJIeTa HJIH Ha O,II;HH onpoc H3 nepBoro 6HJIeTa H Ha YKa3aHHblii ,II;OnOJIHHTeJIbHblii Bonpoc H3 ,II;pyroro 6HJIeTa.

6.4.35. B rpynne 8 qeJIOBeK, rOBopH:W;HX TOJIbKO Ha HeMen;KOM H3bIKe, H 6 qeJIOBeK - TOJIbKO Ha cPHHCKOM. KaKoBa BepOHTHOCTb Toro, qTO H3 ,II;ByX HaY,II;aqy BbI6paHHblx JIIO,II;eii 06a rOBopHT Ha O,II;HOM H3bIKe?

6.4.36. B ypHe HaxO,II;HTCH 3 6eJIbIX, 4 :>KeJITbIX H 2 qepHbIX mapa. 11:3 Hee Hayra,II; BbIHHMalOT (6e3 B03Bpa:w;eHHH) O,II;HH 3a ,II;pyrHM no O,II;HOMy mapy. KaKOBa BepOHTHOCTb TOro, qTO 6eJIblii map nOHBHTCH paHbme :>KeJITOro?

6.4.37. Pa60qHii 06CJIY:>KHBaeT 3 CTaHKa, pa60TalO:W;Hx He3aBHCHMO ,II;pyr OT ,II;pyra. BepoHTHoCTb Toro, qTO 3a CMeHY nepBblii CTaHOK He noTpe6yeT BHHMaHHH pa60qero paBHa 0,9, BTOpoii - 0,8, TpeTHii -

0,75. HaiiTH BepOHTHOCTb TOro, qTO 3a CMeHY: a) TOJIbKO O,II;HH CTaHOK nOTpe6yeT BHHMaHHH;

6) XOTH 6bI O,II;HH CTaHOK nOTpe6yeT BHHMaHHH;

B)TOJIbKO TpeTHii CTaHOK nOTpe6yeT BHHMaHHH pa60qero.

6.4.38..Il,JIH npHeMa napTHH rOTOBbIX H3,II;eJIHii npHMeHHIOT BbI60pOqHblii

KOHTPOJIb • .Il,JIH 9Toro 6epYT Hayra,II; 3 H3,II;eJIHH. ECJIH Cpe,II;H HHX OKa:>KeTCH:

a) XOTH 6bI O,II;HO 6paKoBaHHoe;

6) 60JIee O,II;HOrO 6paKOBaHHoe, TO 6paKyeTcH BCH napTHH. BblqHCJIHTb B 060HX CJIyqMX BepOHTHOCTb Toro, qTO npH TaKOM cnoc06e KOHTPOJIH napTHH, COCTOH:W;M H3 46 CTaH,II;apTHbIX H3,II;e- JIHii H 4 6paKoBaHHblx, 6Y,II;eT npHHHTa.

6.4.39. CTY,II;eHT 3HaeT 30 H3 40 BonpocoB nporpaMMbI. 8K3aMeHaTop 3a- ,II;aeT eMY BonpOCbI ,II;O Tex nop, nOKa He 06HapY:>KHBaeT npo6eJI B

3HaHHHX CTY,II;eHTa. HaiiTH BepOHTHOCTb Toro, qTO 6y)l;yT 3a,II;aHbI:

a) ,II;Ba Bonpoca; 6) 60JIee ,II;ByX BonpOCOB;

B)MeHee nHTH BonpocoB.

6.4.40.8JIeKTpHqeCKM n;enb (pHC. 62 a) COCTOHT H3 4 9JIeMeHTOB, BbIXO,II;

H3 CTPOH KOTOPbIX B 3a,II;aHHOM npOMe:>KYTKe BpeMeHH - He3aBHCHMble C06bITHH, HMelO:W;He BepoHTHoCTH ql = 0,1; q2 = 0,2; q3 = 0,3; q4 = 0,4. HaiiTH BepoHTHoCTb pa3pbIBa n;enH.

