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ომარ ფურთუხია - ალბათობა და...
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omar furTuxia albaToba da statistika magaliTebsa da amocanebSi (Teoria, magaliTebis amoxsnis nimuSebi, amocanebi) Tsu-2011
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ალბათობის და სტასისტიკის ამოცანათა კრებული თეორიით და ნიმუშებით
Transcript of ომარ ფურთუხია - ალბათობა და...
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omar furTuxia
albaToba da statistika
magaliTebsa da amocanebSi
(Teoria, magaliTebis amoxsnis nimuSebi, amocanebi)
Tsu-2011
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s a r C e v i
albaTobis Teoria Tavi I. albaTobis Teoriis elementebi ............................................................................ elementarul xdomilebaTa sivrce. operaciebi xdomilobebze. albaTobis klasi-kuri, statistikuri da geometriuli ganmarteba. kombinatorikis elementebi. al-baTobis gamoTvla kombinatorikis gamoyenebiT.
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Tavi II. rTuli xdomilebis albaTobebi ......................................................................... albaTobaTa Sekrebis kanoni. sxvaobis albaTobis formula. pirobiTi albaToba. namravlis albaToba. xdomilebaTa damoukidebeloba. sruli albaTobis formu-la. baiesis formula. ganmeorebiTi cdebi. bernulisa da puasonis formulebi.
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Tavi III. SemTxveviT sidideTa maxasiaTeblebi .......................................................... SemTxveviTi sidide. ganawilebis kanoni. ganawilebis funqcia. ganawilebis simkv-rive. zogierTi mniSvnelovani ganawileba. kvantili, mediana, moda. maTematikuri lodini, dispersia. organzomilebiani SemTxveviTi sidide. regresiis funqcia.
momentebi, asimetria, eqscesi. kovariacia. korelaciis koeficienti.
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Tavi IV. diskretul ganawilebaTa gamoyenebebi ...................................................... binomialuri da puasonis ganawilebebis gamoyenebebi
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Tavi V. uwyveti tipis ganawilebebi........................................................................................ ganawilebis simkvrive. kvantili, moda, mediana, qveda da zeda kvartili, zeda -kritikuli wertili, momentebi (lodini, dispersia, asimetria, eqscesi).
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Tavi VI. normaluri ganawileba ............................................................................................... 70 Tavi VII. albaTobis Teoriis zRvariTi Teoremebi ............................................. CebiSevis utoloba. did ricxvTa kanoni. CebiSevis Teorema. bernulis Teorema. centraluri zRvariTi Teorema. liapunovis Teorema. muavr-laplasis lokaluri da integraluri Teoremebi.
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statistikuri daskvnebis Teoria Tavi I. wertilovani da intervaluri Sefasebebi. ndobis intervali saSualosaTvis ........................................................................................................................................... wertilovan SefasebaTa saxeebi (Caunacvlebeli, Zalmosili da efeqturi Sefase-bebi). SerCeviTi momentebi (saSualo, dispersia, Sesworebuli dispersia, standa-rtuli gadaxra, asimetria, eqscesi). SerCeviTi korelacia. maqsimaluri dasajer-obis meTodi. momentTa meTodi. ndobis intervali populaciis saSualosaTvis.
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Tavi II. ndobis intervali bernulis sqemaSi ............................................................. asimptoturi ndobis intervali da SerCevis moculoba populaciis proporciis-aTvis. ndobis intervali puasonis ganawilebis ucnobi parametrisaTvis.
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Tavi III. ndobis intervali dispersiisaTvis ............................................................... ndobis intervali normaluri populaciis dispersiisa da standartuli gadaxr-isaTvis cnobili da ucnobi saSualoebisaTvis. didi SerCevebis SemTxveva.
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Tavi IV. hipoTezaTa Semowmeba saSualosaTvis ( 2 cnobilia) ................ ZiriTadi da alternatiuli hipoTezebi. mniSvnelovnebis done. kriteriumis sta-tistika. kritikuli mniSvneloba. kritikuli are. I da II gvaris Secdomebi. krit-eriumis simZlavre. hipoTezaTa Semowmeba normaluri populaciis saSualosaTvis
(dipersia cnobilia). SerCevis minimaluri moculoba. P -mniSvnelobis meTodi.
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Tavi V. hipoTezaTa Semowmeba saSualosaTvis ( 2 ucnobia) ...................... hipoTezaTa Semowmeba normaluri populaciis saSualosaTvis ucnobi dipersiis
SemTxvevaSi. Z da T kriteriumebis gamoyenebis SesaZlo SemTxvevebi.
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Tavi VI. hipoTezaTa Semowmeba dispersiisaTvis ...................................................... 129
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statistikuri hipoTezis Semowmeba normaluri ganawilebis dispersiis Sesaxeb
cnobili da ucnobi saSualoebis SemTxvevaSi. P -mniSvnelobis meTodi. Tavi VII. hipoTezaTa Semowmeba bernulis sqemaSi ................................................. hipoTezaTa Semowmeba bernulis sqemaSi ucnobi albaTobisaTvis (didi da mcire SerCevebi). hipoTezaTa Semowmeba puasonis ganawilebis ucnobi parametrisaTvis
(mcire SerCevebi). P -mniSvnelobis meTodi.
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Tavi VIII. ndobis intervali da hipoTezaTa Semowmeba .................................... 140 Tavi IX. oramokrefiani amocanebi populaciaTa saSualoebisaTvis ... ori damoukidebeli normaluri populaciis saSualoTa Soris gansxvavebis hipoTezaTa Semowmeba cnobili da ucnobi dispersiebis SemTxvevaSi. hipoTezaTa Semowmeba aranormaluri populaciebisaTvis (didi moculobis mqone SerCevebi). ndobis intervalebi saSualoTa sxvaobisaTvis.
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Tavi X. oramokrefiani amocanebi populaciaTa saSualoebisaTvis (mcire SerCevebi). satertvaitis meTodi ............................................................................ ori damoukidebeli aranormaluri populaciis saSualos Soris gansxvavebis hi-poTezaTa Semowmeba mcire SerCevebis SemTxvevaSi. ndobis intervali saSualoTa sxvaobisaTvis. satertvaitis meTodi.
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Tavi XI. oramokrefiani amocanebi populaciaTa dispersiebisaTvis ... hipoTezaTa Semowmeba ori damoukidebeli normaluri populaciis dispersiebis Sesaxeb. dispersiaTa tolobis Sesaxeb hipoTezaTa Semowmebis gamartivebuli procedura. ndobis intervali dispersiaTa fardobisaTvis.
164
Tavi XII. statistikuri daskvnebi dawyvilebuli monacemebisaTvis .... statistikuri daskvnebi dawyvilebul monacemTa saSualoebis sxvaobebisaTvis
(damokidebuli SerCevebi). P -mniSvnelobis meTodi. ndobis intervali.
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Tavi XIII. oramokrefiani amocanebi bernulis sqemaSi ...................................... hipoTezaTa Semowmeba warmatebaTa albaTobebisaTvis bernulis cdaTa ori damo-
ukidebeli mimdevrobisaTvis. ndobis intervali 1 2p p sxvaobisaTvis.
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Tavi XIV. Tanxmobis kriteriumebi ......................................................................................... xi-kvadrat Tanxmobis kriteriumis formula. normalurobis hipoTezis Semowmeba.
186
Tavi XV. damoukideblobis Semowmebis xi-kvadrat kriteriumi .............. 193 Tavi XVI. erTgvarovnebis Semowmebis xi-kvadrat kriteriumi .................. 201 Tavi XVII. dawyvilebuli monacemebi, korelacia ................................................... gabnevis diagrama. misadagebis wiri. regresiis wrfis gantoleba. Sefasebis stan-dartuli Secdoma. prognozi da saprogrozo intervali. daskvnebi korelaciis koeficientis Sesaxeb (hipoTezaTa Semowmeba).
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Tavi XVIII. regresia da korelacia ...................................................................................... martivi wrfivi regresia. regresiis wrfe. umcires kvadratTa meTodi. regresiis wrfis Sefaseba. determinaciis koeficienti. statistikuri daskvnebi regresiis koeficientebis Sesaxeb. mravlobiTi regresia. modelis vargisianobis Semowmeba. statistikuri daskvnebi mravlobiTi modelis parametrebis Sesaxeb. rangobrivi korelacia. spirmenis rangobrivi korelaciis koeficienti.
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Tavi XIX. dispersiuli analizi (ANOVA) ...................................................................... hipoTezaTa Semowmeba sami an meti populaciis saSualoTa tolobis Sesaxeb.
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Tavi XX. sakontrolo werebisa da Sualeduri da saboloo gamocdebis bileTebis nimuSebi 20072009 wlebSi ..............................................
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danarTi (statistikuri cxrilebi) ....................................................................................... 254 damateba (sagamocdo amocanebi xdomilebebisaTvis)............................................. 263 literatura .................................................................................................................................................. 272
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a l b a T o b a
Tavi I albaTobis Teoriis elementebi
eqsperimentis calkeul SesaZlo Sedegs elementaruli xdomileba
ewodeba, xolo maT erTobliobas elementarul xdomilebaTa sivrce da aRiniSneba asoTi: 1 2{ , ,...} .
savarjiSoebi: I. monetis erTxel agdebisas -- ={g, s}; II. monetis orjer agdebisas, an ori monetis erTdroulad agdebisas -- ={gg, gs, sg, ss}; III. monetis samjer agdebisas, an sami monetis erTdroulad agdebisas -- = {ggg, ggs, gsg, sgg, gss, sgs, ssg, sss}; IV. monetis n -jer agdebisas
}),,...,(:{ 1 sang in aaa da Sedegebis saerTo raodenoba tolia n2 -is; V. erTi saTamaSo kamaTlis gagorebisas -- ={1, 2, 3, 4, 5, 6}; VI.
vTqvaT, Tavidan vagdebT monetas. Tu mova gerbi, maSin vagorebT saTamaSo kamaTels; xolo Tu mova safasuri, maSin kidev erTjer vagdebT monetas. am SemTxvevaSi },3,2,1{ sssg,g6,g5,g4,ggg ; VII. ori saTamaSo kamaTlis ga-
gorebisas -- ={(1,1); (1,2); . . . ; (1,6); (2,1); (2,2); . . . ; (2,6); . . . ; (6,1); . . . ; (6,6)} anu }6,...,2,1,:),{( jiji ; VIII. produqciis vargisinobis dadgenisas --
={vargisi, uvargisi}; IX. satelefono sadgurSi gamoZaxebaTa raoden-oba -- = {0, 1, 2, . . . }; X. Zabva qselSi -- ={[0, 220]}. diskretul elementarul xdomilebaTa sivrcis nebismier qvesim-ravles xdomileba ewodeba. Tu eqsperimentis konkretuli Sedegi ekuTv-nis raime xdomilebas, maSin amboben rom es xdomileba moxda, xolo romelsac ar ekuTvnis is xdomileba ar moxda. xdomilebebi aRiniSneba didi laTinuri asoebiT: ,...,,, DCBA . xdomilebas A uwodeben
aucilebel xdomilebas, xolo -- SeuZlebel xdomilebas. A da B xdomilebis gaerTianeba (an jami) ewodeba iseT xdomilebas, romelic xdeba maSin, roca am xdomilebebidan erTi mainc xdeba da aRiniSneba simboloTi BA (an BA ). A da B xdomilebis TanakveTa (an namravli) ewodeba iseT xdomilebas, romelic xdeba maSin, roca es xdomilebebi erTdroulad xdeba da aRiniSneba simboloTi BA (an AB ). A xdomilebis sawinaaRmdego xdomileba ewodeba iseT xdomilebas, romelic
xdeba maSin, roca A ar xdeba da aRiniSneba simboloTi A . A da B xdomilebis sxvaoba ewodeba iseT xdomilebas, romelic xdeba maSin, roca xdeba A magram ar xdeba B da aRiniSneba simboloTi BA \ . A da B xdomilebas ewodeba uTavsebadi Tu BA =.
Tu -s nebismier elementarul xdomilebas i Seesabameba garkveu-
li ricxvebi )( ii Pp , romlebic akmayofileben pirobebs: 10 ip da
1ii p , maSin am ricxvebs ewodebaT i elementaruli xdomilebebis alb-
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4
aTobebi. ( ) : ( )i
iAP A P
. Tu ( )iP const da | | , vRebulobT albaTob-
is klasikur ganmartebas: ( ) | | / | |P A A . xdomilebis statistikur albaTobad iTvleba am xdomilebis fard-
obiTi sixSire ( ) /NW A M N (sadac N cdaTa saerTo ricxvia, ki -- A
xdomilebis moxdenaTa ricxvi) an masTan axlos myofi ricxvi (maTematik-
urad zusti formulireba aseTia: ( ) lim ( )NN
P A W A
).
geometriuli albaToba. Tu L monakveTze SemTxveviT agdeben wert-ils, maSin albaToba imisa, rom agdebuli wertili daecema l L monakve-Tze: ||/|| LlP . analogiuri ganmarteba gvaqvs sibrtyeze da sivrceSi.
kombinatorikis elementebi.
gamravlebis principi: Tu asarCevia m obieqti da arsebobs pirveli
obieqtis arCevis 1n SesaZlebloba, pirveli obieqtis arCevis Semdeg arse-
bobs meore obieqtis arCevis 2n SesaZlebloba, da a. S., 1m obieqtis ar-
Cevis Semdeg arsebobs m -uri obieqtis arCevis mn SesaZlebloba, maSin
arsebobs am m obieqtis am mimdevrobiT arCevis 1 2 mn n n SesaZlebl-
oba. magaliTad, Tu mamakacs aqvs 4 perangi da 2 pijaki, maSin am mamakacs
aqvs perangisa da pijakis SerCevis 4 2 8 SesaZlebloba (varianti). namravlis principTan dakavSirebiT xSirad sasargebloa xis msgavsi
(xisebri) diagramis anu dendrogramis gamoyeneba. magaliTad, dendrogramiT gamosaxuli monetis orjer agdebis Sesa-
bamisi Sedegebis simravle iqneba:
gadanacvlebebi es aris kombinaciebi, romlebic Sedgenilia mocem-uli elementiani simravlis yvela elementisagan da erTamaneTisagan ganxsvavdeba mxolod elementebis ganlagebis rigiT. = !. wyobebi es aris elementiani kombinaciebi gansxvavebuli elem-entis mqone simravlidan, romelebic erTmaneTisagan gansxvavdebian an el-
ementebis SemadgenlobiT an elementebis ganlagebis rigiT. !/( )! n n m .
