Сборник задач по высшей математике 2 том

6.2.36. ,I1;oKa3aTb, qTO:

a) AB + B = B - A;

6)B = A, eCJUI A· B = 0 H A· B = 0.

6.2.37.8JIeKTpHqeCKaH IJ;errb C BbIKJIlOqaTeJIHMH COCTaBJIeHa rro cxeMe, rrpHBe,n:eHHofi Ha pHCYHKe 62. IIycTb Co6bITHe Ai = {BKJIlOqeH BbIKJIIOqaTeJIb C HOMepoM i }, i = 1,2, ... ,5.

a),I1;JIH cxeMbI pHC. 62 a 3arrHcaTb qepe3 Ai Co6bITHe A = {TOK H,n:eT};

6)MH cxeMbI pHC. 626 3arrHcaTb qepe3 Ai C06bITHH AHA.

Puc. 62

6.2.38. IIycTb A, B, C - CJIyqafiHble C06bITHH, rrpHqeM A H B HeCOBMeCT-

HbI. IIoKa3aTb, qTO C06bITHH AC H BC TaK)Ke HeCOBMeCTHbI.

KOHTponbHble BonpOCbl III 60nee CnO>KHble 3aAa'"llil

6.2.39.1:13 KOJIO,n:bI HrpaJIbHbIX KapT (Bcero HX 36) H3BJIeKalOT O,n:Hy. Co-

CTaBHTb He MeHee ,n:ByX rrpOCTpaHCTB 9JIeMeHTapHbIX C06bITHfi MH ,n:aHHoro orrbITa.

6.2.40. CKOJIbKO C06bITHfi MO)KHO COCTaBHTb MH rrpOCTpaHCTBa

6.2.41. IIo,n:6pacbIBalOTcH 3 MOHeTbI. CKOJIbKO HMeeTCH paBHOB03MO)KHbIX HCXO,n:OB ,n:aHHoro orrbITa? CocTaBHTb C06bITHH, 06pa3ylOIu;He rrOJIHylO rpyrrrry. IIpHBecTH rrpHMepbI C06bITHfi, He o6pa3yIOIu;Hx rrOJIHyIO rpyrrrry. YKa3aTb rro,n:MHO)KeCTBa MHO)KeCTBa fl, COOTBeTCTBYIOIu;He C06bITHHM: A - BbIIIaJIO He 60JIee o,n:Hofi peIIIKH; B - BbI-

rraJIO POBHO ,n:Ba rep6a.

6.2.42. M3BecTHo, qTO C06bITHH Al H A2 rrpOH30IIIJIH, a Co6bITHe A3 He

rrpOH30IIIJIO. IIPOH30IIIJIH JIH C06bITHH:

290

6.2.44. 8JIeKTpllqeCKaH u;enb COCTaBJIeHa no cxeMe, npllBe,neHHoil: Ha p"C. 63. lIycTb C06bITllH Ai, i = 1,2,3,4,5, COCTOHT B TOM, qTO o,nHOllMeHHble 9JIeMeHTbI pa6oTaroT 6e30TKa3HO B TeqeHlle BpeMeHll T. C06bITlle B = {cxeMa pa60TaeT 6e30TKa3HO B TeqeHlle Bpe-

Bblpa311Tb C06bITllH B II Ii qepe3 C06bITllH Ai'

a)A·X=A+X;

6.2.48. CnpaBe,nJIllBbI JIll CJIe,nyroIIJ;lle paBeHCTBa:

a) A + A = A; 6) A . A = A;

B) A+B = A·B?

6.2.49.lIPII KaKOM YCJIOBllli cnpaBe,nJIllBO paBeHCTBO (A + B) - B = A?

6.2.50.,Il;oKa3aTb, qTO (A + B) - B = A-B.

6.2.51.lIoKa3aTb, qTO eCJIll B ~ A, TO (A - B) + B = A.

6.2.52. ,Il;oKa3aTb, qTO A - B = 0 ¢:} A ~ B.

§ 3. BEPOHTHOCTb CJ1YYAVlHOrO C06blTlJIH

KnaCCM'·leCKoeonpE!AeneHMe BepOSITHOCTM

Bep0Jl.mxocm'b C06bITHH 'lHCJIeHHOXapaKTepH3yeT CTeneHb B03MO)KHOCTH ero nOHBJIeHHH B paccMaTpHBaeMOM onbITe.

