omar furTuxiaT s u3sarCevialbaTobis Teoria$0. Sesavali ........................................................................................................................................... 5$1. albaTobis Teoriis sagani ........................................................................................... 13$2. elementarul xdomilebaTa sivrce ...................................................................... 16$3. operaciebi xdomilebeze ................................................................................................. 20$4. albaTobis ganmarteba ....................................................................................................... 24$5. geometriuli albaToba .................................................................................................... 28$6. kombinatorikis elementebi .......................................................................................... 31$7. albaTobis gamoTvla kombinatorikis gamoyenebiT ............................. 34$8. jamisa da sxvaobis albaTobis formulebi ................................................ 39$9. pirobiTi albaTobis formula ................................................................................ 41$10. namravlis albaTobis formula ........................................................................... 45$11. damokidebuli da damoukidebeli xdomilebebi .................................... 48$12. sruli albaTobis formula ..................................................................................... 52$13. baiesis formula ................................................................................................................... 56$14. ganmeorebiTi cdebi. bernulis formula .................................................... 59$15. puasonis formula ............................................................................................................... 65$16. SemTxveviTi sidide. ganawilebis kanoni ...................................................... 66$17. SemTxveviTi sididis ganawilebis funqcia da simkvrive ............ 72$18. organzomilebiani SemTxveviTi sidide .......................................................... 79$19. SemTxveviTi sididis maTematikuri lodini .............................................. 83$20. SemTxveviT sidideTa damoukidebloba .......................................................... 91$21. SemTxveviTi sididis dispersia ............................................................................. 94$22. standartuli gadaxra. momentebi ........................................................................ 102$23. kovariacia. korelaciis koeficienti ............................................................. 105$24. CebiSevis utoloba. did ricxvTa kanoni .................................................... 111$25. normaluri ganawileba ................................................................................................... 116$26. centraluri zRvariTi Teorema ............................................................................ 1204maTematikuri statistika$27. maTematikuri statistikis ZiriTadi cnebebi .......................................... 124$28. empiriuli ganawilebis funqcia ........................................................................... 128$29. xi kvadrat, stiudentisa da fiSeris ganawilebebi ......................... 130$30. generaluri erTobliobis parametrebis wertilovaniSefasebebi .................................................................................................................................... 137$31. SerCeviTi parametrebis ganawileba normaluripopulaciisaTvis ................................................................................................................... 141$32. SefasebaTa agebis meTodebi ...................................................................................... 143$33. intervaluri Sefasebebi. ndobis intervali maTematikurilodinisaTvis ........................................................................................................................... 145$34. ndobis intervali dispersiisaTvis da standartuligadaxrisaTvis .......................................................................................................................... 149$35. ndobis intervali bernulis sqemaSi ............................................................... 152$36. hipoTezaTa statistikuri Semowmebis amocanebi ................................. 154$37. hipoTezis Semowmeba lodinis Sesaxeb ........................................................... 159$38. hipoTezis Semowmeba dispersiebis tolobis Sesaxeb ...................... 164$39. hipoTezis Semowmeba SerCeviTi korelaciis koeficientisstatistikuri mniSvnelovnebis Sesaxeb ......................................................... 166$40. hipoTezaTa Semowmeba bernulis sqemaSi ..................................................... 167$41. Tanxmobis kriteriumebi. xi kvadrat kriteriumi ................................ 169$42. kolmogorov-smirnovis kriteriumi .................................................................. 173$43. damoukideblobis hipoTezis Semowmeba ......................................................... 175$44. erTgvarovnebis hipoTezis Semowmeba .............................................................. 179$45. SemTxveviT sidideTa modelireba. monte-karlosmeTodi ............................................................................................................................................. 182danarTi (statistikuri cxrilebi) .............................................................................. 186literatura ......................................................................................................................................... 1985$0. Sesavali albaTobaSiadamianis yoveldRiur cxovrebaSi sityva albaToba xSirad gamo-iyeneba ama Tu im xdomilebis moxdenis an ar moxdenis damajereblobisxarisxis cvlilebis gamosaxatavad, rac garkveuli azriT dakavSirebu-lia Cvens subieqtur survilebTan. aseTia magaliTad, Semdegi Sinaarsismtkicebulebi: xval albaT gamoidarebs, Tvis bolosaTvis lari alb-aT gamyardeba, or weliwadSi saqarTvelo albaT natos wevri gaxdebada a. S. im xdomilebebs Soris, romelTac Cven vaxasiaTebT rogorc nak-lebad albaTuri,an rogorc sakmarisad albaTuri, an rogorc ZalianalbaTuri, SeiZleba ganvasxvavoT sami kategoria: isini, romlebic Seexe-ba Cvens sakuTar qcevas; isini, romlebic Seexeba sxva adamianebis qcevas,da isini, romlebic Seexeba bunebis movlenebs. amasTanave, am sam katego-rias Soris sxvaoba zogjer ganusazRvrelia.rocamevambob:savsebiTalbaTuria,rommexvaldiliTgavalsaxlidan,mezogadformaSivajamebmTelrigmetadTunaklebadrTulmsjelobebs,romelTagannawiliSeexebasxvadasxvaadamianisqcevas, xolo danarCeni bunebis movlenebs. magaliTad, me gadavwyvitegavide saxlidan, Tu ki is Cemi megobari, romelmac me gamafrTxila, romSesaZloa movides Cems sanaxavad, ar mova CemTan; an me gadavwyvite gavi-de saxlidan, Tu ki ar mova Tovli (im SemTxvevaSi, roca dekemberia), --an Tu ar iqneba setyva (im SemTxvevaSi, roca ivlisia). rogorc erT, isemeoreSemTxvevaSi CemisaxlidangasvlisalbaTobadamokidebuliaimalbatobaze, romelsac me mivawer meore piris qcevas an ama Tu im mete-orologiur movlenas. garda amisa, gasaTvaliswinebelia is albaTobac,rasackargijanmrTelobismqoneadamianebiugulebelyofen,rodesacisinigegmavenraimemoklevadianqmedebas:meverSevZlebsaxlidangavide, Tu me Zalian avad gavxdebi, da miT umetes, Tu movkvdebi.maTematikaSi ki sityva albaToba gamoiyeneba mkacrad gansazRru-li azriT, romelsac aranairi kavSiri ara aqvs damajereblobasTan, damiT umetes subieqtur survilebTan. albaTobis Teoria warmoadgens mec-nierebasSemTxveviTobisSesaxeb.misisaSualebiTaRiwerebasamyarosmravali movlena da situacia. jer kidev Soreul warsulSi pirvelyof-ili tomis beladma icoda, rom 10 monadires gacilebiT meti albaTob-iT SeuZlia isari moartyas irems, vidre erT monadires. amitomac, isinikoleqtiurad nadirobdnen. araswori iqneboda gvefiqra, rom morigi omi-saTvis mzadebis procesSi, aleqsandre makedoneli an sxva romelime gam-oCenili mxedarTmTavari mxolod meomarTa mamacobaze da sabrZolo xe-lovnebaze amyarebda imeds. eWvgareSea, rom dakvirvebebisa da samxedroxelmZRvanelobis gamocdilebis safuZvelze maT SeeZloT rogoRac See-fasebinaT TavianTi gamarjvebis an damarcxebis albaToba, icodnen ro-disundaCabmuliyvnenomSidarodisundaaeridebinaTmisTvisTavi.cxadia, rom isini ar iyvnen SemTxveviTobis monebi, magram imavdrouladZalian Sors idgnen albaTobis Teoriisagan.6mogvianebiT,gamocdilebisdagrovebasTanerTad,adamianmasulufro xSirad daiwyo SemTxveviTi movlenebis dakvirvebebisa da cdeb-is (eqsperimentebis) dagegmva, maTi Sedegebis klasificireba, rogorc Se-uZlebeli, SesaZlebeli da aucilebeli Sedegebi. adamianma SeamCnia, romSemTxveviTobebs arc Tu ise iSviaTad safuZvlad udevs (warmarTavs) ob-ieqturi kanonzomierebebi. ganvixiloT umartivesi cda monetis agdeba.gerbis an safasuris mosvla, cxadia SemTxveviTi movlenaa. magram monet-is mravaljeradi agdebisas SesaZlebelia SevamCnioT, rom gerbis mosv-la xdeba daaxloebiT cdaTa ricxvis naxevarjer (me-18 saukuneSi bunebi-smetyvelma biufonma moneta aagdo 4040-jer, saidanac gerbi movida 2048-jer;me-20saukunisdasawyisSimaTematikosmapirsonma monetaaagdo24000-jerdagerbimovida12012-jer). masasadame, miuxedavadimisa,rommonetis calkeuli agdebis Sedegi SemTxveviTi xdomilebaa, monetis mra-valjeradi agdebis Sedegebi obieqtur kanons emorCileba.ganvixiloT meore magaliTi eqsperimenti e. w. galtonis dafiT.gvaqvsvertikalurdafazesamkuTxedisformiTdamagrebulirgolebi,ise rom wveroSi erTi rgolia, meore striqonSi winasgan Tanabr manZil-ebze ori rgoli, mesame striqonSi zeda ori rgolidan Tanabar manZile-bze sami rgoli da a.S. pirveli rgolis Tavze dgas rezervuari, saidan-ac vardebian burTulebi qveviT da grovdebian rgolebis boloSi moTa-vsebul marTkuTxedebSi. TiToeul burTulas morig rgolze dacemisasSeuZlia SemTxveviT gadavardes marjvniv an marcxniv da aRmoCndes bol-oSi ganTavsebul nebismier marTkuTxedSi. aRmoCnda, rom eqsperimentidaneqsperimentamde meordeba burTulebis simetriuli ganlageba marTkuTxe-dbSi,romlisdrosaccentralurmarTkuTxedebSiburTulebibevria,xolo ganapira