6.4.41. 8JIeKTpHqeCKM n;enb COCTOHT H3 5 9JIeMeHTOB:

310

HaiiTH BepOHTHOCTb pa3pbIBa u:enH, npe,nnOJIarM, 'ITOOTKa3bI OT- ,neJIbHbIX 9JIeMeHTOB He3aBHCHMbI. BepOHTHOCTH OTKa30B 9JIeMeHTOB COOTBeTCTBeHHO paBHbI: ql = 0,1; q2 = 0,15; q3 = 0,15; q4 =

=0,15; q5 = 0,2.

6.4.42.)J,Ba HrpoKa nOOqepe,nHO 6pocaIOT:

IIIeCTepKa (B CJIyqae 6). HaiiTH BepOHTHOCTb BbmrpblIIIa nepBoro HrpOKa.

6.4.43.1'13 Ha60pa U:H<PP OT 0 ,LIP 9, HanHcaHHblX no o,nHoii Ha 10 o,nHHa-

KOBbIX KapToHKax, H3BJIeKaIOTCH no o,nHoii 4 U:H<PPbI H CTaBHTCH B pH,n. KaKOBa BepOHTHOCTb Toro, 'ITOnOJIyqHBIIIeeCH qHCJIO:

a) 1957; 6) 2003?

PaccMoTpeTb ,nBa CJIyqM BbI60PKH: 6e3 B03BPallieHHH H C B03Bpa-

ru;eHHeM.

6.4.44. BepoHTHoCTb o,nHoro nOna,J:l;aHHH B u:eJIb npH O,nHOM 3aJIne H3 ,nByX opy,nHii paBHa 0,35. HaiiTH BepOHTHOCTb nOna,J:l;aHHH npH O,nHOM BbICTpeJIe nepBbIM opy,nHeM, eCJIH ,nJIH BToporo Opy,nHH 9Ta Bepo-

HTHOCTb paBHa 0,75.

6.4.45. Pa60Ta HeKOToporo YCTpoiicTBa npeKpaTHJIacb H3-3a Bblxo,na H3 CTPOH o,nHoro H3 qeTbIpex 6JIOKOB. IIPOH3BO,nHTCH nOCJIe,nOBaTeJIbHM 3aMeHa Hay,naqy B3HToro 6JIOKa HOBbIM ,no Tex nop, nOKa YCTpoiicTBO He HaqHeT pa60TaTb (HOBbIe 6JIOKH He 3aMeHHIOTCH).

KaKOBa BepoHTHoCTb Toro, 'ITOnpH,neTCH 3aMeHHTb:

a) O,nHH 6JIOK; 6) ,nBa 6JIOKa;

B)qeTbIpe 6JIOKa?

6.4.46.IIPH aBTOMaTHqeCKOM H3rOTOBJIeHHH 60JITOB ,nonycKaeTcH B cpe,n- HeM 3% 6paKa. KaKoBa BepOHTHOCTb Toro, 'ITOcpe,nH B3HTbIX ,nJIH KOHTPOJIH 5 60JITOB He OKruKeTCH

Ha,J:l;O npH06pecTH, qT06bI C BepOHTHOCTbIO, He MeHbIIIeii 0,9, 6bITb YBepeHHbIM, 'ITOBbmrpaeT XOTH 6bI O,nHH 6HJIeT?

6.4.48. CTpeJIOK npOH3Bo,nHT BbICTpeJI no MHIIIeHH ,no nepBoro nOna,J:l;aHHH. BepoHTHoCTb nona,J:l;aHHH npH O,nHOM BbICTpeJIe paBHa 0,5. KaKOBa BepOHTHOCTb TOro, 'ITOno MHIIIeHH 6y,neT npOH3Be,neHO:

6.4.49.1'13 6YKB A, A, II, JI, M, H pa3pe3Hoii a36YKH BbI6HpaIOT Hay,naqy

no o,nHoii H CTaBHT B pH,n. HaiiTH BepOHTHOCTb Toro, 'ITOnOJIyqHTCH

311

6.4.50. IIoKyrraTeJIb Hm;eT Heo6xo,nHMYIO eMY Bem;b, 06XO,nH TpH Mara3H-; Ha. BepoHTHoCTb HaJIH'"IHHee B Ka>K,nOM Mara3HHe paBHa 0,2. qTO BepOHTHee - Hafi,neT OH HCKOMYIO Bem;b HJIH HeT?

