Taller 1 Probabilidad

 

VKCQFP^CHGH UFHGOÞOCNG S XFNKMJÞOCNG HF NMJML@CG UPM@G@CJCHGH S F^XGHÌ^XCNG XGJJFP 7

KML@PF^9 ◍ Kgtgjcg Xg`gnm ^gjgzgr nþhcom 0570050>1 ◍ Imkgtbgk Hgvch Pmhrìoufz Ngrm nþhcom ◍ Nrcstbcgk Bfrkçkhfz 057>05808 057>05808 ◍ Lysbfjj Lgrcgkg Pmhrcoufz Ocj Nþhcom9 057805722 057805722 ◍ Lcnbgfj Gkhrãs Nglgrom Nglgrom Ug÷m Nþhcom9 057875>>3 057875>>3 Kmtg9 Fj prfsfktf tgjjfr hf`frç rfgjczgrsf fk orupms hf lçxclm 6 cktforgktfs, y sfrç fktrfoghm fk Lmmhjf ùkcnglfktf pmr uk cktforgktf hf nghg orupm.

7. Njgscaquf sfoùk su kgturgjfzg (nugjctgvgs (nugjctgvgs,, nugktgvgs nugktgvgs hcsnrftgs, nugktgvgs nmkkugs) y jg fsngjg hf lfhchg, pgrg jgs j gs scoucfktfs vgrcg`jfs9

K m.

Kml`rf hf jg vgrcg`jf

Nugktgvg Hcsnrf Nmkk tg ug

Fsngjg hf

Nugjctg vg

R

7

Kùlfrm hf gnncmkfs hcgrcgs vfkhchgs fk jg @mjsg Gorìnmjg Kgncmkgj hf Qgjmrfs hf Ougtflgjg.

0

Fqucpm hf ùt`mj hf prffrfkncg

R

>

Mpckcþk sm`rf fj sfrvcncm hf flfrofkncg hf uk bmspctgj

R

8

Kùlfrm hf bcims fk uk kùnjfm glcjcgr

6

Xcflpm (fk hìgs) fk quf ukg rutg lghurg

1

Ngkhgh hf rfstgurgktfs fk jg ncuhgh hf Xukig

?

Hcçlftrm g jg gjturg hfj pfnbm (nl) hf uk çr`mj

2

@grrcm fk fj quf vcvf uk fstuhcgktf.

3

Xflpfrgturg hcgrcg hf jg ncuhgh hf ^moglmsm

7 5

Kùlfrm hf nfktrms hf sgjuh pmr lukcncpcm

7 7 7

Orupm sgkouìkfm hf uk orupm hf pfrsmkgs

R

Lgtfrcgj hf nmkstrunncþk hf jgs pgrfhfs hf jgs ngsgs

R

0 7 > 7 8

Fhgh (fk g÷ms nulpjchms) hf uk orupm hf gjulkms Urfsfkncg hf fkfrlfhghfs rfspcrgtmrcgs fk kc÷ms

R R R R R R R

R R

lfhchg

F^NGJG HF CKXFPQG JM F^NGJG MPHCKGJ F^NGJG MPHCKGJ F^NGJG HF PGYMK F^NGJG HF PGYÞK F^NGJG HF PGYÞK F^NGJG HF PGYÞK F^NGJG  KMLCKGJ F^NGJG HF CKXFPQG JM F^NGJG HF PGYÞK F^NGJG  KMLCKGJ F^NGJG  KMLCKGJ

F^NGJG HF PGYÞK F^NGJG MPHCKGJ

 

7 6 7 1

Xmkfjghgs hf ng÷g prmhunchgs pmr bfntçrfg

R R

Lfhchgs hf gcsjglcfktm9 jgvghm hf lgkms, usm hf ougktfs, usm hf lgsngrcjjg

7 ?

Orghm hf gtgquf hf ukg vcrmscs vfoftgj

R

7 2

Ngkhgh hf nmlputghmrgs pmr bmogr

R

7 3

Mrcfktgncþk hfj vcfktm

R

0 5

Nmktfkchm hf prmtfìkg (gjtm, lfhcm, `gim) fk jfnbf

R

F^NGJG HF PGYÞK F^NGJG  KMLCKGJ

F^NGJG CKXFPQG JM F^NGJG PGYMK F^NGJG MPHCKGJ F^NGJG MPHCKGJ

0. Nmkstruyg jg tg`jg hf rfnufkncgs ghfnughg pgrg jms scoucfktfs hgtms hf ukg lufstrg, nmrrfspmkhcfktfs gj flpm quf sf ujczþ pgrg gtfkhfr g jgs pfrsmkgs fk ukg fstgncþk hf sfrvcncm9