marTkuTxedebSi cota. es damajereblad miuTiTebs bu-rTulebis ganawilebis obieqturi kanonis arsebobis Sesaxeb. roca bur-Tulebi bevria, maSin amboben, rom isini ganawilebulia normaluri kan-onis mixedviT.amrigad, SeiZleba iTqvas, rom SemTxveviToba SeiZleba emorCileb-odes SedarebiT martiv da SedarebiT rTul kanonzomierebas. magram, ib-adeba kiTxva, sad aris aq maTematika da maTematikuri amocanebi?SeiZleba iTqvas, rom albaTobis Teoris ganviTareba daiwyo azar-tuli TamaSebis dros warmoSobili amocanebidan, Tumca misi safuZvle-bis formirebas xeli Seuwyo demografiul monacemebSi aRmoCenilma kan-onzomierebebma (axalSobelTa statistikis, sikvdilianobis statistiki-sa da ubedur SemTxvevaTa statistikis Seswavlam), rac Tavis mxriv, efe-qturad gamoiyeneboda sadazRvevo kompaniebis saqmianobaSi. mogvianebiT,obieqturi kanonzomierebebi aRmoCenul iqna SemTxveviTi movlenebis Ses-wavlisas adamianis moRvaweobis yvela sferoSi. bunebrivia daviwyoT ma-rtivi amocanebis ganxilviT.SuasaukuneebisbolomdeZvlebiTTamaSiyvelazepopularuliazartuliTamaSiiyo.TviTonsityvaazartiasevedakavSirebuliaZvlebiT TamaSTan, ramdenadac is modis arabuli sityvidan alzar, rome-liciTargmnebarogorc saTamaSoZvali. sityvaazararabulad7agreTve niSnavsrTuls.arabebiazartulTamaSsuwodebdnenqulebisiseT kombinacias, romelic SeiZleba gamoCndes erTaderTi gziT, ramod-enime saTamaSo kamaTlis gagorebisas. magaliTad, ori kamaTlis gagore-bisas rTulad (azar) iTvleboda jamSi ori an Tormeti qulis mosvla.aRsaniSnavia, rom sityva hasard frangulad niSnavs SemTxveviTobas, xo-lo jeu de hasard ki azartul TamaSs.miuxedavadimisa,romdResalbaTobisTeoriasimdenivesaerToaqvs azartul TamaSebTan, ramdenic geometrias farTobebis gazomvasTanmiwis samuSaoebis dros, albaTobis Teoriis pirveli paradoqsebi dakav-Sirebulia swored popularul azartul TamaSebTan. 1494 wels italie-lma maTematikosma l. paColim (1445-1514) gamoaqveyna naSromi, romelSicixilavda Semdeg situacias: ori tolZalovani moTamaSe SeTanxmda eTam-aSaT garkveuli TamaSi, manam sanam erTi maTgani ar moigebda npartias.am SemTxvevaSi is iRebda garkveul Tanxas (prizs). magram TamaSi Sewydamas Semdeg rac pirvelma moTamaSem moigo k( k n ), xolo meorem -- m( m n ) partia. mosagebi Tanxis rogori ganawileba iqneba samarTliani?am amocanas SemdgomSi ewoda prizis ganawilebis paradoqsi. miuxedavadimisa, rom sinamdvileSi es amocana ar warmoadgens paradoqss, zogierTiudidesi mecnieris mier am amocanis amoxsnis warumatebelma mcdelobam,da arasworma urTierTsawinaaRmdego pasuxebma warmoqmnes legenda para-doqsis Sesaxeb.TviTon paColim swori amoxsna ver ipova. igi ver xedavda am amoc-anis kavSirs albaTobus TeoriasTan da ixilavda mas rogorc amocanasproporciebze. amitom is Tvlida, rom Tanxa unda ganawilebuliyo prop-orciiT : k m, ar iTvaliswinebda ra partiaTa im raodenobas, romelicunda moigos calkeulma moTamaSem, raTa miiRos mliani Tanxa. arasworiamoxsna ekuTvnis nikolo tartaliasac (1499-1557), miuxedavad imisa, romis iyo sakmarisad genialuri, raTa maTematikur duelSi erTi Ramis gan-mavlobaSi epova kuburi gantolebis amoxsnis formula. 50 wlis Semdeg,meore italielma maTematikosma d. kardanom (1501-1576), samarTlianad ga-akritika paCiolis msjeloba, magram samwuxarod TviTonac mogvca mcda-ri amoxsna. gavida kidev 100-ze meti weli, da mxolod 1654 wels, aRniSn-uli amocana amoxsnil iqna gamoCenili frangi maTematikosebis b. paska-lisa (1623-1662) da p. fermas (1601-1665) mimoweris procesSi erTmaneTisag-andamoukideblad.esaRmoCenaiyoimdenadmniSvnelovani,rombevriTvlis am wels albaTobis Teoriis dabadebis wlad, xolo yvela adrin-del Sedegs winaistoriad. SevxedoT rogor xsnida paskali amocanas,roca 3 n , 2 k da 1 m . paskali da ferma ganixilavdnen am problemasrogorc amocanas albaTobebze. amitom samarTliani iqneba iseTi gayofa,romelic proporciulia TiToeuli moTamaSis mier prizis mogebis Sans-is.davuSvaT,romTamaSiSewyda,rocapirvelmoTamaSesmogebuliaqvs ori partia, xolo meores erTi. jer-jerobiT ucnobia rogor gav-anawiloT Tanxa, magram yvelaferi gamartivdeboda, Tu isini iTamaSebdn-8enkideverTpartias.sinamdvileSi,amSemTxvevaSiSesaZlebeliaoriSedegi:I. Tu am partias moigebs pirveli moTamaSe, maSin mas daugrovdebamogebaTa SeTanxmebuli ricxvi da miiRebs mTlian Tanxas;II. Tu partias moigebs meore moTamaSe, maSin orive eqneba mogebaTaTanabari raodenoba da samarTliani iqneboda Tanxis Tanabrad gayofa.TiToeul am Sedegs moxdenis Tanabari SesaZlebloba aqvs.amrigad,pirvelmoTamaSesSeuZliamoigosanmTeliTanxaanTanxis naxevari, anu saSualod mas SeuZlia moigos Tanxis11322 4nawili. meore moTamaSis SesaZleblobebi ufro mwiria: man SeiZleba anaraferi moigos an Tanxis naxevari moigos, anu igi saSualod igebs Ta-xis10122 4nawils. amitom Tanxa unda ganawildes proporciiT 3:1 (da ara 2:1, rog-orc Tvlida paColi).1718wels londonSigamovidafrangimaTematikosisa.muavris(1667-1754)wignisaxelwodebiT --swavlebaSemTxveviTobaze,romlismTavarimiRwevaaimkanonzomierebisdadgena,romelicZalianxSiradSeimCneva SemTxveviT movlenebSi. muavrma pirvelma aRmoaCina da Teoriu-laddaasabuTanormaluriganawilebisroli(gaixseneTgaltonisdafa). muavrma gazoma 1375 SemTxveviT SerCeuli qalis simaRle. gazomvisSedegebi moyvanilia diagramaze:050100150200250300140145150155160165170175180simaRle santimetrebSiqalTa raodenobazaris msgavsi wiri, romelic daaxloebiT edeba simaRleTa gana-wilebis diagramas, axlosaa2/ 212xy e9funqciis grafikTan, romelsac aqvs Semdegi saxe:normaluri ganawilebis kanons aqvs udidesi praqtikuli mniSvnel-oba. aRmoCnda, rom am kanoniTaa ganawilebuli gazis molekulebis siCqa-re,axalSobilebiswona, gazomviscdomilebaTasidide,damravalisxva fizikuri da biologiuri bunebis mqone SemTxveviTi sidide. aRsani-Snavia, rom normaluri ganawilebis kanoni iZleva Zalian karg miaxloe-bas yovelTvis, roca gansaxilveli sidide warmoadgens bevri damoukid-ebelikomponentiserToblivimoqmedebisSedegsdaamasTanave,jamurefeqtSi TiToeuli komponentis wvlili SedarebiT mcirea.davubrundeTisevprizisganawilebisparadoqss.imSemTxvevaSi,roca 6 n , 5 k da 3 m , paColis pasuxi iyo, rom prizi unda ganawil-ebuliyo mogebuli partiebis proporciulad, anu 5:3-ze. tartalia Tvli-da, rom ganawileba unda momxdariyo 2:1-Tan proporciiT (savaraudod ismsjelobdaSemdegnairad:vinaidan,pirvelmamoTamaSemmoigomeorezeori partiiT meti, rac Seadgens mogebisaTvis aucilebeli 6 partiis me-sameds, amitom pirvelma moTamaSemunda miiRos prizis mesamedi, xolodarCenili nawili 2/3 ganawildes Tanabrad. e. i. pirvelma unda miiRos1 2 2: 23 3 3 , xolo meorem --13, anu gayofa unda moxdes 2:1-ze). vaCvenoT,rom im dros rodesac mogebisaTvis pirvel moTamaSes esaWiroeba mxol-od erTi partiis mogeba, meores ki sami partiisa, prizis samarTlianigayofaa 7:1-ze.fermas ideis Tanaxmad, gavagrZeloT TamaSi sami fiqtiuri parti-iT, im SemTxvevaSic ki roca zogierTi maTgani aRmoCndeba sruliad ze-dmeti (anu, roca erT-erT moTamaSes ukve mogebuli aqvs TamaSi). aseTigagrZelebisasSesaZlebelia 222 8 erTnairadmosalodneliSedegi:mmm, mmw, mwm, wmm, mww, wmw, wwm da www (sadac i -ur adgilzemdgomi m, Sesabamisad, w aRniSnavs, rom i -uri partia moigo, Sesabami-sad, waago pirvelma moTamaSem, 1, 2, 3 i ). vinaidan, mxolod erT SemTxvev-aSiRebulobsmeoremoTamaSeprizs(rocaismoigebssamivepartias),xolo danarCen 7 SemTxvevaSi igebs pirveli moTamaSe, amitom samarTli-ania gayofa 7:1-ze.am amocanis zogad SemTxvevaSi amoxsna agreTve ekuTvniT paskalsadafermas.1654welsmTeliparizilaparakobda axalimecnierebis albaTobis Teoriis warmoSobaze. fermas brwyinvale idea TamaSis gagr-10ZelebisSesaxeb1977welsgamoiyenaandersonma.mandaamtkicaSemdegimniSvnelovani Teorema: moTamaSes, romelic arigebs pirveli, aqvs erTida igive Sansi moigos NpartiaSi Tavis mowinaaRmdegeze ufro adre mi-uxedavad imisa, moTamaSeebi arigeben monacvleobiT, Tu arigebs is vincmoigo wina partia.xdomilebis albaToba ganmartebuli iyo laplasis mier Semdegnai-rad:( )mPAn ,sadac n -- TanabradSesaZlebel xdomilebaTa (SedegTa) saerTo raoden-obaa, xolo m -- im xdomilebaTa raodenoba, romlis drosac xdeba Axdomileba (xelSemwyobi Sedegebis raodenoba). magaliTad, vTqvaT, gam-osaTvlelia albaToba xdomilebis A -- ori saTamaSo kamaTlis gagor-ebisas mosul qulaTa jamia 8. ori kamaTlis gagorebisas SesaZlebeliamiviRoT Semdegi TanabradSesaZlebeli Sedegebi:(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)(6,1) (6,2) (6,3) (6,4) (6,5) (6,6).rogorc vxedavT, sul SesaZlebeli variantebia 36. calke gamoy-ofilia is variantebi, roca moxda A xdomileba. aseTi SemTxvevebia 5,dayvelaisiniTanabradSesaZlebelia.amitomlaplasisganmartebisTanaxmad( ) 5/ 36 PA .bunebrivad ibadeba kiTxva: rodis da romeli SemTxveviTi xdomil-ebebi SeiZleba CaiTvalos TanabradSesaZleblad? ganvixiloT dalamber-is cnobili Secdoma. cnobili frangi maTematikosi da filosofosi J.dalamberi(1717-1783),monetisorjeradiagdebisamocanisganxilvisasTvlida,romgerbisadasafasurismosvlisSansi(albaToba)iyo1/3.sinamdvileSi es aris 1/2. dalamberis Secdomam Tavis droze bevri kama-Ti gamoiwvia da amitom gaxda cnobili. es Secdoma Zalian Wkuis saswav-lebelia. dalamberi msjelobda Semdegnairad. arsebobs am eqsperimentissami SesaZlo Sedegi: 1). movida ori gerbi, 2). movida gerbi da safasuri,3). movida ori safasuri. Tvlida ra dalamberi am Sedegebs tolSesaZl-eblad, askvnida, rom gerbisa da safasuris mosvlis albaTobaa 1/3.cnobili maTematikosis Secdoma mdgomareobda imis daSvebaSi, romaRniSnuli sami Sedegi erTnairad SesaZlebelia. sinamdvileSi ki es asear aris. gerbis mosvla avRniSnoT asoTi g, xolo safasuris asoTi s.maSin monetis orjeradi agdebisas erTnairad SesaZlebeli Sedegebia:gg, gs, sg, ss.am Sedegebis raodenoba oTxia (e. i. 4 n ). aqedan movida gerbi da safa-suri xdomilebas xels uwyobs ori Sedegi: gs da sg, anu 2 m . Sesabam-isad, gerbisa da safasuris mosvlis albaTobaa: 2/4=1/2.albaTobis Teoriis Seswavlis dawyebisas bevri, iseve rogorc od-ezRac dalamberi, uSvebs Secdomas, Tvlis ra tolalbaTurad iseT Sed-egebs, romlebic sinamdvileSi aseTebi ar arian. magaliTad, monetis sam-11jer agdebisas Secdomaa imis daSveba, rom erTnairad SesaZlebelia oT-xi Sedegi: 1). movida sami gerbi, 2). movida ori gerbi da erTi safasuri,3). movida erTi gerbi da ori safasuri, da 4). movida sami safasuri. sin-amdvileSi ki, erTnairadSesaZlebelia Semdegi 8 Sedegi:ggg, ggs, gsg, sgg, gss, sgs, ssg, sss.SevadaroT CamoTvlili oTxi xdomilebis swori da araswori