omar furTuxia albaToba da statistikamagaliTebsa da amocanebSi (Teoria, magaliTebis amoxsnis nimuSebi, amocanebi) Tsu-2011 1 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. 3 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. 15 Tavi III. SemTxveviT sidideTa maxasiaTeblebi .......................................................... SemTxveviTi sidide. ganawilebis kanoni. ganawilebis funqcia. ganawilebis simkv-rive. zogierTi mniSvnelovani ganawileba. kvantili, mediana, moda. maTematikuri lodini,dispersia.organzomilebianiSemTxveviTisidide.regresiisfunqcia. momentebi, asimetria, eqscesi. kovariacia. korelaciis koeficienti. 33 Tavi IV. diskretul ganawilebaTa gamoyenebebi ...................................................... binomialuri da puasonis ganawilebebis gamoyenebebi 50 Tavi V. uwyveti tipis ganawilebebi........................................................................................ ganawilebissimkvrive.kvantili,moda,mediana,qvedadazedakvartili,zeda o -kritikuli wertili, momentebi (lodini, dispersia, asimetria, eqscesi). 61 Tavi VI. normaluri ganawileba ...............................................................................................70 Tavi VII. albaTobis Teoriis zRvariTi Teoremebi ............................................. CebiSevisutoloba.didricxvTakanoni.CebiSevisTeorema.bernulisTeorema. centraluri zRvariTi Teorema. liapunovis Teorema. muavr-laplasis lokaluri da integraluri Teoremebi. 84 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. 94 Tavi II. ndobis intervali bernulis sqemaSi ............................................................. asimptoturi ndobis intervali da SerCevis moculoba populaciis proporciis-aTvis. ndobis intervali puasonis ganawilebis ucnobi parametrisaTvis. 104 Tavi III. ndobis intervali dispersiisaTvis ............................................................... ndobis intervali normaluri populaciis dispersiisa da standartuli gadaxr-isaTvis cnobili da ucnobi saSualoebisaTvis. didi SerCevebis SemTxveva.108 Tavi IV. hipoTezaTa Semowmeba saSualosaTvis (2ocnobilia) ................ 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.114 Tavi V. hipoTezaTa Semowmeba saSualosaTvis (2oucnobia) ...................... hipoTezaTa Semowmeba normaluri populaciis saSualosaTvis ucnobi dipersiis SemTxvevaSi.Zda Tkriteriumebis gamoyenebis SesaZlo SemTxvevebi. 123 Tavi VI. hipoTezaTa Semowmeba dispersiisaTvis ......................................................129 2 statistikurihipoTezisSemowmebanormaluriganawilebisdispersiisSesaxeb 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. 135 Tavi VIII. ndobis intervali da hipoTezaTa Semowmeba ....................................140 Tavi IX. oramokrefiani amocanebi populaciaTa saSualoebisaTvis ... ori damoukidebeli normaluri populaciis saSualoTa Soris gansxvavebishipoTezaTa Semowmeba cnobili da ucnobi dispersiebis SemTxvevaSi. hipoTezaTa Semowmeba aranormaluri populaciebisaTvis (didi moculobis mqone SerCevebi). ndobis intervalebi saSualoTa sxvaobisaTvis. 147 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. 157 Tavi XI. oramokrefiani amocanebi populaciaTa dispersiebisaTvis ... hipoTezaTa Semowmeba ori damoukidebeli normaluri populaciis dispersiebisSesaxeb. dispersiaTa tolobis Sesaxeb hipoTezaTa Semowmebis gamartivebuli procedura. ndobis intervali dispersiaTa fardobisaTvis. 164 Tavi XII. statistikuri daskvnebi dawyvilebuli monacemebisaTvis .... statistikuridaskvnebidawyvilebulmonacemTasaSualoebissxvaobebisaTvis (damokidebuli SerCevebi).P-mniSvnelobis meTodi. ndobis intervali.171 Tavi XIII. oramokrefiani amocanebi bernulis sqemaSi ...................................... hipoTezaTa Semowmeba warmatebaTa albaTobebisaTvis bernulis cdaTa ori damo-ukidebeli mimdevrobisaTvis. ndobis intervali 1 2p p sxvaobisaTvis. 178 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).206 Tavi XVIII. regresia da korelacia ...................................................................................... martivi wrfivi regresia. regresiis wrfe. umcires kvadratTa meTodi. regresiis wrfisSefaseba.determinaciiskoeficienti.statistikuridaskvnebiregresiis koeficientebis Sesaxeb. mravlobiTi regresia. modelis vargisianobis Semowmeba. statistikuridaskvnebimravlobiTimodelisparametrebisSesaxeb.rangobrivi korelacia. spirmenis rangobrivi korelaciis koeficienti. 222 Tavi XIX. dispersiuli analizi (ANOVA) ...................................................................... hipoTezaTa Semowmeba sami an meti populaciis saSualoTa tolobis Sesaxeb. 238 Tavi XX. sakontrolo werebisa da Sualeduri da saboloogamocdebis bileTebis nimuSebi 20072009 wlebSi .............................................. 249 danarTi (statistikuri cxrilebi) .......................................................................................254 damateba (sagamocdo amocanebi xdomilebebisaTvis).............................................263 literatura ..................................................................................................................................................272 3 albaToba Tavi I albaTobis Teoriis elementebi eqsperimentiscalkeulSesaZloSedegselementarulixdomileba ewodeba, xolo maT erTobliobas elementarul xdomilebaTa sivrce da aRiniSneba O asoTi: 1 2{ , ,...} ee O= .savarjiSoebi:I.monetiserTxelagdebisas--O={g,s};II.monetis orjer agdebisas, an ori monetis erTdroulad agdebisas -- O={gg, gs, sg, ss}; III. monetis samjer agdebisas, an sami monetis erTdroulad agdebisas --O= {ggg, ggs, gsg, sgg, gss, sgs, ssg, sss}; IV. monetisn-jer agdebisas } ), ,..., ( : {1s an g = = = Oi na a a e e daSedegebissaerToraodenobatolia n2 -is;V.erTisaTamaSokamaTlisgagorebisas--O={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 { ss sg, g6, g5, g4, g g g = O ; VII. ori saTamaSo kamaTlis ga-gorebisas --O={(1,1); (1,2); . . . ; (1,6); (2,1); (2,2); . . . ; (2,6); . . . ; (6,1); . . . ; (6,6)} anu} 6 ,..., 2 , 1 , : ) , {( = = O j i j i ; VIII. produqciis vargisinobis dadgenisas -- O={vargisi, uvargisi}; IX. satelefono sadgurSi gamoZaxebaTa raoden-oba -- O= {0, 1, 2, . . . }; X. Zabva qselSi -- O={[0, 220]}.diskretulelementarulxdomilebaTasivrcisnebismierqvesim-ravlesxdomilebaewodeba.TueqsperimentiskonkretuliSedegiekuTv-nisraimexdomilebas,maSinambobenromesxdomilebamoxda,xolo romelsac ar ekuTvnis is xdomileba ar moxda. xdomilebebi aRiniSneba didilaTinuriasoebiT:,... , , , D C B A .xdomilebasO = A uwodeben aucilebelxdomilebas,xolo--SeuZlebelxdomilebas.AdaB xdomilebisgaerTianeba(anjami)ewodebaiseTxdomilebas,romelic xdebamaSin,rocaamxdomilebebidanerTimaincxdebadaaRiniSneba simboloTiB A(anB A+ ).A daB xdomilebis TanakveTa (an namravli) ewodebaiseTxdomilebas,romelicxdebamaSin,rocaesxdomilebebi erTdrouladxdebadaaRiniSnebasimboloTiB A (anAB).A xdomilebis sawinaaRmdego xdomileba ewodeba iseT xdomilebas, romelic xdebamaSin,rocaAarxdebadaaRiniSnebasimboloTiA.AdaB xdomilebissxvaobaewodebaiseTxdomilebas,romelicxdebamaSin, roca xdebaA magram ar xdebaB da aRiniSneba simboloTiB A\ .A da B xdomilebas ewodeba uTavsebadiTuB A =.TuO-s nebismier elementarul xdomilebas ieSeesabameba garkveu-liricxvebi) (i iP p e = ,romlebicakmayofilebenpirobebs:1 0 s sip da 1ii p =, maSin am ricxvebs ewodebaT ieelementaruli xdomilebebis alb-4 aTobebi.( ) : ( )iiAP A Peee=. Tu( )iP const e =da | | O 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 tolia6112131= . Sesabamisad, kubis darCenili nawilis moculoba iqneba21613 1 | | = = v . amitom saZiebeli albaToba, ganmartebis Tanaxmad, iqneba .211 :21| || || | = = =VvPmagaliTi 6 (Sexvedris amocana). ori piri SeTanxmda garkveul adgi-lzeSexvdeserTmaneTs6-dan7saaTamde.TiToeulimaTganiSemTxveviT 9 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 x + 6 -iT, xolo meore piris --y + 6 -iT (sadacx daygamosaxulia saaTeb-Si). elementarul xdomilebaTa sivrcis rolSi SegviZlia aviRoT im) , ( y xwertilTa simravle, romlebic ekuTvnian erTeulovan kvadrats. am SemTx-vevaSi CvenTvis saintereso wertilTa (xelSemwyob elementarul xdomile-baTa) simravle iqneba erTeulovani kvadratis im wertilTa simravle, ro-melTakoordinatebierTmaneTisagandaSorebuliaaraumetes3 / 1 60 / 20 = -iT: O s = } 3 / 1 | | ); , {( y x y x A . vinaidan,3 / 1 3 / 1 3 / 1 3 / 1 3 / 1 | | + s s s s s x y x x y y x . amitom adv-iligasagebia,romAsimravleiqnebaerTeulovanikvadratisSigniT 3 / 1 = x yda3 / 1 + = x ywrfeebs Soris moqceuli gaferadebuli are (roca 3 / 1 0 s s x , maSin3 / 1 = x ywrfis nacvlad qveda sazRvris rolSi iqnebax RerZi:3 / 1 0 + s s x y , xolo roca1 3 / 2 s s x , maSin zeda sazRvari3 / 1 + = x ywrfis nacvlad iqneba1 = ywrfe:1 3 / 1 s s y x ). 