 

g) Cktfrprftf Cktfrprftf jgs rfnufkncg rfnufkncgss 0, 0, b>, A> y B6. ◍ 0< kms ckhcng ckhcng quf g ? pfrsmk pfrsmkgs gs jgs gtfkh gtfkhcfrmk cfrmk fk uk cktfrvg cktfrvgjm jm hf flpm flpm hf 0.8> 0.8> g >.51 >.51 ◍ b> b>< < Kms Kms ckhc ckhcng ng quf quf sf rfpct rfpctf f 5. 5.7? 7?6, 6, fs hfnc hfncr, r, fk fj ck cktf tfrv rvgj gjm m hf flpm flpm hf >.51 >.51 g >.13 >.13,, gfkhfk ? pfrsmkgs, jm quf kms ckhcng quf fstf sunfsm sf rfpctf 5.7?6 vfnfs. ◍ B6< kms ckhcng ckhcng quf quf fj 20,6% hf hf jgs pfrsmkgs pfrsmkgs uf uf gtfkhchg gtfkhchg fk uk cktfrvg cktfrvgjm jm hf flpm flpm lfkmr lfkmr g 8.38 `) Chfkaquf ◍ Xcpm Qg Qgrcg rcg`jf `jfvgrcg` 9 ^f trgt trgtg hf ukg vgr vgrcg` jfkug nug nugkt gvg vg ◍ Xcp m hf vgr cg`jf9 jf9gNugk Nu gktg tgvg vgcg`jf nmk nmkku gktg ◍ Fsngjg Fsngjg hf lfhcn lfhcncþk cþk99 Fsngj Fsngjg g hf cktfr cktfrvgj vgjms ms

>. Jg scoucfktf tg`jg lufstrg jg hcstrc`uncþk hf rfnufkncgs hf jg vgrcg`jf prmhunncþk hf pgpg (fk t/bg), sfoùk jg ckmrlgncþk m`tfkchg fk uk lufstrfm gjfgtmrcm hf gjoukms prmhuntmrfs9

Hf gnufrhm nmk jg ckmrlgncþk gktfrcmr, rfspmkhg9 g) ½Nuçj ½Nuçj fs jg vgrc vgrcg` g`jf jf fk fj fstuh fstuhcm= cm= `) ½Wu ½Wuã ã pm pm hf vgrcg` vgrcg`jfs jfs fs= n) ½Fk q quã uã fsngjg fsngjg hf lfhchg lfhchg fstç hghg jg vg vgrcg`j rcg`jf= f= h) ½Fk quã pmrnfktgif hf hf jg lufstrg lufstrg sf m`tuvcfrmk prmhunncmkfs prmhunncmkfs lfkmrfs m cougjfs g 0> t/bg= f) ½Wuã pmrnfkt pmrnfktgif gif hf prmhuntm prmhuntmrfs rfs m`tuvm m`tuvm ukg prmhunncþ prmhunncþk k lgymr g >8 t/bg= t/bg= ) ½Wuã ng ngkhg khgh h hf prmhunt prmhuntmrfs mrfs m`tuvc m`tuvcfrmk frmk prmhun prmhunncmk ncmkfs fs lgymrfs lgymrfs g >3 >3 t/bg= t/bg=

 

g) `) n) h) f) )

Jg vgrcg` vgrcg`jf jf fk fstuhcm fstuhcm fs jg prmhun prmhunncþk ncþk hf pgpg pgpg Jg vgrc vgrcg` g`jf jf fs kulã kulãrcn rcnm m nmkk nmkkug ug Jg vgrcg` vgrcg`jf jf fstç fstç hghg hghg fk tmkfjg tmkfjghg/ hg/ bfntçrfg bfntçrfg Fk uk 72% h hf f jg lufstrg lufstrg sf m`tuvcfrm m`tuvcfrmk k vgjmrfs vgjmrfs lfkmrfs lfkmrfs m cougjfs cougjfs g 0> t/bg Vk >1% hf prmhun prmhuntmrfs tmrfs m`tuv m`tuvcfrmk cfrmk ukg ukg prmhunncþ prmhunncþk k lgymr lgymr g >8 t/bg 70 prmhun prmhuntmrfs tmrfs m`tuvcfrmk m`tuvcfrmk prmhunnc prmhunncmkfs mkfs lgymrfs lgymrfs g >3 t/bg. t/bg.