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jufdebebi es aris elementiani simravlis daulagebeli elem-
entiani qvesimravleebi. !/( !( )!) . cxadia, rom !m m
n nC m A .
albaTobis gamoTvla kombinatorikis gamoyenebiT. amorCeva dabrunebiT: eqsperimentis yovel nabijze yuTidan amoRebu-
li burTi ukan brundeba. vigulisxmoT, rom burTebi gadanomrilia ricx-vebiT 1-dan M -mde. ganasxvaveben ori tipis amorCevebs: dalagebuli amorC-evebi da daulagebeli amorCevebi. dalagebuli amorCevebi aRiniSneba simb-oloTi ( naa ,...,1 ), xolo daulagebeli amorCevebi [ naa ,...,1 ].
dabrunebiT dalagebuli amorCevis SemTxvevaSi:
},...,1;,...,1:),...,(:{ 1 niMaaa in da nM || .
daulagebeli amorCevebi dabrunebiT:
},...,1;,...,1:],...,[:{ 1 niMaaa in , n
nMC 1|| .
amorCeva dabrunebis gareSe: Mn da amoRebuli burTi ukan ar brundeba. dabrunebis gareSe dalagebuli SerCevis SemTxvevaSi:
},...,1;,...,1;,:),...,(:{ 1 niMalkaaaa ilkn da | |n
MA .
daulagebeli amorCevebi dabrunebis gareSe:
},...,1;,...,1;,:],...,[:{ 1 niMalkaaaa ilkn , nMC ||
saboloo suraTi aseTia:
dalagebuli daulagebeli dabrunebiT nM 1
n
M nC
dabrunebis gareSe
nMA n
MC
magaliTad, 3M da 2n -is SemTxvevaSi Sesabamis elementarul
xdomilebaTa sivrceebs eqnebaT Semdegi saxis struqturebi:
SerCeva
(1,1)(1,2)(1,3) (2,1)(2,2)(2,3) (3,1)(3,2)(3,3)
[1,1][2,2][3,3] [1,2][1,3][2,3]
dabrunebiT
(1,2)(1,3) (2,1)(2,3) (3,1)(3,2)
[1,2] [1,3] [2,3] dabrunebis gareSe
erToblioba dalagebuli daulagebeli savarjiSoebi:
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I. kursze, romelzec sami jgufia, jgufxelebis arCevis yvela SesaZ-
lo variantebis ricxvia 321 nnn , sadac in -- i -ur jgufSi studentebis
raodenobaa (viyenebT namravlis princips); II. raodenoba yvela SesaZlo kombinaciebis, ramdennairadac SesaZle-
belia m mgzavri ganvaTavsoT n vagonSi tolia mn (dalagebuli amorCe-va dabrunebiT, sadac M n da n m );
III. m adamianis dabadebis dReebis yvela SesaZlo kombinacia (im pi-robiT, rom TiTeuli dabadebis dRe aris romelime 365 dRidan) tolia
m365 (saqme gvaqvs amorCevasTan dabrunebiT, amasTanave amorCevebi iTvleba dalagebulad, sadac 365M da n m );
IV. raodenoba yvela SesaZlo kombinaciebis, ramdennairadac SesaZl-ebelia 5 burTi ganvaTavsoT 5 yuTSi, ise rom erT yuTSi iyos erTi bur-Ti, tolia 5! (namravlis principis Tanaxmad);
V. partiebis raodenoba, romelic unda Sedges n monawilisagan Sem-
dgar wriul saWadrako turnirSi tolia 2/)1(2 nnCn (daulagebeli amo-
rCeva dabrunebis gareSe).
magaliTebis amoxsnis nimuSebi:
magaliTi 1. vipovoT albaToba imisa, rom simetriuli monetis ori damoukidebeli agdebisas erTjer mainc mova gerbi.
amoxsna. am SemTxvevaSi ={gg, gs, sg, ss}; }{ggP }{gsP = }{sgP
= 4/1}{ ssP ; A={gg, gs, sg} da 4/3)( AP . magaliTi 2 (bednier bileTebze). 25 sagamocdo bileTidan 5 bedn-
ieria, xolo danarCeni 20 ara bednieri. romel students aqvs bedn-ieri bileTis aRebis meti albaToba: vinc pirveli iRebs bileTs, Tu vinc meore iRebs bileTs?
amoxsna. avRniSnoT pirveli studentis mier aRebuli bileTis nome-ri i asoTi, xolo meore studentis mier aRebuli bileTis nomeri j aso-Ti. davuSvaT, rom bednieri bileTebis nomrebia: 1, 2, 3, 4, 5. maSin cxa-dia, rom },25,...,2,1,:),{( jijiji , 6002425|| da bunebrivia CavTval-
oT, rom yvela elementaruli xdomileba tolalbaTuria: 600/1),( jiP .
SemoviRoT xdomilebebi: { }A pirvelma studentma aiRo kargi bileTi ,
{ }B meore studentma aiRo kargi bileTi , maSin am xdomilebebs aqvT Semdegi saxe:
};25,...,1;5,...,1:).{( jijijiA da };5,...,1;25,...,1:),{( jijijiB .
advili dasanaxia, rom 120245|||| BA . amitom albaTobis klasik-uri ganmartebis Tanaxmad:
120 1( ) ( )
600 5P A P B .
e. i. orive students aqvs kargi bileTis aRebis erTi da igive albaToba.
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davaleba. wina amocanaSi vipovoT albaToba imisa, rom mesame stud-enti amoiRebs bednier bileTs. magaliTi 3 (galtonis dafa). gvaqvs dafaze samkuTxedis formiT da-lagebuli rgolebi, ise rom wveroSi erTi rgolia, meore rigSi winasg-an Tanabr manZilebze ori rgoli, mesame rigSi zeda ori rgolidan Tana-bar manZilebze sami rgoli da a.S. boloSi aris eqvsi rgoli. me-7 rigSi ki aris bolo 6 rgolidan Tanabar manZilebze 7 Rrmuli. zeda rgolze agdeben burTs da mas SeuZlia igoraos Tanabari albaTobiT an marjvniv an marcxniv rgolidan rgolze, rac sabolood sruldeba romelime Rrm-ulSi CavardniT. rogoria albaToba imisa, rom burTi Cavardeba mexuTe RrmulSi?
O
O O
O O O
O O O O
O O O O O
O O O O O O
U U U U U U U 1 2 3 4 5 6 7
amoxsna. rogorc vxedavT arsebobs pirvel da me-7 RrmulebSi burT-is Cavardnis erTaderTi gza (traeqtoria), meore da me-6 RrmulebSi bur-Tis Cavardnis -- eqvs-eqvsi gza, mesame da mexuTe RrmulebSi TxuTmet-TxuTmeti gza da bolos, meoTxe RrmulSi burTis Cavardnis 20 gza. gzebis (Sedegebis) sruli raodenobaa 1+6+15+20+15+6+1=64 da yvela es Se-degi Tanabrad mosalodnelia, vinaidan TiToeuli traeqtoriis gavlisas burTi ganicdis eqvs dajaxebas rgolebze da yoveli dajaxebisas is Ta-nabari albaTobebiT gadaadgildeba an marjvniv an marcxniv. Tanabaralb-aTuri 64 Sedegidan mexuTe RrmulSi Cavardnas xels uwyobs 15 Sedegi da Sesabamisad, saZebni albaToba iqneba 15/64. qvemoT moyvanilia galtonis dafaze rgolebis 7 rigis SemTxvevaSi TiToeul poziciaze burTis moxvedris SesaZlo gzebis ricxvi.
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magaliTi 4. vipovoT albaToba imisa, rom wreSi SemTxveviT Cagdebu-li wertili ar Cavardeba am wreSi Caxazul wesier eqvskuTxedSi. amoxsna. davuSvaT, rom wris radiusia R , maSin masSi Caxazuli wes-
ieri eqvskuTxedis gverdic iqneba R . amasTanave, wris farTobia 2|| RS ,
xolo eqvskuTxedis farTobia -- .2
33|| 2Rs amitom saZiebeli albaToba
iqneba:
.174,02
3322
33
||
||||2
22
R
RR
S
sSP
magaliTi 5. AB monakveTze SemTxveviT agdeben sam wertils ,C D
da M . vipovoT albaToba imisa, rom ADAC, da AM monakveTebisagan Sei-
Zleba aigos samkuTxedi? amoxsna. avRniSnoT ADAC, da AM monakveTebis sigrZeebi Sesabamis-
ad yx, da z -iT da elementarul xdomilebaTa sivrcis rolSi ganvixil-
oT sivrcis wertilTa simravle koordinatebiT ),,( zyx . Tu CavTvliT,
rom AB monakveTis sigrZe tolia 1-is, maSin elementarul xdomilebaTa sivrce iqneba kubi, romlis wiboa erTi. amave dros, xelSemwyob elementa-rul xdomilebaTa simravle (samkuTxedis aqsiomis Tanaxmad )Sedgeba im wertilebisagan, romelTa koordinatebisaTvis sruldeba samkuTxedis ut-olobebi: x +y > z, x + z > y, y + z > x. es ki warmoadgens kubis nawils, romel-ic moWrilia misgan sibrtyeebiT: x + y = z, x + z = y, y + z = x z
y
(erT-erTi am sibrtyidan, kerZod x + y = z, moyvanilia naxazze). yoveli ase-Ti sibrtye kubidan moWris piramidas, romlis moculoba tolia
6
11
2
1
3
1 .
Sesabamisad, kubis darCenili nawilis moculoba iqneba
2
1
6
131|| v .
amitom saZiebeli albaToba, ganmartebis Tanaxmad, iqneba
.2
11:
2
1
||
||||
V
vP
magaliTi 6 (Sexvedris amocana). ori piri SeTanxmda garkveul adgi-lze Sexvdes erTmaneTs 6-dan 7 saaTamde. TiToeuli maTgani SemTxveviT
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momentSi midis daTqmul adgilas da meores elodeba 20 wuTis manZilze (Semdeg ki midis). vipovoT albaToba imisa, rom es pirebi Sexvdebian erTm-aneTs?
amoxsna. avRniSnoT, erT-erTi piris daTqmul adgilze misvlis dro x6 -iT, xolo meore piris -- y6 -iT (sadac x da y gamosaxulia saaTeb-
Si). elementarul xdomilebaTa sivrcis rolSi SegviZlia aviRoT im ),( yx wertilTa simravle, romlebic ekuTvnian erTeulovan kvadrats. am SemTx-vevaSi CvenTvis saintereso wertilTa (xelSemwyob elementarul xdomile-baTa) simravle iqneba erTeulovani kvadratis im wertilTa simravle, ro-melTa koordinatebi erTmaneTisagan daSorebulia araumetes 3/160/20 -iT:
}3/1||);,{( yxyxA .
vinaidan, 3/13/13/13/13/1|| xyxxyyx . amitom adv-
ili gasagebia, rom A simravle iqneba erTeulovani kvadratis SigniT 3/1 xy da 3/1 xy wrfeebs Soris moqceuli gaferadebuli are (roca
3/10 x , maSin 3/1 xy wrfis nacvlad qveda sazRvris rolSi iqneba x
RerZi: 3/10 xy , xolo roca 13/2 x , maSin zeda sazRvari 3/1 xy
wrfis nacvlad iqneba 1y wrfe: 13/1 yx ).
0 1/3 2/3 1 x
y 1
1/3
amitom saZiebeli albaToba iqneba:
9
5
1
3/23/21
||
||)(
AAP .
magaliTi 7. ramdegi gansxvavebuli sia SeiZleba Sedgenil iqnes 7 sxvadasxva gvarisagan?
amoxsna. 7 7! 1 2 3 4 5 6 7 5040 .
magaliTi 8. SejibrebaSi monawile 10 sportsmenidan pirvel sam adg-ilze gasuli sami gamarjvebuli ramden sxvadasxvanairad SeiZleba ganTa-vsdes dasajildoebel kvarcxlbekze?
amoxsna. .7208910310
magaliTi 9. SesarCev SejibrebaSi monawileobs 10 adamiani, romelT-agan finalSi gadis sami. finalistebis ramdeni gansxvavebuli sameuli SeiZleba gamovlindes? amoxsna. wina magaliTisagan gansxvavebiT, aq finalistebis rigs (da-lagebas) mniSvneloba ara aqvs. amitom viyenebT jufdebis formulas:
-
10
.1206
1098
!7!3
!10310
magaliTi 10 (damTxvevebze). vipovoT albaToba imisa, rom: a). m Sem-TxveviT arCeuli adamianis dabadebis dReebi ar daemTxveva erTmaneTs (im pirobiT, rom yvela dRe tolalbaTuria); b). m SemTxveviT arCeul adami-anSi moiZebneba ori mainc iseTi, romelTa dabadebis dReebi daemTxveva erTmaneTs.
amoxsna. a). elementarul xdomilebaTa sivrce Seesabameba dalageb-
ul amorCevebs dabrunebiT, sadac 365M da n m , anu | | 365m ; xolo xelSemwyobi elementaruli xdomilebebi ki dalagebul amorCevebs dabru-nebis gareSe, sadac agreTve 365M da n m , amitom maTi raodenobaa --
365
mA . Sesabamisad, albaTobis klasikuri ganmartebis Tanaxmad, gvaqvs
)365
11()
365
21)(
365
11(
365)( 365
mAmP
m
m
.
b). sawinaaRmdego xdomilebis albaTobis gamoTvlis wesis Tanaxmad ki gvaqvs, rom
m
mAmQ
3651)( 365 .
moviyvanoT am albaTobis mniSvnelobebis cxrili zogierTi m -is SemTxvevaSi:
m 4 16 22 23 40 64 70
Q(m) 0.01636 0.28360 0.47569 0.50730 0.89123 0.99711 0.99916
sainteresoa aRiniSnos, rom (molodinis sawinaaRmdegod!) klasis
moswavleTa raodenoba, romelSic 1/2-is toli albaTobiT moiZebneba ori moswavle mainc erTi da igive dabadebis dReebiT, arc ise didia: igi to-lia mxolod 23-is.
magaliTi 11 (mogeba latareaSi). gvaqvs M bileTi gadanomrili ri-cxvebiT erTidan M -mde, romelTagan n bileTi nomrebiT erTidan n -mde momgebiania ( nM 2 ). vyidulobT n cal bileTs. ras udris albaToba imi-sa, rom am n bileTidan erTi mainc iqneba momgebiani (avRniSnoT es xdom-ileba A asoTi)?
amoxsna. vinaidan bileTebis amoRebis (yidvis) Tanmimdevrobas ara aqvs mniSvneloba nayid bileTebSi momgebiani bileTebis arsebobis an ar arsebobis TvalsazrisiT, amitom elementarul xdomilebaTa sivrces eqne-ba Semdegi stuqtura:
},...,1;,:],...,[:{ 1 Malkaaaa ilkn .