01/32/3 1 x y 1 1/3 O amitom saZiebeli albaToba iqneba: 9513 / 2 3 / 2 1| || |) ( = =O=AA P . magaliTi7.ramdegigansxvavebulisiaSeiZlebaSedgeniliqnes7 sxvadasxva gvarisagan? amoxsna. 77! 1 2 3 4 5 6 7 5040 P= = = . magaliTi 8. SejibrebaSi monawile 10 sportsmenidan pirvel sam adg-ilze gasuli sami gamarjvebuli ramden sxvadasxvanairad SeiZleba ganTa-vsdes dasajildoebel kvarcxlbekze? amoxsna.. 720 8 9 10310= = magaliTi 9. SesarCev SejibrebaSi monawileobs 10 adamiani, romelT-aganfinalSigadissami.finalistebisramdenigansxvavebulisameuli SeiZleba gamovlindes? amoxsna. wina magaliTisagan gansxvavebiT, aq finalistebis rigs (da-lagebas) mniSvneloba ara aqvs. amitom viyenebT jufdebis formulas:10 . 120610 9 8! 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-anSimoiZebnebaorimainciseTi,romelTadabadebisdReebidaemTxveva erTmaneTs. amoxsna. a). elementarul xdomilebaTa sivrce Seesabameba dalageb-ul amorCevebsdabrunebiT, sadac365 M =dan m = , anu| | 365mO = ; xolo xelSemwyobi elementaruli xdomilebebi ki dalagebul amorCevebs dabru-nebis gareSe, sadac agreTve365 M =dan m = , amitom maTi raodenobaa -- 365mA . Sesabamisad, albaTobis klasikuri ganmartebis Tanaxmad, gvaqvs )36511 ( )36521 )(36511 (365) (365 = =m Am Pmm. b). sawinaaRmdego xdomilebis albaTobis gamoTvlis wesis Tanaxmad ki gvaqvs, rommmAm Q3651 ) (365 = . moviyvanoTamalbaTobismniSvnelobebiscxrilizogierTim-is SemTxvevaSi: m4162223406470 Q(m)0.016360.283600.475690.507300.891230.997110.99916 sainteresoaaRiniSnos,rom(molodinissawinaaRmdegod!)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). gvaqvsM bileTi gadanomrili ri-cxvebiT erTidanM-mde, romelTagann bileTi nomrebiT erTidann-mde momgebiania ( n M 2 > ). vyidulobTn cal bileTs. ras udris albaToba imi-sa, rom amn bileTidan erTi mainc iqneba momgebiani (avRniSnoT es xdom-ilebaA asoTi)? amoxsna.vinaidanbileTebisamoRebis(yidvis)Tanmimdevrobasara aqvs mniSvneloba nayid bileTebSi momgebiani bileTebis arsebobis an ar arsebobis TvalsazrisiT, amitom elementarul xdomilebaTa sivrces eqne-ba Semdegi stuqtura: } ,..., 1 ; , : ] ,..., [ : {1M a l k a a a ai l k n= = = = = O e e . Sesabamisad, Cvens mier zemoTmoyvanili cxrilis Tanaxmad nMC = O| | . xdomilebas (avRniSnoT igi 0B -iT), rom nayid bileTebSi ar aris mo-mgebiani bileTebi, eqneba Semdegi struqtura:} ,..., 1 ; , : ] ,..., [ : {1 0M n a l k a a a a Bi l k n+ = = = = = e eda nn MC B= | |0. amitom 11 )11 ( )11 )( 1 ( ) (0+ = = = n MnMnMnAACCB PnMnn MnMnn M. Sesabamisad, saZebni albaToba iqneba: 0( ) 1 ( ) 1 (1 )(1 ) (1 )1 1n n nP B P BM M M n= = +. Tu magaliTad, 2n M =da n , maSin 10) (e B P(aqe neperis ric-xvia) da632 , 0 1 ) (1~ e B P , sadac krebadobis siCqare sakmaod didia: ukve roca10 = n--670 , 0 ) ( = B P . davaleba. wina amocanis pirobebSi vipovoT albaToba imisa, rom nay-idin bileTidan zustad m ( n ms ) iqneba momgebiani. magaliTi 12 (urTierTobis upiratesobaze).davuSvaT klasSi, rome-lSic 10 moswavlea tardeba gamokiTxva, sadac TiToeulma moswavlem ank-etaSiundamiuTiTosissamiamxanagi,romelsacaZlevsupiratesobas 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 10 39) (C -is. vinaidan er-Ti anketa SeiZleba Sevsebul iqnes nebismierad, xolo danarCen cxra anke-taSierTipasuxidafiqsirebuliadadanarCenioripasuxikiSeiZleba nebismieradamoirCesrvaSesaZlebelipasuxidan,amitomelementaruli xdomilebebisraodenobaA-Sitolia 3 2 99 8| | ( ) A C C = (Tuwyvilispirvel komponentsSeuZliamiiRosmgansxvavebulimniSvneloba,xolomeore komponents ki pirvelisgan damoukideblad --n gansxvavebuli mniSvnelo-ba, maSin aseTi wyvilebis raodenoba namravlis principis Tanaxmad iqneba --m n ). aqedan gvaqvs 99392810 399 283931) () () () ( = ==CCC C CA P . davaleba. wina amocanaSi ipoveT albaToba imisa, rom erTerTi stu-denti dasaxelebuli iqnebak -jer ( 9 s k ). 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.orikartissxvadasxvakombinaciebisraodenoba,romelic SeiZleba aRmoCndes sayidlebSi tolia496232 = C . kartis SekvraSi oTxi tuziaDdasxvadasxvakombinaciebisraodenoba,romelicmogvcemdaor tuzsAtolia624= C . Sesabamisad, saZiebeli albaToba tolia12 012 , 0496623224= =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 gvaqvsNcali Sem-TxveviT dalagebuli geometriuli figura, romelTa SorisM cali mar-TkuTxedia ( N M s ). moiTxoveba moiZebnos yvela marTkuTxedi, Tu Zebna wa-rmoebs elementebis (figurebis) saTiTaod skanirebiT fiqsaciis moculo-biTerTielementi,amasTanavexdebadakvirvebulielementispoziciis damaxsovreba da mas Tavidan aRar vubrundebiT. vipovoT albaToba imisa, rom damkvirvebeli SeZlebs aRmoaCinos yvelaM marTkuTxedi araumetes n dakvirvebisas ( N M n ,..., = )? amoxsna. yvela SesaZlo elementarul xdomilebaTa raodenoba iqne-ba nNC = O| | . xelSemwyobi elementaruli xdomilebebi iseTi [na a ,...,1] erTo-bliobebia,romlebSicMadgilasganTavsebuliamarTkuTxedebi(amis SesaZleblobaTa raodenobaa MMC ), xolo danarCenM N adgilas ki ara marTkuTxedebi (amis SesaZleblobaTa raodenobaa M nM NC). amitom xelSemwy-obielementarulixdomilebebisraodenobaiqnebaMMCM nM NC.Sesabamis-ad, saZiebeli albaToba tolia MNMnnNM nM NnNM nM NMMMCCCCCC Cn P = ==) ( . 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 irCevenerTs. ipoveT albaToba imisa, rom es ricxvi iqneba: a). 4-is jeradi; b). erTdroulad 4-isa da 6-is jeradi.10.kalaTaSidevs7wiTelida4TeTridiski.dabrunebisgareSe 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,ByuTi Seicavs2 wiT-el burTs.A yuTidan SemTxveviT iReben 1 burTs da gadaaqvTByuTSi. SemdegB yuTidan SemTxveviT iReben 1 burTs da gadaaqvTA yuTSi. ipov-eT albaToba imisa, rom am ori gadatanis Semdeg Savi burTi iqnebaA 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, maSinA yuTidan SemTxveviT iReben 1 burTs, xolo winaaRmdeg SemTxveva-SiB yuTidan SemTxveviT iReben 1 burTs. ipoveT albaToba imisa, rom: a). amoRebuli burTi wiTelia; b). burTi amoRebuli iyoB 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 mowyobilobazeTdrois manZilze SemTxvev-iT momentSi miiReba. mowyobiloba maT ganarCevs, Tu isinitdroiT mainc arian dacilebuli erTmaneTs. ipoveT albaToba imisa, rom orive signali miRebuli iqneba? 18.centrisgarSemombrunaviwregayofiliasamitolifarTobis 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 kartidanAmoTamaSesaqvsori dedofali da erTi mefe.A moTamaSisaganB moTamaSe SemTxveviT iRebs erT karts da ubrunebs isev mas.AmoTamSeerTmaneTSiurevsamsamkartsdaSemdegBmoTamaSe kvlav iRebs erT karts.B moTamaSe igebs Tu mis mier amoRebuli orive karti iqneba mefe. ipoveTB 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. 22.200detalisSemowmebisasstandartulebisfardobiTisixSire aRmoCnda 0.9. ipoveT standartuli detalebis raodenoba. 23. sibrtyeze, romelic dafaruliaa gverdis mqone kvadrtTa bad-iT, SemTxveviT agdeben monetas radiusiT/ 2 r a < . ipoveT albaToba imisa, rom moneta ar gadakveTs arcerTi kvadratis gverds.24. sibrtyeze, romelic dafarulia 6 sm-iT daSorebuli paralelu-riwrfeebiT,SemTxveviTagdebenwresradiusiT1sm.ipoveTalbaToba 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= B A , maSin) ( ) ( ) ( B P A P B A P + = + . sawinaaRmdego xdomilebis albaToba:) ( 1 ) ( A P A P = . sxvaobis albaTobis formula: TuB A c , maSin) ( ) ( ) \ ( B P A P B A P = . Tu= j iA A , rocaj i = , maSin: 1 1( ) ( )n ni ii iP A P A= == . roca xdomilebebi Tavsebadia:) ( ) ( ) ( ) ( B A P B P A P B A P + =.( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P A B C P A P B P C P A B P A C P B C P A B C = + + + sazogadod: 11 11( ) ( ) ( ) ( ) ( 1) ( )nn nni i i j i j k ii ii i j i j kP A P A P A A P A A A P A+= == = == = + + . pirobiTi albaTobis formula A xdomilebis pirobiTi albaToba pirobaSi, rom adgili hqondaB xdomilebas, aRiniSneba) | ( B A P(an( )BP A ) simboloTi da( | ) : ( ) / ( ) P AB P A B P B = ,Tu0 ) ( = B P . 1 ) | ( 0 s s B A P ;B A = 0 ) | ( = B A P ;A Bc1 ) | ( = B A P ; Tu= j iA A , rocaj i = , maSin: 1 1[( ) | ] ( | )n ni ii iP A B P A B= == . namravlis albaTobis formula ( ) ( ) ( | ) ( ) ( | ) P A B P A P BA P B P AB = = . ( ) ( ) ( | ) [ | ( )] P A B C P A P BA 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 AxdomilebasewodebaBxdomilebisagandamoukidebeli,Tu ) ( ) | ( A P B A P = anracigivea) ( ) ( ) ( B P A P B A P = .Tu) ( ) | ( A P B A P = ,maSin gvaqvs damokidebuli xdomilebebi. Tu A daB xdomilebebi damoukideb-elia, maSin xdomilebebiA daB agreTve damoukidebelia. xdomilebaTa erTobliobas nA A A ,..., ,2 1 ewodeba wyvil-wyvilad damo-ukidebeli Tu:j i A P A P A A Pj i j i= = ), ( ) ( ) ( . xdomilebaTaerTobliobas nA A A ,..., ,2 1ewodebaerToblivad damoukidebeliTun k s , 1 2 ki i i = = = : ) ( ) ( ) ( ) (2 1 2 1 k ki i i i i iA P A P A P A A A P = . sruli albaTobis formula xdomilebaTa erTobliobas nA A A ,..., ,2 1 ewodeba xdomilebaTa sruli sistema, Tu= j iA A , rocaj i =daO = =iniA1 (magaliTad,A daA).16 Tu nA A A ,..., ,2 1xdomilebaTasrulisistemaa( ( ) 0,iP A > 1, 2,..., i n = ), maSin adgili aqvs sruli albaTobis formulas:) | ( ) ( ) (1iniiA B P A P B P== .baiesis formula Tu nA A A ,..., ,2 1 xdomilebaTa sruli sistemaa,, 0 ) ( >iA P n i ,..., 2 , 1 = , maSin adgili aqvs baiesis formulas: 1( ) ( | )( | )( ) ( | )i ii nj jjP AP BAP A BP AP BA==. 