6.4.51. HafiTH BepoHTHoCTb Toro, '"ITO3aKa3aHHbIfi (B ,naHHbIfi rrpOMe)KYTOK BpeMeHH) Me)K.nyropo,nHbIfi pa3rOBOp He COCTOHTCH, eCJIH BepoHTHOCTb 3aHHTOCTH Bcex KaHaJIOB CBH3H B 9TOT rrpOMe)KYTOK pasHa

0,7, a BepOHTHOCTb OTCYTCTBHH BbI3bIBaeMoro JIHlla paBHa 0,4. 6.4.52. CTy,neHT MO)KeT ,no6paTbcH ,no HHcTHTYTa HJIli aBTo6ycOM, KOTo-

PbIfi XO,nHT '"Iepe3Ka>K,nbIe 20 MHH, HJIH TpoJIJIefi6ycoM, KOTOPbIfi XO,nHT '"Iepe3Ka>K,nbIe 10 MHH. HafiTH BepOHTHOCTb Toro, '"ITOCTy- ,neHT, rro,nowe,nwHfi K OCTaHOBKe, ye,neT B Te'"leHHe 6JIH)KafiwHX

5 MHH?

KOHTponbHble Bonpocbl III 60nee CnO)l(Hbie 3aACIHIIIJI

6.4.53. ,I1;oKa3aTb cBoficTBa YCJIOBHbIX BepoHTHocTefi:

a) P(O IB) = 1;

6)P(A IB) = 1- P(A IB);

B)P(A + C IB) = P(A IB) + P(C IB) - P(AC IB).

6.4.54.BepHo JIH paBeHCTBO P(A IB) + P(A IB) = I?

6.4.55.B ceMbe ,nBoe ,neTefi. C'"IHTaH,'"ITOpO)K,neHHe MaJIb'"lHKaH ,neBO'"l- KH - He3aBHCHMbIe H paBHOBepOHTHbIe C06bITHH, BbI'"IHCJIHTbBePOHTHOCTb Toro, '"ITO:

a)o6a pe6eHKa - MaJIb'"lHKH;

6)06a pe6eHKa - MaJIb'"lHKH, eCJIH H3BeCTHO, '"ITOB ceMbe eCTb

MaJIb'"lHK. 6.4.56. 3aBHcHMbI JIH:

a) HeCOBMeCTHbIe C06bITHH;

6) C06bITHH, 06pa3YIOm;He rrOJIHYIO rpyrrrry?

6.4.57. 113BeCTHO, '"ITO AB = 0, P(A) > 0, P(B) > 0. ,I1;oKa3aTb, '"ITO C06bITHH A H B 3aBHCHMbI.

6.4.58. Ka>K.nyIO H3 5 rraJIOK pa3JIaMbIBaIOT rrpOH3BOJIbHO Ha ,nBe '"IacTHKOPOTKYIO H ,nJIHHHYIO. 113 rrOJIY'"leHHbIX06JIOMKOB Hay,na'"lyo6pa3yIOT 5 «HOBbIX» rraJIOK. KaKOBa BepOHTHOCTb Toro, '"ITO:

a) 06JIOMKH 06'be,nHHeHbIB TOM BH,ne, B KaKOM OHH 6bIJIH rrepBOHa- '"IaJIbHO;

6) Bce ,nJIHHHble rraJIKH coe,nHHeHbI C KOPOTKHMH?

6.4.59. BpoweHbI 6eJIaH H '"IepHaHHrpaJIbHbIe KOCTH. KaKOBa BepOHTHOCTb Toro, '"ITOHa 6eJIofi KOCTH BbIIIa,neT 60JIbwe O'"lKOB,'"IeMHa '"IepHofi?

6.4.60. A60HeHT 3a6bIJI rrOCJIe,nHIOIO IIH<PPy TeJIe<poHa H Ha6HpaeT ee Hayra,n. KaKOBa BepoHTHoCTb Toro, '"ITO eMY rrpH,neTCH 3BOHHTb He 60JIee '"IeMB 5 MecT?