~ IIycTb npOH3Bo.n;HTCH onbIT C n paBHOB03MO)KHbIMH Hcxo.n;aMH, o6pa3yIOIllHMH nOJIHyIO rpynny HeCOBMeCTHbIX C06bITHiI:. TaKHe HCXO.n;bI Ha3bIBaIOTCH 3.1/,e.MeXmap- H.'b/,.MU ucxoiJa.MU (co6'b/,mUJI..Mu), C.l/,y"taJI..MU, waxca.Mu. CJIY'lail:,KOTOPbIiI: npHBo.n;HT

l( HacTynJIeHHIO C06bITHH A, Ha3bIBaeTCH 6.1/,azonpUJI.mx'b!.M (HJIH 6.1/,azonpUJI.mcm6YIOU{U.M) eMy. ~

291

~BepoSlm'Hocm'b'lO co6'b/,mUSl A Ha3bIBaeTCH OTHOIIIeHHe '1HCJIam CJIY'IaeB,6JIaro-

npHHTcTByIOIIIHX 3TOMY c06b1THIO, K o6IIIeMY '1HCJIYn CJIY'IaeB.

peA) = r;:.

TaKoe onpe.n;eneHHe BepoHTHoCTH Ha3bIBaeTCH lC.I!acCU"tecICUM.

1'13 KJIaCCH'IeCKoroonpe.n;eneHHH CJIe.n;yIOT CBoil:cTBa BepOHTHOCTH: 0 ~ peA) ~

~ 1; P(0) = 0; P(Q) = 1; peA) = 1 - peA); peA + B) = peA) + PCB), eCJIH A·B=0.

reOMeTplII'"IeCKOeonpE!AeneHllle BepORTHOCTIII

0606IIIeHHeM nOHHTHH «KJIaccH'IeCKoil:BepOHTHOCTH» Ha CJIY'Iail:onblTOB C 6ecKOHe'lHblM(Boo6IIIe rOBopH, HeC'IeTHblM)'1HCJIOMHCXO.n;OB HBJIHeTCH nOHHTHe «reoMeTpH'IeCKoil:BepOHTHOCTH». K 3TOMY nOHHTHIO npHBo.n;HT 3a.n;a'lHHa no.n;C'IeTBePOHTHOCTH nona.n;aHHH TO'lKHB HeKYIO 06JIaCTb (OTpe30K, '1aCTbnJIOCKOCTH, '1acTb Tena H T. .n;.).

IIycTb npOCTpaHCTBO 3JIeMeHTapHblX C06b1THiI: Q npe.n;CTaBJIHeT co6oil: HeKOTOpyIO 06JIaCTb ITJIOCKOCTH. Tor.n;a B Ka'leCTBeC06b1THiI: MorYT paCCMaTpHBaTbCH 06JIacTH A, co.n;ep:lKaIIIHeCH B Q.

~ BepoHTHoCTb nona.n;aHHH B 06JIacTb A TO'IKH, Hay.n;a'lYBbl6paHHoil: H3 o6JIaCTH Q, Ha3b1BaeTCH eeoMempu"teclCoi1. 6epOSlm'HOCm'b'lO C06b1THH A H HaxO.n;HTCH no

<popMYJIe

SeA) peA) = S(Q)'

r.n;e SeA) H S(Q) nJIOIIIa.n;H 06JIacTeil: A H Q COOTBeTCTBeHHO.

CJIY'lail:,Kor.n;a Q npe.n;CTaBJIHeT co6oil: OTpe30K HJIH TpeXMepHYIO 06JIacTb, pac-

CMaTpHBaeTCH aHaJIOrH'IHO.

AKCIIIOMaTIII'"IeCKOeonpE!AeneHllle BepORTHOCTIII

IIycTb Q - MHO:lKeCTBO Bcex B03MO:lKHblX HCXO.n;OB HeKOToporo onbITa (3KcnepHMeHTa). COrJIacHo aICCUOMamU"teCICOMY onpeiJe.l!eHU'IO 6epOSlm'HOCmU, Ka:lK.n;OMY CO6b1THIO A (A - no.n;MHO:lKeCTBO MHO:lKeCTBa Q) CTaBHTCH B COOTBeTCTBHH HeKOTOpoe '1HCJIO peA), Ha3b1BaeMOe BepOHTHOCTbIO C06b1THH A, npH'IeMTaK, 'ITOBblnOJIHH-

IOTCH CJIe.n;yIOIIIHe TpH YCJIOBHH (aKCHOMbI BepOHTHocTeil:):

eCJIH Ai .Aj = 0 (i =f. j), T. e. BepOHTHOCTb CYMMbI nonapHO-HeCOBMeCTHblX Co6b1THiI: paBHa CYMMe BepOHTHocTeil: 3THX C06b1THiI:.