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-veuliA xdomilebis moxdena (romelsac pirobiTad warmatebas uwodeb-en) da misi ar moxdena --A (romelsac marcxs uwodeben).A xdomilebis moxdenisalbaTobanebismierieqsperimentisaTvismudmiviadatolia p A P = ) ( , sadac1 0 < < p . Sesabamisad,q p A P A P = = = : 1 ) ( 1 ) (( 1 = +q p ). albaTobasimisa,romneqsperimentSiAxdomilebamoxdak -jer gamoiTvleba e. w. bernulis formulT: !( )!( )!k k n k k n kn nnPk Cp q p qk n k = =.amasTanave, adgili aqvs Tanafardobas: ( )( 1) ( )( 1)n nn kpPk Pkk q+ =+. albaTobebis erTobliobas( )nPk , roca0,1,..., k n =ewodeba albaTob-ebis binomialuri ganawileba.iseT 0kricxvs, romlis Sesabamisi albaToba 0( )nPkudidesia) 0 (nP , ) 1 (nP ,..., ) (n Pn albaTobebs Soris ualbaTesi ricxvi ewodeba. ualbaTesi ri-cxvi gviCvenebsn damoukidebel cdaSi warmatebaTa ra raodenobaa yvela-ze ufro mosalodneli. ualbaTesi ricxvi warmoadgens Semdegi utolob-is mTel amonaxsns:0np q k np p s s + . puasonis formula Tanafardobas lim ( ) / !knnpp k e k=puasonis formula ewodeba. igi sa-SualebasiZlevavipovoTndamoukidebelcdaSiAxdomilebisk -jer moxdenis albaToba (rocan sakmaod didia, xolopsakmaod mcire, amas-Tanave 15 np = < ) puasonis miaxloebiTi formuliT:( ) / !knp k e k~ . magaliTebis amoxsnis nimuSebi: magaliTi1.ganvixiloTojaxebi,sadacor-oribavSvia.rogoria albaTobaimisa,romojaxSiorivebavSvivaJiaTucnobilia,rom:a). ufrosi bavSvi vaJia; b). erTi bavSvi mainc vaJia? 17 amoxsna. aq elementarul xdomilebaTa sivrce aseTia{ } O= vv, vq, qv, qq , sadac v aRniSnavs vaJs, xolo q qals. CavTvaloT, rom oTxive SedegitolalbaTuria.SemoviRoTxdomilebebi:A--iyosxdomileba, romufrosibavSvi--vaJia,xoloB--iyosxdomileba,romumcrosi bavSvivaJia.maSinB A --iqnebaxdomileba,romorivebavSvivaJia, xoloB A --kiiqnebaxdomileba,romerTibavSvimaincvaJia. Sesabamisad,saZiebelialbaTobebiiqneba:a).) | ( A B A P dab). ) | ( B A B A P . advili dasanaxia, rom: 212 / 14 / 1) () () (] ) [() | ( = == = A PB A PA PA B A PA B A P , 314 / 34 / 1) () () ()] ( ) [() | ( = =

=

=B A PB A PB A PB A B A PB A B A P . magaliTi 2 (saukeTesos SerCevaze). mocemuliam obieqti gadanomr-ili ricxvebiTm ,..., 2 , 1 , amasTanave ise, rom vTqvaT, obieqti #1 klasifici-rdeba rogorc saukeTeso, . . . , obieqti #m ki rogorc yvelaze uare-si. igulisxmeba rom obieqtebi Semodian drois momentebSim ,..., 2 , 1SemTx-veviTi rigiT (anu yvela SesaZlom! gadanacvleba tolalbaTuria). damk-virvebelsSeuZliaorimaTganisSedarebiTTqvasromeliaukeTesida 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 SearCiak -ur nabijze ( m k s ), 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, romk -uri ob-ieqti saukeTeoa yvela arsebulm obieqts Soris daB iyos xdomileba, romk -uriobieqtisaukeTeoadakvirvebulk obieqtsSoris.gasagebia, rom mosaZebnia pirobiTi albaToba) | ( B A P .vinaidanB Ac , amitomA B A = da) ( ) ( A P B A P = . Sesabamisad, pir-obiTi albaTobis ganmartebis Tanaxmad ) ( / ) ( ) | ( B P A P B A P =vinaidan obieqtebis yvela SesaZlo gadanacvlebebi tolalbaTuria, amitom albaTobis klasikuri ganmartebis Tanaxmad advili dasanaxia, rom k kkB P1!)! 1 () ( ==da m mmA P1!)! 1 () ( == . Sesabamisad,m k B A P / ) | ( = . strategia. SeiZleba damtkicdes, rom saukeTeso obieqtis amorCevis optimaluristrategiamowyobiliaSemdegnairad.avRniSnoTsimboloTi *miseTi naturaluri ricxvi, romlisTvisac samarTliania utoloba 18 1111111 1* *+ +< s+ +m m m m. saukeTeso obieqtis arCevis optimaluri strategia mdgomareobs ima-Si, rom davakvirdeT da ukuvagdoT pirveli1* mobieqti da Semdeg gava-grZeloT dakvirveba iseT *tmomentamde, roca pirvelad gamoCndeba yvela winamorbedze ukeTes obieqti. magaliTad, Tu10 ,..., 1 = m , maSin *m -is Sesabamisi mniSvnelobebia: m 12345678910 m-optimaluri1111222334 sakmaoddidim-isaTvise m m /*~ (sadace--neperisricxvia, 2.718 e ~ )dasaukeTesoobieqtisarCevisalbaTobadaaxloebiTtolia 1/ 0.368 e ~(Tumca, erTi SexedviT bunebrivia mogvCveneboda, rom gansaxil-veli obieqtebism 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:Aiyos xdomileba, rom pi-rvelma studentma aiRo bednieri bileTi, xoloB iyos xdomileba, rom meore studentma aiRo bednieri bileTi. maSin cxadia, rom elementarul xdomilebaTa sivrce Sedgeba oTxi xdomilebisagan } , , , { B A B A B A B A = O . albaTobisklasikuriganmartebisTanaxmad5 / 1 25 / 5 ) ( = = A P ,xolo( ) 20/ 25 4/ 5 P A= = .meoresmxriv,amocanisSinaarsidangamomdinare,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 / 1 24 / 4 ) | ( = = A B P . analogiurad,6 / 5 24 / 20 ) | ( = = A B P ,24 / 5 24 / 5 ) | ( = = A B P da ( | ) 19/ 24 P BA= .amitomorixdomilebisnamravlisalbaTobisformulis Tanaxmad gvaqvs: 30 / 1 6 / 1 5 / 1 ) | ( ) ( ) ( = = = A B P A P B A P ;6 / 1 6 / 5 5 / 1 ) ( = = B A P ; 19 6 / 1 24 / 5 5 / 4 ) ( = = B A Pda30 / 19 24 / 19 5 / 4 ) ( = = B A P(SevniSnavT, rom Cven aq mxolod sisrulisaTvis gamovTvaleT yvela SesaZlo pirobiTi da namravlis albaTobebi).cxadia, rom) ( ) ( B A B A B + = . amitom albaTobaTa Sekrebis kanonis Tanaxmad )) ( ( 5 / 1 6 / 1 30 / 1 ) ( ) ( ) ( A P B A P B A P B P = = + = + = . magaliTi 4. yuTSim burTia, maT Sorisn 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. iAiyos xdomileba, romi -uri burTi TeTria ( 2 , 1 = i ). maSin albaTobis klasikuri ganmartebis Tanaxmad: a).m n A P / ) (1= .garda amisa,) 1 /( ) 1 ( ) | (1 2 = m n A A Pda) 1 /( ) | (1 2 = m n A A P . amitom namravlis albaTobis formulis Tanaxmad: g).) 1 ( / ) 1 ( ) | ( ) ( ) (1 2 1 2 1 = = m m n n A A P A P A A P .analogiurad,) 1 ( / ) ( ) | ( ) ( ) (1 2 1 2 1 = = m m n m n A A P A P A A P . amitom: b).m n A A P A A P A A A A P A P / ) ( ) ( )] ( ) [( ) (2 1 2 1 2 1 2 1 2= + = + = . magaliTi 5. yuTs aqvsn ujra. albaToba imisa rom burTi aris am ujrebidan erT-erTSi toliap -si. ipoveT albaToba imisa, rom burTi ar-isi -ur ujraSi, Tu cnobilia, rom burTi TiTeul ujraSi SesaZlebelia iyos Tanabari albaTobebiT? amoxsna. iAiyos xdomileba, rom burTi arisi -ur ujraSi.A iyos am xdomilebebis gaerTianeba niiA A1 = = . pirobis Tanaxmadp A P = ) (dan A A Pi/ 1 ) | ( = . amitomn p n p A A P A P A A P A Pi i i/ / 1 ) | ( ) ( ) ( ) ( = = = = . magaliTi6.kartebisnakrebidan(romelSic36kartia)SemTxveviT iReben erT karts. ganvixiloT xdomilebebi:A iyos xdomileba, rom amo-Rebuli karti agurisa, xoloB iyos xdomileba, rom amoRebuli karti mefea. arian Tu ara es xdomilebebi damoukidebeli? cxadia, rom am SemTxvevaSi36 | | = O ,4 / 1 36 / 9 ) ( = = A P ,9 / 1 36 / 4 ) ( = = B Pda ) ( ) ( 9 / 1 4 / 1 36 / 1 ) ( B P A P B A P = = = .e.i. es xdomilebebi damoukidebelia. magaliTi 7. davuSvaT, rom vagorebT or saTamaSo kamaTels. ganvixi-loT xdomilebebi:A -- pirvel kamaTelze movida kenti qula,B -- meore 20 kamaTelze movida kenti qula,C -- orive kamaTelze mosul qulaTa jami kentia. gavarkvioT am xdomilebebis damoukideblobis sakiTxi. cxadia,rom2 / 1 6 / 3 ) ( ) ( = = = B P A P ,xolo4 / 1 36 / 3 3 ) ( = = B A P . amitomA daB xdomilebebi damoukideblia. davaleba. SeamowmeT, rom2 / 1 ) ( = C P . SevniSnoT,romAdaBxdomilebidanerT-erTismoxdenis pirobaSiC xdomileba xdeba maSin da mxolod maSin, roca Sesabamisad, anpirvelanmeorekamaTelzemovidaluwiqula,anugvaqvs Tanafardobebi: B A C A = daB A C B = . A daB xdomilebebis damoukideblobidan gamomdinare xdomilebe-biA daB daA daB agreTve damoukideblebia, amitom4 / 1 ) ( ) ( ) ( = = B P A P B A Pda4 / 1 ) ( = B A P .Sesabamisad,4 / 1 ) ( ) ( = = B A P C A Pda4 / 1 ) ( ) ( = = B A P C B P . es Tanafardobebi ki,2 / 1 ) ( = C PalbaTobis gaTvaliswinebiT, niSn-avs, rom damoukideblebiaA da C daB 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. adviliSesamowmebelia,romaqedannebismieriorixdomileba damoukidebelia,xolosamiveerTaddamokidebulia,vinaidanTuCven gvecodinebarommagaliTad, 1A da 2A moxda,maSinCvenzustadviciT, rom 3AagreTve moxda. davaleba. SeamowmeT, rom xdomilebebi 1A , 2Ada 3Awyvil-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 { a dabadebuli studenti uri =i Ai,150 ,..., 2 , 1 = i ; } 09 . 