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Санкт-Петербургский государственный университет телекоммуникаций им. проф. М.А. Бонч-Бруевича

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Сборник задач по высшей математике 2 том

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<<< < Предыдущая 21 22 23 24 25 26 27 28 29 30 31 3233 / 6033 34 35 36 37 38 39 40 41 42 43 44 45 > Следующая >>>

6.5.27.	Ih 1000 JIaMn 100 npHHa,nJIe)KHT 1-ii napTHH, 250 - 2-ii H OCTaJIb-
	Hb1e - 3-ii IIapTHH. B 1-ii IIapTHH 6%, BO 2-ii - 5%, B 3-ii - 4%
	6paKoBaHHb1x JIaMII. Hay,na'"lYBb16HpaeTcH o,nHa JIaMIIa. KaKOBa
	BepOHTHOCTb TOrO, '"ITOBb16paHHaH JIaMIIa - 6paKOBaHHaH?
6.5.28.	B YCJIOBHHX 3a,na'"lH6.5.18 KOpa6JIb IIpOrneJI 6JIaroIIOJIY'"IHOIIpo-
	JIHB. KaKoBa BepOHTHOCTb Toro, '"ITO OH IIpOrneJI B JIeBoii '"IacTH
	IIpOJIHBa?
6.5.29.	B YCJIOBHHX 3a,na'"lH6.5.21 9JIeKTpH'"IeCKaHu:eIIb pa6OTaeT. KaKoBa
	BepOHTHOCTb TOro, '"ITOBb16paHa 1-H ueIIb?
6.5.30.	B YCJIOBHHX 3a,na'"lH6.5.23 KOpa6JIb 6bIJI IIop~eH. KaKoBa Bepo-
	HTHOCTb TOro, '"ITOOH 6bIJI IIop~eH IIOIIa,naHHeM Tpex paKeT?
6.5.31.	B YCJIOBHHX 3a,na'"lH6.5.25 KYIIJIeHHb1ii TeJIeBH30p He IIOTpe60BaJI
	peMOHTa B Te'"leHHerapaHTHiiHOro cpOKa. KaKaH <l>HpMa BepOHTHee
	Bcero IIOCTaBHJIa ,naHHb1ii TeJIeBH30p?

KOHTponbHble BonpOCbl 1/1 60nee CnO)l(Hbie 3aAa~1/I

6.5.32.Ih IIOJIHOrO Ha60pa ,nOMHHO (28 KocTeii) Hay,na'"lY6epYTcH ,nBe Ko-

CTH. KaKoBa BepOHTHOCTb Toro, '"ITOBTOPYIO KOCTb MO)KHO IIpHCTaBHTb K IIepBoii IIO IIpaBHJIaM ,nOMHHO (pHC. 72)?

~r-:er-l

~ I.!:!.L-.I

Puc. 72

6.5.33. Co06meHHe MO)KeT IIepe,naBaTbCH IIO O,nHOMY H3 10 KaHaJIOB CBH-

3H; H3 HHX 4	KaHaJIa HaxO,nHTCH B OTJIH'"IHOM COCTOHHHH, 3 - B
xoporneM, 2 -	B IIocpe,nCTBeHHOM H 1 - B IIJIOXOM. BepoHTHo-

CTH IIpaBHJIbHOii IIepe,na'"lHco06meHHH .D:JIH pa3Horo BH,na KaHaJIOB paBHb1 COOTBeTCTBeHHO 0,6; 0,4; 0,2; O,l.

KaKoBa BepoHTHoCTb Toro, '"ITOXOTH 6b1 O,nHH pa3 coo6meHHe 6y,neT IIepe,naHO, eCJIH OHO IIepe,naeTCH IIO O,nHOMY H TOMY )Ke Ka-

HaJIY (Bb16paHHOMY Hay,na'"lY):

a) O,nHH pM; 6) ,nBa pa3a?

6.5.34. Ha rnaxMaTHYIO ,nOCKY CTaBHT Hay,na'"lY,nByX CJIOHOB, 6eJIoro H '"IepHOro. KaKOBa BepOHTHOCTb Toro, '"ITOCJIOHb1 6Y.D:YT IIp06HBaTb ,npyr ,npyra?

6.5.35. BepoHTHoCTb OTKa3a IIpH60pa IIpH B03,neiicTBHH Ha Hero TOJIbKO BH6paUHH paBHa 0,1, a TOJIbKO IIeperpeBa - 0,05; BepOHTHOCTl> OTKa3a IIpH B03,neiicTBHH BH6paU:HH H IIeperpeBa paBHa 0,2. Up" 9KCIIJIyaTauHH IIpH60pa BepoHTHoCTb B03HHKHOBeHHH IIeperpeBa paBHa 0,2, BepOHTHOCTb B03HHKHOBeHHH BH6paUHH paBHa 0,3. UeperpeB H BH6paUHH B03HHKalOT He3aBHCHMO. HaiiTH BepoHTHoCTb OTKa3a IIpH60pa.