292

lh aKCHOM (3.1)-(3.3) BhITeKalOT OCHOBHhIe cBoiicTBa BepOHTHOCTH:

1.P(0) = 0, T. e. BepOHTHOCTb HeB03MO)KHOrO C06hITHH paBHa HYJIIO.

2.P(A) + P(A) = 1.

3.0 ~ P(A) ~ 1 .LI.J1H JII060ro C06hITHH A.

4.P(A) ~ P(B), eCJIH A ~ B.

6JIarOnpHHTcTBYIOlllHX C06blTHIO A HCXO,n;OB), n - 'IHCJIO9JIeMeHTOB MHO)KeCTBa n

(,{HCJIO Bcex HCXO,n;OB onbITa).

6.3.1.B ypHe CO,Il;ep:>K11TCH 5 6eJIbIX 11 4 qepHbIX IIIapa, pa3JI11qaI01lI11XCII TOJIbKO IIBeTOM.

1)BbIH11MaeTCH HaY,Il;aqy O,Il;11H IIIap. Hail:T11 BepoHTHoCTb Toro, 'ITO OH 6eJIblil:.

2)BbIH11MaIOTCH HaY,Il;aqy ,Il;Ba IIIapa. Hail:T11 BepoHTHoCTb Toro, 'ITO:a) o6a IIIapa 6eJIble; 6) XOTH 6bI O,Il;11H 113 H11X qepHblil:.

Q 1) IIepeHYMepyeM IIIapbI. IIpocTpaHcTBo 9JIeMeHTapHblx C06bIT11il: MO:>KHO 3almcaTb B B11,Il;e n = {Bl, B 2 , B 3 , B 4 , B 5 , ql, q2, q3, q4,}'IIycTb C06bIT11e

A = {noHBJIeH11e 6eJIoro IIIapa}, TOr,Il;a A = {BI' 2 , B 3 , B 4 , B5}.

TaK KaK Bce 9JIeMeHTapHble 11CXO,Il;bI paBHOB03MO:>KHbI, TO no KJIacC11qeCKOMY onpe,Il;eJIeH111O BepOHTHOCT11 P(A) = r:: = ~.

2) IIp11 BbIH11MaH1111 ,Il;ByX IIIapoB B03MO:>KHbI TaK11e 11CXO,Il;bI: (BI'qr), (B2'B 3), (B3, B 2), (q4, B 5 ) 11 T.,Il;. Q11CJIO Bcex CJIyqaeB paBHo n = A~ =

==9·8 = 72.

a)HCXO,Il;aM11, 6JIarOnp11HTcTBYI01lI11M11 HacTynJIeH1110 C06bIT11H B = {noHBJIeH11e ,Il;ByX 6eJIbIX IIIapoB}, HBJIHIOTCH (BI' 2), (BI' 3), (B3, B 5 ), (B3, B I )

Ii T.,Il;. Q11CJIO TaK11X CJIyqaeB paBHO m = Ag = 5 ·4 = 20. IIo9ToMY P(B) =

== r:: ~~ 5

'= 18 .

6) HCXO,Il;aM11, 6JIarOnp11HTcTBYI01lI11M11 HacTynJIeH1110 C06bIT11H C = {noHBJIeH11e XOTH 6bI O,Il;HOro qepHOrO IIIapa} , HBJIHIOTCH (BI'Qr), (BI'Q2),

(CM. nyHKT a), n09ToMY B OCTa.JIbHbIX CJIyqMX XOTH 6bI O,Il;11H 113 napbI IIIapOB

293

B) Cpe,Il;H HHX 2 CHHHX H 1 3eJIeHblii KapaH,Il;arn.

a CHaqaJIa 3aMeTHM, qTO qHCJIO cnoc060B BbI6paTb 3 KapaH,Il;arna H3 12

HMeroIu;HxcH B HaJIHqHH paBHO n = Cr2 = 220.

a) BbI6paTb 3 CHHHX KapaH,Il;arna H3 5 MO:lKHO Cl cnoc06aMH; 3 KpaCHbIX H3 HMeroIu;HxcH 4 MO:lKHO BbI6paTb Cr cnoc06aMH; 3 3eJIeHbIX H3 3 3eJIeHbIX -

C: cnoc06aMH.