8. ^f tmlgrmk lufstrgs hf nmjcmrlfs tmtgjfs, lfhcgktf fj lãtmhm hf ajtrgncþk pmr lfl`rgkg, pgrg jg fslgncþk hf jg ngjchgh hfj gcrf fk jms prfhcms hf jg Vkcvfrschgh Kgncmkgj hf Nmjml`cg^fhf Ugjlcrg y nmk jms hgtms m`tfkchms sf m`tuvm fj scoucfktf orçanm9

  `

Chfkaquf jg vgrcg`jf, pmjmoìg y fsngjg hf lfhcncþk Ptg9 Qgrcg`jf kulãrcng hcsnrftg,Fsngjg hf rgzþk Nmkstruyg jg tg`jg hf rfnufkncgs nmrrfspmkhcfktf

n

Ptg9 Cktfrprftf jgs scoucfktfs rfnufkncgs9 >, 2, ℋ0, 8, >

 

-  a>4 rfprfsfktg quf fj vgjmr hf 8? VAN/l> tcfkf ukg arfnufkncg hf 8 hfktrm hf jms rfnmjfntghms fk jg

lufstrg, tgl`cãk quf fs fj lçxclm fk ac

-  b0 rfprfsfktg quf fj 71,?5% hfj tmtgj hf jms hgtms hf nmjcamrlfs rfnmochms tcfkfk uk vgjmr hf 0> VAN/l>.

-  A8 rfprfsfktg quf 3 hf jgs lufstrgs hf nmjcamrlfs hftgjjghgs tcfkfk uk vgjmr lçxclm hf 33 VAN/l>.

-  B> rfprfsfktg quf fj 62,>5% hf jgs lufstrgs hf nmjcamrlfs hftgjjghgs tcfkfk uk vgjmr hf 8? VAN/l>.

 

6. Ugrg jgs scoucfktfs sctugncmkfs pjgktfghgs, pjgktfghgs, ckhcquf9 g) `) n) h)

½nuç ½nuçjj fs jg jg pm` pm`jg jgnc ncþk þk= = ½hf ½hfakg akg uk ukg g pmsc` pmsc`jf jf lufs lufstrg trg= = ½nu ½nuçj çj fs jg jg ukch ukchgh gh hf gkçjcsc gkçjcscs= s= Umsc`jf vgrcg`jf hf fstuhcm y su pmjmoìg pmjmoìg (kulãrcng (kulãrcng hcsnrftg, hcsnrftg, kulãrcng nmkkug, nugjctgvg)

^ctugnc þk

Um`jgncþ k

Lufst rg

Vkchgh hf gkçjcscs

^f rfnc`f ukg rflfsg hf pfrkms hf uk hcstrc`uchmr. Ugrg vfrcangr sc jg rflfsg fs gnfptg`jf rfspfntm hf jg ufrzg hf nmrtf, uk ckofkcfrm sfjfnncmkg hcfz pfrkms, ukm trgs mtrm, hfj rfncpcfktf pgrg prm`grjms.

^frìgk ^frìgk jms hcfz tmhms jms pfrkms  pfrkms quf quf sfjfnncm rfnc`fk kg fj hfj hcstrc`uch ckofkcfr  m mr.

Nghg ukm hf jms  pfrkms quf sf vfrcacnþ

Vk ckofkcfrm rfspmksg`jf hfj nmktrmj hf ngjchgh fkf quf

Xmhgs jgs

Jgs 755

Nghg ukg

ngjnujgr fj pmrnfktgif hf pcfzgs g`rcnghgs hffntumsgs fk hftfrlckghm hìg. G jgs 09>5 hf jg tgrhf lufstrfg jgs ùjlgs 755 pcfzgs g`rcnghgs.

 pcfzgs ag`rcnghg s fk uk hìg.

ùjtclgs  pcfzgs hf jgs g jgs  pcfzgs ag`rcngh quf sf jf gs bczm nmktrmj hf ngjchgh

^f bg usg usghm hm hurgkt hurgktf f lunbm lunbm flpm flpm uk prm prmnfs nfsm m pgrg pgrg jg g`rcngncþk hf `mtfjjgs hf pjçsnm y sf sg`f quf 75% hf ãstgs sf fknufktrg hffntumsm. ^f fstç prm`gkhm uk kufvm prmnfsm quf, sf supmkf, rfhunf jg prmpmrncþk hf hffntms. Fk ukg lufstrg hf 755 `mtfjjgs prmhunchgs nmk fj kufvm prmnfsm, hcfz fstg`gk

Xmhgs jgs 755  `mtfjjgs  `mtfjjgs  prmhunchg ag`rcngh s fk fj gs nmk fj kufvm kufvm  prmnfsm.  prmnfsm.

hffntumsgs.