320

6.5.36.	CeMb CTY)1.eHTOB, nOJIyqHB 6HJIeTbI, rOTOBHTCH K OTBeTY 9K3aMe-
	HaTOpy. 3HaHHe 6HJIeTa rapaHTHpyeT C)1.aqy 9K3aMeHa C BepOHTHO-
	CTbIO 0,9, He3HaHHe BepoHTHoCTbIO 0,2. KaKOBa BepoHTHoCTb TOrO,
	"ITO BbI3BaHHblii HaY)1.aqy CTY)1.eHT C)1.aCT 9K3aMeH, eCJIH MBaHOB
	3HaeT 20 6HJIeTOB H3 30, neTpoB - JIHIIIb 15, a OCTaJIbHble CTy-
	)1.eHTbI 3HaIOT Bce 6HJIeTbI?
6.5.37.	M3 MHO>KeCTBa qHCeJI	{I, 2, 3, ... ,99, 100} nOCJIe)1.0BaTeJIbHO (6e3
	B03BpaIII;eHHH) H3BJIeKaIOT )1.Ba qHCJIa. KaKoBa BepOHTHOCTb Toro,
	"ITO nepBoe qHCJIO 60JIbIIIe BToporo:
	a) Ha 20;	6) He MeHee qeM Ha 20?

6.5.38.B aJIb60Me 7 HeraIIIeHblX H 6 raIIIeHblX MapOK. M3 HHX HaY)1.aqy

H3BJIeKaIOTCH 2 MapKH, nO)1.BepraIOTCH raIIIeHHIO H B03BpaIII;aIOTCH B aJIb6oM. nOcJIe qero BHOBb H3BJIeKaIOTCH 3 MapKH. Onpe.n.eJIHTb BepOHTHOCTb Toro, "ITO Bce 3 MapKH qHCTble.

6.5.39. MMeIOTcH TpH ypHbI C 6eJIbIMH H qepHbIMH IIIapaMH. M3BeCTHO, "ITO OTHOIIIeHHe qHCJIa 6eJIbIX IIIapOB K qHCJIY qepHbIX paBHO Cl, C2, C3

)1.JIH I-ii, 2-ii, 3-ii ypH COOTBeTCTBeHHO. HaY)1.aqy BbI6HpaeTCH ypHa H H3 Hee BbITacKHBaeTCH IIIap. KaKOBa BepOHTHOCTb Toro, "ITO OH 6eJIblii?

§ 6. CXEMA lIICnbiTAHlIIl4 6EPHYl1l1l11

<P0pMYlla 6epHYIlll11

TIYCTb npOH3BO):V1TCH n He3aBHCHMbIX HcnbITaHHiI:, B KaJK,n:OM H3 KOTOPbIX MolKeT npoH30il:TH HeKOTopoe c06bITHe A (no TPMHU:HH TaKoil: HCXO,n: onbITa Ha3bIBalOT ycneXO.M) C O,n:HOil: H Toil: :lKe BepOHTHOCTbIO peA) = P HJIH npoH30il:TH npoTMBOnOJIO:lKHOe c06bITHe A (TaKoil: MCXO,n: Ha3bIBaIOT 'HeyiJa"te'iJ.) C BepOHTHOCTbIO peA) = q = 1 - P (TaKoro po,n:a cxeMa HcnbITaHHiI: Ha3bIBaeTCH cxe.Mo'iJ. Bep'Hy/l/lu).

Tor,n:a BepOHTHOCTb Toro, "ITO C06bITHe A HacTynHT POBHO m pa3, HaxO,n:HTCH no

rP0p.MY/le Bep'Hy/l/lu

		Pn(m) = C;:'· pm. qn-m, m = 0,1,2, ... ,n.
OTCIO,n:a, B "IacTHOCTH, CJIe.n:yeT, "ITO BepOHTHOCTb TOrO, "ITO B n HCnbITaHHHX,
Y)]'OBJIeTBOpHIOIu;HXCXeMe BepHYJIJIH, C06bITHe A HacTynHT:
a) .Me'Hee m	pa3 -	paBHa Pn(O) + Pn(1) + ... + Pn(m - 1);
6) 6o/lee m	pa3 -	paBHa Pn(m + 1) + Pn(m + 2) + ... + Pn(n);
B) X011lJl 6'bt oiJu'H pa3 - paBHa Pn(m ~ 1) = 1 _ qn;
r) 'He .Me'Hee ml pa3 U 'He 6o/lee m2 pa3 - paBHa
		m2
Pn(ml ~ m ~ m2) = Pn(m!) + Pn(ml + 1) + ... + Pn(m2) = L			Pn(m).
		m=ml

l.JHCJIO mO (0 ~ mO ~ n) Ha3bIBaeTCH 'HaUaep0.1lm'He'iJ.w,U.M "tUC/lO.M HacTynJIeHHiI: C06bITHH A (HJIH 'Hau6o/lee aepO.1lm'H'bt.M "tUC/lO.M ycnexoa) B cxeMe BepHYJIJIH, eCJIH

11 COOPHHK 38JIII' no BhlcweR MaTCMaTHKe. 2 K)'pC

321

m2 pa3, ••• , co6bITHe Ak - mk pa3 (ml + m2

Pn(mo) ~ Pn(m) .JlJUI BCeX m = 0,1,2, ... , n. ECJIH BepOHTHOCTb pH q OTJIH'IHbIOT

HYJIH, TO 'IHCJIOmO OlIpe,JleJIHTCH H3 ,JlBoitHoro HepaBeHCTBa

np - q ~ mo ~ np +p.