ITo npaBHJIy CJIO:lKeHHH 06Iu;ee qHCJIO m CJIyqaeB, 6JIarOnpHHTcTByroIu;Hx C06bITHro A = {TpH KapaH,Il;arna, BbIHYTbIX H3 Kop06KH, O,Il;HOrO IJ;BeTa}, paBHO

B) ITYCTb C06bITHe C = {H3 Tpex BbI6paHHbIX KapaH,Il;arneii 2 CHHHX H 1

3eJIeHblii}. BbI6paTb 2 CHHHX KapaH,Il;arna H3 HMeroIu;HxcH 5 CHHHX MO:lKHO C~ cnoc06aMH, a 1 3eJIeHblii H3 HMeroIu;HxcH 3 3eJIeHbIX - C~ cnoc06aMH. OTcro,Il;a no npaBHJIy YMHO:lKeHHH HMeeM: m = C~ . C~ = 30. IT03TOMY P(C) = ~ =

KBeHHblX «CJIOB» (HlIJI, OJI5I, OHlI, JI5IM, MlIJI H T. ,Il;.). CJIOBO JIOM npH 3TOM nOHBHTCH JIHrnb O,Il;HH pa3, T. e. m = 1. IT03TOMY BepOHTHOCTb nOHBJIeHHH CJIOBa JIOM (c06bITHe A) paBHa P(A) = ~ = 1~0.

6) IIIecTH6YKBeHHble «CJIOBa» OTJIHqaroTcH ,Il;pyr OT ,Il;pyra JIHrnb nOpH,Il;- KOM pacnOJIO:lKeHHH 6YKB (HOJIM1I5l, 51HOJIlIM, OJIH1I5lM H T.,Il;.). lIx

6.3.4. BporneHbI 2 HrpaJIbHble KOCTH. HaiiTH BepoHTHoCTb Toro, qTO: a) cyMMa BbmaBrnHX OqKOB He npeBOCXO,Il;HT 7;

6) Ha 06eHX KOCTHX Bbma,Il;eT O,Il;HHaKOBoe qHCJ10 OqKOB;

294

B)IIpOH3Be,n;eHHe BbIIIaBIIIHX O'IKOB,n;eJIHTCH Ha 4;

r)XOTH 6bI Ha O,n;HOil: KOCTH BbIIIa,n;eT 6 O'IKOB.

6.3.5.Ko,n; ,n;oMo<poHa COCTOHT H3 8 IJ;H<PP, KOTopble MoryT IIOBTOpHTbCH. KaKoBa BepOHTHOCTb TOro, 'ITO,CJIY'Iail:HoHa6HpM IJ;H<pPbI, MO:>KHO yra,n;aTb Hy:>KHblil: KO,n;?

6.3.6.1h 6YKB A, C, H, H, A, A pa3pe3Hoil: a36YKH COCTaBJIHeTCH Hay- ,n;a'IyCJIOBO, COCTOHmee H3 6 6YKB. KaKOBa BepOHTHOCTb TOro, 'ITO IIO.nY'IHTCHCJIOBO «AHAHAC»?

6.3.7.BOCeMb ,n;pY3eil: paCIIpe,n;eJIHIOT MeCTa 3a KpyrJIbIM CTOJIOM IIO :>Kpe6HIO. KakOBa BepOHTHOCTb TOrO, 'ITO,n;Ba H3 HHX, a HMeHHO A H B,

6y.n;yT CH,n;eTb PH,n;OM?

6.3.8.)l;Boe ,n;py3eil:, A H B, CTOHT B O'Iepe,n;HH3 8 'IeJIOBeK.Hail:TH BepoHTHOCTb Toro, 'ITO:

a) A H B CTOHT pH,n;OM;

6) Me:>K.n;y A H B CTOHT ,n;Ba 'IeJIOBeKa.

6.3.9. Ha 5 KapTO'IKaxHaIIHcaHO IIO O,n;HOil: IJ;H<ppe H3 Ha60pa 1, 2, 3, 4, 5. Hayra,n; BbI6HpalOTcH ,n;Be KapTO'IKH.KaKoBa BepOHTHOCTb TOro, 'ITO'IHCJIOHa BTOPOil: KapTO'IKe60JIbIIIe, 'IeMHa IIepBoil:?