Nghg ukg hf jgs  `mtfjjgs ag`rcnghgs nmk fj kufvm  prmnfsm

Qgrcg`j f

^frìg jg ngjchgh hf jg rflfsg rfspfntm g jg aufrzg hf nmrtf, sf chfktcacng sc fs gnfptg`jf m km.Qgrcg`j f nugjctgtcvg Nmktrmj hf ngjchgh hf jgs pcfzgs, sf chfktcacng sc smk hfafntumsg s m km. Qgrcg`jf nugjctgtcvg  Kùlfrm hf  `mtfjjgs  prmhunchgs nmk fj kufvm  prmnfsm, sf chfktcacngk nuçktgs smk hfafntumsg s y nuçktgs km. Qgrcg`jf nugktctgtcv g hcsnrftg.

1. ^f bg rfgjczghm uk fxpfrclfktm sm`rf jg tmjfrgkncg gj rìm hf jg fspfncf hf pgstm Fnbckmnbjmg nrus-ogjjc. ^f lchcþ jg g`smrncþk hf NM0 hf scftf pjgktgs hf Wuf`fn y scftf pjgktgs hf Lcsscsscppc g vgrcms kcvfjfs hf nmknfktrgncþk hf NM0 gl`cfktgj. Jg lctgh hf jgs pjgktgs hf nghg pm sf fkrcgrmk hurgktf jg kmnbf gktfs hf rfgjczgr fj fxpfrclfktm. Vk gkçjcscs orçanm sm`rf jgs tgsgs hf g`smrncþk hf hcþxchm hf ngr`mkm (ulmj/l[0 sfo) hf nghg pjgktg sfoù sfoùk k fj fstg fstghm hm (Wuf (Wuf`f `fnn m Lcss Lcsscs cssc scpp ppc) c) y, sc sf gp gpjc jcnþ nþ m km fkr fkrcg cglc lcfk fktm tm,, ofkf ofkfrg rg jg scou scoucf cfkt ktf f vcsugjczgncþk9

 

bps9//pu`jcn.tg`jfgu.nml/prmajf/iujcftb.rmhrcoufz#!/vczbmlf/Organm7]717827?66536?5/Bmig0=pu`jc bps9//pu`jcn.tg`jfgu.nml/prmajf/iujcftb.rmhrcoufz#! /vczbmlf/Organm7]717827?66536?5/Bmig0=pu`jc   sb752005/Bmig7=pu`jc bps9//pu`jcn.tg`jfgu.nml/prmajf/iujcftb.rmhrcoufz#! /vczbmlf/Organm0]7178272>752005/Bmig7=pu`jc   sb vgrcg`jfs jgs nugjfs smk jg Xgsg hf g`smrncþk hf hcþxchm hf ngr`mkm, n gr`mkm, fstg fs ukg vgrcg`jf nugktgvg nmkkug, Fj pm hf trgtglcfktm (pjgktgs fkrcghgs m km fkrcghgs), fj nugj fs ukg vgrcg`jf nugjctgvg , juogr hf mrcofk hf jgs pjgktgs, fstç fs ukg vgrcg`jf hf nugjctgvg. Orçang 09 sf fknmktrgrmk 8 vgrcg`jfs jgs nugjfs smk4 fj Xcpm hf trgtglcfktm (pjgktgs fkrcghgs m km fkrcghgs), fstf fs ukg vgrcg`jf nugjctgvg, Nmknfktrgncþk hf NM0 gl`cfktgj, fstg fs ukg vgrcg`jf nugktgvg nmkkug, Xgsg hf g`smrncþk hf hcþxchm hf ngr`mkm, fstg fs ukg vgrcg`jf nugktgvg nmkkug,Ngkhgh hf pjgktgs pgrg nghg trgtglcfktm, fstg f stg fs ukg vgrcg`jf nugktgvg hcsnrftg.

?. Fj hgtgsft Crcs, fs uk nmkiuktm hf hgtms prmpcm cknjuchm fk P. Ugrg fstf fifrncncm, sf jf bgk ghcncmkghm 0 kufvgs vgrcg`jfs gj grnbcvm mrcockgj (Uftgj.kul`fr y nmjmr). Fstf grnbcvm nmkfkf ckmrlgncþk sm`rf jg lfhcncþk hf jms pãtgjms y sãpgjms hf vgrcgs pjgktgs hfj oãkfrm Crcs.