312

6.4.61. IhBecTHo, '"ITO

P(A IB) = P(A IB).

,ILoKa3aTb, '"ITOC06bITlUI A U B He3aBUCUMbI.

6.4.62. IIo,Ll;6pOIIIeHbI3 MOHeTbI. Orrpe,Ll;eJIUTb 3aBUCUMbI UJIU He 3aBUCUMbI {BbIIIa,LI;eHue pelliKU Ha rrepBoii MOHeTe};

B= {BbIIIa,LI;eHue XOTa 6bI o,Ll;Hoii peIIIKu}.

6.4.63.,ILJIa rrOBblIIIeHua Ha,LI;e:lKHOCTU P ,LI;aHHOrO rrpu60pa OH .rry6JIUpyeTca HeCKOJIbKUMU TaKUMU :lKe rrpu60paMu TaK, '"IT06bIrrOJIY'"leHHM CUCTeMa pa60TaJIa (oHa pa6oTaeT, eCJIU pa60TaeT XOTa 6bI O,Ll;UH U3 rrpu60poB). CKOJIbKO rrpu60poB Ha,Ll;O B3aTb, '"IT06bIrrOBblCUTb em Ha,Ll;e:lKHOCTb ,LI;O 3a,L1;aHHOii BepoaTHOCTU PI?

6.4.64.B ypHe ,LI;Ba 6eJIbIX U TpU '"IepHbIXIIIapa. ,ILBa urpoKa rroO'"lepe,Ll;HO BbIHUMaIOT (6e3 B03Bpa.lIl;eHua) rro O,Ll;HOMY IIIapy. BbmrpbIBaeT TOT, KTO rrepBbIM BbITa.IIJ;UT 6eJIblii IIIap. RaitTU BepoaTHocTb Toro, '"ITO Bbm~paeT rrepBblii urpoK.

6.4.65.8JIeKTpU'"IeCKMu:errb COCTOUT U3 5 9JIeMeHTOB (puc. 69), BbIXO,Ll; U3

CTpoa KOTOPbIX B 3a,L1;aHHblii rrpOMe:lKYTOK BpeMeHU - He3aBUCUMble B cOBoKyrrHocTu C06bITua, UMeIO.lIl;Ue COOTBeTCTBeHHO BepoaTHOCTU qi (i = 1,2,3,4,5). RaiiTu BepoaTHOCTb pa3pbIBa u:erru.

Puc. 69

§ 5. QlOPMYJlA nOllHOIf1 BEPOSITHOCTlil. QlOPMYJlA

5EIf1ECA

TeopeMa 6.5. nYCTb Co6blTMe A MO>KeT npoM3oi'iTMTOJlbKO C OAHMM M3 C06bITMi'i

HI, H2, ... , Hn , 06pa3YIO~Mx nOJlHYIO rpynny nonapHo HeCOBMeCTHblX C06bITMi'i, n

T. e. Hi' Hj = 0, i =1= j M E Hi = n.

i=l

TorAa BepO!lTHOCTb C06bITM1I A Bbl'lMCJl1leTC1I no q,0pMyne nonHoiir sepORTHOCTVI

313

TeopeMa 6.6. EC11111 B pe3Y11bTaTe onblTa ocy~ecTBIII110Cb Co6blTille A, TO npe>KHllle,

AOOnblTHbie (11111111 anpVlOpHble) BepOllTHOCTIll rillnOTe3 P(Hl)' . .. , P(Hn ) A011>KHbl

6blTb 3aMeHeHbi Ha HOBbie, nOC11eonblTHble (11111111 anOCTepVlopHble) BepOllTHOCTIll

P(H1 I A), . .. , P(Hn I A), KOTopble Bbl'lIllC11l1IOTClI no q,0pMyne ceifeca:

P(H. I A) = P(Hi) . P(A I Hi)

• P(A)

(i = 1,2, ... , n), rAe BepOllTHOCTb P(A) Bbl'lIllC11l1eTClI no ct>0pMY11e (5.1).