albaTobe-bi:araswori albaToba swori albaTobamovida 3 gerbi 1/4 1/8movida 2 gerbi da 1 safasuri 1/4 3/8movida 1 gerbi da 2 safasuri 1/4 3/8movida 3 safasuri 1/4 1/8saTamaSo kamaTlis paradoqsi. ori saTamaSo kamaTlis gagorebisasmosul qulaTa jami moTavsebulia 2-sa da 12-s Soris. rogorc 9, ise 10SesaZlebelia miviRoT ori sxvadasxva gziT: 9=3+6=4+5 da 10=4+6=5+5. sa-misaTamaSokamaTlisgagorebisaski9da10miiRebaeqvsisxvadasxvagziT. maSin riTi aixsneba is garemoeba, rom ori kamaTlis gagorebisasufro xSirad modis 9, xolo sami kamaTlis gagorebisas ki 10?amocana imdenad martivia, rom Zalian gasakviria, rom Tavis drozeis iTvleboda Zalian rTulad. rogorc kardano, ise galilei aRniSnav-dnen, rom aucilebelia qulaTa mosvlis rigis gaTvaliswineba (winaaRm-degSemTxvevaSiyvelaSedegiariqnebodaTanabradSesaZlebeli). aseTSemTxvevaSi ori kamaTlis gagorebisas 9 da 10 Sesabamisad miiRebian Se-mdegnairad: 9=3+6=6+3=4+5=5+4 da 10=4+6=6+4=5+5. es imas niSnavs, rom orikamaTlis SemTxvevaSi 9 SeiZleba gagordes 4 gziT, xolo 10 3 gziT.Sesabamisad, 9-is miRebis Sansebi metia, vidre 10-is. sami kamaTlis SemTx-vevaSisituaciaicvlebasapirispirod:9SeiZlebamiRebuliqnes25sxvadasxva gziT, xolo 10 26 gziT. ase, rom am SemTxvevaSi 10 ufroalbaTuria, vidre 9.demeresparadoqsi.arsebobsZveliistoriaimisSesaxeb,romXVII saukunis cnobilma frangma moTamaSem Sevalie de merem paskals da-usva azartul TamaSebTan dakavSirebuli Semdegi amocana: erTi saTama-SokamaTlisoTxjergagorebisas albaTobaimisa,romerTxelmaincmova 1, metia 1/2-ze. maSin roca ori saTamaSo kamaTlis 24-jer gagorebi-sas albaToba imisa, rom erTxel mainc mova erTdroulad ori 1, nakle-bia 1/2-ze. es ucnaurad gamoiyureba vinaidan, erTi 1-is mosvlis Sansebi6-jer metia, vidre ori 1-is mosvla, xolo 24 swored 6-jer metia 4-ze.paradoqsisaxsna: TuwesiersaTamaSokamaTelsvagorebT k -jer,maSin SesaZlebel(da tolalbaTur) SedegTa raodenobaa 6k. aqedan 5kSemTxvevaSi ar mova 1, da Sesabamisad, albaToba imisa, rom kamaTlis k -jer gagorebisas erTxel mainc mova 1 tolia(6 5) / 6 1 (5/ 6)k k k k ,12rac metia 1/2-ze Tu 4 k . meores mxriv, analogiurad, miiReba, rom alba-Toba imisa, rom kamaTlis k -jer gagorebisas erTxel mainc mova erTdr-oulad ori 1 tolia(36 35) / 36 1 (35/ 36)k k k k ,rac 24 k -saTvis jer kidev naklebia 1/2-ze da metia 1/2-ze 25 k -dan da-wyebuli.amrigad,erTikamaTlisaTviskritikulimniSvnelobaa 4 k ,xolo ori kamaTlisaTvis -- 25 k .13 $1. albaTobis Teoriis saganiadamianisyoveldRiurcxovrebaSisityvaalbaTobaxSiradgamoiyeneba ama Tu im xdomilebis moxdenis an ar moxdenis damajer-eblobis xarisxis cvlilebis gamosaxatavad, rac garkveuli azriTdakavSirebuliaCvenssubieqtursurvilebTan.aseTiamagaliTad,SemdegiSinaarsismtkicebulebi:xvalalbaTgamoidarebs,TvisbolosaTvislarialbaTgamyardeba,orweliwadSisaqarTveloalbaT natos wevri gaxdeba da a. S. maTematikaSi ki sityva albaT-obagamoiyenebamkacradgansazRruliazriT,romelsacaranairikavSiri ara aqvs damajereblobasTan da miT umetes subieqtur sur-vilebTan.rogorc aRniSnuli iyo albaTobis Teoria warmoadgens maTem-atikis dargs, romelic Seiswavlis iseTi eqsperimentebis (movleneb-is) maTematikur modelebs, romelTa Sedegebi calsaxad ar ganisaz-RvrebacdispirobebiT.aseTeqsperimentebsewodebaT SemTxveviTieqsperimentebi. SemTxveviTi eqsperimentebia: monetis agdebis Sedegi,mizanSi srolisas miznis dazianeba an ardazianeba, xelsawyos muSa-obis xangrZlivoba, satelefono sadgurSi gamoZaxebaTa ricxvi, av-tosagzao SemTxvevaTa raodenoba, arabejiTi studentis mier gamoc-dis Cabarebis Sedegi, valutis kursi da sxva.saWiroa aRiniSnos, rom albaTobis Teoria ikvlevs ara nebism-ierSemTxveviTeqsperiments,aramedmxolodiseTeqsperimentebs,romlebicxasiaTdebian statistikurimdgradobisanusixSireTamdgradobis TvisebiT. es Tviseba xasiaTdeba Semdegnairad. ganvixil-oT SemTxveviTi eqsperimenti da vigulisxmoT, rom am eqsperimentisCatarebis identuri pirobebis SenarCuneba da misi ganmeorebiTi Cat-areba SesaZlebelia (Tu fizikurad ara, azrobrivad mainc) nebismi-er sasurvel raodenoba ricxvjer. aRvniSnoT A-Ti am eqsperimentiserTerTi SesaZlo Sedegi. gavimeoroT es eqsperimenti n -jer da aRv-niSnoT ) ( An -Ti A Sedegis (xdomilebis) moxdenaTa ricxvi am neq-sperimentSi. maSin Sefardebas n An/ ) ( ewodeba A xdomilebis far-dobiTi sixSire, xolo sixSireTa mdgradobis Tviseba mdgomareobsSemdegSi:didi n -ebisaTvis A xdomilebis fardobiTi sixSire mxolodmcired irxeva ( n -is cvlilebisas) garkveuli mudmivi mniSvnelobisirgvliv. rac unda gavigoT Semdegnairad: Tu CavatarebT eqsperimen-tebisramodenimeserias,maSin statistikurimdgradobis Tvisebagulisxmobs, rom: sixSireebii nn Ai/ ) ( (sadacinaris i -ur seriaSieqsperimentebis ricxvia, xolo ) ( Ain -- ki A xdomilebis moxdenaTaraodenoba am seriaSi) axlos iqnebian erTmaneTTan (mcired gansxvav-debian erTmaneTisagan an raime saSualo sidididan) yovelTvis rog-orc kiinricxvebi iqnebian sakmaod didebi. magaliTad, v. feleriswignSi moyvanilia monetis agdebis 10 seriaSi ( 10 ,... 1 i ), sadac TiT-14eul seriaSi 1000 ineqsperimentia, gerbis mosvlaTa ) " ("ginricxv-is Semdegi monacemebi: 501, 485, 509, 536, 485, 488, 500, 497, 494, 484. cxa-dia, rom aq fardobiTi sixSireebi axlosaa erTmaneTTan, da maSasa-dame, eqsperiments, romelic mdgomareobs simetriuli monetis agde-baSi, gaaCnia sixSiris mdgradobis Tviseba.Semdeg cxrilSi mogvyavs is Sedegebi, romlebic eqsperimenta-lurad miRebuli iyo sxvadasxva mkvlevarebis mier me-18 saukunidanmoyolebuli simetriuli monetis n -jer agdebisas gerbTa mosvlis/ mnfardobiTi sixSirisaTvis:mkvlevari agdebis raodenoban fardobiTi sixSirem/nbiufoni4040 0.507de morgani4092 0.5005jevonsi20480 0.5068romanovski80640 0.4923k. pirsoni24000 0.5005feleri10000 0.4979es cxrili gviCvenebs, rom fardobiTi sixSireebi axlosaa 0.5-TTan.moviyvanoTkideverTimagaliTi.g.kramerismiermoyvanilimonacemebi1935welsSveciaSidabadebuliaxalSobilebisSesaxeb(sadacn -- axalSobilTa raodenobaa, xolo / mn -- vaJebis dabadeb-is fardobiTi sixSire) ase gamoiyureba:TveebiI II III IV V VIn 7280 6957 7883 7884 7892 7609m/n 0.515 0.510 0.510 0.529 0.522 0.518TveebiVII VIII IX X XI XIIsuln 7585 7393 7203 6903 6552 7132 88273m/n 0.523 0.514 0.515 0.509 0.518 0.527 0.517miuxedavad imisa, rom axalSobilTa saerTo raodenoba icvl-eba wlis ganmavlobaSi, vaJebis dabadebis fardobiTi sixSire sakma-od mdgradad meryeobs 0.517 saSualo mniSvnelobis irgvliv.aseTitipisstatistikurikanonzomierebebiaRmoCeniliqnaukve me-18 saukuneSi demografiul monacemebSi axalSobilTa sta-tistikis Seswavlisas, sikvdilianobis statistikis Seswavlisas, ub-edurSemTxvevaTastatistikisSeswavlisasdaa.S.(rac,Tavismxriv, sakmaod efeqturad gamoiyebneboda sadazRvevo kompaniebis sa-qmianobaSi). mogvianebiT, me-19 saukunis bolos da me-20 saukunis das-awyisSi axali statistikuri kanonzomierebebi aRmoCenil iqna fizi-kaSi, qimiaSi, biologiaSi, ekonomikaSi da sxva mecnierebebSi.15am kanonzomierebebs mivyavarT albaTobis statistikuri ganma-rtebisaken.ganmarteba. ricxvs,romlisirgvlivacirxeva AxdomilebisfardobiTi sixSire, ewodeba A xdomilebis albaToba da aRiniSneba) ( A PsimboloTi.maTematikuri dasabuTeba imisa, rom fardobiTi sixSire axlo-saa albaTobasTan (fardobiT sixSireTa mimdevrobis zRvaria alba-TobadaSesabamisad,igierTaderTia) moyvaniliaiakobbernuliscnobil TeoremaSi, romelic albaTobis TeoriaSi cnobilia agreT-ve did ricxvTa kanonis saxelwodebiT da igi warmoadgens erT-erTfundamentur Teoremas.aRsaniSnavia, rom yvela SesaZlo eqsperimenti SeiZleba gaiyossam kategoriad:I. eqsperimentebi sruli mdgradobiT, sadac saerTod araaganuzRvreloba;II. eqsperimentebi sadac ara gvaqvs sruli mdgradoba, magramaris statistikuri mdgradoba;III. eqsperimentebi sadac statistikuri mdgradobac ki aragvaqvs.pirveljgufsmiekuTvnebaumravlesobaeqsperimentebis,rom-lebicaRiwerebiansabunebismetyvelomecnierebebis(fizika,qimia)klasikurikanonebiTdamaTiSeswavlaxdebaalbaTobisTeoriisgamoyenebis gareSe. mesame kategoriisaTvis albaTobis Teoria gamo-usadegaria.meorejgufiwarmoadgenssworedalbaTobisTeoriisgamoyenebis ares. TviTon albaTobis Teoriis mier gadasawyveti amo-canebi iyofa or did jgufad. pirveli jgufis amocanebs SeiZlebavuwodoT amocanebi rTuli xdomilebebis albaTobebis gamoTvlaze,rocacnobiliamartivixdomilebebisalbaTobebi.magaliTad,Tucnobilia, rom gerbis mosvlis albaToba } " {"g P =1/2, vipovoT albaT-obebi imisa, rom zemoT moyvanil magaliTSi ) " ("1000g tolia 501-is,an 485-is, an da a. S. 484-is.amocanebis meore jgufi garkveuli azriT pirveli jgufis am-ocanebis Sebrunebulia. aq eqsperimentebis seriis Sedegebis safuZv-elze raime gziT unda SevafasoT martivi xdomilebebis albaTobe-bi. vinaidan, martivi xdomilebebis albaTobebi ucnobia im amocaneb-Si,romlebicpraqtikaSiwarmoiSoba, amitomamocanebisesjgufigansakuTrebiT sainteresoa gamoyenebebis TvalsazrisiT. albaTobisTeoriisimnawils,romelicaxdensmeoregjufisamocanebiszust dasmas da ikvlevs maTi amoxsnis meTodebs, maTematikuri sta-tistika ewodeba.16$2. elementarul xdomilebaTa sivrcealbaTobis maTematikuri Teoriis praqtikuli Rirebuleba damniSvneloba warmoCindeba iseT namdvil Tu warmosaxviT cdebTan damovlenebTan dakavSirebiT, rogoricaa magaliTad, monetis erTjera-di agdeba, monetis agdeba 100-jer, ori saTamaSo kamaTlis gagoreba,kartis darigeba, ruletkis TamaSi, adamianis an radioaqtiuri at-omis sicocxlis xangrZlivobaze dakvirveba, adamianebis garkveuliSemTxveviTi jgufis SerCeva da maTSi caciebis raodenobis daTvla,ori saxeobis mcenaris Sejvareba da Sedegze dakvirveba, satelefo-no sadguris dakavebuli xazebis an satelefono gamoZaxebaTa ricx-vis gansazRvra, eleqtronuli sistemebis SemTxveviTi xmaurebi, sam-rewvelo produqciis xarisxis SerCeviTi kontroli, avtosagzao Se-mTxvevaTa raodenoba, axalSobilis sqesi, ormagi varskvlavebis ri-cxvi cis garkveul ubanze, siTxeSi Cavardnili mtvris mcire nawil-akis mdebareoba, valutis kursi, fasebis indeqsi. jer-jerobiT yve-la am movlenis aRwera sakmaod bundovania, da, imisaTvis, rom Teo-rias mieces zusti azri, Cven unda SevTanxmdeT imaze, Tu ra gvesmisCven gansaxilveli cdis an dakvirvebis SesaZlebeli Sedegebis qveS.monetisagdebisSedegadaraaaucilebelimovidesgerbiansafasuri: moneta SeiZleba dadges wiboze an gagordes Sors (daika-rgos). miuxedavad amisa, Cven vTanxmdebiT ganvixiloT gerbi da saf-asurirogorcmonetisagdebisorad-oriSesaZlebeliSedegi.esSeTanxmeba amartivebs Teorias da ar axdens gavlenas mis