6.3.10.1h KOJIO,n;bI B 36 KapT H3BJIeKalOTCH Hay,n;a'Iy4 KapTbI. KaKoBa

BepOHTHOCTb C06bITHil:: A = {Bce H3BJIe'IeHHble KapTbI IIHKOBOil: MacTH}, B = {cpe,n;H 3THX 'IeTblpexKapT OKa:>KeTCH XOTH 6bI O,n;HH

KOpOJIb}?

6.3.11.Ih 40 BOIIPOCOB, BXO,n;HmHX B 3K3aMeHaIJ;HOHHble 6HJIeTbI, cTy,n;eHT

3HaeT 30. Hail:TH BepOHTHOCTb TOro, 'ITOcpe,n;H Tpex Hayra,n; BbI-

6paHHbIX BOIIPOCOB cTy,n;eHT 3HaeT:

a) 3 BOIIpoca; 6) 2 BOIIpoca;

B)1 BOIIpOC.

6.3.12.TPH 'IeJIOBeKaIIPOH3BOJIbHO pa3MemaroTcH B 8 BarOHax 3JIeKTpH'I- KH. KaKoBa BepOHTHOCTb Toro, 'ITOBce OHH:

B)pa3MeCTHTCH B pa3HbIX BaroHax?

6.3.13.12 'IeJIOBeK,cpe,n;H KOTOPbIX lleTpoB H llBaHoB, pa3MemalOTCH B ro-

CTHHHIJ;e, B KOTOPOil: eCTb O,n;HH 4-MecTHblil:, ,n;Ba 3-MecTHblx H O,n;HH 2-MecTHblil: HOMep. KaKoBa BepoHTHoCTb C06bITHH A, COCTOHmero

B TOM, 'ITOlleTpOB H llBaHoB IIOIIa,IJ;yT B 2-MecTHblil: HOMep?

6.3.14.Ha oTpe30K AB MHHbI a Hay,n;a'IyHaHeceHa TO'IKaC. Hail:TH Bepo-

HTHOCTb Toro, 'ITOMeHbiliHil: H3 OTpe3KOB AC Ii C B HMeeT ,n;JIHHy, 60JIbillYIO, 'IeM~.

Q PaCIIOJIO:>KHM OTpe30K AB Ha 'IHCJIOBOil:OCH OX TaK, KaK 3TO H306pa:>KeHO Ha pHCYHKe 65.

llYCTb x - Koop,n;HHaTa CJIY'Iail:Hoil: TO'IKH C. Tor,n;a IIpOCTpaHCTBO n

3JIeMeHTapHblx C06bITHil: MO:>KHO 3aIIHcaTb B BH,n;e n = {x: 0::;; x ::;; a} . .HcHO, 'ITOHCXO,n;OB OIIbITa (HaHeCeHHe TO'IKHC Ha OTpe30K AB) 6eC'IHCJIeHHOe MHO:>KeCTBO H Bce OHH paBHOB03MO:>KHbI.

295

Puc. 65

Pa306beM OTpe30K AB Ha 6 paBHbIX OTpe3KOB. OqeBH,n;Ho, qTO YCJIOBHe «MeHbIIIHii H3 OTpe3KOB AC H CB HMeeT ,1J.JIHHy, 60JIbIIIYIO, qeM ~» (C06bITHe

A) 6y,n;eT BbIIIOJIHeHO, eCJIH TOqKa Cnona,n;eT Ha OTpe30K M N 5:] .

TaKHM 06pa30M, 06JIaCTbIO, 6JIaronpHHTcTBYIOrn;eii HacTynJIeHHIO C06bITHH A

(Ha pHCYHKe 65 OHa 3aIIITpHXOBaHa), HBJIHeTCH OTpe30K MN, a MHOlKeCTBY Bcex HCXO,n;OB onbITa COOTBeTcTByeT OTpe30K AB. OTCIO,n;a

6.3.15. IIpoTHBHHK B TeqeHHe qaca ,n;eJIaeT O,n;HH ,n;ecHTHMHHYTHbIii HaneT

Ha yqaCTOK IIIocce. B TeqeHHe 3Toro )l(e qaca HY)l(HO npeo,n;OJIeTb 3TOT onaCHbIii yqaCTOK IIIocce. C KaKoii BepoHTHoCTbIO MO)l(HO H36e)l(aTb HaneTa, eCJIH BpeMH npeo,n;OJIeHHH onacHoro yqacTKa nHTb MHHYT?

a 0603HaqHM qepe3 x MOMeHT BpeMeHH, Kor,n;a HaqHHaeTCH BbIXO,n; Ha onacHbIii yqacTOK IIIocce, a qepe3 Y - MOMeHT BpeMeHH Haqana 06CTpeJIa 3Toro yqacTKa IIIocce. .HCHO, qTO 0 ~ X ~ 60, 0 ~ Y ~ 60.