1 { a dabadebuli n studentida 150 mainc erTi = A . naTelia,rom iA xdomilebebierToblivaddamoukidebeliada 365 / 1 ) ( =iA P . garda amisa, iiA A1501 = =da maSasadame, sapovnelia damouki-debelxdomilebaTagaerTianebisalbaToba.gadavideTsawinaaRmdego xdomilebaze da visargebloT de-morganis kanoniT. maSin gvaqvs: 21 ) ( ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) (115011501n iiiiA P A P A P A P A P A P = = = == =. ramdenadac365 / 1 1 ) ( =iA P , Tu visargeblebT niutonis binomis for-muliT == +njj jnnx C x1) 1 ( , miviRebT, rom: + + =4 41503 31502 2150)3651( )3651( )3651(365150) ( C C C A P . vinaidan,41 , 0 365 / 150 ~ , xolo44 4 41501 150 1 (0, 41)( ) ( ) 0.005365 365 24 24C < ~ < , amitommwkrivisniSancvladobisgamo,TugadavagdebTmwkrivis wevrebsdawyebulime-5wevridan(mwkrivebiszogadiTeoriidan gamomdinare), SesaZlebelia vamtkicoT, rom2 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={, , }. SemTxveviTvirCevTerTerTobliobasdaarCeulierTobliobidan erTasos.amorCeulasosZalianmciredroisganmavlobaSivuCvenebT 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 , { = = i i AierToblioba uri amorCeulia ; , , , da iyos xdomileba, rom warmodgenilia Sesa-bamisad , , , da asoebi; } { upasuxa sworad elma damkvirveb = B . amocanis pirobebSi cxadia: 2 / 1 ) ( ) (2 1= = A P A P ; ( 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 1Ada 2Aqmnian xdomilebaTa srul sistemas, amitom TiToe-uliasossworadamocnobisalbaTobaSegviZliagamovTvaloTsruli albaTobis formuliT: ( P )= ) (1A P ( P |1A )+ ) (2A P ( P |2A )=1/8; ( P )= ) (1A P ( P |1A )+ ) (2A P ( P |2A )=1/8; ( P )= ) (1A P ( P |1A )+ ) (2A P ( P |2A )=7/24; ( P )= ) (1A P ( P |1A )+ ) (2A P ( P |2A )=7/24; ( P )= ) (1A P ( P |1A )+ ) (2A P ( P |2A )=1/6. 22 amocanispirobebSicxadiaagreTve,romsworipasuxispirobiTi albaTobebisxvadasxvaasoebiswarmodgenisSemTxvevaSiSesabamisadiq-neba: ( P B|)=( P B|)=( P B|)=0da( P B|)=( P B|)=1. vinaidan, , , , da agreTve xdomilebaTa sruli sist-emaa, amitom sruli albaTobis formula gvaZlevs swori pasuxis albaTo-bas: = ) (B P ( 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 Seadgensx lars da mi-simizaniamiiyvanosesTanxaalaramde.TamaSigrZeldebamanamsanam moTamaSearmiiyvansTavisTanxaswinaswargansazRrulalaramde,an igi ar gakotrdeba (anu waagebs mis xelT arsebul mTelx lars). rogo-ria albaToba imisa, rom moTamaSe gakotrdeba? amoxsna. es albaToba damokidebuli iqneba sawyisx kapitalze. aRv-niSnoTigi) (x p simboloTi.cxadia,romigiganmartebulianebismieri a x s s 0da amasTanave,1 ) 0 ( = Pda0 ) ( = a P . SemoviRoT xdomilebebi: } {1nabijze pirvel moigo moTamaSem = A , } , { gakotrdeba kapitali sawyisi gaaCnia romelsac moTamaSe, x B= . amocanis pirobebSi gvaqvs: 2 / 1 ) ( ) (1 1= = A P A P ,) 1 ( ) | (1+ = x p A B Pda) 1 ( ) | (1 = x p A B P( 1 1 s s a x ). vinaidan, 1Ada 1AxdomilebaTa sruli sistemaa, amitom sruli al-baTobis formula) (x palbaTobisaTvis gvaZlevs Semdeg gantolebas: ) 1 (21) 1 (21) ( + + = x p x p x p ,1 1 s s a x(amtipisgantolebebsmaTematikaSirekurentulgantolebebs uwodeben). SeiZleba Semowmdes, rom am gantolebis amoxsnas aqvs saxe: c bx x p + = ) ( , sadac bdac -- nebismieri mudmivebia. am koeficientebis mosaZebnad unda visargebloTsasazRvropirobebiT1 ) 0 ( = P da0 ) ( = a P .maSinmiviRebT, rom 1 = cda0 = +c ab , saidanac,a b / 1 =da sabolooda x x p / 1 ) ( = ,a x s s 0 . davaleba.davuSvaT, rom Sesamowmebeli jgufis 1%avadmyofia, xo-lodanarCeni99%kijanmrTelia.adamianebisSerCevaxdebaSemTxveviT da amitom( ) 1% 0.01 P = = avadmyofida( ) 99% 0.99 P = = janmrTeli . 23 vigulisxmoT,romimSemTxvevaSi,rocatestirebautardeba adamians,romelsacaraaqvsavadmyofoba,maSin1%-iaalbaTobaimisa, rom miviRoT mcdari dadebiTi Sedegi, e.i. % 1 ) ( = janmrTeli | dadebiTi Pda% 99 ) ( = janmrTeli | uaryofiTi P . dabolos, davuSvaT, rom im SemTxvevaSi, roca testireba utardeba avadmyofadamians,maSin1%-iaalbaTobaimisa,rommiviRoTmcdari uaryofiTi Sedegi, e.i. % 1 ) ( = avadmyofi | uaryofiTi Pda% 99 ) ( = avadmyofi | dadebiTi P . gamoTvaleTalbaTobaimisa,rom:a).adamianijanmrTelia,xolo testmaaCvenauaryofiTiSedegi;b).adamianiavadmyofia,xolotestma aCvenadadebiTiSedegi;g).adamianijanmrTelia,xolotestmaaCvena dadebiTi Sedegi; g). adamiani avadmyofia, xolo testma aCvena uaryofiTi Sedegi.ganvixiloT realuri situacia, romelic gviCvenebs erTi SexedviT moulodnel gansxvavebas) | ( B A Pda) | ( A B PpirobiT albaTobebs Soris. imisaTvis,romgamovavlinoTseriozuliavadmyofobismqoneadamianebi 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 | dadebiTi janmrTeliP P( ) ( ) 0.99 0.01 0.01 0.99 0.0198 P P + = + = avadmyofi dadebiTi| avadmyofi .rogorc cnobilia, magaliTis pirobebSi% 99 ) ( = avadmyofi | dadebiTi P . gamovTvaloT axla Sebrunebuli pirobiTi albaToba, risTvisac vi-sargebloTpirobiTialbaTobisganmartebiTadanamravlisalbaTobis formulebiT. maSin zemoT miRebuli( ) 0.0198 1.98% P = = dadebiTiSedegis Tanaxmad:==(dadebiTi) PdadebiTi avadmyofidadebiTi | avadmyofiPPP) () (( ( ) 1% 99%50%1.98%P P = = =avadmyofi)P dadebiTi| avadmyofiPP1.98%. rogorc vxedavT, pirobiTi albaToba imisa rom testi mogvcems da-debiT Sedegs, pirobaSi rom adamiani avadmyofia tolia 99%-is, maSin ro-desacpirobiTialbaTobaimisaromadamianiavadmyofia,pirobaSirom 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: )} , ( ), . ( ), , ( ), , {(2 2 1 1s g s g A A A A = O , sadac magaliTad,) , (1g A-- niSnavs, rom amoviReT 1Amoneta da misi agdeb-is Sedegad movida gerbi. amocanis pirobebSi gvaqvs: 2 / 1 ) ( ) (2 1= = A P A P ,2 / 1 ) (1= A P | gda3 / 1 ) | (2= A P g . Sesabamisad,namravlisalbaTobisformulisgamoyenebiT gamoviTvliT: 4 / 1 )} , {(1= g A P ,4 / 1 )} , {(1= s A P ,6 / 1 )} , {(2= g A Pda3 / 1 )} , {(2= s A P . amitom baiesis formulis Tanaxmad 53) ( ) ( ) ( ) () ( ) () | (2 2 1 11 11=+=A P A P A P A PA P A PA P| g | g| gg . cxadia, agreTve rom5 / 2 ) | (2= g A P . magaliTi13(keTilgamomcdelzeI).vTqvaT,CvenCasabarebeli gvaqvsgamocdadaSegviZliaavirCioTnebismierisamigamomcdelidan. davuSvaT, CvenTvis cnobilia, rom erTerTi sami gamomcdelidan (ucnobia romeli)--keTiliadaalbaTobaimisa,rommasTanCaabarogamocda tolia 0.4-is, xolo danarCeni ori gamomcdeli avia da maTTan gamocdis Cabarebis albaToba tolia 0.1-is. Cven SemTxveviT avirCieT gamomcdeli da warmatebiTCavabareTgamocda.rogorialabaTobaimisa,romCven avirCieT keTili gamomcdeli? amoxsna. SemoviRoT Semdegi xdomilebebi:A -- amorCeuli gamomcde-li keTilia (maSinA -- iqneba xdomileba, rom amorCeuli gamomcdeli avia) daB -- gamocda Cabarebulia (Sesabamisad,B -- gamocda araa Caba-rebuli). amocanis pirobebSi gvaqvs: 3 / 1 ) ( = A P ,3 / 2 ) ( 1 ) ( = = A P A P ; ( | ) 0.4 P BA= ,( | ) 1 ( | ) 0.6 P BA P BA = = ; ( | ) 0.1 P BA= ,( | ) 1 ( | ) 0.9 P BA P BA = = . cnobilia,rommoxdaBxdomilebadagamosaTvleliapirobiTi albaToba) | ( B A P . vinaidan,A daA xdomilebebi qmnian srul sistemas, baiesis formulis Tanaxmad saZiebeli albaToba iqneba: 32) | ( ) ( ) | ( ) () | ( ) () | ( =+=A B P A P A B P A PA B P A PB A P . 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) (A Pi(Sesabamisad,) (A Pi)simboloTialbaToba (aposterioruli) imisa, rom es gamomcdeli keTilia (Sesabamisad, avia) mas Semdeg rac gamocdil iqnai -uri studenti,3 , 2 , 1 = i . Cven ukve davadgi-neT, rom3 / 2 ) (1= A P . Sesabamisad,3 / 1 ) ( 1 ) (1 1= = A P A P . meoremoswavlisTvalsazrisiTesalbaTobebiwarmoadgenenori SesaZlo hipoTezis apriorul albaTobebs. amitom, baiesis formulis Tan-axmad, meore studentis CaWris Semdeg aposterioruli albaTobebi iqneba: 74) ( ) | ( ) ( ) | () ( ) | () (1 112=+=A P A B P A P A B PA P A B PA P da73) ( 1 ) (2 2= = A P A P . analogiurad,axlamiRebulialbaTobebiukveiqnebaaprioruli albaTobebimesamemoswavlisaTvis,daamitomsaZiebeliaposterioruli albaTobebi, mas Semdeg rac mesame moswavlem ver Caabara gamocda, gamoi-Tvleba isev baiesis formuliT: 178) ( ) | ( ) ( ) | () ( ) | () (2 223=+=A P A B P A P A B PA P A B PA P da ) (179) ( 1 ) (3 3 3A P A P A P > = = . rogorc vxedavT, eqsperimentis (gamocdis) dawyebis win aprioruli albaToba imisa. rom arCeuli gamomcdeli keTilia toli iyo 1/3-is. eqs-perimentebisSemdegamxdomilebisaposteriorulialbaTobagaizarda da gaxda 8/17. miuxedavad amisa, Tu sami eqsperimentis Semdeg misaRebia ga-dawyvetileba am gamomcdelis Sesaxeb, maSin ufro sarwmunoa CavTvaloT igi avad (vinaidan,) ( ) (3 3A P A P > ). magaliTi 15. yuTSi 3 TeTri da 5 Savi burTia. yuTidan SemTxveviTiReben 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 iqneba8 / 3 = p . saZiebeli xdomilebis xelSemwyob uTavsebad SemTxvevebs warmoadgens oTx cdaSi 3 an4TeTriburTisamoReba,romelTaalbaTobebibernulisformulis Tanaxmad Sesabamisad tolia: 1024 / 135 ) 8 / 3 1 ( ) 8 / 3 ( ) 3 (3 4 3 34 4= =C P da 4096 / 81 ) 8 / 5 ( ) 8 / 3 ( ) 4 (0 4 44 4= = C P . 