6.5.1.45% TeJIeBU30pOB, UMeIOm;UXCH B Mara3UHe, U3rOTOBJIeHbI Ha 1-M

JIeBU30p U3rOTOBJIeH Ha 2-M 3aBO.IIe}, H3 = {TeJIeBU30p U3rOTOBJIeH Ha 3-M 3aBO.IIe}.

C06bITUH HI, H2, H3 06pa3YIOT 1I0JIHYIO rpylllly HeCOBMeCTHbIX C06bITUti, IIpU 9TOM: P(HI) = 0,45; P(H2) = 0,15; P(H3) = 0,40. ()J,JIH KOHTPOJIH

MO:lKHO HatiTu CyMMy BepoHTHocTeti rUIIOTe3; OHa .IIOJI)KHa paBHHTbCH e.IIUHU- n

u;e: ~ P(Hi ) = 0,45 + 0,15 + 0,40 = 1).

H4 = {06a IIpueMHUKa He pa6oTaIOT}. C06bITue A MO:lKeT IIpou30tiTU TOJIbKO C O.IIHOti U3 9TUX rUllOTe3. Rati.IIeM BepOHTHOCTb 9TUX rUIIOTe3, paCCMaTpuBM

314

4

(KOHTPOJIb: l: P(Hi ) = 0,09 + 0,09 + 0,81 + 0,01 = 1.) i=l

YCJIOBHbIe BepOHTHOCTH P(A I Hi) COOTBeTCTBeHHO paBHbI: P(A I Hd = =0,8; P(A IH 2 ) = 0,8; P(A I H3) = 0,8+0,8-0,8·0,8= 0,96; P(A I H4 ) = 0.

6.5.3.MMelOTcH )J.Be O)J.HHaKOBbIe ypHbI C mapaMH. B 1-il: HaxO)J.HTCH 3 6eJIbIX H 4 qepHbIX mapa, BO 2-il: - 2 6eJIbIX H 3 qepHbIX. M3 HaY)J.aqy BbI6paHHoil: YPHbI BbIHHMalOT O)J.HH map. KaKOBa BepOHTHOCTb TOro, qTO 9TOT map 6eJIbIil:?

6.5.4.CTY)J.eHT 3HaeT 24 6HJIeTa H3 30. B KaKOM CJIyqae BepOHTHOCTb BbITaIlI,HTb CqacTJIHBbIil: 6HJIeT )J.JIH Hero 60JIbme, eCJIH OH H)J.eT c)J.a-

BaTb 9K3aMeH IIepBbIM HJIH eCJIH - BTOpbIM?

6.5.5.Ha pHcYHKe 70 H306pruKeHa cxeMa )J.opor. TYPHCTbI BbIXO)J.HT H3 "YHKTa A, BbI6HPM Hayra.n. Ha pa3BHJIKe )J.opor O)J.HH H3 B03MO)KHbIX "YTeil:. KaKOBa BepOHTHOCTb Toro, qTO OHH IIOIIa,ITyT B "YHKT B?

Puc. 70

6.5.6.TPH CTpeJIKa "pOH3BeJIH IIO O)J.HOMY BbICTpeJIY IIO HaMeqeHHoil: u;e-

JIH. BepoHTHoCTb IIOIIa)J.aHHH 1-M CTpeJIKOM paBHa 0,6, 2-M - 0,7,

3-M - 0,8. IIPH O)J.HOM IIOIIa.n.aHHH B MHmeHb BepOHTHOCTb IIopa-

315

:lKeHHH u:eJIH paBHa 0,2, IIpH ,nBYX - paBHa 0,6, IIpH Tpex - u:eJIh

3aBe,nOMO IIOpa:lKaeTCH. HaihH BepOHTHOCTh IIOpa:lKeHHH u:eJIH.

6.5.7.IbBecTHo, qTO B cpe,nHeM 95% BhIIIycKaeMoit IIpO.nyKU:HH y,nOBJIeTBopHeT CTaH,napTy. YIIp0III;eHHaH cxeMa KOHTPOJIH "pH3HaeT IIpHro,nHoit IIpO.nyKU:HIO C BepOHTHOCThlO 0,96, eCJIH OHa CTaH,napTHa, H C BepOHTHOCThlO 0,06, eCJIH OHa HeCTaH,napTHa. HaitTH BepOHTHOCTh Toro, qTO B3HToe Hay,naqy H3,neJIHe IIpoit,neT YIIpoIII;eHHhlit KOHTPOJIh.