By,n;eM paCCMaTpHBaTb x H Y KaK ,n;eKapTOBbI Koop,n;HHaTbI Ha nJIOCKOCTH.

Tor,n;a 3JIeMeHTapHbIe HCXO,n;bI B ,n;aHHOM OnbITe (OH COCTOHT B <pHKcaU;HH BpeMeHH Haqana ,n;eiicTBHii 06eHx CTOpOH), H306pa3HTcH TOqKaMH (x, y) KBa,n;paTa co CTOPOHOii T = 60, T.e. 0 = {(x,y): 0 ~ x ~ 60, 0 ~ y ~ 60}.

IIHTepecyIOrn;ee Hac C06bITHe A = {y,n;acTCH H36e)l(aTb HaneTa} HacTynHT Tor,n;a H TOJIbKO Tor,n;a, Kor,n;a HaneT HaqHeTCH cnycTH nHTb (HJIH 60JIbIIIe) MHHyT nOCJIe BbIxo,n;a Ha onacHbIii yqaCTOK JIH60 HaqHeTCH 3a ,n;ecHTb (H 60JIee) MHHyT ,n;o Haqana npeo,n;OJIeHHH yqaCTKa IIIocce, T. e. ,n;OJI)I(HO BbIIIOJIHHTbCH

O,n;HO H3 YCJIOBHii

[ y - x> 5, x - y > 10.

8TH HepaBeHCTBa onpe,n;eJIHIOT 6JIarOnpHHTCTBYIOrn;yIO C06bITHIO A 06JIacTb

D, 3aIIITpHXOBaHHYIO Ha pHCYHKe 66.

IIJIorn;a,n;b 06JIaCTH D paSHa 8(D) = ~ ·50·50 + ~ ·55·55 = 2762,5; nJIOrn;a,n;b KBa,n;paTa 0 paBHa 8(0) = 60·60 = 3600.

296

6.3.16. KaKoBa BepOjiTHOCTb TOro, qTO KOpHH ypaBHeHHjI x 2 +px + q = °

6Y.IU'T,1l.eitcTBHTeJIbHbIMH, eCJIH K09<P<PHIIHeHTbI p H q ypaBHeHHjI BbI6HpalOTCjI HaY,1l.aqy H3 oTpe3Ka [0,1]7

Q BY,1l.eM paCCMaTpHBaTb MHO>KeCTBO Bcex B03MO:>KHbIX rrap qHCeJI (p, q) KaK

p2 _ 4q ~ 0, T. e. q ::;; ~p2. OTCIO,1l.a jlCHO, qTO MHO:>KeCTBO TOqeK KBa,1l.paTa, 6JIarOrrpHjlTCTBYlOIIIHX C06bITHIO A = {KOpHH ypaBHeHHjI ,1l.eitcTBHTeJIbHbI}, eCTb 06JIacTb D (Ha pHCYHKe 67 06JIacTb D 3aIIITpHxoBaHa):

6.3.17.B HeKoTopoit TOqKe C JIHHHH AB ,1l.JIHHbI L rrpOH30IIIeJI pa3pbIB.

KaKoBa BepOjiTHOCTb Toro, qTO TOqKa C y,1l.aJIeHa OT TOqKH A Ha

paccTOjlHHe He MeHbIIIe l?

6.3.18.B Kpyr pa,1l.Hyca r HaY,1l.aqy 6pollieHa TOqKa. KaKoBa BepOjiTHOCTb

Toro, qTO 9Ta TOqKa OKa:>KeTCjI BHyTpH BrrHcaHHoro B Kpyr rrpaBHJIbHOrO TpeyrOJIbHHKa?