SesaZlogamoyenebebze.xSiradaseTiSeanxmebebiaucilebelia.SeuZlebeliauSecdomod gaizomos raime atomis arsebobis xangrZlivoba an rome-lime piris sicocxlis xangrZlivoba. miuxedavad amisa, mecnierulimiznebisaTvis sasargebloa es sidideebi CaiTvalos zust ricxvebad.magram amasTanave, ibadeba kiTxva: romeli ricxvi SeiZleba da rome-li -- ara warmoadgendes adamianis sicocxlis xangrZlivobas? arse-bobs Tu ara maqsimaluri asaki, romlis zemoTac sicocxle SeuZle-belia, Tu asaki SeiZleba iyos nebismieri?Cven cxadia ar SevudgebiT imis mtkicebas, rom adamians SeuZ-lia icocxlos 1000 weli, Tumca Cveulebrivi sadazRvevo praqtikaar awesebs sicocxlis xangrZlivobis aseT sazRvarsac. im formul-ebis Tanaxmad, romelzec dafuZnebulia Tanamedrove sikvdilianob-is cxrilebi im adamianebis wili, romlebmac icocxles 1000 wlamdearis 10 xarisxad (3610 ) rigis. biologiis TvalsazrisiT es mtkice-buleba azrs moklebulia, magram is ar ewinaaRmdegeba cdas: saukun-is ganmavlobaSi ibadeba araumetes1010adamiani da imisaTvis, romzemoT moyvanili mtkicebuloba statistikurad uarvyoT saWiro iqn-eboda 10 xarisxad3510saukune, rac aWarbebs dedamiwis asaks 10 xa-risxad3410 -jer.cxadia,aseTimcireSansebiTavsebadiaSeuZlebe-li Sedegis Cvens warmodgenasTan da SeiZleboda gvefiqra, rom misigaTvaliswinebaabsurdulia,TumcasinamdvileSiigiamartivebs17bevr formulebs. garda amisa, Tu Cven gamovricxavdiT, rom SeuZle-belia 1000 wlamde cxovreba, maSin wavawydebodiT ufro did sirTu-leebs, vinaidan maSin Cven unda dagveSva rom arsebobs maqsimaluriasaki. magramdaSveba, rom adamianma SeiZleba icocxlos xwlamde,magram arSeiZleba icocxlos xweli da 2 wami, arafriT araa uk-eTesi, vidre asakis zeda zRvaris ararseboba.nebismieri Teoria aucileblad gulisxmobs zogierT gamarti-vebas. Cveni pirveli gamartiveba exeba cdis an dakvirvebis SesaZ-lo Sedegebs. maTematikuri Teoriis Sesaswavli oboeqtebi SeiZlebaiyvnenmxolodesSesaZlebeliSedegebi.TuCvengvindaavagoTcdis abstraqtuli modeli, Cven Tavidan unda davadginoT ras warm-oadgens gamartivebuli (idealizirebuli) cdis SesaZlo Sedegi. te-rminologiis erTianobisaTvis eqsperimentis (cdis)an dakvirvebisSedegebs uwodeben xdomilebebs.ganvixiloT eqsperimenti, romlis yvela SesaZlo Sedegebi am-oiwureba Nsxvadasxva mniSvnelobiTN ,..., ,2 1. es mniSvnelobebiararisaucilebladricxviTidamaTifizikuribunebaararisarsebiTi.ganmarteba 1. eqsperimentis calkeul SesaZlo Sedegebs eleme-ntaruli xdomilebebi ewodeba, xolo maT erTobliobas elementa-rul xdomilebaTa sivrce da aRiniSneba asoTi: } ,..., , {2 1 N .moviyvanoT magaliTebi:I. monetis erTxel agdebisas -- ={g, s};II. monetis orjer agdebisas, an ori monetis erTdroulad ag-debisas -- ={gg, gs, sg, ss};III. monetis samjer agdebisas, an sami monetis erTdroulad ag-debisas -- = {ggg, ggs, gsg, sgg, gss, sgs, ssg, sss};IV. monetis n -jer agdebisas } ), ,..., ( : {1s an g i na a a daSedegebis saerTo raodenoba tolian2 -is;V. erTi saTamaSo kamaTlis gagorebisas -- ={1, 2, 3, 4, 5, 6};VI. vTqvaT, Tavidan vagdebT monetas. Tu mova gerbi, maSin vag-orebT saTamaSo kamaTels; xolo Tu mova safasuri, maSin kidev er-Txel vagdebT monetas. am SemTxvevaSi } , 3 , 2 , 1 { ss sg, g6, g5, g4, g g g ;VII. ori saTamaSo kamaTlis gagorebisas --={(1,1); (1,2); . . . ;(1,6); (2,1); (2,2); . . . ; (2,6); . . . ; (6,1); . . . ; (6,6)} anu} 6 ,..., 2 , 1 , : ) , {( j i j i ;18VIII. produqciis vargisinobis dadgenisas --={vargisi, uv-argisi};IX. satelefono sadgurSi gamoZaxebaTa raodenoba --= {0, 1,2, . . . };X. Zabva qselSi -- ={[0, 220]}.ganmarteba 2. elementarul xdomilebaTasivrcisnebismierqvesimravles xdomileba ewodeba.cxadia, elementaruli xdomilebebi agreTve xdomilebebia, is-ini warmoadgenen elementarul xdomilebaTa sivrcis erTelementianqvesimravleebs.yveladanarCenqvesimravles(maTSoriscarielisimravlisa da TviTon sivrcis CaTvliT) xdomilebas uwodeben. zo-gjer (imis aRsaniSnavad, romqvesimravleSi erTze meti elementia)xmarobenagreTve SedgenilianrTulixdomilebiscnebasac.CvenvisargeblebT ubralod xdomilebis cnebiT.monetis orjer agdebisas (ix. magaliTi II) xdomilebis magali-Tebia: a). erTjer mainc movida gerbi (anu movida erTi an meti, maSa-sadame, ori, gerbi). igi warmoadgens simravles -- {gs,sg,gg}; b). gerbimovidaaraumeteserTisa(anumovidaerTiannaklebi,maSasadame,nuli arcerTi, gerbi). igi warmoadgens simravles -- {gs,sg,ss}; g).gerbi movida zustad erTjer (anu pirvelad movida gerbi da meor-ed ki safasuri an piriqiT). es aris Semdegi simravle -- {gs,sg}; daa. S. aRsaniSnavia, rom am SemTxvevaSi sul gveqneba 24=16 xdomileba(rogorc cnobilia nelementiani simravlis yvela SesaZlo qvesim-ravleTa raodenobaa 2n qvesimravle).ganmarteba 3. TueqsperimentiskonkretuliSedegiekuTvnisraimexdomilebas,maSinambobenromes xdomilebamoxda, xoloromelsac ar ekuTvnis is xdomileba ar moxda.albaTobisTeoriaSixdomilebebiaRiniSnebadidilaTinuriasoebiT: ,... , , , D C B A . xdomilebas Auwodeben aucilebel xdomi-lebas, vinaidan is aucileblad xdeba (is SeuZlebelia ar moxdes,radgan eqsperimentis yvela Sedegi mas ekuTvnis); xolo xdomilebas,romelic ar Seicavs arc erTelementarul xdomilebas aRniSnaven simboloTi da uwodeben SeuZlebel xdomilebas, vinaidan misi mo-xdenaSeuZlebelia(radganeqsperimentisarcerTiSedegimasarekuTvnis).SemoviRoTaRniSvnebi: A={monetisorjeragdebisaserTjermainc movida gerbi}={gs,sg,gg}; B ={ monetis orjer agdebisas gerbimovidaaraumeteserTisa}={gs,sg,ss}; C {monetisorjeragdebisasgerbi movida zustad erTjer}={ gs,sg }: D {monetis orjer agdebi-sas orivejer movida safasuri}={ss}; E {monetis orjer agdebisassafasuri movida araumetes erTisa}={gg,sg,gs}. am aRniSvnebSi, Tu mo-netis orjer agdebisas safasuri movida mxolod meored agdebisas,19maSin SegviZlia vTqvaT, rom moxda A, B , Cda Exdomilebebi, xo-lo D xdomileba ki ar moxda.Tu Axdomilebismoxdenasmosdevs B xdomilebismoxdena(simravleTaTeoriisenazeesniSnavs,rom Axdomilebanawilia,qvesimravlea Bxdomilebis), maSinCven davwerT, rom B A da vi-tyviT, rom A xdomileba iwvevs Bxdomilebas. gasagebia, rom nebis-mieri A xdomileba iwvevs aucilebel xdomilebas -- A . Tu Axdomileba iwvevs Bxdomilebas da imavdroulad Bxdomileba iw-vevs A xdomilebas, maSin vityviT, rom A da Bxdomilebi erTmane-Tis tolia da davwerT B A .winaabzacisaRniSvnebSi: C xdomilebaiwvevs A, B da Exdomilebebs; D xdomileba iwvevs Bxdomilebas; A da Exdomil-ebebi erTmaneTis tolia.20$3. operaciebi xdomilebebzexdomilebaTamocemulisistemissaSualebiTSesaZlebebliaaxali xdomilebebis ageba, iseve rogorc simravleTa mocemuli sis-temis saSualebiT igeba axali simravleebi maTi gaerTianebebiT, Ta-nakveTebiTa da damatebebiT.ori A da Bxdomilebis gaerTianeba (an jami) ewodeba iseTxdomilebas, romelic xdeba maSin, roca am xdomilebebidan erTi ma-inc xdeba da aRiniSneba simboloTi B A(an B A+ ). sqematurad esase gamoisaxeba:ori A da Bxdomilebis TanakveTa (an namravli) ewodeba is-eT xdomilebas, romelic xdeba maSin, roca es xdomilebebi erTdr-ouladxdebadaaRiniSnebasimboloTi B A (an AB ).sqematurades ase gamoisaxeba:A xdomilebis sawinaaRmdego xdomileba ewodeba iseT xdomi-lebas, romelic xdeba maSin, roca A ar xdeba da aRiniSneba simbo-loTi A. sqematurad, Tu warmovidgenT, rom elementarul xdomile-baTa sivrce marTkuTxedia, xolo A xdomileba -- wre, maSin sawina-aRmdego xdomileba iqneba oTxkuTxedis gaferadebuli nawili:21ori Ada B xdomilebis sxvaobaewodebaiseTxdomilebas,romelic xdeba maSin, roca xdeba A magram ar xdeba Bda aRiniSn-eba simboloTi B A \ . sqematurad es ase gamoisaxeba:cxadia, rom ori A da Bxdomilebis sxvaoba agreTve SeiZle-ba warmodges, rogorc A xdomilebisa da Bxdomilebis sawinaaRm-degoBxdomilebis TanakveTa: B A B A = \ .gasagebia, rom mas Semdeg rac Cven ganvmarteT ori xdomileb-is gaerTianeba da TanakveTa, bunebrivad SesaZlebelia xdomilebaTanebismieriraodenobisgaerTianebisadaTanakveTisganmarteba.asemagaliTad, sqematurad sami A, Bda CxdomilebisaTvis Tanakve-Ta ABC iqneba:22or A da Bxdomilebas ewodeba araTavsebadi (uTavsebadi, Se-uTavsebeli), TumaTierTdrouladmoxdenaSeuZlebelia,anracigivea maTi TanakveTa aris SeuZlebeli xdomileba: B A =. simrav-leTa Teoriis enaze es niSnavs, rom es ori simravle TanaukveTia.cxadia,romraime AxdomilebadamisisawinaaRmdego AxdomilebauTavsebadia -- A A =. gardaamisa, O =A A .Tu Axdomilebaiwvevs B xdomilebas( B A c ),maSincxadia,romA B A = , B B A =, = B A \ , xolo sxvaoba A B \sqematurad ase gam-oisaxeba:imisaTvis,romdavinaxoTragansxvavebaadaraaqvTsaerTosimravleTa Teoriisa da albaTobis Teoriis tradiciul terminebs,qvemoT Cven moviyvanT Sesabamis cxrils:23AaRniSvnebi simravleTa TeoriisinterpretaciaalbaTobis Teoriisinterpretacia elementi, wertili Sedegi, elementaruli xdomilebaOwertilTa simravle elementarul xdomilebaTasivrce, aucilebeli xdomilebaAwertilTa simravlexdomileba (Tu Sedegi A e ,maSin amboben, rom moxdaAAxdomileba)\ A A = OA simravlis damateba, e.i. imwertilebis simravle, roml-ebic ar Sedian A-Sixdomileba, romelic mdgomareobsA-s ar moxdenaSiA B A da Bsimravleebis gaer-Tianeba, e.i. simravle imwertilebis, romlebicSedian an A-Si an B -Sixdomileba, romelic mdgomareobsA da B xdomilebebidan erTismainc moxdenaSiA B A da Bsimravleebis Tanak-veTa, e.i. simravle im werti-lebis, romlebic Sedian ro-gorc A, ise B xdomilebaSixdomileba, romelic mdgomareobsA da B xdomilebebiserTdroul moxdenaSiCcarieli simravleSeuZlebeli xdomilebaA B = C A da Bsimravleebi arikveTebianA da Bxdomilebebi araTav-sebadia (maTi erTdrouladmoxdena SeuZlebelia)A B +simravleTa jami, e.i.TanaukveTi simravleebisgaerTianebaxdomileba, romelic mdgomareobsori uTavsebadi xdomilebidanerT-is moxdenaSi\ A B A da Bsimravleebis sxva-oba, e. i. simravle im wert-ilebis, romlebic Sedian A-Si, magram ar Sedian B -Sixdomileba, romelic mdgomareobsA xdomilebis moxdenaSi da Bxdomilebisar moxdenaSiAB Asimravleebis simetriulisxvaoba, e.i. simravle( \ ) ( \ ) A B B A xdomileba, romelic mdgomareobsA da Bxdomilebebidan erTismoxdenaSi, magram ara oriveserTdroulad1nnA=