ECJIH B KruK,JlOM H3 n He3aBHCHMbIX HClIbITaHHit BepOHTHOCTb HacTYIIJIeHHH CO6bITHH A paBHapi ('IHCJIapi,Boo6lIIe roBOPH, pa3HbIe), TO BepOHTHOCTb Pn(m) TOro, 'ITOB 3TOit cepHH HClIbITaHHit Co6bITHe A HacTYIIHT m pa3, paBHa K03<P<PHIJ;HeHTY IIpH m-it CTelIeHH (T. e. IIpH zm) MHoro'IJIeHa

'Pn(Z) = (ql +PlZ)(q2 +P2Z)· ... · (qn +Pn . z). cI>YHKIJ;HH 'Pn (z) IIpH 3TOM Ha3bIBaeTCH npou36oiJRtqeit fPY'H!K;,u,ueit.

nOm1HOMII1anbHOe pacnPE!AeneHlI1e

TIyCTb TelIepb KruK,JlOe H3 n HClIbITaHHit MO:lKeT HMeTb TOJIbKO k HCXO,JlOB C06bI-

THit AI, A2, ... , Ak C COOTBeTCTBYIOIIIHMH BepOHTHOCTHMH Pl,P2, ... ,Pk (HCHO, 'ITO k

I: Pi = 1). Tor,Jla BepoHTHoCTb TOro, 'ITOB 3THX OllbITax Co6bITHe Al IIOHBHTCH ml

i=l

pa3, co6bITHe A2 - + ... + mk = n) paBHa

2ha <popMYJIa 3a,JJ.aeT nOJlUHOMUaJlbHOe pacnpeiJeJleHUe BepOHTHocTeit (Ha3BaHHe 06'bHCHHeTCHTeM, 'ITOBblpruKeHHe ,JlJIH Pn(ml, m2, ... , mk) HBJIHeTCH 06lIIHM 'IJIeHOMIIOJIHHOMa (PI +P2 + ... +Pkt). 3aMeTHM, 'ITOcxeMa BepHYJIJIH HBJIHeTCH 'IacTHbIMCJIY'IaeMIIOJIHHOMHaJIbHOrO paClIpe,JleJIeHHH IIpH k = 2, P2 = 1 - PI = ql.

6.6.1.	I1rPaJIbHYIO KOCTb IIo.D:6paCblBalOT 10 pa3. RailTH BepoHTHoCTb Ta-
	ro, qTO rneCTepKa BbIIIa.D:eT:
	a) .D:Ba pa3a;	6) He 60JIee BOCbMH pa3;
	B) XOTH 6bI O.D:HH pa3.
a IIpoBo.D:HTCH 10 He3aBHCHMbIX HCIIbITaHHil. KroK.D:Oe HCIIbITaHHe HMeeT .D:Ba

HCX0.D:a: rneCTepKa BbIIIa,D;eT, rnecTepKa He BbIIIa,D;eT. BepOHTHOCTb BbIIIa,D;eHHH rneCTepKH B KroK.D:OM HCIIbITaHHH IIOCTOHHHa H pasHa i, T. e. p = i. TaKHM

06pa30M, MbI HMeeM .D:eJIO co cxeMoil BepHYJIJIH. )J;JIH HaxOlK.D:eHHH HCKOMbIX

BepOHTHOCTeil IIpHMeHHM <POpMyJIY BepHYJIJIH.
a) 3.D:eCb n = 10, m = 2,	1	1	5
a) 3.D:eCb n = 10, m = 2,	p = 6'q = 1 -	6	= 6·OTCIO.D:a

Plo (2)	2 (1)2 (5)10-2	1	(5)8
Plo (2)	= ClO · 6 . 6	= 45· 36·	6 :::::: 0,291.

6) I1CKOMaH BepOHTHOCTb paBHa

PlO(O) +Plo (l) +PlO (2) +Plo (3) +PlO (4) +Plo (5) +Plo (6) +Plo (7) +Plo (8).

322

O,UHaKO B 9TOM CJIy'laenpOrn;e HaiiTH BepOflTHOCTb npOTHBOnOJIO:>KHOrO C06bI-

TYl5i - IIIeCTepKa Bbma,n,eT 60JIee 8 pM, T. e. Bbma,n,eT 9 HJIH 10 pM. MMeeM:

P!O(9)+PlO(1O)=e~o·(~t(~)\etg·(~)lO. (~)o=10.6~O +1· 6!O = :l~·

MTaK, BepOflTHOCTb Toro, 'ITOIIIeCTepKa Bblna,n,eT He 60JIee BOCbMH pa3,

paBHa 1 -		(PlO (9) + PlO (10)) = 1 - 5l~.
		6
	B) MCKOMaf! BepmlTHOCTb paBHa PlO (rn ~ 1) = 1 -		(~)10. Ee MO:>KHO
HafiTH H TaK ('ITO,KOHe'lHO,ropa3,UO CJIO:>KHee): PlO (l) + PlO (2) + ... + PlO (10)
I1JHf	1- PlO(O) = 1- ero . (~f .(~)10 = 1- (~)10.		•
6.6.2.		BCXO:>KeCTb ceMflH ,UaHHoro copTa pacTeHHii COCTaBJIfleT 70%. Haii-
		TH HaHBepoflTHeiiIIIee '1HCJIOBCXO:>KHX ceMflH B napTHH H3 240 ce-
a		MflH.
	HaHBepoflTHeiiIIIee '1HCJIOrno BCXO:>KHX ceMflH HaXO,UHM H3 yCJIOBHfI

np - q ~ rno ~ np + p.