26 Sesabamisad,saZiebelialbaTobaalbaTobaTaSekrebiskanonis Tanaxmad iqneba: 4096 / 621 4096 / 81 1024 / 135 = + . magaliTi 16. davuSvaT vamowmebT defeqturobaze saqonlis partias, romelic Sedgeba30nawarmisagan. 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 ricxvia30 = n , xolo albaT-obis klasikuri ganmartebis Tanaxmad5/100 0.05 p = =(Sesabamisad,0.95 q = ). Sesabamisad, gvaqvs: 0 0 30 0 3030 30(0) 0.05 0.95 0.95 0.2146 P C= = = , garda amisa, 30 30 30 3030 30 0.05 30( 1) ( ) ( ) ( )1 1 0.95 19( 1)kp k kP k P k P k P kk q k k + = = =+ + +, saidanac roca0 k = : 3030 0(0 1) 0.2146 0.338919 (0 1)P+ = = +; roca1 k = : 3030 1(1 1) 0.3389 0.258619 (1 1)P+ = = + da a. S. sabolood gveqneba Semdegi cxrili: defeqturi nawa- rmis ricxvi k albaToba( )nPkkumulatiuri albaToba( ) n P k00.21460.2146 10.33890.5535 20.25860.8122 30.12700.9392 40.04510.9844 50.01240.9967 60.00270.9994 70.00050.9999 80.00010.999998 90.0000010.999999 am cxrilis Sesabamisi albaTobebis ganawilebis grafiki iqneba: 27 00.050.10.150.20.250.30.350.40 2 4 6 8 10 kumulatiuri albaTobebis Sesabamisi ganawilebis grafiki iqneba: 00.20.40.60.811.20 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? savarjiSo2.ramdeniSemTxveviTicifriundaaviRoT,romcifri 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, maSinA yuTidan SemTxveviT iReben 1 burTs, xolo winaaRmdeg Sem-TxvevaSiByuTidanSemTxveviTiReben1burTs.ipoveTalbaTobaimisa, rom: a). amoRebuli burTi mwvanea; b). burTi amoRebuli iyoA yuTidan, Tu cnobilia, rom amoRebuli burTi mwvanea. 2.( ) 3/ 4 P A= ,( | ) 1/ 5 P BA= ,( | ) 4/ 7 P BA= . ipoveT: a).( ) P A B ; b).( ) P B ; g). ( | ) P AB .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-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/ 25 P A= ,( ) 9/ 25 P B= ,( | ) 5/ 9 P AB= . ipoveT: a).( ) P A B ; b).( | ) P BA ; g).( ) P A B ; d).( | ) P AB ; e).( ) P A B .5. 12 waxnagian wesier mravalwaxnagaze aRniSnulia ricxvebi 1, 2, . . . , 12. mocemulia 9 danayofiani firfita, romlis centrSi devs moneta. LR agdeben mravalwaxnagas: Tu movida 2, 3, 5, 7 an 11 qula, maSin monetas gad-aadgileben TiTo danayofze R-is mimarTulebiT (marjvniv), xolo winaaRm-degSemTxvevaSiL-ismimarTulebiT(marcxniv).TamaSimTavrdeba,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-armoedanzeigebsalbaTobiT0.9.TuIgundisyvelamoTamaSeararis 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. CanTaSidevs25diski, romelTagan nawili TeTriada 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.( ) 0.6 P A= ,( ) 0.8 P B= ,( | ) 0.45 P BA= ,( ) 0.28 P B C= . ipoveT: a).( ) P A B ; b).( | ) P CB ; g).( | ) P AB .10.( ) 0.75 P A= ,( | ) 0.8 P BA= ,( | ) 0.6 P BA= . ipoveT( ) P Bda( | ) P AB .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 BA ; b).( | ) P BA ; g).( | ) P BA ; d).) | ( A B P ; e) ( ) P B .12. albaToba imisa, rom momavali wlis ivnisis pirveli dRe iqneba mSraliaris0.4.TuivnisisromelimekonkretulidRemSralia,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 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 caciastudentebidan1uWersmxarsargentinas,xolo10kibrazilias. 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.( ) 0.3 P A= .BxdomilebadamoukidebeliaAxdomilebisagan. ( ) 0.4 P B= .CarisxdomilebaromarcAmoxdadaarcB.ipoveT:a). ( \ ) P AB ; b).( ) P A B ; g).) | ( A C P . 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.ipoveTalbaTobaimisa,romSemTxveviTSerCeulistudenti:a). swavlobs orives: mTematikas da mecnierebas; b). swavlobs maTematikas, ma-gram ar swavlobs mecnierebas.18.( | ) 1/ 5 P DC= ,( | ) 1/ 4 P CD= ,( ) P C D p = . ipoveT: a).( ) P C ; b).( ) P D .19.( | ) 1/ 5 P DC= ,( | ) 1/ 4 P CD= ,( ) 1/ 5 P C D= , ( ) P C D p = . ipoveTp . 20. stadionis 40 bileTidan 10 aris CrdiloeTis mxare, 14 aRmosavl-eTis da 16 ki dasavleTis mxare. SemTxveviT iReben erT bileTs da aZlev-enX -s (adamians saxeladX ). Semdgom, SemTxveviT iReben meore bileTs da aZleven Y -s da analogiurad mesame bileTs aZlevenZ -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-naanmotocikleti;e).anammoigomanqanapirobaSi,romekammoigo manqana an motocikleti.30 22. ( ) A B=C ,( | ) 1/ 3 P AB= ,( ) 6/ 7 P 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-Tobaimisa,romdinamomoigebs;b).gasuloTxSabaTsdinamommatCi 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.TuromelimedRemSralia,maSinalbaTobaimisa,rommomdevno 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,romelTaganIdaIIIdReiyosveli.ipoveTalbaTobaimisa,romII dRe iyo mSrali.26. kazinoSi dgas oriA 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 airCieTA magida; g). Tqven SemTxveviT airCieT magida da waag-eT. ras udris albaToba imisa, rom Tqven airCieT 'A magida. 27.kazinoSidgas samisaTamaSomagidaAda 'Atipis,romlebic sruliad erTnairad gamoiyureba. amasTanave, 'Atipismagida ori calia. A tipis magidaze mogebis albaToba aris 1/3, xolo 'Atipis magidaze 1/4. a). ipoveT SemTxveviT arCeul magidaze mogebis albaToba; b). Tqven Se-mTxveviT airCieT magida da moigeT. ras udris albaToba imisa, rom Tqven airCieTA 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 gvancaadre 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 erTnairiA,B da C yuTi Sedgeba or-ori ujrisagan.A yu-TisTiToeulujraSidevsTiTolari,ByuTisTiTeulujraSidevs 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.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. albaTobaimisa,romismiuswrebs8saaTianmatarebelsorSabaTsaris 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 albaTobaimisa,romzafxulisSemTxveviTarCeuldResbaTumSi iwvimebs,Tucnobilia,romamdRismaqsimaluritemperatura25 gradusze metia.34.sakarnavaloTamaSSimonawilejeragdebsmonetas,xolo SemdegisvriskamaTels.isigebsprizsimSemTxvevaSi,Tumonetaze 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.vipovoTalbaTobaimisa,romkamaTlisoTxjeragdebisas samjer mova oTxi qula. 37.A daB damoukidebelia.( ) 0.4 P A= ,( ) 0.7 P A B= . ipoveT:( ) P B .38. agdeben erT wesier kamaTels da meore iseT kamaTels, romelzec 6qulismosvlisalbaTobaa1/4.ipoveTalbaTobaimisa,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.agdebenerTwesiermonetasdameoreiseTmonetas,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 daB damoukidebeli xdomilebebia.( ) 0.4 P A= ,( ) 0.88 P A B= . ip-oveT: a).( ) P B ;b). albaToba imisa, rom romelimeA daBxdomilebebidan 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. 32 42. albaToba imisa, rom oTx damoukidebel cdSiA xdomileba erT-jer mainc moxdeba aris 15/16. ipoveT calkeul cdaSiA 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 erTSiki 5 wiTeli da 1mwvane 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. 33 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: xix1 x2 xn pip1 p2 pn aq1ii p =. albaTobis statistikuri ganmartebidan gamomdinare: Teoriuli sixSire~ erToblivi sixSire albaToba. ,( , ) ( )x a bP a b P x e( )e( ) = =. Tu mocemulia diskretuli tipis SemTxve-viTi sidide da raime ricxviTig funqcia, maSin( ) g isev iqneba disk-retuli tipis SemTxveviTi sidide ganawilebis kanoniT: ( )igx1( ) gx2( ) gx ( )ngx