1 2, ,... A Asimravleebis gaer-Tianeba, e.i. im wertilebissimravle, romlebic SedianerTerTSi maincxdomileba, romelic mdgomareobs1 2, ,... A AxdomilebebidanerTerTis mainc moxdenaSi1nnA=1 2, ,... A Asimravleebis jami,e.i. gaerTianeba wyvil-wyvil-ad TanaukveTi simravleebisxdomileba, romelic mdgomareobs1 2, ,... A AuTavsebadi xdomilebeb-idan erTis moxdenaSi1nnA=1 2, ,... A Axdomilebebis Tanak-veTa, e.i. simravle im werti-lebis, romlebic Sedian yve-la xdomilebaSixdomileba, romelic mdgomareobs1 2, ,... A Axdomilebebis erTdro-ul moxdenaSi24$4. albaTobis ganmartebaganmarteba 1. Tu elementaruli xdomilebaTa sivrcis nebismi-erelementarulxdomilebasi Seesabamebagarkveuliricxvebi) (i iP p = ,romlebicakmayofilebenpirobebs: 1 0 s sip da 11==Niip ,maSin am ricxvebs ewodebaTielementaruli xdomilebebis albaT-obebi ( P -- aris pirveli aso inglisuri sityvisProbability, rome-lic niSnavs albaTobas).monetiserTjeragdebisas( O={g,s})cxadiaundadavuSvaT,rom: p P = } {g , p P =1 } {s ,sadac 1 0 s sp .SemTxvevas,roca 2 / 1 = pewodeba simetriulimonetisagdeba. monetisorjeragdebisas( O={gg, gs, sg, ss}) --1} { p P = gg ,2} { p P = gs ,3} { p P = sg ,4} { p P = ss , sad-ac yvela 1 0 s sipda 141== iip . SemTxvevas, roca yvela 4 / 1 =ip , CvenvuwodebT simetriuli monetis or damoukidebel agdebas.ganmarteba 2. A xdomilebis albaToba ) ( A P ewodeba masSi Sem-avali elementaruli xdomilebebis albaTobebis jams:e=AiiP A P ) ( ) ( .aq da yvelgan SemdgomSi simboloTi A e aRiniSneba is faq-ti, rom elementaruli xdomileba ekuTvnis A xdomilebas.cxadia, rom 1 ) ( = O P da ( P )=0.Tu yvela elementaruli xdomileba tolalbaTuria anu yvelaN Pi/ 1 ) ( = , N i ,..., 2 , 1 = , maSin amboben, rom gvaqvs klasikuri albaTu-ri modeli (praqtikul situaciaSi es niSnavs, rom SemTxveviTi eqsp-erimentis yvela Sedegi erTnairad mosalodnelia da yvelas ganxo-rcielebiserTidaigiveSansiaqvs).amSemTxvevaSiganmarteba 2gvaZlevs, rom klasikur modelSi xdomilebis albaToba daemTxvevaalbaTobisklasikurganmartebas, romelicSemoRebuliiyo1812wels laplasis mier:albaTobis klasikur ganmarteba. Tu SemTxveviTi eqsperimentisSesaZlo SedegTa raodenoba sasrulia da calkeul Sedegs aqvs gan-xorcielebis Tanabari Sansi, maSin xdomilebis albaToba tolia ma-sSi Semaval elementarul xdomilebaTa raodenoba gayofili yvelaSesaZlo elementarul xdomilebaTa raodenobaze:| || |) (O=AA P ,sadac | | B -- aRniSnavs BxdomilebaSi Semaval elementarul xdomi-lebaTa raodenobas.mocemul xdomilebaSi Semaval elementarul xdomilebebs uw-odeben xelSemwyob elementarul xdomilebebs. Sesabamisad, albaTo-25bis klasikuri ganmarteba ase gamoiTqmis: klasikur modelSi xdomi-lebis albaToba tolia xelSemwyob elementarul xdomilebaTa ra-odenoba gayofili yvela SesaZlo elementarul xdomilebaTa raod-enobaze.amocana1. simetriulimonetisoridamoukidebeliagdebisasvipovoT albaToba imisa, rom gerbi mova erTjer mainc.amoxsna. amSemTxvevaSi O={gg,gs,sg,ss}; = } {gg P } {gs P == = } {sg P 4 / 1 } { = ss P ; A={gg, gs, sg} da 4 / 3 ) ( = A P .amocana2(bednierbileTebze). 25sagamocdobileTidan5bednieria,xolodanarCeni20 arabednieri.romelstudentsaqvs bednieri bileTis aRebis meti albaToba: vinc pirveli iRebsbileTs, Tu vinc meore iRebs bileTs?amoxsna. avRniSnoT pirveli studentis mier aRebuli bileTisnomeri iasoTi, xolo meore studentis mier aRebuli bileTis nom-eri jasoTi. davuSvaT, rom bednieri bileTebis nomrebia: 1, 2, 3, 4,5.maSincxadia,rom } , 25 ,..., 2 , 1 , : ) , {( j i j i j i = = = O , 600 24 25 | | = = O dabunebriviaCavTvaloT,romyvelaelementarulixdomilebatola-lbaTuria: 600 / 1 ) , ( = j i P .SemoviRoT xdomilebebi:{ } A = pirvelma studentma aiRo kargi bileTi ,{ } B = meore studentma aiRo kargi bileTi ,maSin am xdomilebebs aqvT Semdegi saxe:} ; 25 ,..., 1 ; 5 ,..., 1 : ) . {( j i j i j i A = = = =da } ; 5 ,..., 1 ; 25 ,..., 1 : ) , {( j i j i j i B = = = = .advilidasanaxia,rom 120 24 5 | | | | = = =B A .amitomalbaTobisklasikuri ganmartebis Tanaxmad:120 1( ) ( )600 5PA PB = = = .e. i. orive moswavles aqvs kargi bileTis aRebis erTi da igive alb-aToba.davaleba. wina amocanaSi vipovoT albaToba imisa, rom mesamestudenti amoiRebs bednier bileTs.amocana 3 (galtonis dafa). gvaqvs dafaze samkuTxedis formiTdalagebuli rgolebi, ise rom wveroSi erTi rgolia, meore rigSiwinasgan Tanabr manZilebze ori rgoli, mesame rigSi zeda ori rgo-lidan Tanabar manZilebze sami rgoli da a.S. boloSi aris eqvsirgoli. me-7 striqonSi ki aris bolo 6 rgolidan Tanabar manZileb-ze 7 Rrmuli. zeda rgolze agdeben burTs da mas SeuZlia igoraosTanabari albaTobiT an marjvniv an marcxniv rgolidan rgolze,rac sabolood sruldeba romelime RrmulSi CavardniT. rogoriaalbaToba imisa, rom burTi Cavardeba mexuTe RrmulSi?26OO OO O OO O O OO O O O OO O O O O OU U U U U U U1 2 3 4 5 6 7amoxsna. rogorc vxedavT arsebobs pirvel da me-7 RrmulebSiburTis Cavardnis erTaderTi gza (traeqtoria), meore da me-6 Rrmu-lebSiburTis Cavardnis -- eqvs-eqvsi gza, mesame da mexuTe Rrmule-bSi TxuTmet-TxuTmeti gza da bolos, meoTxe RrmulSi burTisCavardnis 20 gza. gzebis (Sedegebis) sruli raodenobaa 1+6+15+20+15++6+1=64 da yvela es Sedegi Tanabrad mosalodnelia, vinaidan TiTo-euli traeqtoriis gavlisas burTi ganicdis eqvs dajaxebas rgole-bze da yoveli dajaxebisas is Tanabari albaTobebiT gadaadgilde-baanmarjvnivanmarcxniv.TanabaralbaTuri64SedegidanmexuTeRrmulSiCavardnasxelsuwyobs15 SedegidaSesabamisad,saZebnialbaToba iqneba 15/64.qvemoT moyvanilia galtonis dafaze rgolebis 7 rigis SemTx-vevaSi TiToeul poziciaze burTis moxvedris SesaZlo gzebis ricx-vi.albaTobisklasikuriganmartebagamosadegiaamocanaTamxo-lod Zalian viwro klasisaTvis, sadac yvela SesaZlo SedegTa sim-ravle sasrulia da TiToeuli SesaZlo Sedegi Tanabrad SesaZlebe-lia. umravles realur amocanaSi es piroba irRveva, da Sesabamisad,klasikuri ganmartebis gamoyeneba SeuZlebelia. aseT SemTxvevaSi sa-27Wiroa xdomilebis albaTobis sxva gziT ganmarteba. am mizniT, wina-swar SemoviRoT A xdomilebis fardobiTi sixSirisW(A) cneba, ro-gorc cdaTa im ricxvis Sefardeba, romelSic dafiqsirda (ganxorc-ielda) A xdomileba, Catarebuli eqsperimentebis saerTo raodenob-asTan:( )MWAN= ,sadacN cdaTa saerTo ricxvia, -- A xdomilebis moxdenaTa ri-cxvi.eqsperimentebis didma raodenobam aCvena, rom Tu cdebi tarde-ba erTi da igive pirobebSi, maSin cdaTa (dakvirvebaTa) didi ricxv-isaTvis, fardobiTi sixSire umniSvnelod icvleba, meryeobs (irxeva)ra garkveuli mudmivi ricxvis irgvliv. es ricxvi SeiZleba CaiTva-los gansaxileveli xdomilebis albaTobad.ganmarteba. xdomilebis statistikuralabaTobadiTvlebaamxdomilebis fardobiTi sixSire an masTan axlos myofi ricxvi (ma-Tematikurad zusti formulireba aseTia: ( ) lim ( )NNPA W A= ).SeniSvna. imisaTvis, romarsebobdes xdomilebis statistik-uri albaToba, saWiroa:a). eqsperimentebis usasrulod didi ricxvis Catarebis SesaZ-lebloba;b). sxvadasxva seriebSi xdomilebis moxdenis fardobiTi si-xSireebis mdgradoba sakmarisad didi raodenoba cdebisaTvis.28$5. geometriuli albaTobaalbaTobisklasikuriganmartebiserT-erTinakliisaa,rommisi gamoyeneba ar SeiZleba im eqsperimentebisaTvis, romlebsac gaa-CniaT Sedegebis usasrulo raodenoba. aseT SemTxvevaSi SesaZlebe-lia visargebloT geometriuli albaTobis cnebiT.davuSvaT rom L monakveTze SemTxveviT agdeben wertils. racimas niSnavs, rom wertili aucileblad daecema L monakveTze da am-asTanave,TanabariSesaZleblobebiTisSesaZlebeliadaemTxvesammonakveTis nebismier wertils. amave dros, albaToba imisa, rom wer-tili daecema L monakveTis nebismier nawilze araa damokidebuli amnawilis ganlagebaze monakveTis SigniT da proporciulia misi sig-rZis. maSin albaToba imisa, rom agdebuli wertili daecema l monak-veTze, romelic aris nawili L monakveTis, gamoiTvleba formuliT:| | / | | L l P = , (1)sadac | | l -- lmonakveTis sigrZea, xolo | | L -- L monakveTis sigrZe.analogiurad,SesaZlebeliaamocanisdasmawertilisaTvis,romelsac vagdebT brtyel Sareze da albaToba imisa, rom is mox-vdeba am aris snawilSi ganimarteba rogorc:| | / | | S s P = , (2)sadac | | s -- saris farTobia, xolo | | S -- Saris farTobi.samganzomilebianSemTxvevaSialbaTobaimisa,romSemTxvevi-Ti gziT VsxeulSi Cavardnili wertili aRmoCndeba am sxeulis vnawilSi, gamoiTvleba formuliT:| | / | | V v P = ,(3)sadac | | v -- vsxeulis moculobaa, xolo | | V -- Vsxeulis mocul-oba.amocana 1. vipovoT albaToba imisa, rom wreSi SemTxveviT Ca-gdebuli wertili ar Cavardeba am wreSi Caxazul wesier eqvskuTxe-dSi.amoxsna. davuSvaT, rom wris radiusia R , maSin masSi Caxazu-li wesieri eqvskuTxedis gverdic iqneba R . amasTanave, wris radiu-sia2| | R S = , xolo eqvskuTxedis farTobia -- .23 3| |2R s= amitom saZ-iebeli albaToba iqneba:. 174 , 023 323 3| || | | |22 2~===RR RSs SPamocana 2. AB monakveTze SemTxveviT agdeben sam wertils , CD da M . vipovoT albaToba imisa, rom AD AC,da AMmonakveTebi-sagan SeiZleba aigos samkuTxedi?amoxsna. avRniSnoT AD AC,da AMmonakveTebis sigrZeebi Ses-abamisad y x,da z -iT da elementarul xdomilebaTa sivrcis rolSi29ganvixiloT samganzomilebiani sivrcis wertilTa simravle koord-inatebiT ) , , ( z y x . Tu CavTvliT, rom AB monakveTis sigrZe tolia 1-is, maSin elementarul xdomilebaTa sivrce iqneba kubi, romlis wi-boa erTi. amave dros, xelSemwyob elementarul xdomilebaTa simra-vle (samkuTxedis aqsiomis Tanaxmad )Sedgeba im wertilebisagan, ro-melTa koordinatebisaTvis sruldeba samkuTxedis utolobebi: : x++y > z, x + z > y,y + z > x. es ki warmoadgens kubis nawils, romelic mo-Wrilia misgan sibrtyeebiT: x + y = z, x + z = y,y + z = xz y(erT-erTi am sibrtyidan, kerZod x+ y= z, moyvanilia naxazze). yove-li aseTi sibrtye kubidan moWris piramidas, romlis moculoba to-lia6112131= .Sesabamisad, kubis darCenili nawilis moculoba iqneba21613 1 | | = = v .amitom saZiebeli albaToba, ganmartebis Tanaxmad, iqneba.211 :21| || || | = = =VvPamocana3(Sexvedrisamocana). oripiriSeTanxmdagarkveuladgilze Sexvdes erTmaneTs 6-dan 7 saaTamde. TiToeuli maTgani Se-mTxveviTmomentSimidisdaTqmuladgilasdameoreselodeba20wuTis manZilze (Semdeg ki midis). vipovoT albaToba imisa, rom espirebi Sexvdebian erTmaneTs?amoxsna. avRniSnoT, erT-erTi piris daTqmul adgilze mosvl-is dro x + 6 -iT, xolo meore piris -- y + 6 -iT (sadac xda ygamos-axulia saaTebSi). elementarul xdomilebaTa sivrcis rolSi Segvi-Zlia aviRoT im ) , ( y xwertilTa simravle, romlebic ekuTvnian erT-eulovankvadrats.amSemTxvevaSiCvenTvissainteresowertilTa(xelSemwyob elementarul xdomilebaTa) simravle iqneba erTeulov-ani kvadratis im wertilTa simravle, romelTa koordinatebi erTm-aneTisagan daSorebulia araumetes 3 / 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 . amit-om advili gasagebia, rom A simravle iqneba erTeulovani kvadrat-30is SigniT 3 / 1 = x y da 3 / 1 + = x y wrfeebs Soris moqceuli gaferad-ebuli are(roca 3 / 1 0 s s x , maSin 3 / 1 = x y wrfis nacvlad qvedasazRvris rolSi