ITOCKOJIbKY n = 240, p = 0,7 H q = 0,3, TO 240·0,7 -0,3 ~ rno ~ 240·0,7 +0,7,

T. e.	167,7 ~ rno ~ 168,7. OTcIO,Ua CJIe.n.yeT, 'ITOrno = 168.			•
6.6.3.		lIpH60p COCTOHT H3 3	He3aBHCHMO pa6oTaIOrn;HX 9JIeMeHTOB. Be-
		POflTHOCTH OTKa30B 9JIeMeHTOB 3a BpeMfI t pa3JIH'IHbIH COOTBeT-
		CTBeHHO paBHbI: Pl = 0,1, P2 = 0,2, P3 = 0,3. HaiiTH BepOflTHOCTH
		Toro, 'ITO3a BpeMfI t OTKa:>KYT:
		a) Bce 9JIeMeHTbI;	6) ,UBa 9JIeMeHTa;
a		B) O,UHH 9JIeMeHT,	r) HOJIb 9JIeMeHTOB.
a	TaK KaK Pl = 0,1, P2 = 0,2, P3 = 0,3, TO BepOflTHOCTH Toro, 'ITO9JIeMeHTbI

He OTKa:>KYT, COOTBeTCTBeHHO paBHbI: ql = 0,9, q2 = 0,8, q3 = 0,7. CocTaBHM npOH3BO.D.flrn;yIO <PYHKII.HIO:

'P3(Z) = (O,9+0,lz)(0,8+0,2z)(O,7+0,3z) = 0,006z3+O,092z2+O,398z+0,504.

OTclO,Ua CJIe.n.yeT, 'ITO:

a) P3(3) = 0,006;

6)P3 (2) = 0,092;

B)P3 (1) = 0,398;

r)	P3 (0) = 0,504.	•
6.6.4.	lIo MHIIIeHH npOH3Be,UeHO 3 BbICTpeJIa. BepoflTHoCTb nona,n,aHHfI
	npH Ka:>K,UOM BblCTpeJIe paBHa 0,7. HaiiTH BepOflTHOCTb n nona,n,a-
	HHii B MHIIIeHb, r,Ue n = 0, 1,2,3.
6.6.5.	TecT co,Uep:>KHT 10 BonpocoB, Ha KOTopble CJIe.n.yeT OTBe'laTb,HC-
	nOJIb3YfI O,UHO H3 ,UByX CJIOB: ,Ua, HeT. KaKOBa BepoflTHoCTb nOJIy-
	'1eHHfI He MeHee 80% npaBHJIbHbIX OTBeTOB, eCJIH HCnOJIb30BaTb
	«MeTO,U yra,n,bIBaHHfI»?
6.6.6.	lIycTb BepoflTHoCTb Toro, 'ITOcTy,UeHT on03,UaeT Ha JIeKII.HIO, paBHa
	0,08. HaiiTH HaH60JIee BepOflTHoe '1HCJIOon03,UaBIIIHX H3 96 cTy,UeH-
	TOB.

323

6.6.7. B Hrn:HKe HaxOAHTCH 70% CTaHAapTHbIX H 30% HeCTaHAapTHblX AeT8.lIeit. RaitTH BepOHTHOCTb Toro, qTO H3 5 B3HTbIX HaYAaqy AeTaJIeit He 60JIee OAHOit OKroKeTCH HeCTaHAapTHbIMH.

6.6.8. KOpa6JIb BblXOAHT H3 CTPOH, eCJIH IIOJIyqHT He MeHee 5 IIolIMaHHA B HMBOAHYIO qacTb HJIH 2 IIOIIMaHHH B IIOABOAHYIO qacTb. RaitTH BepOHTHOCTb BbIXOAa H3 CTPOH Kopa6JIH IIpH 5 IIOIIMaHHHX, eCJIH BepOHTHOCTH IIOlIaAaHHH B HMBOAHYIO H IIOABOAHYIO qacTH IIpHIIOIIMaHHH B KOpa6JIb OTHOCHTCH KaK ceMb K TpeM.

6.6.9.B ceMbe 6 AeTeit. RaitTH BepOHTHOCTb Toro, qTO B AaHHoit ceMbe

He MeHee ABYX M8.lIbqHKOB, HO He 60JIee qeTblpex. CqHTaTb Bepo-

HTHOCTH pmKAeHHH M8.lIbqHKa H AeBOqKH paBHblMH 0,5.

6.6.10.B IIOMern:eHHH 6 9JIeKTpOJIaMIIOqeK. BepoHTHoCTb Toro, qTO Ka-

:lKAaH JIaMIIOqKa OCTaHeTCH HCIIPaBHOit B TeqeHHe rOAa, paBHa 0,7.

RaitTH:

a) BepOHTHOCTb Toro, qTO B TeqeHHe rOAa IIpHAeTCH 3aMeHHTb

2 JIaMIIOqKHj

6) HaHBepoHTHeitlIlee qHCJIO JIaMIIOqeK, KOTopble 6YAYT pa60TaTb B TeqeHHe rOAa.

6.6.11. IIpH60p COCTOHT H3 5 He3aBHCHMO pa6oTalOrn:Hx 9JIeMeHTOB. BepoHTHOCTb OTKa3a 9JIeMeHTa B MOMeHT BKJIlOqeHHH paBHa 0,2. RaitTH: a) BepOHTHOCTb OTKa3a IIpH60pa, eCJIH AJIH 9Toro AOCTaTOqHO, qTO6bl OTKa38.lIO He MeHee 4 9JIeMeHTOBj

6) HaHBepoHTHeitlIlee qHCJIO rno OTKa3aBlIlHX 9JIeMeHTOBj

B)BepOHTHOCTb P5 (rno).

6.6.12.BepoHTHoCTb IIpHeMa PMHOCHrH8.lIa IIpH KroKAoit IIepeAaqe paBHa

0,86. RaitTH BepOHTHOCTb Toro, qTO IIpH IIHTHKpaTHoit IIepeAaqe CHrH8.lI 6YAeT IIpHHHT:

a) 4 pa3aj 6) He MeHee 4 pa3.