Sesabamisad, Cvens mier zemoTmoyvanili cxrilis Tanaxmad nMC || .
xdomilebas (avRniSnoT igi 0B -iT), rom nayid bileTebSi ar aris mo-
mgebiani bileTebi, eqneba Semdegi struqtura:
},...,1;,:],...,[:{ 10 MnalkaaaaB ilkn da n
nMCB || 0 .
amitom
-
11
)1
1()1
1)(1()( 0
nM
n
M
n
M
n
A
A
C
CBP
nM
nnM
nM
nnM .
Sesabamisad, saZebni albaToba iqneba:
0( ) 1 ( ) 1 (1 )(1 ) (1 )1 1
n n nP B P B
M M M n
.
Tu magaliTad, 2nM da n , maSin 10)(eBP (aq e neperis ric-
xvia) da 632,01)( 1 eBP , sadac krebadobis siCqare sakmaod didia: ukve
roca 10n -- 670,0)( BP .
davaleba. wina amocanis pirobebSi vipovoT albaToba imisa, rom nay-idi n bileTidan zustad m ( nm ) iqneba momgebiani. magaliTi 12 (urTierTobis upiratesobaze). davuSvaT klasSi, rome-lSic 10 moswavlea tardeba gamokiTxva, sadac TiToeulma moswavlem ank-etaSi unda miuTiTos is sami amxanagi, romelsac aZlevs upiratesobas cxra amxanagidan. A iyos xdomileba, rom erTerTi moswavle dasaxeleb-ul iqna yvela SesaZlo cxra anketaSi. ras udris misi albaToba, Tu ank-etis Sevseba iyo SemTxveviTi, anu anketis Sevsebis nebismieri kombinacia tolalbaTuria. amoxsna. calkeuli moswavlisaTvis anketis Sevsebis sxvadasxva kom-
binaciaTa raodenoba tolia 39C -is, xolo 10 anketis Sevsebis variantebis
raodenoba pirveli magaliTis analogiurad tolia 1039 )(C -is. vinaidan er-
Ti anketa SeiZleba Sevsebul iqnes nebismierad, xolo danarCen cxra anke-taSi erTi pasuxi dafiqsirebulia da danarCeni ori pasuxi ki SeiZleba nebismierad amoirCes rva SesaZlebeli pasuxidan, amitom elementaruli
xdomilebebis raodenoba A -Si tolia 3 2 99 8| | ( )A C C (Tu wyvilis pirvel
komponents SeuZlia miiRos m gansxvavebuli mniSvneloba, xolo meore komponents ki pirvelisgan damoukideblad -- n gansxvavebuli mniSvnelo-ba, maSin aseTi wyvilebis raodenoba namravlis principis Tanaxmad iqneba -- m n ). aqedan gvaqvs
9
9
39
28
1039
928
39
3
1)(
)(
)()(
C
C
C
CCAP .
davaleba. wina amocanaSi ipoveT albaToba imisa, rom erTerTi stu-denti dasaxelebuli iqneba k -jer ( 9k ). magaliTi 13 (or tuzze). ganvixiloT preferansis TamaSi, rodesac kartis Sekvris maRali 32 karti SemTxveviT nawildeba (rigdeba) sam moTa-maSes Soris, ise rom TiTeuli Rebulobs 10 karts da ori karti inaxeba sayidlebSi. rogoria albaToba imisa, rom sayidlebSi aRmoCndeba ori tuzi?
amoxsna. ori kartis sxvadasxva kombinaciebis raodenoba, romelic
SeiZleba aRmoCndes sayidlebSi tolia 496232 C . kartis SekvraSi oTxi
tuziaDda sxvadasxva kombinaciebis raodenoba, romelic mogvcemda or
tuzsAtolia 624 C . Sesabamisad, saZiebeli albaToba tolia
-
12
012,0496
6232
24
CC .
davaleba. davuSvaT wina amocanaSi erTerTma moTamaSem, naxa ra Tav-isi kartebi, icis, rom mas tuzi ara aqvs. Seicvleba Tu ara maSin alba-Toba imisa, rom sayidlebSi ori tuzia? gamoTvaleT es albaToba. magaliTi 14 (mxedvelobiT moZebnaze). davuSvaT gvaqvs N cali Sem-TxveviT dalagebuli geometriuli figura, romelTa Soris M cali mar-TkuTxedia ( NM ). moiTxoveba moiZebnos yvela marTkuTxedi, Tu Zebna wa-rmoebs elementebis (figurebis) saTiTaod skanirebiT fiqsaciis moculo-biT erTi elementi, amasTanave xdeba dakvirvebuli elementis poziciis damaxsovreba da mas Tavidan aRar vubrundebiT. vipovoT albaToba imisa, rom damkvirvebeli SeZlebs aRmoaCinos yvela M marTkuTxedi araumetes n dakvirvebisas ( NMn ,..., )?
amoxsna. yvela SesaZlo elementarul xdomilebaTa raodenoba iqne-
ba nNC || . xelSemwyobi elementaruli xdomilebebi iseTi [ naa ,...,1 ] erTo-
bliobebia, romlebSic M adgilas ganTavsebulia marTkuTxedebi (amis
SesaZleblobaTa raodenobaa MMC ), xolo danarCen MN adgilas ki ara
marTkuTxedebi (amis SesaZleblobaTa raodenobaa Mn MNC ). amitom xelSemwy-
obi elementaruli xdomilebebis raodenoba iqneba MMCMnMNC
. Sesabamis-
ad, saZiebeli albaToba tolia
MN
Mn
nN
MnMN
nN
MnMN
MM
MC
C
C
C
C
CCnP
)( .
amocanebi
1. saTamaSo ruleti Sedgeba 38 toli farTobis mqone seqtorisag-
an, romelTagan 18 Savia, 18 wiTelia da 2 ki mwvane. agdeben burTs, romel-ic sabolood Cerdeba erT-erT seqtorSi. ipoveT albaToba imisa, rom bu-rTi: a). gaCerdeba wiTel seqtorSi; b). ar gaCerdeba Sav seqtorSi.
2. ipoveT albaToba imisa, rom SemTxveviT SerCeuli erTi cifri iq-neba 3-is jeradi.
3. ipoveT albaToba imisa, rom SemTxveviT SerCeuli ori cifri: a). ar daemTxveva erTmaneTs; b). daemTxveva erTmaneTs.
4. ipoveT albaToba imisa, rom SemTxveviT SerCeuli sami cifridan: a). samive gansxvavebulia; b). samive daemTxveva erTmaneTs; g). ori mainc da-emTxveva erTmaneTs; d). mxolod ori daemTxveva erTmaneTs.
5. vipovoT albaToba imisa, rom ori saTamaSo kamaTlis gagorebisas qulebis jami iqneba: a). 9 qula; b). 10 qula.
6. vipovoT albaToba imisa, rom sami saTamaSo kamaTlis gagorebisas qulebis jami iqneba: a). 9 qula; b). 10 qula.
7. 60 sagamocdo sakiTxidan studentma icis 50 sakiTxi. ipoveT alba-Toba imisa, rom studenti upasuxebs orsakiTxiani bileTis orive sakiTxs.
-
13
8. yuTSi aris 11 wiTeli, 12 yviTeli, 10 TeTri da 17 Savi burTi. ip-oveT albaToba imisa, rom SemTxveviT amoRebuli burTi iqneba wiTeli an Savi.
9. 1000 dadebiTi mTeli ricxvidan SemTxveviT irCeven erTs. ipoveT albaToba imisa, rom es ricxvi iqneba: a). 4-is jeradi; b). erTdroulad 4-isa da 6-is jeradi.
10. kalaTaSi devs 7 wiTeli da 4 TeTri diski. dabrunebis gareSe irCeven or disks. ipoveT albaToba imisa, rom: a). orive diski wiTelia; b). orive diski erTi da igive ferisaa; g). diskebi sxvadasxva ferisaa; d). me-ore diski wiTelia; e). meore diski TeTria.
11. CanTaSi devs 6 wiTeli da 4 mwvane kalkulatori. iReben or kal-kulators dabrunebis gareSe. ipoveT albaToba imisa, rom: a). orive kalk-ulatori wiTelia; b). orive mwvanea; g). zustad erTi kalkulatori wiTe-lia; d). erTi mainc kalkulatori wiTelia; e). meore kalkulatori wiTe-lia.
12. 52 kartidan iReben or karts dabrunebis gareSe. ipoveT albaTo-ba imisa, rom: a). orive karti suraTebiania ( , ,K Q J ); b). arcerTi ar aris suraTebiani; g). erTi mainc suraTebiania; d). erTi mainc wiTelia.
13. CanTaSi devs 6 wiTeli da 4 mwvane kalkulatori. SemTxveviT iR-eben erT kalkulators, mis fers iniSnaven da mas abruneben CanTaSi. Sem-deg SemTxveviT iReben meore kalkulators. ipoveT albaToba imisa, rom: a). orive kalkulatori wiTelia; b). orive mwvanea; g). zustad erTi wiTe-lia; d). erTi mainc wiTelia; e). meore wiTelia.
14. A yuTi Seicavs 1 wiTel da 1 Sav burTs, B yuTi Seicavs 2 wiT-el burTs. A yuTidan SemTxveviT iReben 1 burTs da gadaaqvT B yuTSi. Semdeg B yuTidan SemTxveviT iReben 1 burTs da gadaaqvT A yuTSi. ipov-eT albaToba imisa, rom am ori gadatanis Semdeg Savi burTi iqneba A yu-TSi.
15. A yuTi Seicavs 6 wiTel da 2 mwvane burTs, B yuTi Seicavs 4 wi-Tel da 3 mwvane burTs. agoreben wesier kamaTels. Tu movida luwi qula, maSin A yuTidan SemTxveviT iReben 1 burTs, xolo winaaRmdeg SemTxveva-Si B yuTidan SemTxveviT iReben 1 burTs. ipoveT albaToba imisa, rom: a). amoRebuli burTi wiTelia; b). burTi amoRebuli iyo B yuTidan, Tu cnob-ilia, rom amoRebuli burTi wiTelia.
16. ipoveT albaToba imisa, rom erTeulovan gverdian kvadratSi Sem-TxveviT SerCeuli wertili am kvadratis uaxloesi gverdidan daSorebu-li iqneba araumetes 0.15-iT.
17. ori signali mimReb mowyobilobaze T drois manZilze SemTxvev-iT momentSi miiReba. mowyobiloba maT ganarCevs, Tu isini t droiT mainc arian dacilebuli erTmaneTs. ipoveT albaToba imisa, rom orive signali miRebuli iqneba?
18. centris garSemo mbrunavi wre gayofilia sami toli farTobis mqone seqtorad da gadanomrilia cifrebiT 1, 2 da 3. mas atrialeben or-jer da yoveli gaCerebisas iniSnaven im seqtoris nomers, romelSic moxv-deba am mowyobilebaze uZravad mimagrebuli isari. ipoveT albaToba imi-
-
14
sa, rom: a). orive nomeri iqneba erTidaigive; b). arcerTi nomeri ar iqneba 2; g). erTi nomeri mainc iqneba 3; d). arcerTi nomeri araris 2 da orive erTidaigivea; e). arcerTi nomeri araris 2 an orive erTidaigivea.
19. 36 kartidan A moTamaSes aqvs ori dedofali da erTi mefe. A moTamaSisagan B moTamaSe SemTxveviT iRebs erT karts da ubrunebs isev mas. A moTamSe erTmaneTSi urevs am sam karts da Semdeg B moTamaSe kvlav iRebs erT karts. B moTamaSe igebs Tu mis mier amoRebuli orive karti iqneba mefe. ipoveT B moTamaSis mogebis albaToba.
20. wesieri tetraedris waxnagebi gadanomrilia cifrebiT 1, 2, 3 da 4. mas agdeben orjer da iweren im cifrs, romliTac tetraedri daecema iatakze. ipoveT albaToba imisa, rom: a). orive cifri iqneba erTidaigive; b). arcerTi cifri ar iqneba 3; g). erTi cifri mainc iqneba 3; d). arcerTi cifri ar aris 4 da orive erTidaigivea; e). arcerTi cifri ar aris 4 an orive erTidaigivea.
21. yuTSi devs 10 burTi gadanomrili 1-dan 10-mde ricxvebiT. ipoveT albaToba imisa, rom SemTxveviT amoRebul 6 burTSi aRmoCndeba: a). burTi # 9; b). burTi # 9 da #10.
22. 200 detalis Semowmebisas standartulebis fardobiTi sixSire aRmoCnda 0.9. ipoveT standartuli detalebis raodenoba.
23. sibrtyeze, romelic dafarulia a gverdis mqone kvadrtTa bad-iT, SemTxveviT agdeben monetas radiusiT / 2r a . ipoveT albaToba imisa, rom moneta ar gadakveTs arcerTi kvadratis gverds.
24. sibrtyeze, romelic dafarulia 6 sm-iT daSorebuli paralelu-ri wrfeebiT, SemTxveviT agdeben wres radiusiT 1 sm. ipoveT albaToba imisa, rom wre ar gadakveTs arcerT wrfes.
25. SemTxveviT iReben or dadebiT ricxvs, romelTagan TiToeuli ar aRemateba 2-s. ipoveT albaToba imisa, rom maTi namravli ar aRemateba 1-s, xolo ganayofi ki ar aRemateba 2-s.
pasuxebi: 1. 9/19; 10/19. 2. 3/10. 3. 9/10; 1/10. 4. 18/25; 1/100; 14/50; 27/100. 5. 1/9; 1/12. 6. 25/216; 1/8. 7. 245/354. 8. 14/25. 9. 1/4; 83/1000. 10. 21/55; 27/55; 28/55; 7/11; 4/11. 11. 1/3; 2/15; 8/15; 13/15; 3/5. 12. 11/221; 10/17; 7/17; 77/102. 13. 9/25; 4/25; 12/25; 21/25; 3/5. 14. 2/3. 15. 37/56; 16/37. 18. 1/3; 4/9; 5/9; 2/9; 5/9. 19. 1/9. 20. 1/4; 9/16; 7/16; 3/16; 5/8. 21. 0.6; 1/3.
22. 180. 23. 2 2( 2 ) /a r a . 24. 2/3. 25. (1 3ln2) /8 0.38 .