iqneba x RerZi: 3 / 1 0 + s s x y , xolo roca1 3 / 2 s s x , maSin zeda sazRvari 3 / 1 + = x y wrfis nacvlad iqneba 1 = ywrfe: 1 3 / 1 s s y x ).01/32/3 1 xy11/3O amitom saZiebeli albaToba iqneba:9513 / 2 3 / 2 1| || |) ( = =O=AA P .davaleba 1. ipoveT albaToba imisa, rom erTeulovan gverdiankvadratSi SemTxveviT SerCeuli wertili am kvadratis uaxloesigverdidan daSorebuli iqneba araumetes 0,15-iT?davaleba 2. ori signali mimReb mowyobilobaze Tdrois manZ-ilzeSemTxveviTmomentSimiiReba.mowyobilobamaTganarCevs,Tuisini tdroiT mainc arian dacilebuli erTmaneTs. ipoveT albaTo-ba imisa, rom orive signali miRebuli iqneba?31$6. kombinatorikis elementebiZalianbevrSemTxvevaSi elementarulxdomilebaTasivrcisada xdomilebis albaTobis gansazRvra pirdapiri gadaTvliT SeuZl-ebelia. xdomilebaTa albaTobebis gamoTvlis dros xSirad saWiroxdeba kombinatorikis formulebis gamoyeneba. kombinatorika arismecniereba, romelic swavlobs kombinaciebs, romelTa Sedgenac Ses-aZlebelia garkveuli sasruli simravlis elementebidan gansazRru-li wesebis gamoyenebiT. ganvsazRvroT ZiriTadi kombinaciebi.pirvel rigSi CamovayaliboT kombinatoruli analizis ZiriTa-di principi, romelsac gamravlebis principi ewodeba:Tu asarCevia m obieqti da arsebobs pirveli obieqtis arCevis1n SesaZlebloba, pirveli obieqtis arCevis Semdeg arsebobs meoreobieqtisarCevis2n SesaZlebloba,daa.S., 1 m obieqtisarCevisSemdegarsebobs m-uriobieqtisarCevismn SesaZlebloba,maSinarsebobs am m obieqtis am mimdevrobiT arCevis1 2 mn n n SesaZ-lebloba.magaliTad, Tu mamakacs aqvs 4 perangi da 2 pijaki, maSin am ma-makacs aqvs perangisa da pijakis SerCevis 4 2 8 SesaZlebloba (va-rianti).namravlisprincipTandakavSirebiTxSiradsasargebloaxismsgavsi (xisebri) diagramis anudendrogramis gamoyeneba.magaliTad, dendrogramiT gamosaxuli monetis orjer agdebisSesabamisi Sedegebis simravle iqneba:ganmarteba 1. gadanacvlebebi es aris kombinaciebi, romleb-ic Sedgenilia mocemulielementiani simravlis yvela element-isagan da erTamaneTisagan ganxsvavdeba mxolod elementebis ganla-gebis rigiT. elementiani simravlis yvela SesaZlo gadanacvleb-aTa ricxvia: = !.32marTlac,TuamamocanasSevxedavT rogorc n obieqtidan obieqtis arCevis amocanas, maSin arsebobs pirveli obieqtis arCevisn SesaZlebloba,pirveliobieqtisarCevisSemdegarsebobsmeoreobieqtis arCevis 1 n SesaZlebloba, da a. S., me- nobieqtis arCeviserTaderTi SesaZlebloba. Sesabamisad, gamravlebis principis Tana-xmad:( 1) 1 !nP n n n magaliTi 1. ramdegi gansxvavebuli sia SeiZleba Sedgenil iqn-es 7 sxvadasxva gvarisagan?amoxsna. 7 = 7! = 234567 = 5040.ganmarteba 2. wyobebi es aris elementiani kombinaciebi ganxsvavebuli elementis mqone simravlidan, romelebic erTmaneTis-agangansxvavdebiananelementebisSemadgenlobiTanelementebisganlagebis rigiT. sxva sityvebiT, wyoba ariselementiani simrav-liselementian, dalagebul qvesimravleTa raodenoba. yvela Ses-aZlo wyobebis ricxvia:!/( )!n n m .amamocanasCvenSegviZliaSevxedoT,rogorc n obieqtidanm obieqtis SerCevis amocanas. radganac, arsebobs pirveli obieqt-isarCevis n SesaZlebloba,pirveliobieqtisarCevisSemdegarsebobs meore obieqtis arCevis 1 n SesaZlebloba, da a. S., me- mobieqtis arCevis 1 n m SesaZlebloba, amitom gamravlebis princi-pis safuZvelze gvaqvs:( 1)( 2)...( 1) !/( )! n n m magaliTi2. SejibrebaSimonawile10sportsmenidanpirvelsam adgilze gasuli sami gamarjvebuli ramden sxvadasxvanairad Se-iZleba ganTavsdes dasajildoebel kvarcxlbekze?amoxsna. . 720 8 9 10310 ganmarteba3. jufdebebi esaris elementianisimravlisdaulagebelielementiani qvesimravleebi (anu iseTi erTobliobe-bi,romlebicerTmaneTisagangansxvavdebianmxolodelementebisSemadgenlobiT). jufdebaTa ricxvi tolia:!!( )! .vinaidan, elementiani simravlis yvela elementiani, dala-gebuli qvesimravlis misaRebad, Cven SegviZlia aviRoTelementia-ni simravlis yvela daulagebeli elementiani qvesimravle (rome-lTa raodenobaa) da TiToeulSi movaxdinoT yvelanairi gadana-cvlebebi (amis SesaZleblobaa ! m ), amitom adgili aqvs Tanafardo-bas:!m mn nC m A .aqedan cxadia, rom33magaliTi 3. SesarCev SejibrebaSi monawileobs 10 adamiani, ro-melTagan finalSi gadis sami. finalistebis ramdeni gansxvavebulisameuli SeiZleba gamovlindes?amoxsna. winamagaliTisagangansxvavebiT,aqfinalistebisrigs(dalagebas)mniSvnelobaaraaqvs.amitomviyenebTjufdebisformulas. Sesabamisad, saZiebeli ricxvia:. 120610 9 8! 7 ! 3! 10310 !! !( )!mm nnC nAm mn m 34$7. albaTobis gamoTvla kombinatorikis gamoyenebiTganvixiloT magaliTebi, romlebic dakavSirebulia yuTidan, romel-Sic Mraodenobis burTia, sxvadasxva gzebiTn raodenobis burTis amo-RebasTan.amorCevadabrunebiT. amSemTxvevaSieqsperimentisyovelnabijzeyuTidan amoRebuli burTi ukan brundeba. vigulisxmoT, rom burTebi ga-danomrilia ricxvebiT erTidan M -mde. Sesabamisad, elementarul xdomi-lebaTasivrceSeiZlebawarmodgeniliqnesiqnebarogorcricxvTa n -eulebina a ,...,1, sadaciaaRniSnavs i -ur nabijze amoRebuli burTis no-mers. ganasxvaveben ori tipis amorCevebs: dalagebuli amorCevebi da dau-lagebeli amorCevebi. pirvel SemTxvevaSi amorCevebi, romlebic SedgebianerTi da igive elementebisgan, magram gansxvavdebian am elementebis dal-agebisrigiT,cxaddebagansxvavebulad. meoreSemTxvevaSielementebisdalagebis ricxvi ar miiReba mxedvelobaSi da aseTi amorCevebi cxaddebaidenturad. imisaTvis, rom ganvasxvavoT es amorCevebi dalagebuli amorC-evebiaRiniSnebasimboloTi(na a ,...,1),xolodaulagebeliamorCevebi [na a ,...,1].amgvarad, dabrunebiT dalagebuli amorCevis SemTxvevaSi elementar-ul xdomilebaTa sivrces aqvs Semdegi struqtura} ,..., 1 ; ,..., 1 : ) ,..., ( : {1n i M a a ai n= = = = O da gansxvavebul SedegTa (amorCevaTa) raodenoba, romelsac kombinatori-kaSi uwodeben wyobebs M -dan n dabrunebebiT, tolianM = O| | .Tu ki Cven vixilavT daulagebel amorCevebs dabrunebiT, romelsackombinatorikaSi uwodeben jufdebas M -dan n , maSin} ,..., 1 ; ,..., 1 : ] ,..., [ : {1n i M a a ai n= = = = O .gasagebia, rom gansxvavebul daulagebel amorCevaTa ricxvi naklebia vid-re dalagebulebis ricxvi. vaCvenoT, romnn MC1| | += O , sadac)! ( !!:l k lkClk= --aris jufdebaTa ricxvi k -dan l .damtkicebas Cven CavatarebT maTematikuri induqciis principiT. aRv-niSnoT ) , ( n M N -iT CvenTvis saintereso Sedegebis ricxvi. cxadia, rom TuM k s , maSin1) 1 , (kC k k N = = . davuSvaT axla, romnn kC n k N1) , ( += , M k sda va-CvenoT, rom es formula ZalaSi darCeba n -is 1 + n -iT Secvlisas. SevniSn-aT, rom daulagebeli [1 1,...,+ na a ] SerCevebis ganxilvisas SeiZleba vigulis-xmoT, rom misi elementebi dalagebulia klebis mixedviT:1 2 1 +s s sna a a .advili dasanaxia, rom im daulagebeli1 2 1 +s s sna a aSerCevebis raode-noba,romelTaTvisac 11 = a tolia ) , ( n M N -is,romelTaTvisac 21 = a --) , 1 ( n M N -is da a. S. amitom= + + + = + ) , 1 ( ) , 1 ( ) , ( ) 1 , ( n N n M N n M N n M N+ = + + =+ +++ + +) (1111 1 1nn Mnn Mnnnn Mnn MC C C C C1 1111 111) ( ) (+++++ + ++ = + + + +nn Mnnnnnnnn Mnn MC C C C C C ,sadac Cven visargebleT binomis koeficientebis Semdegi TvisebiT:lklklkC C C11+= + .35amorCeva dabrunebis gareSe. vigulisxmoT, rom M n sda amoRebuliburTi ukan ar brundeba. am SemTxvevaSic agreTve ganixileba ori SesaZl-ebloba dakavSirebuli amorCevis dalageba -- daulageblobasTan.dabrunebis gareSe dalagebuli SerCevis SemTxvevaSi, romelsac ko-mbinatorikaSi uwodeben wyobebs M -dan n dabrunebebis gareSe, Sedegebissimravles aqvs saxe:} ,..., 1 ; ,..., 1 ; , : ) ,..., ( : {1n i M a l k a a a ai l k n= = = = = = O ,xolo am simravlis elementTa raodenoba tolia ) 1 ( ) 1 ( | | + = O n M M M .es ricxvi aRiniSneba simboloTinM) (annMAda ewodeba wyobaTa ricxviM -dan n.daulagebeli amorCevisas dabrunebis gareSe, romelsac kombinator-ikaSi uwodeben jufdebas M -dan n dabrunebebis gareSe, SedegTa simrav-les aqvs saxe} ,..., 1 ; ,..., 1 ; , : ] ,..., [ : {1n i M a l k a a a ai l k n= = = = = = O damisielementebis raodenobatolianMC = O| | .marTlac,daulagebeli[na a ,...,1] erTobliobidan, romelic Sedgeba gansxvavebuli elementebisagan,SeiZleba miviRoT zustad ! nraodenoba dalagebuli erTobliobebi. amit-omnMA n = O ! | |, da maSasadame,nMnMC n A = = O ! / | | .sabolood Cven gvaqvs yuTidan, romelSic Mraodenobis burTia nraodenobis burTis amoRebis ricxvis Semdegi cxrili:dalagebuli daulagebelidabrunebiTnM1nM nC+ dabrunebisgareSenMAnMCmagaliTad, 3 = M da 2 = n -isSemTxvevaSiSesabamiselementarulxdomilebaTa 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]dabrunebisgareSeerToblioba dalagebuli daulagebelinawilakebis ganlageba danayofebSi. ganvixiloT sakiTxi elementar-ul xdomilebaTa sivrcis struqturis Sesaxeb n nawilakis (burTis) MdanayofSi(yuTSi)ganlagebisamocanaSi.aseTiamocanebTansaqmegvaqvsstatistikurfizikaSi,rocaswavloben n nawilakis(esSeiZlebaiyosprotonebi, neitronebi, da sxva) MmdgomareobaSi (es SeiZleba iyos ener-getikulidoneebi)ganawilebissakiTxebs.vigulisxmoT,romdanayofebigadanomrilia nomrebiT1, 2,..., Mda Tavidan davuSvaT, rom nawilakebi ga-rCevadia(gansxvavebulebia,aqvTnomrebi 1, 2,..., n ).maSin n nawilakis MdanayofSiganawilebissruliadaRiwerebadalagebulierTobliobiT(na a ,...,1), sadacia -- warmoadgens im danayofis nomers, romelSic moxvda36nawilaki nomriTi . Tu ki ganvixilavT ganurCevel nawilakebs, maSin maTiganawileba MdanayofSi sruliad aRiwereba daulagebeli erTobliobiT[na a ,...,1], sadacia -- im danayofis nomeria, romelSic moxvda nawilaki i -ur nabijze.magaliTebi:I. kursze, romelzec sami jgufia, jgufxelebis arCevis yvela SesaZ-lovariantebisricxvia3 2 1n n n ,sadacin -- i -urjgufSistudentebisraodenobaa (viyenebT namravlis princips);II. raodenoba yvela SesaZlo kombinaciebis, ramdennairadac SesaZle-belia m mgzavri ganvaTavsoT nvagonSi toliamn(dalagebuli amorCe-va dabrunebiT, sadac M n =da n m = );III. m adamianis dabadebis dReebis yvela SesaZlo kombinacia (im pi-robiT,romTiTeulidabadebisdRearisromelime365dRidan)toliam365(saqme gvaqvs amorCevasTan dabrunebiT, amasTanave amorCevebi iTvlebadalagebulad, sadac 365 M =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 nmonawilisagan Sem-dgar wriul saWadrako turnirSi tolia 2 / ) 1 (2 = n n Cn (daulagebeli am-orCeva dabrunebis gareSe).amocana 1 (damTxvevebze). vipovoT albaToba imisa, rom: a). m SemTx-veviT arCeuli adamianis dabadebis dReebi ar daemTxveva erTmaneTs (im pi-robiT, rom yvela dRe tolalbaTuria); b). m SemTxveviT arCeul adamian-Si moiZebneba ori mainc iseTi, romelTa