6.6.13. '(IToBepOHTHee BbmrpaTb Y paBHOCHJIbHOrO IIPOTHBHHKa:

a) OAHY H3 ABYX IIapTHit HJIH ABe H3 qeTblpeXj

6) He MeHee ABYX H3 Tpex IIapTHit HJIH He MeHee qeTblpex H3 BOCbMH?

6.6.14. BepoHTHoCTb C06blTHH A B OAHOM HClIblTaHHH paBHa 0,1. KaKoe MHHHM8.lIbHOe qHCJIO HClIblTaHHit AOCTaTOqHO IIpOBeCTH, qT06bl C BepOHTHOCTblO, He MeHblIleit, qeM 0,95, C06blTHe A HacTYIIHJIO XOTH 6bl OAHH pa3?

6.6.15. ,I:LecHTb qeJIOBeK IIPHlIlJIH Ha H36HpaTeJIbHbiit yqaCTOK H CJIyqaitHblM 06pa30M OTA8.lIH CBOH rOJIOca 3a OAHoro H3 IIHTH KaHAHAaTOB B IIpe3HAeHTbI. KaKOBa BepOHTHOCTb Toro, qTO 3a IIepBOrO 110 CIIHCKY KaHAHAaTa IIporOJIOCOB8.lIO 3 qeJIOBeKa?

324

6.6.16.B ypHe 8 6eJIbIX H 4 '1epHbIXIIIapa. HaYAa'lYBbIHHMaIOTClI C B03-

Bparn;eHHeM 12 IIIapOB. HaitTH BepOllTHOCTb TOrO, 'ITO6eJIbIX IIIapOB

6YAeT BbIHYTO:

a)1;

6)He MeHee 10 6eJIbIX IIIapoB?

6.6.17.BepOllTHOCTb BbIHrpblIIIa no 6HJIeTY JIOTepeH paBHa 0,125. HaitTH BepOllTHOCTb BbIHrpaTb He MeHee '1eMno AByM 6HJIeTaM H3 nllTH.

6.6.1S. OTMe'leHO,'ITOB ropOAe D B cpeAHeM 10% 3aKJIIO'IeHHbIX6paKoB B Te'leHHerOAa 3aKaH'IHBaIOTClIpa3BOAOM. KaKoBa BepOllTHOCTb Toro, 'ITOH3 8 CJIY'laitHooTo6paHHblx nap, 3aKJIIO'IHBIIIHX6paK, B Te'leHHerOAa:

a) HH OAHa napa He pa3BeAeTClI;

6) pa3Be.n:yTClI 2 napbI?

6.6.19.113 '1eTblpex0PYAHit npOH3BeAeH 3aJIn no IJ;eJIH. BepOllTHOCTb no-

naAaHHlI B IJ;eJIb AJIlI 1-ro 0PYAHlI paBHa 0,8, 2-ro - 0,7, 3-ro -

0,6, 4-ro - 0,5. HaitTH BepOllTHOCTb Toro, 'ITOB IJ;eJIb nona.n:yT: a) ABa 0PYAHlI; 6) TpH 0PYAHlI;

B)'1eTblpe0pYAHlI.

6.6.20.BepOllTHOCTb BbIXOAa Ha JIHHHIO KroKAoro H3 18 aBTo6ycOB paBHa 0,9. KaKoBa BepOllTHOCTb HOPMaJIbHOit pa60TbI aBTo6a3bI B Te'leHHe AHlI, eCJIH AJIlI 3Toro Heo6xoAHMO HMeTb Ha JIHHHH He MeHee 15 aBTo6ycOB?

6.6.21.B 3JIeKTpH'IKyH3 4 BarOHOB CaA1lTClI HaYAa'lY8 naCCroKHpOB. KaKOBa BepOllTHOCTb Toro, 'ITOB KroKAblit BarOH BOIIIJIO no 2 '1eJIOBeKa?

6.6.22.Ha KroKAblit JIOTepeitHblit 6HJIeT C BepOllTHOCTbIO 0,19 MO)J{eT BbI-

naCTb KpynHblit BbIHrpbIIII, C BepOllTHOCTbIO 0,11 - MeJIKHit	BbI-
HrpbIIII H C BepOllTHOCTbIO 0,70 6HJIeT MO)J{eT OKa3aTbClI 6e3	BbI-

HrpblIIIa. KynJIeHO 15 6HJIeToB. HaitTH BepOllTHOCTb nOJIY'leHHlI Tpex KpynHblx BbIHrpblIIIeit H OAHoro MeJIKOro.

6.6.23.B paBHocTopoHHHit TpeyroJIbHHK co CTOPOHOit, paBHoit a, BnHcaH

Kpyr. BHYTPH TpeyrOJIbHHKa He3aBHCHMO APyr OT APyra HaYAa- '1yBbI6HpaIOTClI 5 TO'leK.HaitTH BepOllTHOCTb Toro, 'ITO3 H3 HHX OKroKYTClI BHyTpH Kpyra.

6.6.24.CKOJIbKO pa3 Ha,D:O nOA6pocHTb HrpaJIbHyIO KOCTb, '1To6bIHaHBepOllTHeitIIIee '1HCJIOBblIIa,D:eHHit IIIeCTepKH 6bIJI'OpaBHo 5?

6.6.25.C'IHTa1I,'ITOB cpeAHeM 15% OTKpbIBaIOrn;HXClI MaJIbIX npe.n:npHlITHit CTaHOBlITClI B Te'leHHerOAa 6aHKpoTaMH, HaitTH BepOllTHOCTb Toro, 'ITOH3 10 HOBbIX MaJIbIX npeAnpHlITHit 3a 3TO BpeMlI 6aHK-

pOTaMH cTaHYT:
a) O,II;HO npeAnpHlITHe;	6) 60JIee Tpex npeAnpHlITHit.