H4 = {oTKa:lKeT O,nHH 9JIeMeHT: I-it, He OTKa:lKYT 2-it, 3-it}j

H5 = {OTKa:lKeT O,nHH 9JIeMeHT: 2-it, He OTKa:lKYT I-it, 3-it}j

H6 = {oTKa:lKeT O,nHH 9JIeMeHT: 3-it, He OTKa:lKYT I-it, 2-it}j

H7 = {Bce 9JIeMeHThI, 6y.nyT pa6oTaTh}.

nOJIh3YHCh "PaBHJIOM YMHO:lKeHHH BepoHTHocTeit ,nJIH He3aBHCHMhIX co6hITHit, Hait,neM BepoHTHoCTH 9THX rHIIOTe3:

P(Ho) = 0,2·0,4·0,3= 0,024j

P(HI) = 0,2 . 0,4 . 0,7 = 0,056j

P(H2 ) = 0,2·0,3·0,6= 0,036j

P(H3) = 0,4·0,3. 0,8 = 0,096j

P(H4 ) = 0,2·0,6·0,7= 0,084j

P(H5) = 0,4·0,8·0,7= 0,224j

P(H6) = 0,3·0,8·0,6= 0,I44j

P(H7 ) = 0,8·0,6·0,7= 0,336.

7

(KOHTPOJIh: E P(Hi ) = 0,024 + 0,056 + ... + 0,336 = 1.) i=O

YqHThIBaH, qTO B pe3YJIhTaTe OIIhITa "POH30IIIJIO C06hITHe A, KOTopoe HeB03MO:lKHO IIpH rHIIoTe3ax H4, H5 , H6 , H7 H ,nOCTOBepHO IIpH rHIIoTe3ax

Ho, HI, H2 , H3 , Hait,neM YCJIOBHhle BepoHTHoCTH C06hITHit P(A I Hi):

P(A I Ho) = 1, P(A IHI) = 1, P(A IH2 ) = 1, P(A IH3 ) = 1, P(A IH4) = 0, P{A I H5 ) = 0, P(A I H6 ) = 0, P(A I H7) = 0.

Hait,neM BepoHTHoCTh rHIIOTe3hI HI IIpH YCJIOBHH, qTO C06hITHe A IIpoH30IIIJIO (T.e. P(HI I A)) IIO ¢opMYJIe Beiteca. )];JIH 9Toro IIpe,nBapHTeJIhHO

316

Haii,neM BepOaTHOCTb C06bITHa A IIO <popMYJIe (5.1)

7

P(A) = L P(Hi ) . P(A IHd =

i=O

= 0,024 . 1 + 0,056 . 1 + 0,036 . 1 + 0,096 . 1 + °+ °+ °+ °= 0,212.

6.5.9.IIpe,nIIoJIO)KHM, qTO 5% MY)KqHH H 0,25% Bcex )KeHIIIHH ,naJIbTOHHKH. Hayra,n BbI6paHHoe JIHll;o OKa3aJIOCb ,naJIbTOHHKOM. CqHTM, qTO MY)KqHH H )KeHIIIHH o,nHHaKOBoe KOJIHqeCTBO, HaiiTH BepoaTHOCTb Toro, qTO 9TOT qeJIOBeK:

r,na B KaqeCTBe rHIIOTe3 "pHMeM C06bITHa HI = {BbI6paHHblii qeJIOBeK -

,ILrra HaxO)K,neHHa HCKOMbIX BepoaTHOCTeii, T. e. YCJIOBHbIX BepoaTHOCTeii

P(HI I A) H P(H2 I A), BOCIIOJIb3yeMca <p0pMYJIoii Beiieca. CHaqaJIa no <p0PMYJIe IIOJIHOii BepoaTHOCTH Haii,neM P(A): T. K. IIO YCJIOBHIO P(A IHd = 0,05 H. P(A I H 2) = 0,0025, TO P(A) = 0,5 . 0,05 + 0,5 ·0,0025 = 0,02625. CJIe,no-