dabadebis dReebi daemTxveva er-TmaneTs.amoxsna. a). elementarul xdomilebaTa sivrce Seesabameba dalageb-ul amorCevebs dabrunebiT, sadac 365 M =da n m = , anu | | 365mO = ;xoloxelSemwyobi elementaruli xdomilebebi ki dalagebul amorCevebs dabr-unebis gareSe, sadac agreTve 365 M =da n 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 Tanaxmadki gvaqvs, rommmAm Q3651 ) (365 = .moviyvanoTamalbaTobismniSvnelobebiscxrilizogierTi m-isSemTxvevaSi:m 4 16 22 23 40 64 70Q(m) 0.01636 0.28360 0.47569 0.50730 0.89123 0.99711 0.99916sainteresoaaRiniSnos,rom(molodinissawinaaRmdegod!)klasismoswavleTa raodenoba, romelSic 1/2-is toli albaTobiT moiZebneba orimoswavle mainc erTi da igive dabadebis dReebiT, arc ise didia: igi to-lia mxolod 23-is.37amocana 2 (mogeba latareaSi). gvaqvs MbileTi gadanomrili ricxv-ebiT erTidan M -mde, romelTagan n bileTi nomrebiT erTidan n-mde mom-gebiania ( n M 2 > ). vyidulobT ncal bileTs. ras udris albaToba imisa,rom am nbileTidan erTi mainc iqneba momgebiani (avRniSnoT es xdomile-ba A-Ti)?amoxsna. vinaidanbileTebisamoRebis(yidvis)Tanmimdevrobasaraaqvs mniSvneloba nayid bileTebSi momgebiani bileTebis arsebobis an ararsebobis TvalsazrisiT, amitom elementarul xdomilebaTa sivrces eqn-eba Semdegi stuqtura:} ,..., 1 ; , : ] ,..., [ : {1M a l k a a a ai l k n= = = = = O .Sesabamisad, Cvens mier zemoTmoyvanili cxrilis TanaxmadnMC = O| | .xdomilebas (avRniSnoT igi0B -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+ = = = = = dann MC B= | |0.amitom)11 ( )11 )( 1 ( ) (0+ = = = n MnMnMnAACCB PnMnn MnMnn M.Sesabamisad, saZebni albaToba iqneba0( ) 1 ( ) 1 (1 )(1 ) (1 )1 1n n nPB PBM M M n= = +.Tu magaliTad,2n M =da n , maSin10) (e B P(aq eneperis ric-xvia) da 632 , 0 1 ) (1~ e B P , sadac krebadobis siCqare sakmaod didia: ukveroca 10 = n -- 670 , 0 ) ( = B P .davaleba. winaamocanispirobebSivipovoTalbaTobaimisa,romnayidi n bileTidan zustadm ( n m s ) iqneba momgebiani.amocana 3 (urTierTobis upiratesobaze). davuSvaT klasSi, romelS-ic10 moswavlea tardeba gamokiTxva, sadac TiToeulma moswavlem anket-aSi unda miuTiTos is sami amxanagi, romelsac aZlevs upiratesobas cxraamxanagidan. A iyos xdomileba, rom erTerTi moswavle dasaxelebul iqnayvela SesaZlo cxra anketaSi. ras udris misi albaToba, Tu anketis Sev-seba iyo SemTxveviTi, anu anketis Sevsebis nebismieri kombinacia tolalb-aTuria.amoxsna. calkeuli moswavlisaTvis anketis Sevsebis sxvadasxva kom-binaciaTa raodenoba tolia39C -is, xolo 10 anketis Sevsebis variantebisraodenoba pirveli magaliTis analogiurad tolia10 39 ) (C -is. vinaidan er-Ti anketa SeiZleba Sevsebul iqnes nebismierad, xolo danarCen cxra anke-taSi erTi pasuxi dafiqsirebulia, xolo danarCeni ori pasuxi ki SeiZle-banebismieradamoirCesrvaSesaZlebelipasuxidan,amitomelementaru-li xdomilebebis raodenoba A-Si tolia9 2839) ( | | C C A =(Tu wyvil-is pirvel komponents SeuZlia miiRos m gansxvavebuli mniSvneloba, xo-lo meore komponents ki pirvelisgan damoukideblad -- n gansxvavebulimniSvneloba, maSin aseTi wyvilebis raodenoba namravlis principis Tana-xmad iqneba -- mn ). aqedan gvaqvs99392810 399 283931) () () () ( = ==CCCC CA P .davaleba. wina amocanaSi ipoveT albaToba imisa, rom erTerTi stu-denti dasaxelebuli iqneba k -jer ( 9 s k ).38amocana4(ortuzze). ganvixiloTpreferansisTamaSi,rodesackartis Sekvris maRali 32 karti SemTxveviT nawildeba (rigdeba) sam moT-amaSes Soris, ise rom TiTeuli Rebulobs 10 karts da ori karti inaxebasayidlebSi. rogoria albaToba imisa, rom sayidlebSi aRmoCndeba orituzi?amoxsna. orikartissxvadasxvakombinaciebisraodenoba,romelicSeiZleba aRmoCndes sayidlebSi tolia 496232 = C . kartis SekvraSi oTxituziaDdasxvadasxvakombinaciebisraodenoba,romelicmogvcemdaortuzsAtolia 624 = C . Sesabamisad, saZiebeli albaToba tolia012 , 0496623224= =CC.davaleba. davuSvaT wina amocanaSi erTerTma moTamaSem, naxa ra Ta-visi kartebi, icis, rom mas tuzi ara aqvs. Seicvleba Tu ara maSin alb-aToba imisa, rom sayidlebSi ori tuzia? gamoTvaleT es albaToba.amocana 5 (mxedvelobiT moZebnaze). davuSvaT gvaqvs N cali SemTxv-eviT dalagebuli geometriuli figura, romelTa Soris Mcali marTku-Txedia ( N M s ). moiTxoveba moiZebnos yvela marTkuTxedi, Tu Zebna warm-oebs elementebis (figurebis) saTiTaod skanirebiT fiqsaciis moculobiTerTi elementi, amasTanave xdeba dakvirvebuli elementis poziciis damax-sovreba da mas Tavidan aRar vubrundebiT. vipovoT albaToba imisa, romdamkvirvebeliSeZlebsaRmoaCinosyvela M marTkuTxediaraumetes ndakvirvebisas ( N M n ,..., = )?amoxsna. yvela SesaZlo elementarul xdomilebaTa raodenoba iqne-banNC = O| | . xelSemwyobi elementaruli xdomilebebi iseTi [na a ,...,1] erTo-bliobebia,romlebSic M adgilasganTavsebuliamarTkuTxedebi(amisSesaZleblobaTa raodenobaaMMC ), xolo danarCen M N adgilas ki aramarTkuTxedebi(amisSesaZleblobaTaraodenobaaM nM NC).amitompirvelimagaliTis analogiurad xelSemwyobi elementaruli xdomilebebis raode-noba iqneba MMCM nM NC . Sesabamisad, saZiebeli albaToba toliaMNMnnNM nM NnNM nM NMMMCCCCCC Cn P = ==) ( .39$8. jamisa da sxvaobis albaTobis formulebigavixsenoT, rom A xdomilebis albaToba ewodeba masSi Semav-vali elementaruli xdomilebebis albaTobebis jams:e=AiiP A P ) ( ) ( .aqedan Cven SegviZlia miviRoT e. w. albaTobaTa Sekrebis kano-ni: Tu A da Bxdomilebebi uTavsebadia ( = B A ), maSin xdomileb-aTa jamis albaToba albaTobebi jamis tolia) ( ) ( ) ( B P A P B A P + = + . (1)marTlac, gvaqvs:) ( ) ( ) ( ) ( ) ( ) () (B P A P P P P B A PBiAiB Aii i i+ = + = = + e e + e .Sedegi 1. sawinaaRmdego xdomilebis albaToba tolia erTs ga-moklebuli TviTon am xdomilebis albaToba:P ) ( 1 ) ( A P A P = .damtkiceba. vinaidan A A = da O =A A , amitom albaToba-Ta Sekrebis kanonis Tanaxmad gvaqvs:) ( ) ( ) ( ) ( 1 A P A P A A P P + == O = ,saidanac vrwmundebiT Sedegis sisworeSi.Sedegi 2 (sxvaobis albaTobis formula). Tu Bxdomileba iw-vevs A xdomilebas ) ( A B c , maSin A da Bxdomilebebis sxvaobisalbaToba albaTobaTa sxvaobis tolia: ) ( ) ( ) \ ( B P A P B A P = .damtkiceba. am SemTxvevaSi A xdomileba SeiZleba warmodgen-il iqnes rogorc uTavsebadi Bda B A \xdomilebebis jami:) \ ( B A B A + = .amitom albaTobaTa Sekrebis kanonis Tanaxmad gvaqvs:) \ ( ) ( ) ( B A P B P A P + = ,saidanac Cans Sedegis marTebuloba.cxadia, rom albaTobaTa Sekrebis kanonis ganzogadoeba SesaZ-lebelia wyvil-wyvilad uTavsebad xdomilebaTa nebismieri raoden-obisaTvis:TunA A A ,..., ,2 1xdomilebaTawyvil-wyviladuTavsebadisiste-maa ( = j iA A=, roca j i = 0, maSin:40 = ==niiniiA P A P1 1) ( ) ( . (2)visargebloTmaTematikuriinduqciismeTodiT.(2)formula2 = n -is SemTxvevaSi samarTliania (ix. (1)). davuSvaT, rom igi samarT-liania n -saTvis da vaCvenoT, rom samarTliani iqneba 1 + n -is SemTx-vevaSi. gvaqvs: +=+=+= =++== + = + = + =1111) 2 (11) 1 (1111) ( ) ( ) ( ) ( ) ( ) ( ) (nii nnii nniinin iniiA P A P A P A P A P A A P A P .amiT (2) Tanafardoba damtkicebulia.Teorema 1. Tu A da Bnebismieri xdomilebebia, maSin) ( ) ( ) ( ) ( B A P B P A P B A P + =.(3)damtkiceba. ganmartebis Tanaxmad e= ) () ( ) (B AiiP B A P da e e+ = +BiAii iP P B P A P ) ( ) ( ) ( ) ( .magram jamSi ) ( ) ( B P A P +albaTobebi ) (iP , roca B Ai e , gaTvali-swinebuliaorjer,xoloam ) (iP -ebisjamiaris ) ( B A P .amitomTu ( ) ( ) PA PB +jams gamovaklebT ) ( B A P -s, miviRebT swored saZieb-el ) ( B A P albaTobas.davaleba. daamtkiceT Teorema 1 albaTobaTa Sekrebis kanonisada sxvaobis albaTobis formulis gamoyenebiT.sami xdomilebis SemTxvevaSi (Tu visargeblebT e. w. de morga-nis kanoniT ) ( ) ( ) ( C B C A C B A = ) gveqneba:) 3 ( ) 3 (] ) [( ) ( ) ( ) ) (( ) ( = += = C B A P C P B A P C B A P C B A P) 3 ( ) 3 ()] ( ) [( ) ( ) ( ) ( ) ( = + + = C B C A P C P B A P B P A P= + + + = )]} ( ) [( ) ( ) ( { ) ( ) ( ) ( ) () 3 (C B C A P C B P C A P B A P C P B P A P) ( ) ( ) ( ) ( ) ( ) ( ) ( C B A P C B P C A P B A P C P B P A P + + + = .zogad SemTxvevaSi samarTliania Semdegi Tanafardoba:) ( ) 1 ( ) ( ) ( ) ( ) (1111inink j ik j ij ij inii iniA P A A A P A A P A P A P=+= = = == + + + = .41$9. pirobiTi albaTobis formulaalbaTobisTeoriaSialbaTobiscnebasTanerTadSemodise.w. pirobiTi albaTobis cneba. imisaTvis, rom davinaxoT misi saWiro-ebamoviyvanoTSemdegimartivimagaliTi:davuSvaT,romadamiansdaaviwydatelefonisnomrisoricifri. vipovoTalbaTobaimisa,rom SemTxveviT akrefili ori cifriT moxerxdeba sasurvel abonen-tTan dakavSireba?cxadia, rom am SemTxvevaSi } 9 ,..., 1 , 0 , : ) , {( = = O j i j ida namravlisprincipisTanaxmad 100 10 10 | | = = O .vinaidansasurvelabonentTanSeerTebas xels uwyobs cifrebis erTaderTi ) . ( j iwyvili, amitom sa-Ziebeli albaToba tolia 1/100-is.vTqvaT, adamians gaaxsenda rom es cifrebi iyo sxvadasxva. raiqneba maSin sasurvel abonentTan dakavSirebis albaToba? am SemTx-vevaSi elementarul xdomilebaTa sivrce iqneba {( , ) : 0,1,..., 9; i j i O = =0,1,..., 9; } j i j = = . Sesabamisad, 90 9 10 | | = = O , xolo saZiebeli albaTo-ba iqneba 1/90. rogorc vxedavT damatebiTi informaciis gaCenam Secv-ala (kerZod, gazarda) albaToba.moviyvanoT kidev erTi magaliTi. vipovoT albaToba imisa, romerTikamaTliserTjergagorebisasmovasamisjeradiqula,Tucnobilia, rom movida luwi qula? am SemTxvevaSi } 6 , 4 , 2 { = O , 3 | | = O ,xolo xelSemwyobi elenmentaruli xdomileba erTaderTia (kerZod,6-ianis mosvla) dasaZiebeli albaToba iqneba 1/3.SevxedoTigivemagaliTssxvanairad.davuSvaT,SemTxveviTimovlena mdgomareobs erTi kamaTlis erTjer gagorebaSi. SemoviRoTxdomilobebi: A -- movida samis jeradi qula da B -- movida luwiqula.naTelia,romamdros B A iqneba movidasamisjeradiluwiqula.cxadia,romamSemTxvevaSi: } 6 , 5 , 4 , 3 , 2 , 1 { = O ; } 6 , 3 { = A ;} 6 , 4 , 2 { = B ; B A ={6}, xolo albaTobis klasikuri ganmartebis Tanax-mad ki: 3 / 1 6 / 2 ) ( = = A P ; 2 / 1 6 / 3 ) ( = = B Pda 6 / 1 ) ( = B A P . advili das-anaxia, rom zemoT gamoTvlili albaToba 1/3 (romelsac bunebriviadavarqvaT A xdomilebis albaToba pirobaSi, rom adgili hqonda Bxdomilebas) formalurad SegveZlo migveRo Semdegnairad:) () (2 / 16 / 13 / 1B PB A P = = .ama Tu im movlenis analizis dros xSirad ibadeba kiTxva ragavlenas axdens raime A xdomilebis moxdenis SesaZleblobaze ra-imesxva B xdomilebismoxdena.umartivesimagaliTebia,roca Bxdomilebis moxdena aucileblad iwvevs A xdomilebis ganxorcie-lebas,e.i. A B c ,anpiriqiT, B xdomilebismoxdenagamoricxavsA xdomilebis ganxorcielebis