6.6.26.IIpOBOAlITClI HcnbITaHHlI no cxeMe BepHYJIJIH. BepOllTHOCTb ycnexa B OAHOM HcnbITaHHH paBHa p. HaitTH BepOllTHOCTb C06bITHlI

A = {Bce m ycnexoB B n HcnbITaHHlIX nOllBlITClI nOAPllA}.

325

6.6.27. MOHeTa 6pocaeTcH

a) 2 pa3a;

6) 4 pa3a.

KaKOBa BepoHTHoCTb BblIIa.D:eHHH O.D:HOrO rep6a B cJIY'Iaea), .D:BYX rep60B B cJIY'Iae6)?

KOHTponbHble Bonpocbl III 60nee cnQ)KHble 3aACIHIIIH

6.6.28.3aaa"ta Ba7l,axa. HeKHit KYPHrn;Hit HOCHT C c060it .D:Be Kop06KH CIIH-

	'IeK.BCHKHit pa3, KOr.D:a He06xo.D:HMa CIIH'IKa,OH BbI6HpaeT Hayra.D:
	O.D:HY H3 Kop060K. B KaKoit-To MOMeHT O.D:Ha H3 Kop060K OKaJKeTCH
	IIYCTOit. KaKOBa BepOHTHOCTb Toro, 'ITOB 9TOT MOMeHT B .D:Pyroit
	Kop06Ke OKaJKeTCH m CIIH'IeK(m = 0,1,2, •.. , n; n - 'IHCJIOCIIH-
	'IeK,6bIBIIIHX IIepBOHa'IaJIbHOB KaJK.D:Oit H3 Kop060K)?
6.6.29.	B 3a.D:a'IeHa cxeMY BepHYJIJIH HaitTH 3Ha'IeHHep, IIpH KOTOPOM Be-
	POHTHOCTb P3(2) .D:OCTHraeT MaKCHMYMa, H BbI'IHCJIHTb9TOT MaK-
	CHMYM.
6.6.30.	IIo MHIIIeHH, COcToHrn;eit H3 «H6JIo'IKa» H .D:BYX KOJIeu:, IIpOH3Be.D:eHO
	5 BbICTpeJIOB. BepoHTHoCTb IIOIIa.D:aHHH B H6JIo'IKOpaBHa 0,2, B l-e

KOJIbU:O - 0,3, BO 2-e - 0,5. HaitTH BepOHTHOCTb Toro, 'ITO6Y,D;yT

.D:Ba IIOIIa,D;aHHH BO BTopoe KOJIbU:0, .D:Ba B IIepBoe KOJIbU:0 H O.D:HO -

B «H6JIo'IKO».

6.6.31.MrpaJIbHYro KOCTb IIo.D:6pacbIBaroT 8 pa3. KaKoBa BepOHTHOCTb TOro, 'ITO5 pa3 BblIIa,D;yT O.D:HHaKOBble 'IHCJIa?

§ 7. npVl61W1>KEHHbIE tlJOPMYllbl B CXEME

6EPHYllllVl

HenOCpe,ll;CTBeHHoe npHMeHeHlre <P0PMYJIbI BepHYJIJIH npH 60JIbiliOM 'IHCJIeHCnbITaHHiI: CBH3aHO C rpOM03,l1;KHMH BbI'IHCJIeHHHMH.TI03TOMY npH 60JIbIIIHX n BMeCTO Hee, KaK npaBHJIO, HCnOJIb3YIOT npH6JIHlKeHHble <P0PMYJIbI TIyaccoHa H MyaBpaJIanJIaca.

Be3,l1;e ,lI;aJIee pe'lbH,lI;eT 0 cepHH n He3aBHCHMbIX HcnbITaHHiI: no cxeMe BepHYJIJIH, Pn (m) 03Ha'laeTBepOHTHOCTb m ycnexoB B 3TOil: cepHH.

ECJIH 'IHCJIOHcnbITaHHiI: n )J,OCTaTO'IHO BeJIHKO, a BepOHTHOCTb P ,lI;OCTaTO'lHO MaJIa, npH'IeMHX npOH3Be,ll;eHHe a = np He MaJIO H He BeJIHKO (06bI'IHO,lI;OCTaTO'lHO YCJIOBHiI: p < 0,1; npq < 10), TO BepOHTHOCTb Pn(m) MOlKHO npH6JIHlKeHHO Hail:TH no <popMYJIe TIyaccoHa

326

J1oKal1bHaH liJopMYl1a MyaBpa-llanl1aca

TeopeMa 6.7. Ecml YII1CJ10 II1cnblTaHlI1iii n AOCTaTOYHO Bel1l11KO, a BepmlTHOCTII1 p 111 q He OyeHb 6111113KII1 K HyJ110 (06bIYHO AOCTaTOYHO yCI10BII1iii n > 100, npq > 20),

TO BepOHTHOCTb Pn(m) MO>KHO npIl16J1I11>KeHHO HaiiiTII1 no nOKanbHoilr q,opMyne

Myaspa-flannaca

Pn(m) ~ ~. <p(X),

				ynpq
m - np	1	x2	-
rAe x = r,n;n;;' <p(x) =	--e- T		-	cPYHKlIlI1H raycca.
ynpq	vf:2-rr
Ta6JIHIJ;a 3HaqeHHil: <PYHKIJ;HH <p(x)				rrpHBO,LlHTCH B rrpHJIO:lKeHHHx.

lI1HTerpal1bHaH liJopMYl1a MyaBpa-Jlanl1aca

TeopeMa 6.S. B YCI10BII1HX 110Kal1bHOiii cP0PMYl1bl MyaBpa-nanl1aca BepOHTHOCTb

Pn(ml ::;;; m ::;;; m2) Toro, YTO YII1CJ10 ycnexoB m 3aKJ1lOyeHO Me>KAY ml 111 m2,

MO>KHO np1l1611111>KeHHO HaiiiTII1 no VlHTerpanbHoilr q,opMyne Myaspa-flannaca

Xl =	ml-np	X2 =	m2 -np
	..;nPii ,		..;nPii ,

Ta6JIHIJ;a 3HaqeHHil: <PYHKIJ;HH <I> (x)o rrpHBO,LlHTCH B rrpHJIO:lKeHHHX.