ip 1p2pnphipergeometriuliganawileba.davuSvaT,romyuTSiN burTiada maT SorisM TeTria. SemTxveviT, dabrunebis gareSe yuTidan viRebTn burTs. albaToba imisa, rom amoRebuln burTs Soris zustadkcali iqneba TeTri gamoiTvleba formuliT: ( ; ; ; ) : ( )m n mM N Mn nNC CP NMn m P mC= = = ,0,1,..., min( , ) m Mn = . ricxvTa am mimdevrobas hipergeometriuli ganawileba ewodeba. geometriuli ganawileba:1( )kP k pq = =( 1 q p = ),1, 2,... k = . puasonis ganawileba parametriT :{ } , 0,1, 2,...!kP k e kk= = = . SemTxveviTi sididis ganawilebis funqcia:( ) : ( ) Fx P x = , maSin 0 / 2/ 200 / 2( ) 0 cos 0 sin | 1xFx du udu du xttt = + + = =} } }.sabolood gvaqvs: 0, 0,( ) sin , 0 / 2,0, / 2.xFx x xxtts = < s> amocanebi 1. SemTxveviTi sididis ganawilebis simkvrivea:1(1 ), 0 8;( ) 80,c x xf x s s=Tusxvagan. ipoveT: a).c mudmivis mniSvneloba; b).{ 6} P > ; g).{4 6} P s s .2. SemTxveviTi sididis ganawilebis simkvrivea:2, 0 3;( )0,cx xf x s s=Tusxvagan. ipoveT: a).c mudmivismniSvneloba; b).{ 2} P s ; g).{1.5 2.5} P s s ; d).x-is mniSvneloba, Tu cnobilia, rom{ } 0.2 P x s = .3. SemTxveviTi sididis ganawilebis simkvrivea:2, 0 3;( )0,cx xf x s s=Tusxvagan. ipoveT: a).c mudmivis mniSvneloba; b).{ 1.5} P s .4. SemTxveviTi sididis ganawilebis simkvrivea:2(4 ), 2 2;( )0,c x xf x s s=Tusxvagan. ipoveT: a).c mudmivis mniSvneloba; b).{ 0} P > ; g).{ 1} P > ; d).{| | 1} P > ; e). { 0.5 0.5} P s s .5. kompiuteris kartrijis muSaobis xangrZlivobaa saaTi. Sem-TxveviTi sididis ganawilebis simkvrivea: >=. , 0; 400 ,) (2sxvaganTu x cxx f 67 gamoTvaleTc mudmivis mniSvneloba. ipoveT albaToba imisa, rom: a). ka-rtriji imuSavebs sul cota 500 saaTi; b). kartriji Sesacvleli gax-deba manam sanam is imuSavebs600 saaTi; g). ori kartriji Sesacvleli iqneba manam sanam TiToeuli imuSavebs 600-600 saaTi; d). oTxi kartriji-dan ori imuSavebs 600 saaTze mets, xolo ori 600 saaTze naklebs.6. SemTxveviTi sididis ganawilebis simkvrivea:2( ) , 0 ;( )0,c x a x af x s s=Tusxvagan. ipoveT: a).a da c mudmivis mniSvnelobebi, Tu33= c a ; b).{ / 2} P a > .7. SemTxveviTi sididis ganawilebis simkvrivea:21, 0 3;( ) 90,x xf xs s=Tusxvagan. ipoveT SemTxveviTi sididis: a). mediana; b). qveda da zeda kvartilebi.8. kompiuteris kartrijis saRebavis muSaobis xangrZlivoba , ga-zomili saaTebSi, modelirdeba albaTuri ganawilebis simkvriviT:2400 , 400;( )0,x xf x >=Tusxvagan. ipoveT kartrijis muSaobis xangrZlivobis mediana.9. SemTxveviTi sididis ganawilebis simkvrivea:2 4, 2 3;( )0,x xf x s s =Tusxvagan. a). daxazeT simkvrivis grafiki; b). ipoveTSemTxveviTi sididis mediana da moda; g). ipoveT kvartilTSorisi gabnevis diapazoni. 10. SemTxveviTi sididis ganawilebis simkvrivea:2 1(1 ), 0 5;( ) 5 50,x xf x s s=Tusxvagan. a). daxazeT simkvrivis grafiki; b). ipoveTSemTxveviTi sididis mediana da modaluri mniSvneloba.11. SemTxveviTi sididis ganawilebis simkvrivea:3(2 ), 0 2;( ) 40,x x xf x s s=Tusxvagan. ipoveT SemTxveviTi sididis modaluri mniSvneloba.12. SemTxveviTi sididis ganawilebis simkvrivea:2(1 ), 0 1;( )0,cx x xf x s s=Tusxvagan. 68 a). ipoveTc; b). gamoTvaleT moda; g). dawereT gantoleba, romelsac akma-yofilebs medianaM.13. SemTxveviTi sididis ganawilebis simkvrivea:21, 0 3;( ) 90,x xf xs s=Tusxvagan. ipoveT saSualo da dispersia.14. SemTxveviTi sididis ganawilebis simkvrivea:2 4, 2 3;( )0,x xf x s s =Tusxvagan. ipoveT saSualo da dispersia.15. SemTxveviTi sididis ganawilebis simkvrivea:1 1(1 ), 0 8;( ) 4 80,x xf x s s=Tusxvagan. a). daxazeT simkvrivis grafiki; b). ipoveT saSualo da dispersia.16. sawarmos mier gamoSvebuli silikonis masa , gazomili kilogr-amebSi, modelirdeba ganawilebis simkvriviT:23(4 ), 0 4;( ) 320,x x xf x s s=Tusxvagan. a).daxazeTsimkvrivisgrafiki;b).ipoveTsilikonismasissaSualoda dispersia.17. elementebis muSaobis xangrZlivoba , gazomili saaTebSi, mode-lirdeba ganawilebis simkvriviT:43000 , 10;( )0,x xf x >=Tusxvagan. a).daxazeTsimkvrivisgrafiki;b).ipoveTelementebismuSaobis xangrZlivobis saSualo da dispersia.18.kompiuteriskartrijismuSaobisxangrZlivoba ,gazomili saaTebSi, modelirdeba albaTuri ganawilebis simkvriviT:2, 400 900;( )0,cx xf x s s=Tusxvagan. a). gamoTvaleTc-s mniSvneloba; b). daxazeT simkvrivis grafiki; g). ipoveT kartrijis muSaobis xangrZlivobis saSualo da dispersia.19. SemTxveviTi sididis ganawilebis funqciaa: 0, 2,( ) 0.5 , 2 4,1, 4.xFx x xxs = < s> 69 ipoveT albaToba imisa, romSemTxveviTi sidide miiRebs mniSvnelobas, romelic: a). naklebia 0.2-ze; b). naklebia 3-ze; g). ar aris naklebi 3-ze; d). ar aris naklebi 5-ze.20. SemTxveviTi sididis ganawilebis funqciaa: 20, 0,( ) , 0 1,1, 1.xFx x xxs = < s> ipoveT albaToba imisa, rom 4 damokidebeli dakvirvebis Sedegad SemT-xveviTi sidide 3-jer miiRebs mniSvnelobas (0.25, 0.75) Sualedidan. 21. ipoveT ganawilebis simkvrive, Tu ganawilebis funqciaa: 0, 0,( ) sin , 0 / 2,1, / 2.xFx x xxtts = < s> 22. ipoveT ganawilebis simkvrive, Tu ganawilebis funqciaa: 0, 0,( ) sin2 , 0 / 4,1, / 4.xFx x xxtts = < s> 23. ipoveT ganawilebis funqcia, Tu ganawilebis simkvrivea:sin , 0 / 2,( )0,x xf xt < s =sxvagan. 24. ipoveT ganawilebis funqcia, Tu ganawilebis simkvrivea: 1/ 2, 1 2,( )0,x xf x < s =sxvagan. 25. ipoveT [2, 8] monakveTze Tanabrad ganawilebuli SemTxveviTi sid-idis ganawilebis simkvrive da standartuli gadaxra. pasuxebi: 1. 1/4; 1/16; 3/16. 2. 1/9; 8/27; 147/324; 1.75. 3. 1/15; 11/40. 4. 3/32; 1/2; 5/32; 5/16; 47/128. 5. 400; 4/5; 1/3; 1/9; 8/27. 6. 3; 1/9; 1/8. 7. 2.38; 2.73; 1.89. 8. 800. 9. --; 12 22+ , 2; 1( 3 1)2 . 10. --; 55 22 ,0 . 11. 1. 12. 12; 2/3; 4 36 8 1 0 M M + = . 13. 9/4, 27/80. 14. 8/3, 1/18. 15. --; 8/3, 32/9. 16. --; 2, 4/5. 17. --; 15, 75. 18. 720; --; 584, 19100. 19. 0; 0.5; 0.5;0.20.0.25(4(4, 0.5) : (3) 0.25 Bi P = ).21. (0, / 2]cos ( ) x xt_ .22. (0, / 4]2cos2 ( ) x xt_ .23. ( ) 0, 0; 1 cos , 0 / 2; 1, / 2 Fx x x x x t t = s < s > .24.( ) 0, 1 Fx x = s ; 21/ 2( ) x x , 1 2 x < s ; 1, 2 x > . 25. [2,8]1( )6x _ ;3 . 70 Tavi VI normaluri ganawileba SemTxveviTsididesewodebanormaluridaaRiniSnebasimboloTi 2( , ) N a o , Tu mis ganawilebis simkvrives (Sesabamisad, ganawilebis funqci-as) aqvs saxe:222( )2( , )1( )2x aNaf x eooto= (Sesabamisad, 222( )2( , )1( )2t a xNaF x e dtooto=}).standartuli normaluri ganawilebis ( (0,1) N ) simkvrive (Sesabamis-ad, ganawilebis funqcia) aRiniSneba simboloTi: 22(0,1)1( ) : ( )2xNx f x e t= =(Sesabamisad, 22(0,1)1( ) : ( )2x tNx F x e dttu = =}; gardaamisa, ixmareba aRniSvna: 22001( ) :2x tx e dttu =}).cxadia, rom:( ) ( ) x x = ;( ) 1 ( ) x x u = u ; 0( ) 0.5 ( ) x x u = +u ,0 x > ; 2( , )(0,1)N a aNoo= ;{ (0,1) , } ( ) ( ) P N c d d c e( ) =u u ; 2{ ( , ) , } ( ) ( )d a c aP N a c d oo o e( ) =u u ; 2, 0.1 ap px x aoo = + ; 2,0.1appx axoo= , sadac2, apxoaris 2( , ) N a o -isp -kvantili. Tu5 . 0 s o ,maSinoo = u 5 . 0 ) (0z ;xoloTu5 . 0 > o ,maSin 5 . 0 ) (0 = u ooz , sadac oz standartuli normaluri ganawilebis zedao -kritikuli wertilia. magaliTebis amoxsnis nimuSebi: magaliTi 1. vipovoT albaToba imisa, rom standartuli normaluri SemTxveviTi sidide naklebia 1.18-ze,( (0,1) 1.18) P N < ? amoxsna. davxazoT standartuli normaluri ganawilebis simkvrivis wiri da abscisTa RerZze avRniSnoT 1.18-is Sesabamisi wertili (am SemTxv-evaSi1.18 z = ). 71 normaluriganawilebiscxrilispirvelsvetSimovZebnoTricxvi 1.1, xolo pirvel striqonSi ki ricxvi .08.1.1-is Sesabamisi striqonisa da .08-is Sesabamisi svetis gadakveTaze vpoulobT ricxvs 0.8810. amitom saZiebeli albaToba iqneba ( (0,1) 1.18) 0.8810 P N < = anu 88.10%. magaliTi 2. vipovoT albaToba imisa, rom standartuli normaluri SemTxveviTi sidide naklebia -0.63-ze,( (0,1) 0.63) P N ? amoxsna. am SemTxvevaSi1.48 z =da gamosaTvlelia normaluri gana-wilebis wiris qveS -1.48-is marjvniv moTavsebuli aris farTobi. Tu visargeblebT normaluri ganawilebis simetriulobiTa da( ) z ufunqciis cxrilebiT, miviRebT: ( (0,1) 1.48) ( (0,1) 1.48) (1.48) 0.9306 P N P N > = < =u = . SeniSvna. SegviZlia visargebloT sawinaaRmdego xdomilebis albaT-obis gamosaTvleli formuliT da maSin magaliTi daiyvneba wina magali-Tze: ( (0,1) 1.48) 1 ( (0,1) 1.48) 1 ( (0,1) 1.48) 1 0.0694 0.9306 P N P N P N > = s = =u ,saidanac gvaqvs: 61.6451.3t > . aqedan advilad miviRebT, rom8.1385 8.14 t > ~ . maSasadame, nika ar unda dabrundes 8.14 wuTze adre. magaliTi 11. biologi agrovebs monacemebs konkretuli saxeobis ka-qtusis simaRlis Sesaxeb. misi dakvirvebiT kaqtusebis 34.25%-is sigrZe 12 sm-zenaklebia,xolo18.4%-issigrZe16sm-zemetia.biologmadauSva, rom simaRle ganawilebulia normalurad. vipovoT am ganawilebis saSua-lo da standartuli gadaxra. amoxsna.kaqtusissimaRleavRniSnoTH , 2( , ) H N a o ~ .mosaZebniaa da o . biologis dakvirvebis Tanaxmad:{ 12} 0.342 P H < =da{ 16} 0.184 P H > = . Sesabamisad,12( ) 0.342aou =da 16( ) 1 0.184 0.816aou = = . 76 amitom 120.407ao=da 160.900ao= . saidanac vaskvniT, rom:13.2 a =da3.06 o = . magaliTi 12. supermarketSi dRis ganmavlobaSi gayiduli citruseb-is raodenoba modelirdeba normaluri ganawilebiT. aRmoCnda, rom xangr-Zlivi periodis manZilze dReSi saSualod iyideboda 35 kg. citrusi, xo-lo 15 kg-ze naklebi gayidul iqna saSualod yoveli 20 dRidan erT dRe-Si. a). gamoTvaleT gayidvebis standartuli gadaxra o ; b). cnobilia, rom konkretul dRes gaiyida 53 kg-ze meti citrusi. ipoveT albaToba imisa, rom am dRes gaiyida 56 kg-ze meti citrusi. amoxsna. dReSi gayiduli citrusebis raodenoba aRvniSnoT -Ti. ma-Sin: 2(35, ) N o ~da 1{ 15} 0.0520P < = = .a).15 35{ 15} 0.05 ( ) 0.05 P o< = u = 15 351.6448 12.2 oo = = ; b). { 56 53} { 56}{ 56| 53}{ 53} { 53}P PPP P > > >> > = = => > 56 351 ( )1 0.957412.20.60853 351 0.92991 ( )12.2u= = =u. amocanebi 1. cnobilia, rom(0,1) Z N ~ . normaluri ganawilebis funqciis cxri-lisgamoyenebiTipoveTalbaTobebi:a).{ 1.23} P Z < ;b).{ 2.47} P Z s ;g). { 0.16} P Z < ;d).{ 1.24} P Z > ;e).{ 2.38} P Z > ;v).{ 0.59} P Z > ;z).{ 1.83} P Z > ;T). { 2.06} P Z s ; i).