-
15
Tavi II rTuli xdomilebis albaTobebi
albaTobaTa Sekrebis kanoni: Tu BA , maSin
)()()( BPAPBAP .
sawinaaRmdego xdomilebis albaToba: )(1)( APAP .
sxvaobis albaTobis formula: Tu B A , maSin )()()\( BPAPBAP .
Tu ji AA , roca ji , maSin: 1 1( ) ( )n n
i ii iP A P A
.
roca xdomilebebi Tavsebadia: )()()()( BAPBPAPBAP .
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )P A B C P A P B P C P A B P A C P B C P A B C sazogadod:
1
1 11
( ) ( ) ( ) ( ) ( 1) ( )nn n
n
i i i j i j k ii i
i i j i j k
P A P A P A A P A A A P A
.
pirobiTi albaTobis formula A xdomilebis pirobiTi albaToba pirobaSi, rom adgili hqonda B
xdomilebas, aRiniSneba )|( BAP (an ( )BP A ) simboloTi da
( | ) : ( ) / ( )P A B P A B P B , Tu 0)( BP .
1)|(0 BAP ; BA = 0)|( BAP ; AB 1)|( BAP ;
Tu ji AA , roca ji , maSin: 1 1[( ) | ] ( | )n n
i ii iP A B P A B
.
namravlis albaTobis formula ( ) ( ) ( | ) ( ) ( | )P A B P A P B A P B P A B .
( ) ( ) ( | ) [ | ( )]P A B C P A P B A P C A B .
sazogadod: 1 2 1 2 1 1 1( ) ( ) ( | ) ( | )n n nP A A A P A P A A P A A A .
damokidebuli da damoukidebeli xdomilebebi A xdomilebas ewodeba B xdomilebisagan damoukidebeli, Tu
)()|( APBAP an rac igivea )()()( BPAPBAP . Tu )()|( APBAP , maSin
gvaqvs damokidebuli xdomilebebi. Tu A da B xdomilebebi damoukideb-
elia, maSin xdomilebebi A da B agreTve damoukidebelia.
xdomilebaTa erTobliobas nAAA ,...,, 21 ewodeba wyvil-wyvilad damo-
ukidebeli Tu: jiAPAPAAP jiji ),()()( .
xdomilebaTa erTobliobas nAAA ,...,, 21 ewodeba erToblivad
damoukidebeli Tu nk , 1 2 ki i i :
)()()()(2121 kk iiiiii
APAPAPAAAP .
sruli albaTobis formula
xdomilebaTa erTobliobas nAAA ,...,, 21 ewodeba xdomilebaTa sruli
sistema, Tu ji AA , roca ji da
i
n
iA
1 (magaliTad, A da A ).
-
16
Tu nAAA ,...,, 21 xdomilebaTa sruli sistemaa ( ( ) 0,iP A 1,2,...,i n ),
maSin adgili aqvs sruli albaTobis formulas: )|()()(1
i
n
ii ABPAPBP
.
baiesis formula
Tu nAAA ,...,, 21 xdomilebaTa sruli sistemaa, ,0)( iAP ni ,...,2,1 ,
maSin adgili aqvs baiesis formulas:
1
( ) ( | )( | )
( ) ( | )
i ii n
j jj
P A P B AP A B
P A P B A
.
ganmeorebiTi cdebi. bernulis formula ganvixiloT erTi da igive eqsperimentebis seria, romlebic tardeba
erTi da igive pirobebSi erTmaneTisagan damoukideblad. amasTanave, yov-el konkretul eqsperimentSi Cven ganvasxvavebT mxolod or Sedegs: gark-veuli A xdomilebis moxdena (romelsac pirobiTad warmatebas uwodeb-
en) da misi ar moxdena -- A (romelsac marcxs uwodeben). A xdomilebis moxdenis albaToba nebismieri eqsperimentisaTvis mudmivia da tolia
pAP )( , sadac 10 p . Sesabamisad, qpAPAP :1)(1)( ( 1 qp ).
albaTobas imisa, rom n eqsperimentSi A xdomileba moxda k -jer gamoiTvleba e. w. bernulis formulT:
!
( )!( )!
k k n k k n k
n n
nP k C p q p q
k n k
.
amasTanave, adgili aqvs Tanafardobas: ( )
( 1) ( )( 1)
n n
n k pP k P k
k q
.
albaTobebis erTobliobas ( )nP k , roca 0,1,...,k n ewodeba albaTob-
ebis binomialuri ganawileba.
iseT 0k ricxvs, romlis Sesabamisi albaToba 0( )nP k udidesia )0(nP ,
)1(nP ,..., )(nPn albaTobebs Soris ualbaTesi ricxvi ewodeba. ualbaTesi ri-
cxvi gviCvenebs n damoukidebel cdaSi warmatebaTa ra raodenobaa yvela-ze ufro mosalodneli. ualbaTesi ricxvi warmoadgens Semdegi utolob-
is mTel amonaxsns: 0np q k np p .
puasonis formula
Tanafardobas lim ( ) / !knnp
p k e k
puasonis formula ewodeba. igi sa-
Sualebas iZleva vipovoT n damoukidebel cdaSi A xdomilebis k -jer moxdenis albaToba (roca n sakmaod didia, xolo p sakmaod mcire, amas-
Tanave 15np ) puasonis miaxloebiTi formuliT: ( ) / !knp k e k .
magaliTebis amoxsnis nimuSebi:
magaliTi 1. ganvixiloT ojaxebi, sadac or-ori bavSvia. rogoria
albaToba imisa, rom ojaxSi orive bavSvi vaJia Tu cnobilia, rom: a). ufrosi bavSvi vaJia; b). erTi bavSvi mainc vaJia?
-
17
amoxsna. aq elementarul xdomilebaTa sivrce aseTia { } vv, vq, qv, qq ,
sadac v aRniSnavs vaJs, xolo q qals. CavTvaloT, rom oTxive Sedegi tolalbaTuria. SemoviRoT xdomilebebi: A -- iyos xdomileba, rom ufrosi bavSvi -- vaJia, xolo B -- iyos xdomileba, rom umcrosi bavSvi vaJia. maSin BA -- iqneba xdomileba, rom orive bavSvi vaJia, xolo BA -- ki iqneba xdomileba, rom erTi bavSvi mainc vaJia. Sesabamisad, saZiebeli albaTobebi iqneba: a). )|( ABAP da b).
)|( BABAP . advili dasanaxia, rom:
2
1
2/1
4/1
)(
)(
)(
])[()|(
AP
BAP
AP
ABAPABAP ,
3
1
4/3
4/1
)(
)(
)(
)]()[()|(
BAP
BAP
BAP
BABAPBABAP .
magaliTi 2 (saukeTesos SerCevaze). mocemulia m obieqti gadanomr-ili ricxvebiT m,...,2,1 , amasTanave ise, rom vTqvaT, obieqti #1 klasifici-rdeba rogorc saukeTeso, . . . , obieqti #m ki rogorc yvelaze uare-si. igulisxmeba rom obieqtebi Semodian drois momentebSi m,...,2,1 SemTx-veviTi rigiT (anu yvela SesaZlo m ! gadanacvleba tolalbaTuria). damk-virvebels SeuZlia ori maTganis SedarebiT Tqvas romelia ukeTesi da romeli uaresi. saWiroa saukeTesos SerCeva im pirobiT rom obieqtebi wa-rmoidgineba saTiTaod da ukugdebulis damaxsovreba xdeba damkvirvebl-is mier. ar SeiZleba saukeTesod miCneul iqnes is obieqti, romelic dakv-irvebuli obieqtebidan erTze mainc uaresia an romelic ukve iqna ukugd-ebuli. vTqvaT, damkvirvebelma obieqti SearCia k -ur nabijze ( mk ), anu daTvalierebuli obieqtebidan ukanaskneli aRmoCnda yvela winaze ukeTe-si da amitom moxda misi SerCeva. rogoria albaToba imisa, rom amorCeu-li obieqti iqneba saukeTeso mTeli erTobliobidan rogorc ukve ganx-i-lul, ise jer kidev ganuxilav obieqtebs Soris?
amoxsna. SemoviRoT xdomilebebi: A iyos xdomileba, rom k -uri ob-ieqti saukeTeoa yvela arsebul m obieqts Soris da B iyos xdomileba, rom k -uri obieqti saukeTeoa dakvirvebul k obieqts Soris. gasagebia, rom mosaZebnia pirobiTi albaToba )|( BAP .
vinaidan BA , amitom ABA da )()( APBAP . Sesabamisad, pir-obiTi albaTobis ganmartebis Tanaxmad
)(/)()|( BPAPBAP vinaidan obieqtebis yvela SesaZlo gadanacvlebebi tolalbaTuria,
amitom albaTobis klasikuri ganmartebis Tanaxmad advili dasanaxia, rom
kk
kBP
1
!
)!1()(
da
mm
mAP
1
!
)!1()(
.
Sesabamisad, mkBAP /)|( .
strategia. SeiZleba damtkicdes, rom saukeTeso obieqtis amorCevis optimaluri strategia mowyobilia Semdegnairad. avRniSnoT simboloTi
*m iseTi naturaluri ricxvi, romlisTvisac samarTliania utoloba
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18
1
1
1
11
1
11**
mmmm
.
saukeTeso obieqtis arCevis optimaluri strategia mdgomareobs ima-
Si, rom davakvirdeT da ukuvagdoT pirveli 1* m obieqti da Semdeg gava-
grZeloT dakvirveba iseT * momentamde, roca pirvelad gamoCndeba yvela winamorbedze ukeTes obieqti.
magaliTad, Tu 10,...,1m , maSin *m -is Sesabamisi mniSvnelobebia: m 1 2 3 4 5 6 7 8 9 10 m-optimaluri 1 1 1 1 2 2 2 3 3 4
sakmaod didi m -isaTvis emm /* (sadac e -- neperis ricxvia,
2.718e ) da saukeTeso obieqtis arCevis albaToba daaxloebiT tolia 1/ 0.368e (Tumca, erTi SexedviT bunebrivia mogvCveneboda, rom gansaxil-veli obieqtebis m raodenobis zrdasTan erTad, saukeTeso obieqtis arC-evis albaToba unda wasuliyo nulisaken). e. i. saukeTeso obieqtis arCev-is optimaluri strategia mdgomareobs imaSi, rom unda ukuvagdoT obieq-tebis saerTo raodenobis daaxloebiT mesamedi da Semdeg avirCioT pirve-li iseTi obieqti, romelic yvela winaze ukeTesia.
magaliTi 3 (bednier bileTebze). 25 sagamocdo bileTidan 5 bedn-ieria, xolo danarCeni 20 ara bednieri. romel students aqvs bedni-eri bileTis aRebis meti albaToba: vinc pirveli iRebs bileTs, Tu vinc meore iRebs bileTs?
amoxsna. es amocana Cven ukve amovxseniT albaTobis klasikuri ganm-artebis gamoyenebiT. amovxsnaT axla igi elementarul xdomilebaTa sivr-cis axleburi SemotaniTa da namravlis albaTobis formulis gamoyeneb-
iT. winaswar SemoviRoT Semdegi xdomilebebi: A iyos xdomileba, rom pi-
rvelma studentma aiRo bednieri bileTi, xolo B iyos xdomileba, rom meore studentma aiRo bednieri bileTi. maSin cxadia, rom elementarul xdomilebaTa sivrce Sedgeba oTxi xdomilebisagan
},,,{ BABABABA .
albaTobis klasikuri ganmartebis Tanaxmad 5/125/5)( AP , xolo
( ) 20/25 4/5P A . meores mxriv, amocanis Sinaarsidan gamomdinare, Tu cnobilia, rom pirvelma studentma aiRo bednieri bileTi, maSin albaTo-ba imisa rom meore studenti aiRebs bednier bileTs isev SeiZleba gamov-iTvaloT albaTobis klasikuri ganmartebiT: am SemTxvevaSi yvela SesaZ-lo SedegTa raodenobaa 24, xolo xelSemwyob elementarul xdomilebaTa raodenoba ki mxolod 4 (radgan erTi bednieri bileTi ukve aRebulia) da Sesabamisad,
6/124/4)|( ABP .
analogiurad, 6/524/20)|( ABP , 24/524/5)|( ABP da
( | ) 19/24P B A . amitom ori xdomilebis namravlis albaTobis formulis Tanaxmad gvaqvs:
30/16/15/1)|()()( ABPAPBAP ; 6/16/55/1)( BAP ;
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19
6/124/55/4)( BAP da 30/1924/195/4)( BAP (SevniSnavT, rom Cven aq mxolod sisrulisaTvis gamovTvaleT yvela
SesaZlo pirobiTi da namravlis albaTobebi).
cxadia, rom )()( BABAB . amitom albaTobaTa Sekrebis kanonis
Tanaxmad
))((5/16/130/1)()()( APBAPBAPBP .
magaliTi 4. yuTSi m burTia, maT Soris n TeTria. vipovoT albaT-oba imisa, rom yuTidan ori burTis mimdevrobiT dabrunebis gareSe amoR-ebisas: a). pirveli burTi TeTria; b). meore burTi TeTria; g). orive bur-Ti TeTria.
amoxsna. iA iyos xdomileba, rom i -uri burTi TeTria ( 2,1i ). maSin
albaTobis klasikuri ganmartebis Tanaxmad:
a). mnAP /)( 1 . garda amisa,
)1/()1()|( 12 mnAAP da )1/()|( 12 mnAAP . amitom namravlis albaTobis formulis Tanaxmad:
g). )1(/)1()|()()( 12121 mmnnAAPAPAAP . analogiurad,
)1(/)()|()()( 12121 mmnmnAAPAPAAP .
amitom:
b). mnAAPAAPAAAAPAP /)()()]()[()( 212121212 . magaliTi 5. yuTs aqvs n ujra. albaToba imisa rom burTi aris am
ujrebidan erT-erTSi tolia p -si. ipoveT albaToba imisa, rom burTi ar-is i -ur ujraSi, Tu cnobilia, rom burTi TiTeul ujraSi SesaZlebelia iyos Tanabari albaTobebiT?
amoxsna. iA iyos xdomileba, rom burTi aris i -ur ujraSi. A iyos
am xdomilebebis gaerTianeba n
iiAA
1
. pirobis Tanaxmad
pAP )( da nAAP i /1)|( .
amitom
npnpAAPAPAAPAP iii //1)|()()()( .
magaliTi 6. kartebis nakrebidan (romelSic 36 kartia) SemTxveviT iReben erT karts. ganvixiloT xdomilebebi: A iyos xdomileba, rom amo-Rebuli karti agurisa, xolo B iyos xdomileba, rom amoRebuli karti mefea. arian Tu ara es xdomilebebi damoukidebeli?
cxadia, rom am SemTxvevaSi 36|| , 4/136/9)( AP , 9/136/4)( BP da
)()(9/14/136/1)( BPAPBAP . e.i. es xdomilebebi damoukidebelia.
magaliTi 7. davuSvaT, rom vagorebT or saTamaSo kamaTels. ganvixi-loT xdomilebebi: A -- pirvel kamaTelze movida kenti qula, B -- meore
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20
kamaTelze movida kenti qula, C -- orive kamaTelze mosul qulaTa jami kentia. gavarkvioT am xdomilebebis damoukideblobis sakiTxi.
cxadia, rom 2/16/3)()( BPAP , xolo 4/136/33)( BAP .
amitom A da B xdomilebebi damoukideblia. davaleba. SeamowmeT, rom 2/1)( CP .