6.3.19. Ha rrJIOIIIa,rr,KY, rrOKpbITYIO Ka<PeJIbHoit rrJIHTKoit co CTOPOHOit a =

= 6 CM, CJIyqaitHo rra,rr,aeT MOHeTa pa,rr,Hyca r = 2 CM. HaitTH BepojlTHOCTb Toro, qTO MOHeTa IIeJIHKOM OKa:>KeTCjI BHyTpH KBa,rr,paTa.

6.3.20. Ha oTpe3Ke [0,3] HaY,1l.aqy BbI6paHbI ,1l.Ba qHCJIa x H y. HaitTH BepOjiTHOCTb Toro, qTO 9TH qHCJIa y,1l.0BJIeTBOpjlIOT HepaBeHCTBaM

x 2 ::;; 3y ::;; 3x.

6.3.21.B CHrHaJIH3aTOp rrocTyrraIOT CHrHaJIbI OT ,1l.ByX YCTPOitCTB, rrpHqeM

rrocTyrrJIeHHe Ka:>K,1l.0ro H3 CHrHaJIOB He 3aBHCHT ,1l.pyr OT ,1l.pyra H

297

paBHOB03MO)KHO B JIIo6oil: IIpOMe)KYTOK BpeMeHH )J,JIHTeJIbHOCTblO

3'Iaca.CHrHaJIH3aTOp cpa6aTbIBaeT, eCJIH HHTepBaJI Me)K)J.y MoMeHTaMH IIOCTYIIJIeHHH CHrHaJIOB MeHee 0,15 'I. Hail:TH BepOHTHOCTb TOrO, 'ITOCHrHaJIH3aTOp cpa60TaeT B Te'IeHHe3 'IaCOB,eCJIH KroK)J,oe H3 YCTPOil:CTB IIOIIIJIeT IIO O)J,HOMY CHrHaJIY.

6.3.22.MHHHoe 3arproK)J,eHHe COCTOHT H3 MHH, paCIIOJIO)KeHHbIX B O)J,HY JIHHHIO Ha paCCTOHHHH 50 M O)J,Ha OT )J,Pyroil:. IIIHpHHa KOpa6JIH

20M. KaKoBa BepoHTHoCTb TOro, 'ITOKopa6JIb 6JIarOlIOJIY'IHOIIpOil:- )J,eT 'Iepe33arproK)J,eHHe?

6.3.23.B map BIIHcaH Ky6. Hail:TH BepoHTHoCTb Toro, 'ITOBbI6paHHM Ha- y)J,a'IyBHyTpH mapa TO'IKaOKroKeTCH BHyTpH Ky6a.

6.3.24.OIIHPMCb Ha aKCHOMbI TeopHH BepoHTHocTeil:, )J,OKa3aTb cJIe)J.ylOIII;He YTBep)K)J,eHHH:

a) P(0) = 0; 6) P(A) = 1 - P(A).

a a) TaK KaK 0 + 0 = 0, TO P(0 + 0) = P(O). ITo aKCHOMe (3.3):

P(0 + 0) = P(0) + P(O),

T.K. 0·0 = 0. 1haK, P(0) + P(O) = P(O), oTKY)J,a P(0) = O.

6)TaK KaK A + A = 0 H A· A = 0, TO IIO aKCHOMaM (3.2)-(3.3):

P(A + A) = P(A) + P(A) = P(O) = 1.

=P(B) - P(AB). CJIe)J,oBaTeJIbHO, P(A + B) = P(A) + P(B) - P(AB). •

6.3.26.)l;oKa3aTb, 'ITO)J.JIH JII06bIX C06bITHil: A H B BbIIIOJIHeHO HepaBeH-

CTBO P(A + B) ~ P(A) + P(B).

6.3.27.)l;oKa3aTb, 'ITO)J.JIH JII06bIX C06bITHil: A, B H C

P(A + B + C) = P(A) + P(B) + P(C)-

-P(AB) - P(AC) - P(BC) + P(ABC).

6.3.28.ITYCTb P(A) = P(B) = !. )l;oKa3aTb, 'ITOP(AB) = P(A . B).

6.3.29.)l;oKa3aTb, 'ITOeCJIH A ;2 B, TO P(A - B) = P(A) - P(B).

AononHIIITenbHble 3aAaHIIIH

6.3.30. CKOJIbKO pa3 HY)KHO 6pOCHTb HrpaJIbHylO KOCTb, 'ITo6bIC BepOHTHOCTblO He MeHbme 0,6 XOTH 6bI O)J,HH pa3 BbIIIaJIO 6 O'IKOB?