6.7.1.	BepOjiTHOCTb «c6ojl»	B pa60Te TeJIe<poHHoit CTaHIJ;~Hl: npH KroK,n:OM
	BbI30Be paBHa 0,007. IIocTynHJIo 1000 BbI30BOB. Onpe,n:eJIHTb Bepo-
	jlTHOCTb 9 «c6oeB».
Q IIo YCJIOBHIO n = 1000, m		= 9, p = 0,007. IIocKOJIbKY n - ,n:OCTaTO'lHO
BeJIHKO, p - MaJIO (npq < 7),		TO ,n:JIjI BLI'IHCJIeHHjIPlOoo(9) MQ)KHO HCnOJIb-

30BaTb <P0PMYJIY IIyaccoHa. IIMeeM a =					[np] = 1000·0,007 = 7, oTKy,n:a
P	()	~	79 . e-7	~ 0,1014.	•
1000	9	~	9!	~ 0,1014.
6.7.2.			3aBo,n:-H3roTOBHTeJIb OTnpaBHJI		Ha 6a3y 12000 ,n:06pOKa'leCTBeH-
			HbIX H3,n:eJIHit. qHCJIO H3,n:eJIHit		nOBpe)l{,n:eHHblx npH TpaHcnopTH-

327

		pOBKe, COCTaBJUleT B Cpe.D:HeM 0,05%. HaiiTH BepmlTHOCTb Toro,'
		qTO Ha 6a3Y rrocTyrrHT:
		a) He 60JIee 3 rrOBpeJK.D:eHHbIX H3.D:eJIHii;
		6) XOTH 6bI 2 rrOBpe:lK.D:eHHbIX.
<)	,nJIH HaXO:lK.D:eHHH HCKOMbIX BepoHTHocTeii rrpHMeHHM <POPMYJIY ITyacco-
Ha. IIMeeM n = 12000, p = 0,0005, a = 12000 . 0,0005 = 6.
	a) B 3TOM cJIyqae HMeeM
	P12000(0 :::; m		:::; 3) = PI2000(0) + PI2000(1) + PI2000(2) + P12000(3) ~
'""	60	. e-6 + 61	.e-6	+ 62 .e-6 + 63	.e-6 = e-6	(1 + 6 + 18 + 36) = °151	.
'""		O!	I!	2!	3!	'	.
	6) IIcKoMM BepOHTHOCTb P12000(m ~ 2) =					PI2000(2) + P l2 000 (3) + ...

... +P l2 000(12 000) BbIqHCJIHeTCH .D:OBOJIbHO rpOM03.D:KO, rro3ToMY Haii.D:eM Be-

POHTHOCTb rrpoTHBOrrOJIO:lKHOrO C06bITHH: rrocTyrrHJIO MeHee 2-x rrOBpe:lK.D:eHHbIX .D:eTaJIeii:

PI2000(0:::; m:::; 1) = PI2000(0) + P12000(1) ~ e-6 + 6e-6 = 0,0174.
CJIe.D:OBaTeJIbHO, P12000(m ~ 2) = 1- 0,0174 = 0,9826.	•

6.7.3.BepoHTHoCTb BbIXO.D:a H3 CTPOH O.D:HOrO 3JIeMeHTa YCTpoiicTBa, B TeqeHHe t qaCOB pa6oTbI, paBHa 0,002. KaKoBa BepOHTHOCTb TOro, qTO 3a BpeMH t H3 1500 He3aBHCHMO pa6oTalOIIIHx 3JIeMeHTOB

BbIii.D:eT H3 CTPOH:
a) 43JIeMeHTa;	6) He 60JIee 2 3JIeMeHToB?

6.7.4.BepoHTHoCTb .D:OrrYCTHTb orrm6KY rrpH Ha60pe HeKOToporo TeKCTa, COCTOHIIIero H3 1200 3HaKOB, paBHa 0,005. HaiiTH BepOHTHOCTb TO-

ro, qTO rrpH Ha60pe 6Y.D:eT .D:OrrYIIIeHO:

a) 6orrm6oK;

6) XOTH 6bI O.D:Ha orrm6Ka.

6.7.5.KaKoBa BepOHTHOCTb TOro, qTO Cpe.D:H 730 rraCCa:lKHpOB rrOe3.D:a:

a)qeTBepO PO.D:HJIOCb 23 <peBPaJIH;

6) .D:BOe PO.D:HJIOCb 1 MapTa;

B)HHKTO He PO.D:HJICH 22 HIOHH? (CqHTaTb, qTO B ro,D;y 365 .D:Heii.)

6.7.6.HeKaqeCTBeHHble H3.D:eJIHH COCTaBJIHIOT 2% Bceii rrpO,D;yKU;HH u;exa. KaKoBa BepOHTHOCTb TOro, qTO Cpe.D:H 200 HaY.D:aqy B3HTbIX H3.D:e- JIHii OKa:lKeTCH:

a) He 60JIee 5 HeKaqecTBeHHbIX H3.D:eJIHii;

6) .D:Ba HJIH TpH HeKaqeCTBeHHbIX H3.D:eJIHH.