SevniSnoT, rom A da B xdomilebidan erT-erTis moxdenis pirobaSi C xdomileba xdeba maSin da mxolod maSin, roca Sesabamisad, an pirvel an meore kamaTelze movida luwi qula, anu gvaqvs Tanafardobebi:
BACA da BACB . A da B xdomilebebis damoukideblobidan gamomdinare xdomilebe-
bi A da B da A da B agreTve damoukideblebia, amitom
4/1)()()( BPAPBAP da 4/1)( BAP . Sesabamisad,
4/1)()( BAPCAP da 4/1)()( BAPCBP .
es Tanafardobebi ki, 2/1)( CP albaTobis gaTvaliswinebiT, niSn-
avs, rom damoukideblebia A da C da B da C xdomilebaTa wyvilebic. davaleba. SeamowmeT, rom xdomilebebi magaliTi 7-dan ar arian erT-
oblivad damoukidebeli. magaliTi 8. davuSvaT, vagdebT sam monetas. SemoviRoT xdomilebebi:
1A -- pirveli da meore moneta daeca erTi da igive mxareze;
2A -- meore da mesame moneta daeca erTi da igive mxareze;
3A -- pirveli da mesame moneta daeca erTi da igive mxareze.
advili Sesamowmebelia, rom aqedan nebismieri ori xdomileba damoukidebelia, xolo samive erTad damokidebulia, vinaidan Tu Cven
gvecodineba rom magaliTad, 1A da 2A moxda, maSin Cven zustad viciT,
rom 3A agreTve moxda.
davaleba. SeamowmeT, rom xdomilebebi 1A , 2A da 3A wyvil-wyvilad
damoukidebelia. magaliTi 9 (dabadebis dReebze). vipovoT albaToba imisa, konkretu-
li skolis rom 150 moswavlidan erTi mainc dabadebulia mocemul fiqsi-rebul dRes, magaliTad pirvel seqtembers?
amoxsna. SemoviRoT xdomilebebi:
}09.1{ adabadebulistudentiuri iAi , 150,...,2,1i ;
}09.1{ adabadebulinstudentida150maincerTiA .
naTelia, rom iA xdomilebebi erToblivad damoukidebelia da
365/1)( iAP . garda amisa, ii
AA150
1 da maSasadame, sapovnelia damouki-
debel xdomilebaTa gaerTianebis albaToba. gadavideT sawinaaRmdego xdomilebaze da visargebloT de-morganis kanoniT. maSin gvaqvs:
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21
)()(1)(1)(1)(1)( 1150
1
150
1ni
ii
iAPAPAPAPAPAP
.
ramdenadac 365/11)( iAP , Tu visargeblebT niutonis binomis for-
muliT
n
j
jjn
n xCx1
)1( , miviRebT, rom:
4415033
15022
150 )365
1()
365
1()
365
1(
365
150)( CCCAP .
vinaidan, 41,0365/150 , xolo 4
4 4 4
150
1 150 1 (0,41)( ) ( ) 0.005365 365 24 24
C ,
amitom mwkrivis niSancvladobis gamo, Tu gadavagdebT mwkrivis wevrebs dawyebuli me-5 wevridan (mwkrivebis zogadi Teoriidan gamomdinare), SesaZlebelia vamtkicoT, rom
2 3(0.41) (0.41)( ) 0.41 0.41 0.08 0.01 0.34.
2 6P A
magaliTi 10 (asoebis amocnoba). gvaqvs asoebis ori erToblioba: I={, , , } da II={, , }.
SemTxveviT virCevT erT erTobliobas da arCeuli erTobliobidan erT asos. amorCeul asos Zalian mcire drois ganmavlobaSi vuCvenebT damkvirvebels (ise rom mas ar SeuZlia mTlianad aRiqvas aso). rogoria asos sworad gamocnobis albaToba, Tu damkvirvebelis pasuxia , roca is asos gamosaxulebaSi dainaxavs vertikalur xazs da pasuxia , roca asos gamosaxulebaSi vertikaluri xazi ar aris?
amoxsna. SemoviRoT xdomilebebi:
}2,1,{ iiAi erTobliobauriamorCeulia ;
, , , da iyos xdomileba, rom warmodgenilia Sesa-bamisad , , , da asoebi;
}{ upasuxasworadelmadamkvirvebB . amocanis pirobebSi cxadia:
2/1)()( 21 APAP ;
(P | 1A )= (P | 1A )= (P | 1A )= (P | 1A )=1/4, (P | 1A )=0;
(P | 2A )= (P | 2A )= (P | 2A )=1/3, (P | 2A )= (P | 2A )=0.
vinaidan 1A da 2A qmnian xdomilebaTa srul sistemas, amitom TiToe-uli asos sworad amocnobis albaToba SegviZlia gamovTvaloT sruli albaTobis formuliT:
(P )= )( 1AP (P | 1A )+ )( 2AP (P | 2A )=1/8;
(P )= )( 1AP (P | 1A )+ )( 2AP (P | 2A )=1/8;
(P )= )( 1AP (P | 1A )+ )( 2AP (P | 2A )=7/24;
(P )= )( 1AP (P | 1A )+ )( 2AP (P | 2A )=7/24;
(P )= )( 1AP (P | 1A )+ )( 2AP (P | 2A )=1/6.
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22
amocanis pirobebSi cxadia agreTve, rom swori pasuxis pirobiTi albaTobebi sxvadasxva asoebis warmodgenis SemTxvevaSi Sesabamisad iq-neba:
(P B |)= (P B |)= (P B |)=0 da (P B |)= (P B |)=1.
vinaidan, , , , da agreTve xdomilebaTa sruli sist-emaa, amitom sruli albaTobis formula gvaZlevs swori pasuxis albaTo-bas:
)(BP (P ) (P B |)+ (P ) (P B |)+ (P ) (P B |)+
+ (P ) (P B |)+ (P ) (P B |)= (P )+ (P )=7/12. magaliTi 11 (moTamaSis gakotrebaze). ganvixiloT e. w. gerbi-safas-
uris TamaSi: Tu monetis agdebisas mova moTamaSis mier winaswar dasaxe-lebuli monetis mxare, maSin igi igebs 1 lars, winaaRmdeg SemTxvevaSi ki agebs 1 lars. vTqvaT, moTamaSis sawyisi kapitali Seadgens x lars da mi-si mizania miiyvanos es Tanxa a laramde. TamaSi grZeldeba manam sanam moTamaSe ar miiyvans Tavis Tanxas winaswar gansazRrul a laramde, an igi ar gakotrdeba (anu waagebs mis xelT arsebul mTel x lars). rogo-ria albaToba imisa, rom moTamaSe gakotrdeba?
amoxsna. es albaToba damokidebuli iqneba sawyis x kapitalze. aRv-niSnoT igi )(xp simboloTi. cxadia, rom igi ganmartebulia nebismieri
ax 0 da amasTanave, 1)0( P da 0)( aP . SemoviRoT xdomilebebi:
}{1 nabijze pirvel moigo moTamaSemA ,
},{ gakotrdebakapitali sawyisi gaaCnia romelsac moTamaSe, xB . amocanis pirobebSi gvaqvs:
2/1)()( 11 APAP , )1()|( 1 xpABP da )1()|( 1 xpABP ( 11 ax ).
vinaidan, 1A da 1A xdomilebaTa sruli sistemaa, amitom sruli al-
baTobis formula )(xp albaTobisaTvis gvaZlevs Semdeg gantolebas:
)1(2
1)1(
2
1)( xpxpxp , 11 ax
(am tipis gantolebebs maTematikaSi rekurentul gantolebebs uwodeben). SeiZleba Semowmdes, rom am gantolebis amoxsnas aqvs saxe:
cbxxp )( ,
sadac b da c -- nebismieri mudmivebia. am koeficientebis mosaZebnad unda visargebloT sasazRvro pirobebiT 1)0( P da 0)( aP . maSin miviRebT, rom
1c da 0 cab , saidanac, ab /1 da sabolood axxp /1)( , ax 0 .
davaleba. davuSvaT, rom Sesamowmebeli jgufis 1% avadmyofia, xo-lo danarCeni 99% ki janmrTelia. adamianebis SerCeva xdeba SemTxveviT da amitom
( ) 1% 0.01P avadmyofi da ( ) 99% 0.99P janmrTeli .
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23
vigulisxmoT, rom im SemTxvevaSi, roca testireba utardeba adamians, romelsac ara aqvs avadmyofoba, maSin 1%-ia albaToba imisa, rom miviRoT mcdari dadebiTi Sedegi, e.i.
%1)( janmrTeli|dadebiTiP da %99)( janmrTeli|uaryofiTiP . dabolos, davuSvaT, rom im SemTxvevaSi, roca testireba utardeba
avadmyof adamians, maSin 1%-ia albaToba imisa, rom miviRoT mcdari uaryofiTi Sedegi, e.i.
%1)( avadmyofi|uaryofiTiP da %99)( avadmyofi|dadebiTiP . gamoTvaleT albaToba imisa, rom: a). adamiani janmrTelia, xolo
testma aCvena uaryofiTi Sedegi; b). adamiani avadmyofia, xolo testma aCvena dadebiTi Sedegi; g). adamiani janmrTelia, xolo testma aCvena dadebiTi Sedegi; g). adamiani avadmyofia, xolo testma aCvena uaryofiTi Sedegi.
ganvixiloT realuri situacia, romelic gviCvenebs erTi SexedviT moulodnel gansxvavebas )|( BAP da )|( ABP pirobiT albaTobebs Soris. imisaTvis, rom gamovavlinoT seriozuli avadmyofobis mqone adamianebi adreul stadiaze, xdeba adamianebis didi jgufis testireba. miuxedavad winaswari Semowmebis sargeblobisa, am midgomas gaaCnia uaryofiTi mxare: Tu adamians sinamdvileSi ar gaaCnia avadmyofoba da sawyisma testma aCve-na dadebiTi Sedegi (daudgina avadmyofoba), is iqneba stresul mdgomare-obaSi (rac Tavis mxriv uaryofiTad moqmedebs mis cxovrebaze) sanam uf-ro warmatebuli testi ar aCvenebs, rom is janmrTelia. am problemis mni-Svneloba SesaZlebelia kargad gavigoT pirobiTi albaTobebis termineb-Si.
wina davalebis monacemebSi gamovTvaloT albaToba imisa, rom tes-ti aCvenebs dadebiT Sedegs. sruli albaTobis formulis Tanaxmad:
( )P dadebiTi )()( janmrTeli|dadebiTijanmrTeliPP
( ) ( ) 0.99 0.01 0.01 0.99 0.0198P P avadmyofi dadebiTi| avadmyofi . rogorc cnobilia, magaliTis pirobebSi
%99)( avadmyofi|dadebiTiP . gamovTvaloT axla Sebrunebuli pirobiTi albaToba, risTvisac vi-
sargebloT pirobiTi albaTobis ganmartebiTa da namravlis albaTobis formulebiT. maSin zemoT miRebuli
( ) 0.0198 1.98%P dadebiTi Sedegis Tanaxmad:
(dadebiTi)PP
dadebiTiavadmyofidadebiTi|avadmyofi
P
PP
)()(
( ( ) 1% 99%50%
1.98%
P P
avadmyofi)P dadebiTi | avadmyofi
PP1.98%.
rogorc vxedavT, pirobiTi albaToba imisa rom testi mogvcems da-debiT Sedegs, pirobaSi rom adamiani avadmyofia tolia 99%-is, maSin ro-desac pirobiTi albaToba imisa rom adamiani avadmyofia, pirobaSi rom testma mogvca dadebiTi Sedegi aris mxolod 50%. aq SerCeuli monacemeb-
-
24
is SemTxvevaSi ukanaskneli Sedegi SeiZleba CaiTvalos miuRebelad: naxe-vari adamianebis, romelTa testirebam aCvena dadebiTi Sedegi, faqtiurad aris janmrTeli.
magaliTi 12. yuTSi moTavsebulia ori moneta: 1A -- simetriuli mon-
eta gerbis mosvlis albaTobiT 1/2, da 2A -- arasimetriuli moneta gerbis mosvlis albaTobiT 1/3. SemTxveviT viRebT erT monetas da vagdebT. davu-SvaT, rom movida gerbi. rogoria albaToba imisa, rom amoRebuli moneta iyo simetriuli?
amoxsna. am SemTxvevaSi elementarul xdomilebaTa sivrce iqneba:
)},(),.(),,(),,{( 2211 sgsg AAAA ,
sadac magaliTad, ),( 1 gA -- niSnavs, rom amoviReT 1A moneta da misi agdeb-is Sedegad movida gerbi. amocanis pirobebSi gvaqvs:
2/1)()( 21 APAP , 2/1)( 1 AP |g da 3/1)|( 2 AP g . Sesabamisad, namravlis albaTobis formulis gamoyenebiT
gamoviTvliT:
4/1)},{( 1 gAP , 4/1)},{( 1 sAP , 6/1)},{( 2 gAP da 3/1)},{( 2 sAP . amitom baiesis formulis Tanaxmad
5
3
)()()()(
)()()|(
2211
111
APAPAPAP
APAPAP
|g|g
|gg .
cxadia, agreTve rom 5/2)|( 2 gAP . magaliTi 13 (keTil gamomcdelze I). vTqvaT, Cven Casabarebeli
gvaqvs gamocda da SegviZlia avirCioT nebismieri sami gamomcdelidan. davuSvaT, CvenTvis cnobilia, rom erTerTi sami gamomcdelidan (ucnobia romeli) -- keTilia da albaToba imisa, rom masTan Caabaro gamocda tolia 0.4-is, xolo danarCeni ori gamomcdeli avia da maTTan gamocdis Cabarebis albaToba tolia 0.1-is. Cven SemTxveviT avirCieT gamomcdeli da warmatebiT CavabareT gamocda. rogoria labaToba imisa, rom Cven avirCieT keTili gamomcdeli?
amoxsna. SemoviRoT Semdegi xdomilebebi: A -- amorCeuli gamomcde-
li keTilia (maSin A -- iqneba xdomileba, rom amorCeuli gamomcdeli
avia) da B -- gamocda Cabarebulia (Sesabamisad, B -- gamocda araa Caba-rebuli). amocanis pirobebSi gvaqvs:
3/1)( AP , 3/2)(1)( APAP ;
( | ) 0.4P B A , ( | ) 1 ( | ) 0.6P B A P B A ;
( | ) 0.1P B A , ( | ) 1 ( | ) 0.9P B A P B A .
cnobilia, rom moxda B xdomileba da gamosaTvlelia pirobiTi
albaToba )|( BAP . vinaidan, A da A xdomilebebi qmnian srul sistemas,
baiesis formulis Tanaxmad saZiebeli albaToba iqneba:
3
2
)|()()|()(
)|()()|(
ABPAPABPAP
ABPAPBAP .