 

^f pchf, rfgjczgr uk gkçjcscs hfsnrcpvm hfj hgtgsft, hmkhf hã nufktg hf ukg hfsnrcpncþk tg`ujgr pgrg ukg vgrcg`jf nugjctgvg, y ukg hfsnrcpncþk orçang pgrg ukg vgrcg`jf kulãrcng. M`tfkfr fj hgtgsft fk9 bps9//www.dgoojf.nml/iujcftbdrmhrcoufz/crcs-pjgkt

Kmtg9 Gpmygrsf hf gjoukg bfrrglcfktg pgrg rfgjczgr fj gkçjcscs quf sf smjcnctg (P^tuhcm m Fxnfj). Xfkog fk nufktg jg pmjmoìg hf jgs vgrcg`jfs hghgs fk fj hgtgsft, pgrg quf fj gkçjcscs g rfgjczgr sfg nmrrfntm.

^f ghiuktg grnbcvm hf fxfj nmk fj hfsgrrmjjm hf fstf puktm

2. Vk prmnfsm pgrg prmhunncþk hf lgtfrcgjfs hf nmkstrunncþk fstç hcsf÷ghm pgrg ofkfrgr nmktfkfhmrfs hf trfs pfsms hcfrfktfs9 75 jc`rgs, 77 jc`rgs y 70 jc`rgs. Vk fxglfk hf 85 nmktfkfhmrfs rfocstrg sus pfsms rfgjfs y sus pfsms hfsfghms. ^f nmkschfrg quf uk nmktfkfhmr fs ‚hffntumsm“ sc su pfsm rfgj hcafrf fk lçs hf 5.6 jc`rgs hf su pfsm hfsfghm. Pfgjcnf ukg tg`jg hf nmkkofkncg hf fstms hgtms fk jg nugj ckhcquf nuçktms nmktfkfhmrfs, fk nghg ukm hf jms trfs orupms, fstçk hfktrm hf jg hcfrfkncg pfrlcsc`jf. Pfocstrf jgs mpfrgncmkfs nmlm ‚hffntumsm“ y ‚km hffntumsm“.

½Uufhf hftfntgr gjoùk pgtrþk= ½G quã nmknjuscmkfs pufhf jjfogr= P9 prclfrm quf tmhm bgnflms jg tg`jg y sgnglms jg hcfrfkncg pgrg sg`fr nugjfs smk hffntumsms y nugjfs smk km hffntumsms. Km sf pufhf vfr kckoùk pm hf pgtrþk gpgrfktf gj gkgjczgr jg tg`jg y jms hgtms m`tfkchms.

 

Jufom sgnglms jg tg`jg hckçlcng pgrg vfr nugjfs nmktfkfhmrfs pgsgk m km fj nmktrmj

Ugrg fj gkçjcscs hf rfsujtghms pmhflms vfr quf9

 

◍ ◍ ◍

76 nmktfkfhmr nmktfkfhmrfs fs quf fkfk fkfk uk pfsm pfsm hf 75 j` hf jjms ms nugjfs nugjfs pmhflms pmhflms vfr qu quf f smjm 6 nufktgk nufktgk nmk jgs jgs rfojg rfojgss fstg`jfnchgs. 77 nmktf nmktfkfhmr kfhmrfs fs nmk uk pfsm pfsm hf 77 j` hf hf jms nug nugjfs jfs smjm smjm 0 nufktgk nufktgk nmk jgs jgs rfojgs fstg`j fstg`jfnchgs fnchgs.. 78 nmktf nmktfkfhmr kfhmrfs fs nmk uk pfsm pfsm hf 70 j` hf hf jms nug nugjfs jfs smjm smjm 0 nufktgk nufktgk nmk jgs jgs kmrlgs kmrlgs fstg`jfnch fstg`jfnchgs, gs, pgr pgrg g uk tmtgj hf 85 nmktfkfhmrfs.

Mtrg hf jgs nmknjuscmkfs g jgs quf pmhflms jjfogr fs vfr quf hf jms 85 nmktfkfhmrfs smjm 3 nulpjfk nmk jgs nmkhcncmkfs quf sf kfnfsctgk ,quf su pfsm km hcafrg fk lçs hf 5,6 j` ,pfrm >7 hf jms nmktfkfhmrfs n mktfkfhmrfs fkfk quf sfr rfnbgzghms pmrquf fxnfhfk fj pfsm hf hcfrfkncg ,fstm fk nugktm g prmhunncþk hf nmktfkfhmrfs pgrg ukg flprfsg ,sf vfrìg rffighm fk ogstms ckiusanghms fk lgkm hf m`rg,flpm,lgquckgrcg fktrf mtrgs.