6.7.7.BepoHTHoCTb Toro, qTO rrpH aBTOMaTHqeCKOii lIlTaMrrOBKe H3.D:eJIHii

OT.D:eJIbHOe H3.D:eJIHe OKa:lKeTCH 6paKoBaHHbIM (T. e. C OTKJIOHeHHeM OT CTaH.D:apTa), rrOCTOHHHa H paBHa 0,05. KaKoBa BepOHTHOCTb Toro, qTO B rrapTHH H3 1000 H3.D:eJIHii BCTpeTHTCH POBHO 40 6paKo-

BaHHbIx?

<) ITo YCJIOBHIO 3a,Il;aqH n = 1000, m = 40, p = 0,05, q = 0,95. TeOpeTHqeCKH MO:lKHO HCrrOJIb30BaTb <P0PMYJIY BepHYJIJIH, TOr.D:a

P IOOO (40) = ct800 . (0,05)40 . (0,95)960.

328

xO,95 =

O,UHaKO IIOJIY'-IeHHOe BblproKeHHe CJIHWKOM rpOM03,UKO, II03TOMY y,no6Hee UpHMeHHTb JIOKaJIbHYIO <POPMYJIY MyaBpa-JIaIIJIaCa (n . p' q = 1000·0,05x

47,5> 20). TaK KaK

Vnpq = V47,5 ~ 6,892, m -			np = 40 -	1000·0,05= 40 - 50 = -10,
TO		-10
X =	m-np		cp{x) = cp{ -1,45) = cp{1,45) ~ 0,1394
	vnpq	= 6,892 ~ -1,45,
(3Ha'-leHHe<PYHKU;HH cp{x) HaxO,nHM IIO Ta6JIHu;e). CJIe,noBaTeJIbHO,					•
		1		0,1394
		PlOoo (40) ~ vnpq . cp{x) =		6,892 = 0,02.
6.7.8.	IIcIIOJIb3YH YCJIOBHe 3a,na'-lH6.7.7, BbIHCHHTb, CKOJIbKO He6paKo-
	BaHHblX H3,neJIHit CJIe,nyeT O:lIm,naTb C BepoHTHoCTblO 0,042.
o IIMeeM: n =		1000, p = 0,95 (BepoHTHoCTb CTaH,napTHoro H3,neJIHH), q =

= 0,05; Tpe6yeTcH HaitTH m. CHoBa HCIIOJIb3yeM JIOKaJIbHYIO <POPMYJIY My- aBpa-JIaIIJIaca. TaK KaK vnpq = VlOOO . 0,95 . 0,05 ~ 6,892, TO

PlOOO{m) ~ 6892'cp{x) = 0,042.

OTCIO,na HaxO,nHM cp{x) = 6,892·0,042 ~ 0,289.

ITo Ta6JIHu;e 3Ha'-leHHit<PYHKU;HH cp{x) HaxO,nHM X ~ ±0,80. Y'-IHTbIBM,

'-ITO X =	m - np	m - 1000 . 0,95	m - 950	m -	950
'-ITO X =	vnpq =	6,892	= 6,892'IIOJIY'-IaeM	6,892
= ±0,80. OTclO,na m = 950 ± 5,51, T.e. m = 955 HJIH m = 945 (m -					u;eJIoe
'-II1CJlO.II03TOMY oKPYrJIHeM).					•

6.7.10.BepoHTHoCTb OTKa3a KroK,noro IIpH60pa IIpH HCIIblTaHHH paBHa 0,2. ITpH60Pbl HCIIblTblBalOTCH He3aBHCHMO ,npyr OT ,npyra. qTO BepoHTHee: OTKa3 10 IIPH6opOB IIpH HCIIblTaHHH 80, HJIH OTKa3 15 IIpH HCIIblTaHHH 120?

6.7.11.MOHeTa IIo,n6pacblBaeTcH 2020 pa3. KaKOBa BepOHTHOCTb TOrO, '-ITO rep6 BblIIa,neT 1000 pa3?

6.7.12.BcxmKecTb ceMHH ,naHHoro copTa pacTeHHit COCTaBJIHeT 70%. RaitTH BepOHTHOCTb TOrO, '-ITOH3 700 IIOCroKeHHblX ceMHH 6y,neT 500 IIpOpOCWHX.

6.7.13.B ropo,ne N H3 KroK,nbiX 100 ceMeit 85 HMelOT U;BeTHble TeJIeBH30pbl. KaKoBa BepOHTHOCTb TOrO, '-ITOH3 400 ceMeit 340 HMelOT TaKHe TeJIeBH30pbl?

6.7.14.BepoHTHoCTb IIOIIa,naHHH B u;eJIb H3 CKopOCTpeJIbHOrO Opy,nHH IIpH OT,neJIbHOM BblCTpeJIe paBHa 0,75. RaitTH BepOHTHOCTb TOrO, '-ITO IIpH 300 BblCTpeJIaX '-IHCJIOIIoIIa,naHHit 6y,neT He MeHee 210, HO He 60JIee 230 pa3.

~no YCJIOBHIO n = 300, p = 0,75, q = 0,25, ml = 210, m2 = 230. .il:JIH HaXO)I(,neHHH BepOHTHOCTH P30o {210 ~ m ~ 230) BOCIIOJIb3yeMcH HHTerpaJIbHoit

¢OPMYJIOit MyaBpa-JIaIIJIaca.

329

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