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25
davaleba. magaliT 13-is pirobebSi gamoTvaleT albaToba imisa, rom arCeul iqna avi gamomcdeli, Tu cnobilia, rom gamocda Cabarebul iqna warmatebiT?
magaliTi 14 (keTil gamomcdelze II). davuSvaT, rom gamomcdelTan, romelTanac warmatebiT Caiara gamocdam (ix. magaliTi 13) gamosacdelad rig-rigobiT mivida kidev ori moswavle. jer gamocda ver Caabara meore moswavlem, Semdeg mivida mesame da manac ver Caabara gamocda. am faqtis Semdeg romeli hipoTezaa ufro dasajerebeli: es gamomcdeli keTilia Tu avi?
amoxsna. avRniSnoT )(APi (Sesabamisad, )(APi ) simboloTi albaToba
(aposterioruli) imisa, rom es gamomcdeli keTilia (Sesabamisad, avia) mas Semdeg rac gamocdil iqna i -uri studenti, 3,2,1i . Cven ukve davadgi-
neT, rom 3/2)(1 AP . Sesabamisad,
3/1)(1)( 11 APAP . meore moswavlis TvalsazrisiT es albaTobebi warmoadgenen ori
SesaZlo hipoTezis apriorul albaTobebs. amitom, baiesis formulis Tan-axmad, meore studentis CaWris Semdeg aposterioruli albaTobebi iqneba:
7
4
)()|()()|(
)()|()(
11
12
APABPAPABP
APABPAP da
7
3)(1)( 22 APAP .
analogiurad, axla miRebuli albaTobebi ukve iqneba aprioruli albaTobebi mesame moswavlisaTvis, da amitom saZiebeli aposterioruli albaTobebi, mas Semdeg rac mesame moswavlem ver Caabara gamocda, gamoi-Tvleba isev baiesis formuliT:
17
8
)()|()()|(
)()|()(
22
23
APABPAPABP
APABPAP da )(
17
9)(1)( 333 APAPAP .
rogorc vxedavT, eqsperimentis (gamocdis) dawyebis win aprioruli albaToba imisa. rom arCeuli gamomcdeli keTilia toli iyo 1/3-is. eqs-perimentebis Semdeg am xdomilebis aposterioruli albaToba gaizarda da gaxda 8/17. miuxedavad amisa, Tu sami eqsperimentis Semdeg misaRebia ga-dawyvetileba am gamomcdelis Sesaxeb, maSin ufro sarwmunoa CavTvaloT
igi avad (vinaidan, )()( 33 APAP ).
magaliTi 15. yuTSi 3 TeTri da 5 Savi burTia. yuTidan SemTxveviT iReben 4 burTs dabrunebiT. ipoveT albaToba imisa, rom amoRebuli bur-Tebidan TeTri burTebis raodenoba meti iqneba Savi burTebis raodenoba-ze?
amoxsna. Tu warmatebad CavTvliT TeTri burTis amoRebas, maSin pi-robis Tanaxmad erT cdaSi warmatebis albaToba iqneba 8/3p . saZiebeli xdomilebis xelSemwyob uTavsebad SemTxvevebs warmoadgens oTx cdaSi 3 an 4 TeTri burTis amoReba, romelTa albaTobebi bernulis formulis Tanaxmad Sesabamisad tolia:
1024/135)8/31()8/3()3( 343344 CP da
4096/81)8/5()8/3()4( 04444 CP .
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26
Sesabamisad, saZiebeli albaToba albaTobaTa Sekrebis kanonis Tanaxmad iqneba:
4096/6214096/811024/135 . magaliTi 16. davuSvaT vamowmebT defeqturobaze saqonlis partias,
romelic Sedgeba 30 nawarmisagan. cnobilia, rom defeqturi produqciis wili Seadgens 5%-s. rogoria saqonlis am partiaSi defeqturi produqci-is ama Tu im ricxvis aRmoCenis albaTobebi?
amoxsna. am SemTxvevaSi eqsperimentebis ricxvia 30n , xolo albaT-obis klasikuri ganmartebis Tanaxmad 5/100 0.05p (Sesabamisad, 0.95q ). Sesabamisad, gvaqvs:
0 0 30 0 30
30 30(0) 0.05 0.95 0.95 0.2146P C ,
garda amisa,
30 30 30 30
30 30 0.05 30( 1) ( ) ( ) ( )
1 1 0.95 19( 1)
k p k kP k P k P k P k
k q k k
,
saidanac roca 0k : 3030 0
(0 1) 0.2146 0.338919 (0 1)
P
;
roca 1k : 3030 1
(1 1) 0.3389 0.258619 (1 1)
P
da a. S. sabolood gveqneba
Semdegi cxrili:
defeqturi nawa- rmis ricxvi k
albaToba
( )nP k
kumulatiuri
albaToba ( )nP k
0 0.2146 0.2146 1 0.3389 0.5535 2 0.2586 0.8122 3 0.1270 0.9392 4 0.0451 0.9844 5 0.0124 0.9967 6 0.0027 0.9994 7 0.0005 0.9999 8 0.0001 0.999998 9 0.000001 0.999999
am cxrilis Sesabamisi albaTobebis ganawilebis grafiki iqneba:
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0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 2 4 6 8 10
kumulatiuri albaTobebis Sesabamisi ganawilebis grafiki iqneba:
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10
savarjiSo 1. eqsperimenti mdgomareobs sami saTamaSo kamaTlis gag-
orebaSi. ipoveT alabaToba imisa, rom eqsperimentis 10-jer gameorebisas, zustad 4 eqsperimentSi mova zustad or-ori 6?
savarjiSo 2. ramdeni SemTxveviTi cifri unda aviRoT, rom cifri 5 movides erTjer mainc aranakleb 0.9-is toli albaTobiT?
amocanebi
1. A yuTi Seicavs 8 wiTel da 7 mwvane burTs, B yuTi Seicavs 6 wi-
Tel da 4 mwvane burTs. agoreben wesier kamaTels. Tu movida 3-is jeradi qula, maSin A yuTidan SemTxveviT iReben 1 burTs, xolo winaaRmdeg Sem-TxvevaSi B yuTidan SemTxveviT iReben 1 burTs. ipoveT albaToba imisa, rom: a). amoRebuli burTi mwvanea; b). burTi amoRebuli iyo A yuTidan, Tu cnobilia, rom amoRebuli burTi mwvanea.
2. ( ) 3/ 4P A , ( | ) 1/5P B A , ( | ) 4/7P B A . ipoveT: a). ( )P A B ; b). ( )P B ; g). ( | )P A B .
3. studentma unda gaiaros testireba laboratoriaSi muSaobis das-awyebad. testirebis gavlis Semdeg studenti tests ukan ar abrunebs. al-baToba imisa, rom SemTxveviT SerCeuli studenti testirebas gaivlis pi-rvel cdaze aris 1/3. ganmeorebiTi testirebis SemTxvevaSi CaWris albaT-
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28
oba aris wina cdaze CaWris albaTobis naxevari. ipoveT albaToba imisa, rom studenti: a). gaivlis testirebas araumetes 3 mcdelobis Sedegad; b). gaivlis testirebas pirvel mcdelobaze, Tu cnobilia, rom man testireba gaiara araumetes 3 mcdelobiT.
4. ( ) 13/ 25P A , ( ) 9/ 25P B , ( | ) 5/9P A B . ipoveT: a). ( )P A B ; b). ( | )P B A ;
g). ( )P A B ; d). ( | )P A B ; e). ( )P A B . 5. 12 waxnagian wesier mravalwaxnagaze aRniSnulia ricxvebi 1, 2, . . .
, 12. mocemulia 9 danayofiani firfita, romlis centrSi devs moneta. L R
agdeben mravalwaxnagas: Tu movida 2, 3, 5, 7 an 11 qula, maSin monetas gad-aadgileben TiTo danayofze R-is mimarTulebiT (marjvniv), xolo winaaRm-deg SemTxvevaSi L-is mimarTulebiT (marcxniv). TamaSi mTavrdeba, roca moneta miaRwevs an Rs an L (firfitis romelime bolos). ipoveT albaTo-ba imisa, rom: a). TamaSi damTavrdeba 4 gadaadgilebis Semdeg R-ze; b). Tam-aSi damTavrdeba 4 gadaadgilebis Semdeg; g). TamaSi damTavrdeba 5 gadaa-dgilebis Semdeg; d). TamaSis damTavrebas dasWirdeba 6-ze meti gadaadgi-leba.
6. moyvaruli sinoptikosis Teoriis Tanaxmad Tu erT wels iyo wya-ldidoba, maSin albaToba imisa, rom igi ganmeordeba momdevno wels aris 0.7, xolo Tu erT wels ar iyo wyaldidoba, maSin albaToba imisa, rom igi ar iqneba momdevno wels aris 0.6. gasul wels wyaldidoba ar yofi-la. ipoveT albaToba imisa, rom wyaldidoba iqneba: a). momdevno sam wel-iwads zedized; b). zustad erTjer momdevno sami wlis ganmavlobaSi.
7. albaToba imisa, rom safexburTo klubis I gundis yvela moTamaSe formaSia aris 0.7. roca I gundis yvela moTamaSe formaSia, klubi sakuT-ar moedanze igebs albaTobiT 0.9. Tu I gundis yvela moTamaSe ar aris formaSi, maSin sakuTar moedanze mogebis albaTobaa 0.4. ipoveT albaToba imisa, rom: a). klubi moigebs sakuTar moedanze; b). I gundis yvela moTama-Se iyo formaSi, Tu klubma ukanaskneli Sexvedra ver moigo.
8. CanTaSi devs 25 diski, romelTagan nawili TeTria da danarCeni Savi. CanTidan erTdroulad SemTxveviT iReben or disks. albaToba imisa, rom amoRebuli diskebi erTi da igive ferisaa emTxveva albaTobas imisa, rom es diskebi sxvadasxva ferisaa. ramdeni TeTri diskia CanTaSi?
9. ( ) 0.6P A , ( ) 0.8P B , ( | ) 0.45P B A , ( ) 0.28P B C . ipoveT: a). ( )P A B ; b). ( | )P C B ; g). ( | )P A B .
10. ( ) 0.75P A , ( | ) 0.8P B A , ( | ) 0.6P B A . ipoveT ( )P B da ( | )P A B . 11. klasSi aris 7 biWi da 9 gogona. SemTxveviT SerCeulia am klasis
ori moswavle. A{pirveli moswavle gogonaa}, B {meore moswavle gogo-
naa}. ipoveT: a). ( | )P B A ; b). ( | )P B A ; g). ( | )P B A ; d). )|( ABP ; e) ( )P B .
12. albaToba imisa, rom momavali wlis ivnisis pirveli dRe iqneba mSrali aris 0.4. Tu ivnisis romelime konkretuli dRe mSralia, maSin albaToba imisa, rom momdevno dRe iqneba mSrali aris 0.6. sxva SemTxveva-Si albaToba imisa, rom momdevno dRe iqneba mSrali aris 0.3. ipoveT alb-aToba imisa, rom: a). ivnisis pirveli ori dRe iqneba mSrali; b). ivnisis
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29
meore dRe iqneba mSrali g). sul cota erTi ivnisis pirveli sami dRidan iqneba mSrali.
13. albaToba imisa, rom momavali wlis ivnisis I dRe iqneba mSrali aris 0.5. Tu ivnisis romelime konkretuli dRe mSralia, maSin albaToba imisa, rom momdevno dRe iqneba mSrali aris 0.4. sxva SemTxvevaSi albaTo-ba imisa, rom momdevno dRe iqneba mSrali aris 0.6. ipoveT albaToba imi-sa, rom: a). ivnisis pirveli ori dRe iqneba sveli; b). ivnisis meore dRe iqneba sveli; g). sul cota erTi ivnisis pirveli sami dRidan iqneba sve-li.
14. 30 students dausves kiTxvebi: uWers is mxars argentinas, brazi-lias, Tu arcerTs da is caciaa Tu ara. 12 studentma upasuxa, rom is ar-is cacia da maTgan 4 uWers mxars argentinas, xolo 7 ki brazilias. ara cacia studentebidan 1 uWers mxars argentinas, xolo 10 ki brazilias. jgufidan SemTxveviT aarCies 1 studenti. ipoveT albaToba imisa, rom: a). studenti caciaa; b). studenti arcerT gunds ar uWers mxars; g). studen-ti caciaa, Tu cnobilia, rom studenti arcerT gunds ar uWers mxars.
15. ( ) 0.3P A . B xdomileba damoukidebelia A xdomilebisagan.
( ) 0.4P B . C aris xdomileba rom arc A moxda da arc B . ipoveT: a).
( \ )P A B ; b). ( )P A B ; g). )|( ACP . 16. isvrian wiTel da lurj kamaTels da maTze mosuli qulebi ikr-
ibeba. vipovoT albaToba imisa, rom: a). qulaTa jami iqneba aranakleb 10-isa, Tu wiTel kamaTelze movida 6 qula; b). qulaTa jami iqneba aranakl-eb 10-isa, Tu erT kamaTelze mainc movida 6 qula.
17. kolejis studentebis naxevari swavlobs mecnierebas, xolo 30% ki maTematikas. maT Soris vinc swavlobs mecnierebas 40% swavlobs maTe-matikas. ipoveT albaToba imisa, rom SemTxveviT SerCeuli studenti: a). swavlobs orives: mTematikas da mecnierebas; b). swavlobs maTematikas, ma-gram ar swavlobs mecnierebas.