6.5.10.CHcTeMa 06HapY)KeHHa CaMOJIeTa H3-3a HaJIHqHa IIOMex B 30He ,neiicTBHa JIOKaTOpa MO)KeT ,naBaTb JIO)KHble IIOKa3aHHa C BepoaTHOCTbIO 0,05, a IIpH HaJIHqHH u;eJIH B 30He CHCTeMa 06Hap~HBaeT ee C BepoaTHOCTbIO 0,9. BepoaTHocTb IIOaBJIeHHa "POTHBHHKa B 30He paBHa 0,25. OIIpe,neJIHTb BepoaTHOCTb JIO)KHoii TpeBorH.

6.5.11.B YCJIOBHaX 3a,naqH 6.5.7 B3aToe H3,neJIHe_ "POIllJIO y"poIIIeHHblii KOHTPOJIb. HaiiTH BepoaTHOCTb TOro, qTO OHO CTaH,napTHoe. A eCJIH H3,neJIHe ,nBa)K,nbI "POIllJIO YIIpoIIIeHHblii KOHTPOJIb?

6.5.12.B yCJIOBHaX 3a,naqH 6.5.5 TypHCTbI IIPHIllJIH B "YHKT B. KaKOBa BepoaTHOCTb TOro, qTO OHH IIOIllJIH IIO ,nopore N! 3?

6.5.13.B Mara3HH "ocTy"aIOT o,nHHaKOBble H3,neJIHa C Tpex 3aBo,nOB, IIpH-

qeM l-ii 3aBO,n IIOCTaBHJI 50 H3,neJIHii, 2-ii - 30, 3-ii - 20 H3,neJIHii.

Cpe,nH H3,neJIHii l-ro 3aBo,na 70% IIepBocopTHbIX, a cpe,nH H3,ne.rmii

317

2-1'0- 80%,3-1'0- 90% rrepBocopTHbIX. KyrrJIeHO O,II.HO H3,II.eJIHe. OHO OKa3aJIOCb rrepBocopTHbIM. KaKOBa BepOHTHOCTb Toro, qTO 9TO H3,II.eJIHe BbIlIyrn;eHo 1-M 3aBO,II.OM?

6.5.14.IIepe,II. rroceBOM 80% Bcex ceMHH 6bIJIO 06pa6oTaHo H,II.OXHMHKaTaMH. BepoHTHoCTb rropruKeHHH pacTeHHil:, rrpopOCWHX H3 9THX ceMHH, Bpe,II.HTeJIHMH pasHa 0,06, a pacTeHHil:, rrpopOCWHX H3 Heo6pa60TaHHbIX ceMHH - 0,3. KaKoBa BepOHTHOCTb Toro, qTO B3HToe

HaY,II.aqy pacTeHHe OKruKeTCH rropruKeHHbIM? ECJIH OHO rropruKeHHoe, TO KaKOBa BepOHTHOCTb Toro, qTO OHO Bblparn;eHO H3 06pa6oTaHHoro ceMeHH?

6.5.15.B cTY,II.eHqeCKoil: rpyrrrre 70% - IOHOWH. 20% IOHoweil: H 40% ,II.eByweK HMeIOT COTOBblil: TeJIeq,oH. IIocJIe 3aHHTHil: B ay,II.HTOpHH 6bIJI, Hail:,II.eH KeM-TO 3a6bITblil: TeJIeq,oH. KaKOBa BepOHTHOCTb TOro, qTO OH rrpHH8,II,JIe)KaJI

6.5.16.,I1;Ba CTpeJIKa He3aBHCHMO ,II.pyr OT ,II.pyra C,II.eJIaJIH rro O,II.HOMY BbICTpeJIY B MHweHb. BepoHTHoCTH HX rrOrra,II.aHHH B MHweHb COOTBeTCTBeHHO paBHbI 0,75 (1-il: CTpeJIOK) H 0,80 (2-il: CTpeJIOK). IIocJIe CTpeJIb6bI B MHweHH 06HapY)KeHa O,II.Ha rrp060HHa. KaKoBa BepoHTHOCTb Toro, qTO B MHweHb rrorraJI 2-il: CTpeJIOK?