6.3.31. 113 IIOCJIe)J,OBaTeJIbHOCTH 'IHCeJI1,2,3,4, ••• ,600 Hay)J,a'IyBbI6HpaIOTCH )J,Ba 'IHCJIa.KaKOBa BepOHTHOCTb Toro, 'ITOO)J.HO H3 HHX MeHbme 126, a )J,pyroe 60JIbme 126?

298

6.3.32. B JIOTepee pa3blrpbIBaeTCH 100 6I1JIeToB. BbmrpbIIIII1 BblIIaJIl1 Ha 20 6I1JIeToB. HeKTo npl106peJI 5 6I1JIeToB. Hail:TI1 BepOHTHOCTI1 CJIeAYIOml1X C06bITI1il::

a) BbmrpbIIII BblIIa,II;eT Ha Bce 5 6I1JIeTOB;

6) BbmrpbIIII BblIIa,II;eT XOTH 6bI Ha 1 6I1JIeT;

B)BbmrpbIIII BblIIa,II;eT Ha 2 611JIeTa.

6.3.33.BoceMb IIIaXMaTI1CTOB, cpe,II;11 KOTOPbIX Tpl1 rpoccMeil:cTepa, nYTeM :lKepe6beBKI1 ,II;eJIHTCH Ha ,II;Be KOMaH,II;bI no 4 'IeJIOBeKa.KaKoBa BePOHTHOCTb Toro, 'ITO,II;Ba rpoccMeil:cTepa nona,u,yT B O,II;HY KOMaHAY,

HOCTb Toro, 'ITOcpe,II;11 6 Hayra,II; B3HTbIX ,II;eTaJIeil: HeCTaH,II;apTHbIX

He OKa:lKeTCH?

6.3.35. )KeJIe3HO,II;OpO:lKHblil: COCTaB 113 9 BaroHoB 11 BaroHa-pecTopaHa<pop- Ml1pyeTcH np0l13BOJIbHbIM 06pa30M. KaKoBa BepOHTHOCTb Toro,

'ITO:

a) BarOH N~ 7 11 BarOH-peCTopaH pacnOJIO:lKeHbI pH,II;OM;

6) Me:lKAY BaroHoM N~ 7 11 BarOHOM-peCTopaHOM OKa:lKeTCH 5 Bara-

HOB?

6.3.36. ,nBe O,II;HOTl1nHble pa,II;110CTaHIII1I1 I1MeIOT 8 <pI1KCl1pOBaHHblx O,II;I1HaKOBbIX 'IaCTOT.KaKoBa BepOHTHOCTb Toro, 'ITOnpl1 He3aBI1CI1MOM 11

np0l13BOJIbHOM BbI60pe 'IaCTOTOHI1 OKa:lKYTCH HacTpoeHHbIMI1 Ha:

a) O,II;Hy 'IacTOTY; 6) pa3Hble 'IaCToTbI?

6.3.37. Hail:TI1 BepOHTHOCTb TOro, 'ITO30 CTY,II;eHTOB O,II;HOil: rpynnbI PO,II;I1-

JII1Cb:

a) B pa3Hble ,II;HI1 rO,II;a (B roAY 365 ,II;Heil:);

e)B pa3Hble ,II;HI1 ceHTH6pH.

6.3.38.HaY,II;a'IYBbI611paIOT 5 BOeHHOCJIY:lKamI1X 113 rpynnbI, COCTOHmeil: 113

4 o<pI1IIepoB 11 12 COJI,II;aT. KaKoBa BepoHTHoCTb TOro, 'ITOB rpynne 6Y,II;eT He 60JIee ,II;ByX o<pI1IIepoB?

6.3.39.HaiiTI1 BepOHTHOCTb TOro, 'ITOY'IaCTHI1KJIOTepel1 «CnOpTJIOTO - 6 113 49», KynI1BIIII1il: O,II;I1H 6I1JIeT, yra,II;aeT npaBI1JIbHO:

6.3.42. Ha 9 O,II;I1HaKOBbIX KapTO'IKaxHanl1CaHbI 6YKBbI E, E, P, P, C, C, H, r, 11. 8TI1 KapTO'IKI1 BhIKJIa,II;bIBaIOT HaY,II;a'IY B pH,II;. KaKoBa BepmITHOCTb TOro, 'ITOnpl1 3TOM nOJIY'Il1TCHCJIOBO PErPECCIIH?

299