18. ( | ) 1/5P D C , ( | ) 1/ 4P C D , ( )P C D p . ipoveT: a). ( )P C ; b). ( )P D . 19. ( | ) 1/5P D C , ( | ) 1/ 4P C D , ( ) 1/5P C D , ( )P C D p . ipoveT p . 20. stadionis 40 bileTidan 10 aris CrdiloeTis mxare, 14 aRmosavl-
eTis da 16 ki dasavleTis mxare. SemTxveviT iReben erT bileTs da aZlev-en X -s (adamians saxelad X ). Semdgom, SemTxveviT iReben meore bileTs da aZleven Y -s da analogiurad mesame bileTs aZleven Z -s. ipoveT alba-Toba imisa, rom: a). X -s Sexvdeba CrdiloeTis mxare; b). X -s da Y -s Sexv-deba CrdiloeTis mxare; g). samives Sexvdeba erTi da igive mxare; d). ors Sexvdeba erTi mxare, xolo mesames sxva mxare.
21. 24 prizidan 4 avtomanqanaa, 8 motocikleti da 12 saaTi. anam, ekam da elzam moiges TiTo prizi. ipoveT albaToba imisa, rom: a). anam moigo manqana, xolo ekam -- motocikleti an saaTi; b). anam da ekam (orivem) moi-go manqana, Tu cnobilia, rom elzam moigo manqana; g). anasa da elzas So-ris erTma mainc moigo manqana; d). anam moigo manqana, xolo ekam ki manqa-na an motocikleti; e). anam moigo manqana pirobaSi, rom ekam moigo manqana an motocikleti.
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30
22. ( )A B , ( | ) 1/3P A B , ( ) 6/7P A . ipoveT ( )P B . 23. albaToba imisa, rom konkretuli dRe mSralia aris 3/10. albaT-
oba imisa, rom dinamo moigebs mSral dRes aris 3/8, xolo svel dRes 3/11. a). dinamom Semdegi matCi orSabaTs unda Caataros. ras udris alba-Toba imisa, rom dinamo moigebs; b). gasul oTxSabaTs dinamom matCi moigo. ras udris albaToba imisa, rom oTxSabaTs mSrali amindi iyo; g). gasul SabaTs dinamom matCi waago. ras udris albaToba imisa, rom Sab-aTs sveli amindi iyo.
24. dedamiwis garkveul nawilSi mSrali dReebi ufro metia, vidre sveli. Tu romelime dRe mSralia, maSin albaToba imisa, rom momdevno dRe iqneba isev mSrali aris 0.8. Tu romelime dRe svelia, maSin albaTo-ba imisa, rom momdevno dRe iqneba isev sveli aris 0.6. cnobilia, rom or-SabaTi sveli dRe iyo. ipoveT albaToba imisa, rom imave kviris: a). samSa-baTi da oTxSabaTi orive iqneba mSrali; b). oTxSabaTi iqneba mSrali.
25. Tu romelime dRe mSralia, maSin albaToba imisa, rom momdevno dRe iqneba isev mSrali aris 0.8. Tu romelime dRe svelia, maSin albaTo-ba imisa, rom momdevno dRe iqneba isev sveli aris 0.6. wlis ganmavlobaSi Catarda 44 kriketis matCi, TiToeuli grZeldeboda sami dRe mimdevrob-iT, romelTagan I da III dRe iyo sveli. ipoveT albaToba imisa, rom II dRe iyo mSrali.
26. kazinoSi dgas ori A da 'A saTamaSo magida, romlebic sruliad
erTnairad gamoiyureba. A magidaze mogebis albaToba aris 1/3, xolo 'A magidaze 1/4. a). ipoveT SemTxveviT arCeul magidaze mogebis albaToba; b). Tqven SemTxveviT airCieT magida da moigeT. ras udris albaToba imisa, rom Tqven airCieT A magida; g). Tqven SemTxveviT airCieT magida da waag-
eT. ras udris albaToba imisa, rom Tqven airCieT 'A magida.
27. kazinoSi dgas sami saTamaSo magida A da 'A tipis, romlebic
sruliad erTnairad gamoiyureba. amasTanave, 'A tipis magida ori calia.
A tipis magidaze mogebis albaToba aris 1/3, xolo 'A tipis magidaze 1/4. a). ipoveT SemTxveviT arCeul magidaze mogebis albaToba; b). Tqven Se-mTxveviT airCieT magida da moigeT. ras udris albaToba imisa, rom Tqven airCieT A magida; g). Tqven SemTxveviT airCieT magida da waageT. ras ud-
ris albaToba imisa, rom Tqven airCieT 'A magida. 28. albaToba imisa, rom gvanca adre gaiRviZebs aris 1/3. Tu gvanca
adre gaiRviZebs, maSin albaToba imisa, rom skolaSi droulad miva aris 3/4. Tu gvanca gvian gaiRviZebs, maSin skolaSi droulad misvlis albaTo-ba aris 1/5. erT SemTxveviT arCeul dRes gvanca droulad mivida skola-Si. ipoveT albaToba imisa, rom man gvian gaiRviZa.
29. sami erTnairi A , B da C yuTi Sedgeba or-ori ujrisagan. A yu-Tis TiToeul ujraSi devs TiTo lari, B yuTis TiTeul ujraSi devs xuT-xuTi lari, xolo C yuTis erT ujraSi devs 1 lari da meoreSi ki 5 lari. SemTxveviT irCeven yuTs, xsnian mis erT ujras da pouloben masSi 5 lars. ras udris albaToba imisa, rom am yuTis meore ujraSi agreTve devs 5 lari.
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31
30. ojaxSi aris ori biWi da ori gogo, romelTagan erTs hqvia eka. SemTxveviT arCeven or bavSvs. ipoveT albaToba imisa, rom: a). orive bavS-vi gogoa, Tu erTi maTgani gogoa; b). orive gogia, Tu erTi maTgani ekaa.
31. magidas aqvs sami ujra. I ujraSi devs 3 oqros moneta, II-Si 2 oqrosa da 1 vercxlis moneta, xolo III-Si 3 oqrosa da 2 vercxlis mo-neta. SemTxveviT irCeven ujras da iqidan SemTxveviT iReben monetas. ipov-eT albaToba imisa, rom: a). amoRebuli moneta iqneba oqrosi; b). amorCeu-li iyo III ujra, Tu amoRebuli moneta oqrosia.
32. kviris samuSao dReebSi juanSeri samsaxurSi dadis matarebliT. albaToba imisa, rom is miuswrebs 8 saaTian matarebels orSabaTs aris 0.66, xolo sxva samuSao dReebSi ki es albaToba 0.75-ia. SemTxveviT virC-evT kviris erT samuSao dRes. vipovoT albaToba imisa, rom: a). am dRes juanSeri miuswrebs 8 saaTian matarebels; b). arCeuli dRe iyo orSabaTi, Tu cnobilia, rom am dRes juanSerma miuswro 8 saaTian matarebels.
33. albaToba imisa, rom zafxulis romelime dRes baTumSi iwvimebs aris 0.2. albaToba imisa, rom dRis maqsimaluri temperatura wvimian dRes 25 graduss gadaaWarbebs aris 0.3, xolo Tu wvima ar aris 0.6. vipovoT albaToba imisa, rom zafxulis SemTxveviT arCeul dRes baTumSi iwvimebs, Tu cnobilia, rom am dRis maqsimaluri temperatura 25 gradusze metia.
34. sakarnavalo TamaSSi monawile jer agdebs monetas, xolo Semdeg isvris kamaTels. is igebs prizs im SemTxvevaSi, Tu monetaze movida gerbi, xolo kamaTelze samze naklebi qula. vipovoT albaToba imisa, rom is moigebs prizs.
35. vipovoT albaToba imisa, rom kamaTlis samjer agdebisas orjer mova sami qula.
36. vipovoT albaToba imisa, rom kamaTlis oTxjer agdebisas samjer mova oTxi qula.
37. A da B damoukidebelia. ( ) 0.4P A , ( ) 0.7P A B . ipoveT: ( )P B . 38. agdeben erT wesier kamaTels da meore iseT kamaTels, romelzec
6 qulis mosvlis albaTobaa 1/4. ipoveT albaToba imisa, rom: a). wesier kamaTelze mova 6 qula da arawesierze ar mova 6 qula; b). sul cota erT kamaTelze mainc mova 6 qula; g). zustad erT kamaTelze mova 6 qula, Tu cnobilia, rom sul cota erT kamaTelze mainc movida 6 qula.
39. agdeben erT wesier monetas da meore iseT monetas, romelzec gerbis mosvlis albaTobaa 1/3. ipoveT albaToba imisa, rom: a). wesier mo-netaze mova gerbi da arawesierze ar mova gerbi; b). sul cota erT mone-taze mainc mova gerbi; g). zustad erT monetaze mova gerbi, Tu cnobilia, rom sul cota erT monetaze mainc movida gerbi.
40. A da B damoukidebeli xdomilebebia. ( ) 0.4P A , ( ) 0.88P A B . ip-oveT: a). ( )P B ; b). albaToba imisa, rom romelime A da B xdomilebebidan moxdeba, magram ara orive.
41. ipoveT alabaToba imisa, rom SemTxveviT SerCeuli naTura iqneba umaRlesi xarisxis, Tu cnobilia, rom naTurebis 4% arastandartulia da standartuli naTurebis 75% umaRlesi xarisxisaa.
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32
42. albaToba imisa, rom oTx damoukidebel cdSi A xdomileba erT-jer mainc moxdeba aris 15/16. ipoveT calkeul cdaSi A xdomilebis mox-denis albaToba.
43. erT yuTSi aris 18 TeTri da 2 wiTeli burTi, meore yuTSi 9 TeTri da 1 wiTeli burTi. II yuTidan SemTxveviT amoRebul burTs deben I yuTSi. ipoveT albaToba imisa, rom amis Semdeg I yuTidan amoRebuli bu-rTi iqneba TeTri.
44. erT yuTSi aris 3 TeTri da 2 lurji burTi, meoreSi ki 4 TeTri da 4 lurji burTi. I yuTidan SemTxveviT gadaitanes 2 burTi meoreSi. ip-oveT albaToba imisa, rom amis Semdeg II yuTidan amoRebuli burTi iqneba TeTri.
45. 10 yuTidan cxraSi devs 2 wiTeli da 2 mwvane kalkulatori, xo-lo erTSi ki 5 wiTeli da 1 mwvane kalkulatori. SemTxveviT SerCeuli yuTidan SemTxveviT SerCeuli kalkulatori aRmoCnda wiTeli. ipoveT al-baToba imisa, rom kalkulatori amoRebuli iyo yuTidan, romelSic 5 wi-Teli kalkulatoria.
pasuxebi: 1. 19/45; 7/19. 2. 3/20; 9/35; 7/12. 3. 26/27; 21/25. 4. 1/5; 5/13; 17/25; 1/2; 21/25. 5. 0.030; 0.146; 0, 0.712. 6. 0.196; 0.288. 7. 0.75; 0.28. 8. 10 an 15. 9. 0.27; 0.35; 0.3375. 10. 0.75; 0.8. 11. 8/15; 7/15; 3/5; 2/5; 9/16. 12. 0.24; 0.42; 0.706. 14. 2/5; 4/15; 1/8. 15. 0.18; 0.58; 0.6. 16. 1/2; 5/11. 17. 1/5; 1/10. 18. 5p ; 4p . 20. 0.25; 0.0577; 0.1057; 0.6676. 21. 10/69; 3/253;
43/138; 11/138; 11/69. 22. 3/7. 24. 0.32; 0.56. 25. 2/11. 28. 8/23. 29. 2/3. 30. 1/5; 1/3. 31. 34/45; 9/34. 32. 183/250; 44/183. 33. 1/9. 34. 1/6. 35. 5/72. 36. 5/324. 37. 1/2. 38. 1/8; 3/8; 8/9. 39. 1/3; 2/3; 3/4. 40. 0.8; 0.56. 41. 0.72. 42. 0.5. 43. 9/10. 44. 0.52. 45. 5/32.
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Tavi III SemTxveviT sidideTa maxasiaTeblebi
elementarul xdomilebaTa sivrceze gansazRrul ricxviT funqcias
SemTxveviTi sidide ewodeba. SemTxveviT sidides ewodeba diskretuli ti-pis Tu is Rebulobs calkeul, izolirebul SesaZlo mniSvnelobebs. Sem-TxveviT sidides ewodeba uwyveti tipis Tu misi SesaZlo mniSvnelobebis simravle mTlianad avsebs raime ricxviT Sualeds.
cxrils, romelSic CamoTvlilia diskretuli tipis SemTxveviTi si-didis SesaZlo mniSvnelobebi da maTi Sesabamisi albaTobebi, ganawileb-is kanoni (an mwkrivi) ewodeba:
xi x1 x2 xn pi p1 p2 pn
aq 1ii p . albaTobis statistikuri ganmartebidan gamomdinare: Teoriuli sixSire erToblivi sixSire albaToba.
,( , ) ( )
x a bP a b P x
. Tu mocemulia diskretuli tipis SemTxve-
viTi sidide da raime ricxviTi g funqcia, maSin ( )g isev iqneba disk-retuli tipis SemTxveviTi sidide ganawilebis kanoniT:
( )ig x 1( )g x 2( )g x ( )ng x
ip 1p 2p np
hipergeometriuli ganawileba. davuSvaT, rom yuTSi N burTia da maT Soris M TeTria. SemTxveviT, dabrunebis gareSe yuTidan viRebT n burTs. albaToba imisa, rom amoRebul n burTs Soris zustad k cali iqneba TeTri gamoiTvleba formuliT:
( ; ; ; ) : ( )m n m
M N Mn n
N
C CP N M n m P m
C
, 0,1,...,min( , )m M n .
ricxvTa am mimdevrobas hipergeometriuli ganawileba ewodeba.
geometriuli ganawileba: 1( ) kP k pq ( 1q p ), 1,2,...k .
puasonis ganawileba parametriT : { } , 0,1,2,...!
k
P k e kk
.
SemTxveviTi sididis ganawilebis funqcia: ( ) : ( )F x P x ( Sesabamisad, ( ) : ( )F x P x )
( ) ( ) ( )P a b F b F a (Sesabamisad, ( ) ( 0) ( 0)P a b F b F a ). ganawlebis funqciis Tvisebebi: 1). nebismieri x -saTvis 0 ( ) 1F x ;
2). ganawilebis funqcia araklebadia; 3). ganawilebis funqcia uwyvetia marcxnidan (Tu ganawilebis funqc-
ias ganvmartavT rogorc: ( ) : ( )F x P x , maSin is iqneba marjvnidan uwyve-ti);
4). ( ) { }k
kx x
F x P x
(Sesabamisad, ( ) { }k
kx x
F x P x
);
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34
5). { } ( 0) ( )k k kP x F x F x (Sesabamisad,
{ } ( ) ( 0)k k kP x F x F x , roca ( ) : ( )F x P x ).