{ 0.07} P Z > ; k).{ 1.83} P Z s ; l).{ 2.76} P Z < ; m).{ 0.21} P Z s .2. cnobilia, rom(0,1) Z N ~ . ipoveT albaTobebi: a).{1.15 1.35} P Z < < ; b). {1.11 2.22} P Z s s ;g).{0.39 2.42} P Z < < ;d).{0 1.55} P Z s < ;e).{ 1.82 2.33} P Z < < ; v).{ 0.85 2.03} P Z < s ;z).{ 2.51 1.09} P Z < < ;T).{ 0.55 0} P Z s s ;i). { 2.82 1.82} P Z < = ;z).{ } 0.01 P Z u > = ;T).{ } 0.22 P Z v > = ;i). { } 0.99 P Z s > = ; k).{ } 0.97 P Z t > = ; l).{ } 0.85 P Z u > = ; m).{ } 0.5 P Z v > = .4. mocemulia(20,16) N ~ . ipoveT Semdegi albaTobebi: a).{ 26} P s ; b). { 30} P > ; g).{ 17} P > ; d).{ 13} P < .77 5. mocemulia(24,9) N ~ . ipoveT Semdegi albaTobebi: a).{ 29} P s ; b). { 31} P > ; g).{ 22} P > ; d).{ 16} P < .6.mocemulia(50,16) N ~ .ipoveTSemdegialbaTobebi:a). {54 58} P s s ;b).{40 44} P < s ;g).{47 57} P < < ;d).{39 53} P s < ;e). {44 56} P s s .7. SemTxveviTisidideganawilebulianormaluradsaSualoTi3 da dispersiiT 4. ipoveT albaToba imisa, rom miiRebs uaryofiT mniSvn-elobas.8.SemTxveviTi sidide ganawilebulia normalurad saSualoTia ( 0) a >da dispersiiT 2/ 4 a . a). ipoveT{ 1.5 } P a > ; b). ipoveT albaToba imi-sa, rom uaryofiTia.9.mocemulia(44, 25) N ~ .ipoveTSesabamisads ,t ,uanv,Tu:a). { } 0.98 P s s = ; b).{ } 0.77 P t > = ; g).{ } 0.05 P u > = ; d).{ } 0.33 P v s = .10. mocemulia(15, 4) N ~ . ipoveT Sesabamisads ,t , u,v anw, Tu: a). { } 0.91 P s s = ;b).{ } 0.57 P t > = ;g).{ } 0.10 P u s = ;d).{ } 0.39 P v s = ;e). {15 15 } 0.9 P w w < < + = .11. mocemulia(35.4,12.5) N ~ . ipoveT Sesabamisads , t ,u anv, Tu: a). { } 0.96 P s < = ; b).{ } 0.94 P t > = ; g).{ } 0.29 P u > = ; d).{ } 0.15 P v < = .12. SemTxveviTi sidide ganawilebulia normalurad saSualoTi 32 da dispersiiT 2o . albaToba imisa, rom naklebia 33-ze aris 0.64. ipoveT 2o .13. SemTxveviTi sidide ganawilebulia normalurad dispersiiT 18 da{ 73} 0.03 P > = . ipoveT saSualo.14. SemTxveviTi sidide ganawilebulia normalurad,{ 59} 0.02 P > =da{ 29} 0.93 P > = . ipoveTSemTxveviTi sididis saSualo da standartu-li gadaxra.15. mocemulia 2( , ) N a o ~ ,{ 9.81} 0.16 P > =da{ 8.82} 0.01 P s = . ipoveT a da o .16.afTiaqSiwamlisdamzadebazedaxarjulidroganawilebulia normalurad saSualoTi 15 wuTi da standartuli gadaxriT 2.8 wuTi. ipo-veT albaToba imisa, rom wamlis damzadebaze daxarjuli dro: a). 20 wuT-ze metia; b). 8 wuTze naklebia; g). moTavsebulia 10 wuTsa da 18 wuTs Sor-is.17. jgufSi 16 wlis gogonebis simaRle ganawilebulia normalurad saSualoTi 161.2 sm da standartuli gadaxriT 4.7 sm. ipoveT albaToba im-isa, rom am jgufidan erTi gogonas simaRle: a). 165 sm-ze metia; b). 150 sm-ze naklebia; g). moTavsebulia 165 sm-sa da 170 sm-s Soris; d). moTavsebu-lia 150 sm-sa da 163 sm-s Soris. 16 wlis 500 SerCeuli gogonasaTvis Seaf-aseT raodenoba im gogonebis, romelTa simaRle gava zemoT moyvanili 4 diapazonidan.18. avtomobilis SuSis sawmendi rezinis sigrZe ganawilebulia nor-malurad saSualoT 25 sm da standartuli gadaxriT 0.2 sm. SeafaseT 200 78 cali SuSis sawmendidan ramdenis sigrZes unda movelodeT, rom iqneba: a). 25.3 sm an meti; b). moTavsebuli 24.89 sm-sa da 25.11 sm-s Soris; g). moTavseb-uli 24.89 sm-sa da 25.25 sm-s Soris.19. avtomobilis gacveTili samuxruWe xundis Secvlis dro ganawi-lebulia normalurad saSualoTi 90 wuTi da standartuli gadaxriT 5.8 wuTi. ipoveT albaToba imisa, rom xundis Secvlas dasWirdeba: a). 105 wu-Tze meti; b). 85 wuTze naklebi. daadgineT saSualos mimarT simetriuli ( , ) a bintervali, romelSic moxvdeba xundis Secvlis dro 90%-iani saime-doobiT.20. kompaniis mier warmoebuli dRis ganaTebis muSaobis xangrZlivo-baganawilebulianormaluradsaSualoTi2010saaTidastandartuli gadaxriT 20 saaTi. kompaniam gadawyvita gazardos gayidvebis ricxvi, ris-Tvisacman valdebulebaaiRo sagarantio drois gasvlamde gafuWebuli nebismieridRisganaTebaufasodSecvalosaxliT.vipovoTissagaran-tio dro, romlis dawesebis SemTxvevaSi kompanias mouwevs ufasod Secva-los dRis ganaTebis mxolod 3%.21. yvavilis foTolis sigrZe ganawilebulia normalurad saSualo-Ti18.2smdastandartuligadaxriT2.3sm.a).ipoveTalbaTobaimisa, rom yvavilis foTlis sigrZe moTavsebulia 16 sm-sa da 20 sm-s Soris; b). yvavilis foTlebis 12%hsm-ze grZelia, xolo 20%lsm-ze moklea. ipo-veTh dal ; g). SeafaseT yvavilis 500 foTlidan ramdeni iqneba 14 sm-ze mokle.22. boTls, romelSic asxia ualkoholo sasmeli, aweria 330 ml. sin-amdvileSiCamosxmuliualkoholosasmelismoculobaganawilebulia normalurad standartuli gadaxriT 2.5 ml. ramdeni unda iyos Camosxmu-liualkoholosasmelismoculobissaSualo,raTadarwmunebuliviy-oT, rom boTlebis sul cota 99%-Si sasmeli meti iqneba 330 ml-ze. 23. fuTas, romelSic iyideba Saqari, aqvs warwera 1 kg. Saqari. fa-qtiurad,fuTaSimoTavsebuliSaqriswonaganawilebulianormalurad saSualoTi 1.08 kg. fuTebis Semowmebam aCvena, rom fuTebis 2.5% naklulia (Seicavs miTiTebul 1 kg-ze nakleb Saqars). a). ipoveT am ganawilebis sta-ndartuligadaxra;b).TumocemulifuTanaklulia,maSingamoTvaleT albaToba imisa, rom misi wona saSualoze 3 standartuli gadaxriT nak-lebia.24. generatoris muSaobis xangrZlivoba ganawilebulia normalurad saSualoTi210sT.aRmoCnda,romgeneratorebis4%muSaobs222saaTze mets. ipoveT am ganawilebis dispersia. 25.albaTobisgamocdazestudentebis15%Rebulobs63qulaze mets, xolo 10% -- 32 qulaze naklebs. vigulisxmoT, rom qulebi ganawil-ebulia normalurad da vipovoT qulebis saSualo da standartuli gada-xra.26. mziani saaTebis raodenoba TveSiHgarkveul kurortze ganawi-lebulia normalurad saSualoTi 130 sT. cnobilia, rom{ 179} 0.975 P H < = . a).gamoTvaleTH -isstandartuligadaxra;b).gamoTvaleT {100 150} P H < < .79 amocanebi gameorebisaTvis 27. uwyveti SemTxveviTi sididis ganawilebis simkvrivea: (4 ), 0 4,( )0, .k x xf x s s =Tusxvagan gamoTvaleT: a).k ; b).{ 2.5} P > ; g).E . 28. uwyveti SemTxveviTi sididis ganawilebis simkvrivea: 1, 0 2,( ) 20, .x xf xs s=Tusxvagan ipoveT: a).SemTxveviTi sididis mediana da qveda da zeda kvartili; b). E ; g).D .29. uwyveti USemTxveviTi sidide Tanabradaa ganawilebuli [0.5, 2.5] segmentze. ipoveT am SemTxveviTi sididis: a). ganawilebis simkvrive; b). ma-Tematikuri lodini; g). dispersia.30. xis naWrebis sigrZe modelirdeba uwyveti Tanabari ganawilebiT, romlis simkvrivea: 1, 0.2 0.8,( )0, .Uxf x ks s=Tusxvagan ipoveT: a).k ; b).EU ; g).DU . d). SemTxveviT SeirCa sami naWeri. risi to-lia albaToba imisa, rom maTgan mxolod oris sigrZe iqneba 0.6-ze meti. 31. uwyveti SemTxveviTi sididis ganawilebis simkvrivea: 3, 0 2,( )0, .kx xf x s s=Tusxvagan ipoveT: a).k ; b).EU ; g).DU ; d). ganawilebis mediana; e). albaToba imisa, rom dakvirveba moTavsebuli iqneba saSualodan erTi standartuli gada-xris farglebSi; v). albaToba imisa, rom 4 dakvirvebidan 2 iqneba saSual-oze meti da 2 saSualoze naklebi. 32. uwyveti SemTxveviTi sididis ganawilebis simkvrivea: ( )(2 ), 0 2,( )0, .x a a x xf x s s =Tusxvagan ipoveT: a).a; b).E .33. cnobilia, rom(10,8) N ~ . ipoveT{ 6} P > .34. cnobilia, rom(140,56.25) N ~ . ipoveT albaToba imisa, romme-tia 128.75-ze.35. garkveuli modelis avtomobili 56 mili/sT siCqariT moZraobis-as erTi galoni benziniT gadis manZils, romlis saSualo mniSvnelobaa 32.4 mili da standartuli gadaxra 1.4 mili. CavTvaleT, rom saqme gvaqvs normalur ganawilebasTan da gamoTvaleT albaToba imisa, rom es avtom-80 obili56mili/sTsiCqariTmoZraobisaserTigalonibenziniTgaivlis 30 milze mets.36. normalurad ganawilebuliSemTxveviTi sididis saSualoa 20, xolo dispersia 4.15. ipoveT albaToba imisa, rom 18 21 < < .37. farTobi, romlis SeRebvac SeiZleba 1 litri bunebrivi saRebav-iT ganawilebulia normalurad saSualoTi 13.2 kv. metri da standartuli gadaxriT 0.197 kv. metri. farTobi, romlis SeRebvac SeiZleba 1 litri xe-lovnuri saRebaviT ganawilebulia normalurad saSualoTi 13.4 kv. metri da standartuli gadaxriT 0.343 kv. metri. SesaRebia 12.9 kv. metri farTo-bi.gansazRvreT,romelisaRebavis1litriiqnebasakmarisiufrometi albaTobiT am farTobis SesaRebad.38. kvercxis wona , gazomili gramebSi, ganawilebulia normalur-ad:(85, 0.36) N ~ . kvercxebi iyofa sam kategoriad: didi, saSualo da pata-ra. didi kvercxis wona metia an toli 90 gramze. kvercxebis 25% pataraa. gamoTvaleT:a).kvercxebisramdeniprocentiadidi;b).patarakvercxis maqsimaluri wona.39. SemTxveviTi sidide ganawilebulia normalurad saSualoTia da dispersiiTa. cnobilia, rom{ 3} 0.2 P < = . ipoveT a.40. ori firma,A daB, awarmoebs gamoZaxebis SeCerebis mowyobile-bas sakomunikacio organizaciisaTvis. gamoZaxebis SeCerebis dro izomeba nanowamebSi (nwm). a). SeCerebis droA firmis mier warmoebuli SeCerebis mowyobilebisaTvis modelirdeba normaluri ganawilebiT saSualoTi 283 nwm da standartuli gadaxriT 8 nwm. ipoveT albaToba imisa, romA fir-mis mowyobilebisaTvis SeCerebis dro moTavsebulia 275 nwm-sa da 286 nwm-s Soris; b).B firmis mowyobilebisaTvis SeCerebis dro isev igulisxme-ba, rom normaluria. aq SeCerebis drois 10% naklebia 274.6 nwm-ze da 7.5% metia 288.2 nwm-ze. vipovoT am ganawilebis saSualo da standartuli gad-axra.41. SemTxveviTi sidide ganawilebulia normalurad saSualoTia da dispersiiT 2o . mocemulia, rom{ 81.89} 0.01 P > =da{ 27.77} 0.1 P < = . ip-oveTa da 2o .42. cnobilia, rom( , 4) N a ~ . ipoveTa, Tu cnobilia, rom Sesabamisi ganawilebis simkvrivis wiris wves moqceuli aris farTobi 12-dan+-mde aris 0.8.43. danadgari Wris grZel milebs patar-patara milebad. patara mil-is sigrZe ganawilebulia normalurad saSualoTia sm da standartuli gadaxriT 0.25 sm.a-s mniSvnelobis Secvla SesaZlebelia danadgaris reg-ulirebis xarjze. a). ipoveTa, romlisTvisac albaToba imisa, rom SemT-xveviT SerCeuli patara milis sigrZe naklebia 6.5 sm-ze aris 0.1; b). dana-dgari daregulirda ise, rom6.4 a = , xolo standartuli gadaxra ar Sec-vlila.ipoveTalbaTobaimisa,romSemTxveviTSerCeulipataramilis sigrZe moTavsebulia 6.3 sm-sa da 6.6 sm-s Soris. 