6.5.17.B ypHy, cO,II.ep)Karrr,yIO 100 wapOB, orryrn;eH KpacHblil: wap, rrOCJIe qero H3 Hee HaY,II.aqy BbIHHMaIOT wap. 1) KaKOBa BepOHTHOCTb Toro, qTO OH KpacHblil:? 2) lhBecTHo, qTO H3 ypHbI BbIHyT KpacHblil: wap. KaKOBa BepoHTHoCTb Toro, qTO B Heil: CO,II.ep)KaJIOCb 44 KpacHbIX wapa?

(Bce rrpe,II.rrOJIO)KeHHH 0 rrepBOHaqaJIbHOM KOJIHqeCTBe KpacHbIX wapOB B ypHe paBHOB03MO)KHbI.)

6.5.18.BoeHHblil: KOpa6JIb MO)KeT rrpoil:TH B,II.OJIb rrpOJIHBa WBpIfHOil: 1 KM e MHHHbIM 3arpruK,II.eHHeM B JII060M MeCTe. BepoHTHoCTb ero rrO,II.pbIBa Ha MHHe B rrpasoil: qaCTH 3arpruK,II.eHHH wHPHHOil: 200 M paBHa 0,3, aHa OCTaJIbHOil: qacTH - 0,8. Hail:TH BepOHTHOCTb Toro, qTO KOpa6JIb 6JIarOrrOJIyqHo rrpoil:,II.eT rrpOJIHB.

6.5.19.Ha C60PKY rrorra,naIOT ,II.eTaJIH C Tpex aBToMaToB.lhBecTHo, qTO 1-il:

aBTOMaT ,II.aeT 0,25% 6paKa, 2-il: - 0,40%, 3-il: - 0,60%. KaKOBa BepOHTHOCTb rrorra,naHHH Ha C60PKY 6paKOBaHHoil: ,II.eTaJIH, eCJIH C 1-1'0aBTOMaTa rrocTyrrHJIO 2000, co 2-ro - 1500 H C 3-1'0- 1300 ,II.eTaJIeil:?

rn;aIOTCH. KaKOBa BepOHTHOCTb Toro, qTO BTOPOil: wap - 6eJIblil:?

318

6.5.21. ,ILBe 9JIeKTpHqeCKHe I.J;eIIH CO,ll;ep:lKaT COOTBeTCTBeHHO 3 H 49JIeMeH-

Ta (pHC. 71). BbIXO,ll; H3 CTPOH 9THX 9JIeMeHTOB - He3aBHCHMble C06bITHH, HMeIOm;He BepOHTHOCTH PI = 0,1, P2 = 0,2, P3 = 0,3 (l-H I.J;errb); P4 = P5 = P6 = P7 = 0,4 (2-H I.J;errb). HaY,ll;aqy BbI6HpaeTcH O,ll;Ha I.J;errb. KaKOBa BepoHTHoCTb Toro, qTO OHa pa6oTaeT?

Puc. 71

6.5.22. B l-il: ypHe HaXO,ll;HTCH 7 6eJIbIX H 5 qepHbIX rrrapOB, a BO 2-il: -

4 6eJIbIX H 8 qepHbIX. 1'13 rrepBoil: YPHbI HaY,ll;aqy rrepeKJIa,ll;bIBaIOT BO BTOpyIO 2 rrrapa, a 3aTeM H3 2-il: YPHbI H3BJIeKaIOT O,ll;HH rrrap. KaKOBa BepOHTHOCTb Toro, qTO OH OKa:lKeTCH 6eJIbIM?

6.5.23. llJIaHHPyeTcH paKeTHblil: 3aJIrr rro KOpa6JIIO rrpOTHBHHKa. BepoHT-

xaMH, TO YCTPOil:CTBO perHcTpHpyeT HaJIHqHe CHrHaJIa C BepOHTHOCTbIO PI; eCJIH TOJIbKO rrOMeXH - C BepOHTHOCTbIO P2. KaKOBa BepOHTHOCTb Toro, qTO YCTPOil:CTBO 3aperHcTpHpyeT KaKoil:-To CHrHaJI?

319