SemTxveviT sidides, romelsac aqvs uwyveti ganawilebis funqcia, uwodeben uwyvet SemTxveviT sidides. Tu uwyveti SemTxveviTi sididis ga-nawilebis funqcia ( )F x warmoebadia, maSin mis warmoebuls SemTxveviTi
sididis ganawilebis simkvrive ewodeba da aRiniSneba ( )f x -iT: '( ) ( )f x F x .
( ) 0f x ; ( ) 1f x dx
; ( , ) ( )
b
aP a b f x dx ; ( ) ( )
x
F x f y dy
uwyveti ganawilebis funqciis p rigis kvantili ewodeba iseT px
ricxvs, romlisTvisac ( )pF x p . diskretuli ganawilebis SemTxvevaSi, Tu
1 2 1 2 1i i ip p p x p p p p , maSin 1p ix x . 1/ 2p rigis kvantils
SemTxveviTi sididis an misi ganawilebis funqciis mediana ewodeba da aR-
iniSneba x . e. i. 1/2x x . moda ewodeba SemTxveviTi sididis im mniSvnelob-as (an mniSvnelobebs), romelic Seesabameba ganawilebis simkvrivis loka-lur maqsimums uwyveti SemTxveviTi sididis SemTxvevaSi an albaTobis lokalur maqsimums diskretuli SemTxveviTi sididis SemTxvevaSi.
organzomilebiani SemTxveviTi sidide. diskretuli organzomilebiani ( , ) SemTxveviTi sididis ganawileb-
is kanons (anu da SemTxveviTi sidideebis erTobliv ganawilebis kan-ons) aqvs organzomilebiani cxrilis saxe, romelic gvaZlevs SesaZlo mniSvnelobebis calkeuli komponentebis CamonaTvals da im p(xi, yj) albaT-obebs, ra albaTobebiTac miiReba mniSvneloba (xi, yj):
x1 x2 xi xn y1 p(x1, y1) p(x2, y1) p(xi, y1) p(xn, y1)
yj p(x1, yj) p(x2, yj) p(xi, yj) p(xn, yj)
ym p(x1, ym) p(x2, ym) p(xi, ym) p(xn, ym)
,( , ) 1i ji j p x y ; ( ) ( , )i i jjP x p x y ; ( ) ( , )j i jiP y p x y .
organzomilebiani ( , ) SemTxveviTi sididis ganawilebis funqcia
(an da SemTxveviTi sidideebis erToblivi ganawilebis funqcia) ewod-eba: F( , ) = p ( x, y ).
1). 0 F(x, y) 1; 2). F(x, y) aris TiToeuli argumentis mimarT araklebadi, marjvnidan
uwyveti funqcia; 3). adgili aqvs zRvrul Tanafardobebs: F(-, y) = 0; F(x, - ) = 0;
F(- , -) = 0; F( , ) = 1;
4). F(x, ) = F1(x); F( , y) = F2(y). SemTxveviT sidideebs ewodeba damoukidebeli, Tu
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35
F(x, y) = F1(x) F2(y). diskretuli SemTxveviTi sididebi damoukidebelia maSin da mxol-
od maSin, roca p(xi, yj)= p(xi) p( yj), ,i j .
uwyveti organzomilebiani SemTxveviTi sididis erToblivi ganawil-ebis simkvrive (anu organzomilebiani simkvrive) ewodeba erToblivi gan-awilebis funqciis Sereul meore rigis kerZo warmoebuls:
yx
yxFyxf
),(),(
2
.
1). f(x, y) 0; 2).
y x
dxdyyxfyxF ),(),( ;
3).
1),( dxdyyxf ; 4). D
dxdyyxfDYXp .),()),((
5). 11
( , )( ) ( , )
( ) ( , )
x
d f x ydF x dF x
f x f x y dydx dx dx
,
analogiurad,
.),()(2 dxyxfyf
6). uwyveti SemTxveviTi sidideebi damoukidebelia maSin da mxolod maSin, roca f(x, y)= f1(x) f2( y).
diskretuli SemTxveviTi sididis maTematikuri lodini aRiniSne-
ba E simboloTi da ewodeba ricxvs:
( ) ( )E P
, anu { }i i i ii iE x P x x p . uwyveti tipis SemTxveviTi sididis maTematikuri lodini ewodeba
ricxvs
( )E xf x dx
, (Tu | | ( )x f x dx
).
a). Ec c const ; b). ( )E E E ; g). ( )E c cE ; d). ( ) 0E E ;
e). 2 2 2( ) ( ) ( )E c E E c E ; v). 2 2( , )min ( ) ( )
cE c E E
;
z). Tu da damoukidebeli SemTxveviT sididebia, maSin ( )E E E (Sebrunebuli debuleba mcdaria).
SemTxveviTi sididis funqciis maTematikuri lodini:
( ) ( ) { } ( )i i i ii iEg g x P x g x p . Tu SemTxveviTi sididis SesaZlo mniSvnelobebia 1 2, ,x x ..., nx , xo-
lo B raime xdomilebaa ( ( ) 0P B ), maSin SemTxveviTi sididis pirobiTi
maTematikuri lodini B xdomilebis mimarT aRiniSneba ( | )E B simbolo-Ti da ganimarteba Semdegnairad:
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36
1
( | ) ( | )n
i ii
E B x P x B
.
cxadia, rom ( | )E E da 1
( | ) ( )( )
BE B E IP B
.
SemTxveviTi sididis SemTxveviT sidideze regresiis mrudi
(funqcia) ewodeba funqcias ( ) ( | )R y E y .
SemTxveviTi sidides ( )R (regresiis funqciaSi Casmulia SemTxv-
eviTi sidide) SemTxveviTi sididis pirobiT maTematikur lodins uwod-
eben pirobiT da ( | )E simboloTi aRniSnaven. SemTxveviTi sididis dispersia.
SemTxveviTi sididis dispersia D ganimarteba Semdegnairad: 2 2 2( ) ( )DX E E E E .
E -s ewodeba SemTxveviTi sididis meore rigis momenti (TviTon dispersias uwodeben agreTve meore rigis centralur moments).
Tu diskretuli tipis SemTxveviTi sididis ganawilebis kanonia
xi x1 x2 xn pi p1 p2 pn
maSin 2
1 1
( )n n
i j j ii j
DX x x p p
, anu 2 21 1
( )n n
i i j ji j
DX x p x p
.
I. mudmivis dispersia nulis tolia -- 0Dc ;
II. 2( )D a b a D .
III. Tu da damoukidebeli SemTxveviTi sidideebia, maSin
( )D D D .
SemTxveviTi sididis pirobiTi dispersia B xdomilebis mimarT: 2 2 2( | ) : {[ ( | )] | } ( | ) [ ( | )]D B E E B B E B E B .
bernulis ganawilebas (anu bernulis SemTxveviT sidides) aqvs saxe: 1 0
P p 1 p
aq E p da (1 )D p p .
binomialuri ganawilebis SemTxvevaSi: E np da (1 )D np p . hipergeometriuli ganawilebis SemTxvevaSi:
n ME
N
da
2
( ) ( )
( 1)
n M N M N nD
N N
.
puasonis ganawilebis SemTxvevaSi: E D .
geometriuli ganawilebis SemTxvevaSi: 1/E p da 2(1 ) /D p p .
eqsponencialuri ganawileba. SemTxveviT sidides ewodeba eqsponen-
cialurad ganawilebuli parametriT ( 0 ), Tu:
.0,
,0,0)(
xe
xxf x
am SemTxvevaSi: 1/E da 21/D .
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37
standartuli gadaxra. momentebi. SemTxveviTi sididis saSualo kvadratuli gadaxra (an standart-
uli gadaxra): D .
SemTxveviTi sididis standartizacia: E
( 0E , 1D ).
momentebi, asimetria da eqscesi. SemTxveviTi sididis n rigis sawyisi momenti ewodeba sidides
: nn E ( 1:a E ).
SemTxveviTi sididis n rigis centraluri momenti ewodeba sidi-
des : ( )nn E a (2
2: D ).
eqscesis koeficienti ewodeba sidides: 4
4
4 2 4
( )3 3
[ ( ) ]
E Ee
E E
.
asimetriis koeficienti ewodeba sidides: 3
3
3 2 3
( )
[ ( ) ]
E E
E E
.
kovariacia. korelaciis koeficienti. kovariaciis koeficienti an ubralod kovariacia ewodeba sidides:
cov( , ) [( )( )] ( )E E E E E E .
1). Tu da damoukidebelia, maSin cov( , ) 0 (piriqiT, ara);
2). cov( , ) D ;
3). cov( , ) cov( , ) ;
4). cov( , ) cov( , )c c ;
5). cov( , ) cov( , ) cov( , ) ;
6). ( ) 2cov( , )D D D ;
7). | cov( , ) | D D . korelaciis koeficienti ewodeba sidides:
cov ;( ; ) :
E EE
D D
.
a). 1(;)1.
b). Tu (;)=1, maSin =k+b, sadac k da b mudmivebia, k>0.
g). Tu (;)= 1, maSin =k+b, sadac k da b mudmivebia, k0; (;)= 1 roca ki
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38
}{ sssssg,sgs,gss,sgg,gsg,ggs,ggg,
da, Sesabamisad, saZiebeli SemTxveviTi sidide iqneba -ze gansazRruli Semdegi ricxviTi funqcia:
( ggg) 3 ; (ggs) (gsg) (sgg) 2;
(gss) (sgs) (ssg) 1 da (sss) 0 . cxadia es SemTxveviTi sidide diskretuli tipisaa, is Rebulobs
izolirebul mniSvnelobebs, magaliTad, 1-sa da 2-s Soris is ar Rebulobs arcerT mniSvnelobas.
magaliTi 2. SemTxveviTi sidide iyos ori saTamaSo kamaTlis gagor-ebisas mosul qulaTa jami. am SemTxvevaSi elementarul xdomilebaTa si-vrce Sedgeba 36 elementaruli xdomilebisagan:
}6,...,2,1,:),{( jiji ,
xolo SemTxveviTi sidide calkeul elementarul xdomilebas ),( ji (sad-ac i -- pirvel kamaTelze mosuli qulaa, xolo j -- meore kamaTelze mo-
suli qula) Seusabamebs: ( , )i j i j (pirvel da meore kamaTelze mosuli
qulebis jami). magaliTad, (1,3) (2,2) (3,1) 4 . aRniSnuli SemTxveviTi sididis SesaZlo mniSvnelobebia: 2, 3, . . . , 12. is dikretuli tipisaa.
magaliTi 3. ori msroleli TiTojer esvris samiznes. maT mier sami-znis dazianebis (mizanSi moxvedris) albaTobebia Sesabmisad 0.6 da 0.7. Se-mTxveviTi sidide iyos dazianebul samizneTa raodenoba. SevadginoT misi ganawilebis mwkrivi.
amoxsna. cxadia, rom SemTxveviTma sididem SeiZleba miiRos Semd-egi mniSvnelobebi: 0 (verc erTma msrolelma ver daaziana samizne), 1 (mxolod erTma msrolelma daaziana samizne) da 2 (orive msrolelma daaziana samizne). vipovoT Sesabamisi albaTobebi.
bunebrivia SegvZlia vigulisxmoT rom pirveli da meore msrolel-is srolis Sedegebi erTmaneTisagan damoukidebelia. SemoviRoT xdomil-ebebi: A -- pirvelma msrolelma daaziana samizne da B -- meore msrole-lma daaziana samizne. mocemulia, rom ( ) 0.6P A da ( ) 0.7P B . Sesabamisad,
( ) 0.4P A da ( ) 0.3P B . garda amisa, A da B damoukidebeli xdomilebebia.
damoukidebeli xdomilebebia agreTve: A da B , A daB , A da B . advili dasanaxia, rom xdomileba verc erTma msrolelma ver daa-
ziana samizne iqneba BA , xdomileba -- mxolod erTma msrolelma daaz-
iana samizne iqneba )()( BABA da xdomileba -- orive msrolelma daa-
ziana samizne iqneba BA . gasagebia, rom )( BA da )( BA uTavsebadi
xdomilebebia )()( BABA .
amitom, damoukidebel xdomilebaTa namravlis albaTobisa da uTav-sebad xdomilebaTa jamis albaTobis formulebis Tanaxmad gveqneba:
12,03,04,0)()()()0( BPAPBAPXP ;
( 1) {( ) ( )} ( ) ( )P X P A B A B P A B P A B
( ) ( ) ( ) ( ) 0,6 0,3 0,4 0,7 0,46P A P B P A P B ;
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39
( ) ( ) ( ) 0,6 0,7 0,42P A B P A P B
Sesabamisad, SemTxveviTi sididis ganawilebis mwkrivi iqneba:
ix 0 1 2
ip 0.12 0.46 0.42
magaliTi 4. auditoriaSi myofi 15 studentidan 5 vaJia. vipovoT al-
baToba imisa, rom SemTxveviT SerCeul 6 students Soris 3 vaJia? amoxsna. Tu mivusadagebT hipergeometriul ganawilebas, gasagebia,
rom: 15, 5, 6N M n da 3k . amitom saZiebeli albaToba iqneba: 3 6 3 3 3
10 15 10 10 5
6 6
15 15
120 10(15;5;6;3) 0.239
5005
C C C CP
C C
.
magaliTi 5. saqonlis partiaSi defeqtur nawarmTa ricxvi Rebu-lobs mniSvneloba 0-s albaTobiT 0.3; mniSvneloba 1-s albaTobiT 0.4; mniSvneloba 2-s albaTobiT 0.2 da mniSvneloba 3-s albaTobiT 0.1 anu -s ganawilebis mwkrivs aqvs saxe:
ix 0 1 2 3
ip 0.3 0.4 0.2 0.1
gamovTvaloT -s ganawilebis funqcia da avagoT misi grafiki.
Tu 0x , maSin ( ) ( ) ( ) 0F x P x P ;
Tu 0 1x , maSin ( ) ( ) ( 0) 0.3F x P x P ;
Tu 1 2x , maSin ( ) ( ) {( 0) ( 1)}F x P x P
( 0) ( 1) 0.3 0.4 0.7P P ;
Tu 32 x , maSin ( )