81 amocanebi gamocdisaTvis 44. SemTxveviTisidideganawilebulianormaluradsaSualoTi 250 da dispersiiT 50. ipoveT misi simkvrivis wiris qveS 180-dan 280-mde mo-Tavsebuli aris farTobi.45. garkveuli saxeobis Citis wona ganawilebulia normalurad saS-ualoTi 0.8 kg da standartuli gadaxriT 0.12 kg. ipoveT albaToba imisa, rom SemTxveviT SerCeuli am saxeobis Citis wona moTavsebuli iqneba 0.74 kg-sa da 0.95 kg-s Soris.46. SemTxveviTisidideganawilebulianormaluradda { 10} 0.67 P s = ,{ 12} 0.937 P s = . ipoveTE . 47. SemTxveviTi sididis ganawilebis simkvrivea: 2, 0 2,( )0, .kx xf x s s=Tusxvagan a). aCveneT, rom3/ 8 k = ; b). gamoTvaleTE ; g). ipoveT SemTxveviTi sidi-dis mediana.48. saqmiani qali dRis ganmavlobaSi telefonze laparaks andomebs TsaaTs. TSemTxveviTi sididis ganawilebis simkvrivea: 31(8 ), 0 2,( ) 120, .Tx x xf x s s=Tusxvagan ipoveT SemTxveviTisididis:a).maTematikurilodini;b).mediana;g). moda.49. uwyveti SemTxveviTi sididis ganawilebis simkvrivea: 21(4 ), 0 2,( ) 40, .x x xf x s s=Tusxvagan ipoveT SemTxveviTi sididis mediana.50. uwyveti SemTxveviTi sididis ganawilebis simkvrivea: 2(2 ), 0 2,( )0, .c x x xf x s s=Tusxvagan ipoveT: a).c mudmivis mniSvneloba; b). albaToba{0.25 0.5} P s s .51. uwyveti SemTxveviTi sididis ganawilebis funqciaa: 0, 0,( ) sin , 0 / 2,1, / 2.xF x x xxtts = s s>TuTuTu ipoveT SemTxveviTi sididis: a). ganawilebis simkvrive: b). mediana.82 52.uwyveti SemTxveviTisididisganawilebissimkvrivea: ( ) sin f x x= , Tu0 / 2 x t s s , =0 sxvagan. ipoveT SemTxveviTi sididis media-na.53. SemTxveviTi sidide ganawilebulia normalurad saSualoTia dadispersiiT 2o .cnobilia,rom{ 6.2} 0.9474 P > = da{ 9.8} 0.6368 P < = . ipoveTa da 2o .54. uwyvetiSemTxveviTi sididis ganawilebis simkvrive ganmarteb-ulia [0, ] aintervalze TanafardobiT: 2/ 2, 0 3,( )27/ 8 , 3 .x xf xx x as s =< sTuTu ipoveTa-s mniSvneloba.55. birTvis srolis SejibrebaSi monawileobs ori aTleti, ivane da erekle.TiToeuliaTletisaTvisbirTvissrolismanZiliganawilebu-lia normalurad. ivanesaTvis gasul wels birvTis srolis manZilis saS-ualo iyo 60.33 m, xolo standartuli gadaxra 1.95 m. a). gasul wels ivan-esaTvis birvTis srolis manZilis 80% meti iyo vidrex metri. ipoveTx; b). gasul wels ereklesaTvis birvTis srolis manZilis 80% meti iyo vid-re 59.5 metri. cnobilia, rom gasul wels ereklesaTvis birvTis srolis manZilis saSualo iyo 59.39 m. ipoveT misi standartuli gadaxra; mimdina-re wels ereklesaTvis birvTis srolis manZilis saSualoa 59.5 m, xolo standartuli gadaxra 3 m. ivanesaTvis Sesabamisi parametrebia 60.33 m da 1.95 m. Sejibrebis dros, Semdeg turSi gasasvlelad aTletma birTvi unda isrolos sul cota 65 metrze. pirvel turSi TiToeuli aTleti birTvs isvris 3-jer; g). gansazRvreT romeli aTletis gasvlaa ufro mosalodne-li Semdeg turSi pirveli srolis Sedegis mixedviT; d). ipoveT albaToba imisa, rom orive aTleti gava Semdeg turSi.56. SemTxveviTi sidide ganawilebulia normalurad saSualoTia da dispersiiT 2o , ise, rom{ 50.32} 0.119 P > =da{ 43.56} 0.305 P > = . ipoveT a da 2oda gamoTvaleT{| | 5} P a < .57. sawarmo yidulobs WanWikebis 44%-sA firmidan, xolo danarCens B firmidan. TiToeuli firmis mier damzadebuli WanWikebis diametri ga-nawilebulia normalurad standartuli gadaxriT 0.16 mm.A firmis mier damzadebuli WanWikebis diametris saSualoa 1.56 mm.B firmis mier damz-adebuli WanWikebis 24.2%-is diametri naklebia 1.52 mm-ze. a). ipoveTB fi-rmis mier damzadebuli WanWikebis diametris saSualo; b). ipoveT albaTo-ba imisa, rom sawarmos sawyobidan SemTxveviT SerCeuli WanWikis diamet-ri naklebi iqneba 1.52 mm-ze; g). SemTxveviT SerCeuli WanWikis diametri na-klebia 1.52 mm-ze. vipovoT albaTobaimisa, rom es WanWiki damzadebulia B firmaSi; d).B firma dReSi amzadebs 8000 WanWiks. TiTeuli WanWiki iZ-leva 1.5 larian mogebas, Tu misi diametri moTavsebulia 1.52 mm-sa da 1.83 mm-sSoris.WanWiki,romlisdiametrinaklebia1.52mm-zeundagadagdon 83 da iZleva 0.85 larian danakargs. dabolos, WanWiki, romlis diametri me-tia 1.83 mm-ze iyideba, magram mogebas amcirebs 0.5 laramde. gamoTvaleTB firmis mosalodneli mogeba.58. albaToba imisa, rom mSeneblobaze miitanen samSeneblo masalebs ( ) 0.85 P A= ,albaTobaimisa,rommSeneblobadasruldebadroulad 17 / 11 ) ( = B P . albaToba imisa, rom samSeneblo masalebs miitanen da mSeneb-loba droulad dasruldeba aris 0.55. a). aCveneT, romA daB xdomile-bebidamoukidebelia;b).cnobilia,romsamSeneblomasalebidroulad miitanesmSeneblobaze.ipoveTalbaTobaimisa,rommSeneblobaar dasruldeba droulad.59. mSeneblobaze muSaobs 10 kaciani jgufi, romelTa Soris 3 eleq-trikiada2santeqnikosi.arqiteqtormadaibarasaTaTbirod5muSada movida SemTxveviT SerCeuli 5 maTgani. a). ipoveT albaToba imisa, rom am xuTSi moxvdeba 2 eleqtriki da 1 santeqnikosi; b). mSeneblobaze kviraSi muSaobis saaTebis raodenoba ganawilebulia normalurad saSualoTi 42 sT. jgufis 10% muSaobs kviraSi 48 an met saaTs. ipoveT albaToba imisa, rom orive santeqnikosi konkretul kviraSi imauSavebs 40 saaTze mets. pasuxebi: 1.0.8907;0.9932;0.5636;0.1075;0.0087;0.2776;0.9664;0.0197;0.5279;0.0336;0.0029; 0.4168.2.0.0366;0.1203;0.3405;0.4394;0.9557;0.7816;0.8561;0.2088;0.0320;0.1186; 0.4472; 0.9500. 3. 0.4399; 1.1750; 2.0537; 1.0364; 0.2275; 1.1750; 2.3263; 0.7722; -2. 3263; -1.8808; -1.0364; 0. 4. 0.9332; 0.0062; 0.7734; 0.0401. 5. 0.9522; 0.0098; 0.7475; 0.0038. 6. 0.1359; 0.0606; 0.7333; 0.7704; 0.8664. 7. 0.0668. 8. 0.1587; 0.0228. 9. 54.27; 40.31; 52.22; 41.80. 10. 17.68; 14.65; 12.44; 14.44; 3.29. 11. 41.6; 29.9; 37.4; 31.7. 12. 7.78. 13. 65.0. 14. 41.5;8.50.15.9.51;0.298.16.0.0371;0.0062;0.8209.17.0.2094;0.0086;0.1788;0.6405; 105; 4; 89; 320. 18. 13; 84; 121. 19. 0.0049; 0.1943; (80, 100). 20. 1972 sT. 21. 0.614; 20.9, 16.3; 17. 22. 336 ml. 23. 0.041 kg; 0.054. 24. 47. 25. 49.1; 13.4. 26. 25; 0.673. 27. 1/8; 0.141; 4/3. 28.2 , 1,3 ; 4/3; 2/9. 29.( ) 1/ 2Uf x= , Tu[0.5, 2.5] xe , da =0 sxvagan; 3/2; 1/3. 30. 0.6; 0.5; 0.03; 2/9. 31. 1/4; 1.6; 0.107; 1.68; 0.697; 0.351. 32. 36 a = ; 2.73. 33. 0.921. 34. 0.933. 35. 0.957. 36. 0.525. 37. bunebrivi: 0.936 > 0.928. 38. 20.2%; 81 gr. 39. 18.94. 40.0.488; 281, 5.41.47; 15. 42.10.3.43. 6.82; 0.444. 44. 0.645.45.0.5858. 46. 9.193. 47. 3/2; 2/32 . 48. 56/45; 1.294; 1.633. 49. 1.082. 50. 3/4; 0.1133. 51.( ) cos f x x= , Tu 0 / 2 x t s s ,=0sxvagan;/ 6 t .52./ 3 t .53.9.161;1.827.54.54/11.55.58.69;3.41; ereklesaTvis(0.0334>0.0083);0.119.56.38.41;10.09;0.3798.57.1.632;0.312;0.775; 6587. 58. 6/17. 59. 5/21; 0.44 84 Tavi VII albaTobis Teoriis zRvariTi Teoremebi CebiSevis utoloba. CebiSevisutolobaafasebsSemTxveviTisididisgadaxrasTavisi maTematikurilodinidan.Tu raimeSemTxveviTisididea,maSin nebismieri0 c >ricxvisaTvis samarTliania utoloba: 2(| | ) 1 / P E D c c < > . did ricxvTa kanoni.CebiSevisTeorema.vTqvaT,SemTxveviTisidideebi 1 2, ,... wyvil-wyviladdamoukidebelia(iEricxvisaTvis samarTliania utoloba 2(1 ){| | }m p pP pn ncc > s( 0 , rocan). erTTan ragind axlos myofi albaTobiT SeiZleba vamtkicoT, romdamoukidebel cdaTa sakmaod didi ricxvisaTvis dakvirvebadi xdomileb- is moxdenaTa sixSire ragind umniSvnelod gansxvavdeba misi moxdenis al-baTobisagan calkeul cdaSi. centraluri zRvariTi Teorema. did ricxvTa kanoni ar ikvlevs SemTxveviT sidideTa jamis ganawi-lebis kanonis saxes. es sakiTxi Seiswavleba Teoremebis jgufSi, romleb-saccentralurizRvariTiTeoremaewodeba.esTeoremaamtkicebs,rom SemTxveviT sidideTa jamis ganawilebis kanoni, romelTag