SesaZleblobas, e. i. = B A . alba-Tobis TeoriaSi A da Bxdomilebebs Soris kavSiris dasaxasiaTe-42blad Semodis A xdomilebis pirobiTi albaTobis cneba Bxdomi-lebis mimarT.ganmarteba 1. A xdomilebis pirobiTi albaToba pirobaSi, romadgili hqonda B xdomilebas aRiniSneba ) | ( B A PsimboloTi da gan-imarteba Semdegnairad) ( / ) ( ) | ( B P B A P B A P = ,(1)Tu 0 ) ( = B P .zemoT moyvanili ukanaskneli magaliTi garkveuli azriT SeiZ-leba CaiTvalos pirobiTi albaTobis ganmartebis motivaciad. movi-yvanoT axla mosazreba, romelic gaamarTlebs A xdomilebis piro-biTi albaTobis (pirobaSi, rom moxda Bxdomileba) ) | ( B A P -s ganma-rtebas (1) TanafardobiT. davuSvaT, rom eqsperimentis Sedegad Sesa-Zlebelia moxdes A da Bxdomilebi. A xdomilebis pirobiTi six-Sire, pirobaSi rom moxda Bxdomileba vuwodoT A xdomilebis si-xSires gamoTvlils ara yvela eqsperimentis mimarT, aramed im eqsp-erimentebis mimarT, romlebSic adgili hqonda Bxdomilebas. sxvasityvebiT,romvTqvaT,Tu n --yvelaeqsperimentisraodenobaa,) (Bn -- Bxdomilebis moxdenaTa ricxvia, ) ( B An -- B Axdomi-lebis moxdenaTa ricxvia (anu im eqsperimentebis raodenoba, sadacerTdroulad moxda Ada B ), maSin pirobiTi sixSire arisn Bn B ABB Annnn/ ) (/ ) () () ( =.n -isdidimniSvnelobebisaTvisamgamosaxulebismarcxenamxares SeiZleba SevxedoT rogorc Bxdomilebis moxdenis piroba-Si A xdomilebis moxdenis pirobiTi albaTobis ) | ( B A Pmiaxloeb-iTmniSvnelobas,Sefardeba n B An/ ) ( -- aris ) ( B A P albaTobismiaxloebiTimniSvneloba,xoloSefardeba n Bn/ ) ( kiaris ) (B PalbaTobismiaxloebiTimniSvneloba.esmsjelobaudevssworedsafuZvlad pirobiTi alabaTobis (1) ganmartebas.pirobiT albaTobebs gaaCniaT Cveulebrivi albaTobebis anal-ogiuri Tvisebebi:I. nebismieri Ada BxdomilebebisaTvis ( 0 ) ( = B P ):1 ) | ( 0 s s B A P ;II. Ada BuTavsebadia ( B A =), maSin 0 ) | ( = B A P ;III. Tu Biwvevs A -s ( A B c ), maSin 1 ) | ( = B A P .Teorema 1. TunA A A ,..., ,2 1 wyvil-wyvilad uTavsebadi xdomileb-ebia,maSinnebismieri B xdomilebisaTvis( 0 ) ( = B P )samarTlianiatoloba: = ==niiniiB A P B A P1 1) | ( ] | ) [( .43damtkiceba. cxadia, rom = = = niiniiB A B A1 1) ( ) ( . garda amisa, vin-aidan xdomilebebinA A A ,..., ,2 1 wyvil-wyvilad uTavsebadi xdomilebe-bia,wyvil-wyviladuTavsebadiiqnebaxdomilebebi1 2, ,..., A BA nA B . amitom pirobiTi albaTobis ganmartebis Tanaxmad, albaTob-aTa Sekrebis kanonis gaTvaliswinebiT, vwerT:==== = = ==) () () ()] ( [) (] ) [(] | ) [(1 1 11B PB A PB PB A PB PB A PB A Pniiniiniinii = ===niiniiB A PB PB A P1 1) | () () (.amocana1. ganvixiloTojaxebi,sadacor-oribavSvia.rogo-ria albaToba imisa, rom ojaxSi orive bavSvi vaJia pirobaSi, rom:a). ufrosi bavSvi vaJia; b). erTi bavSvi mainc vaJia?amoxsna. aq elementarul xdomilebaTa sivrce aseTia} { qq qv, vq, vv, = O ,sadac v aRniSnavs vaJs, xolo q qals. CavTvaloT, rom oTxiveSedegi tolalbaTuria. SemoviRoT xdomilebebi: A -- iyos xdomile-ba, rom ufrosi bavSvi -- vaJia, xolo B -- iyos xdomileba, rom um-crosi bavSvi vaJia. maSin B A -- iqneba xdomileba, rom orive ba-vSvi vaJia, xolo B A -- ki iqneba xdomileba, rom erTi bavSvi ma-incvaJia.Sesabamisad, saZebnialbaTobebiiqneba: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 .amocana 2 (saukeTesos SerCevaze). mocemulia m obieqti gadan-omriliricxvebiT m ,..., 2 , 1 ,amasTanaveise,romvTqvaT,obieqti#1klasificirdeba rogorc saukeTeso, . . . , obieqti #m ki rogorcyvelaze uaresi. igulisxmeba rom obieqtebi Semodian drois momen-tebSi m ,..., 2 , 1SemTxveviTi rigiT (anu yvela SesaZlo m! gadanacvl-eba tolalbaTuria).damkvirvebels SeuZlia ori maTganis Sedareb-iTTqvasromeliaukeTesidaromeliuaresi.saWiroasaukeTesosSerCeva im pirobiT rom obieqtebi warmoidgineba saTiTaod da ukug-debulis damaxsovreba xdeba damkvirveblis mier. ar SeiZleba sauke-Tesod miCneul iqnes is obieqti, romelic dakvirvebuli obieqtebid-an erTze mainc uaresia an romelic ukve iqna ukugdebuli. vTqvaT,damkvirvebelma obieqti SearCia k -ur nabijze ( m k s ), anu daTvalie-rebuli obieqtebidan ukanaskneli aRmoCnda yvela winaze ukeTesi daamitom moxda misi SerCeva. rogoria albaToba imisa, rom amorCeuli44obieqti iqneba saukeTeso mTeli erTobliobidan rogorc ukve ganx-ilul, ise jer kidev ganuxilav obieqtebs Soris?amoxsna. SemoviRoTxdomilebebi: Aiyosxdomileba,rom k -uri obieqti saukeTeoa yvela arsebulm obieqts Soris da Biyosxdomileba,rom k -uriobieqtisaukeTeoadakvirvebul k obieqtsSoris. gasagebia, rom mosaZebnia pirobiTi albaToba ) | ( B A P .vinaidan B A c , amitom A B A = da ) ( ) ( A P B A P = . Sesabamisad,pirobiTi albaTobis ganmartebis Tanaxmad) ( / ) ( ) | ( B P A P B A P =vinaidan obieqtebis yvela SesaZlo gadanacvlebebi tolalba-Turia,amitomalbaTobisklasikuriganmartebisTanaxmadadvilidasanaxia, romk kkB P1!)! 1 () ( ==dam mmA P1!)! 1 () ( == .Sesabamisad, m k B A P / ) | ( = .strategia. SeiZleba damtkicdes, rom saukeTeso obieqtis amo-rCevis optimaluri strategia mowyobilia Semdegnairad. avRniSnoTsimboloTi*miseTi naturaluri ricxvi, romlisTvisac samarTli-ania utoloba1111111 1* *+ +< s+ +m m m m.saukeTesoobieqtisarCevisoptimaluristrategiamdgomare-obsimaSi,romdavakvirdeTdaukuvagdoTpirveli 1* m obieqtida Semdeg gavagrZeloT dakvirveba iseT*momentamde, roca pirve-lad gamoCndeba yvela winamorbedze ukeTes obieqti.magaliTad,Tu 10 ,..., 1 = m ,maSin*m -isSesabamisimniSvnelobe-bia:m 1 2 3 4 5 6 7 8 9 10m-optimaluri1 1 1 1 2 2 2 3 3 4sakmaod didi m-isaTvis e m m /*~(sadace -- neperis ricxvia,2.718 e~ ) da saukeTeso obieqtis arCevis albaToba daaxloebiT to-lia 1/ 0.368 e~(Tumca, erTi SexedviT bunebrivia mogvCveneboda, romgansaxilveliobieqtebis mraodenobiszrdasTanerTad,saukeTe-so obieqtis arCevis albaToba unda wasuliyo nulisaken). e. i. sauk-eTesoobieqtisarCevisoptimaluristrategiamdgomareobsimaSi,rom unda ukuvagdoT obieqtebis saerTo raodenobis daaxloebiT me-samedi da Semdeg avirCioT pirveli iseTi obieqti, romelic yvelawinaze ukeTesia.45$10. namravlis albaTobis formulavigulisxmoT, rom 0 ) ( B P , maSin pirobiTi albaTobis formu-lidan) ( / ) ( ) | ( B P B A P B A P SegviZlia davweroT, rom) | ( ) ( ) ( B A P B P B A P . (1)analogiurad, Tu vigulisxmebT, rom 0 ) ( A P , maSin ) | ( A B Ppi-robiTi albaTobis formulidan miviRebT, rom) | ( ) ( ) ( A B P A P B A P .e. i. ori xdomilebis namravlis albaToba tolia erT-erTis albaT-oba gamravlebuli meoris pirobiT albaTobaze pirobaSi, rom mox-da pirveli.cxadia,romsamixdomilebisSemTxvevaSi(Tu 0 ) ( B P da0 ) ( C P ) gveqneba:) 1 ( ) 1 (] | ) [( ) ( ] ) [( ) ( C B A P B A P C B A P C B A P] | [ ) | ( ) () 1 (B A C P A B P A P .analogiurad,nA A A ,..., ,2 1 xdomilebebisaTvis gveqneba, rom:) | ( ) | ( ) ( ) (1 1 1 2 1 2 1 n n nA A A P A A P A P A A A P . (2)davaleba 1. miuTiTeT ra SemTxvevaSia samarTliani (2) Tanafa-rdoba da daamtkiceT igi.amocana 1 bednier bileTebze). 25 sagamocdo bileTidan 5 be-dnieria, xolo danarCeni 20 ara bednieri. romel students aqvsbednieri bileTis aRebis meti albaToba: vinc pirveli iRebs bil-eTs, Tu vinc meore iRebs bileTs?amoxsna. es amocana Cven ukve amovxseniT albaTobis klasikuriganmartebis gamoyenebiT. amovxsnaT axla igi elementarul xdomile-baTa sivrcis axleburi SemotaniTa da namravlis albaTobis formu-lis gamoyenebiT. winaswar SemoviRoT Semdegi xdomilebebi: Aiyosxdomileba,rompirvelmastudentmaaiRibednieribileTi,xoloBiyos xdomileba, rom meore studentma aiRi bednieri bileTi. ma-Sincxadia,romelementarulxdomilebaTasivrceSedgebaoTxixdomilebisagan} , , , { B A B A B A B A .albaTobisklasikuriganmartebisTanaxmad 5 / 1 25 / 5 ) ( A P ,xolo ( ) 20/ 25 4/ 5 PA . meores mxriv, amocanis Sinaarsidan gamomd-inare, Tu cnobilia, rom pirvelma studentma aiRo bednieri bile-Ti, maSin albaToba imisa rom meore studenti aiRebs bednier bil-eTsisevSeiZlebagamoviTvaloTalbaTobisklasikuriganmarteb-iT: am SemTxvevaSi yvela SesaZlo SedegTa raodenobaa 24, xolo xe-46lSemwyob elementarul xdomilebaTa raodenoba ki mxolod 4 (rad-gan 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 PB A . amitom ori xdomilebis namravlis albaTobis formu-lis 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 ;6 / 1 24 / 5 5 / 4 ) ( B A Pda 30 / 19 24 / 19 5 / 4 ) ( B A P(SevniSnavT, rom Cven aq mxolod sisrulisaTvis gamovTvaleT yvelaSesaZlo pirobiTi da namravlis albaTobebi).cxadia,rom ) ( ) ( B A B A B .amitomalbaTobaTaSekrebiskanonis Tanaxmad)) ( ( 15 6 / 1 30 / 1 ) ( ) ( ) ( A P B A P B A P B P .amocana 2. yuTSi m burTia, maT Soris nTeTria. vipovoT al-baToba imisa, rom yuTidan ori burTis mimdevrobiT dabrunebis ga-reSe amoRebisas: a). pirveli burTi TeTria; b). meore burTi TeTria;g). orive birTvi TeTria.amoxsna.iA iyos xdomileba, rom i -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 .davaleba 2. davuSvaT, rom Sesamowmebeli jgufis 1% avadmyo-fia, xolo danarCeni 99% ki janmrTelia. adamianebis SerCeva xdebaSemTxveviT da amitom( ) 1% 0.01 P avadmyofida ( ) 99% 0.99 P janmrTeli .vigulisxmoT, rom im SemTxvevaSi, roca testireba utardeba adami-ans,romelsacaraaqvsavadmyofoba,maSin1%-iaalbaTobaimisa,rom miviRoT mcdari dadebiTi Sedegi, e.i.% 1 ) ( janmrTeli | dadebiTi Pda % 99 ) ( janmrTeli | uaryofiTi P .da bolos, davuSvaT, rom im SemTxvevaSi, roca testireba utardebaavadmyof adamians,maSin 1%-ia albaToba imisa, rom miviRoT mcda-ri uaryofiTi Sedegi, e.i.% 1 ) ( avadmyofi | uaryofiTi Pda % 99 ) ( avadmyofi | dadebiTi P .47gamoTvaleTalbaTobaimisa,rom:a).adamianijanmrTelia,xolotestma aCvena uaryofiTi Sedegi; b). adamiani avadmyofia, xolo tes-tma aCvena dadebiTi Sedegi; g). adamiani janmrTelia, xolo testmaaCvena dadebiTi Sedegi; g). adamiani avadmyofia, xolo testma aCvenauaryofiTi Sedegi.amocana 3. yuTs aqvs nujra. albaToba imisa rom burTi arisamujrebidanerT-erTSitolia p -si.ipoveTalbaTobaimisa,romburTi aris i -ur ujraSi, Tu cnobilia, rom burTi TiTeul ujraSiSesaZlebelia iyos Tanabari albaTobebiT?amoxsna.iA iyos xdomileba, rom burTi aris i -ur ujraSi. Aiyos am xdomilebebis gaerTianebaniiA A1 . pirobis Tanaxmadp A P ) (da n A A Pi/ 1 ) | ( .amitomn p n p A A P A P A A P A Pi i i/ / 1 ) | ( ) ( ) ( ) ( .48$11. damokidebuli da damoukidebeli xdomilebebialbaTobis TeoriaSi or A da B xdomilebas ewodeba damouk-idebeli,TuerT-erTimaTganismoxdenaarcvlismeoremaTganismoxdenis albaTobas. winaaRmdeg SemTxvevaSi am xdomilebebs ewode-baT damokidebuli. im SemTxvevaSi, roca erT-erTi xdomilebis alb-aToba aranulovania, vTqvaT, 0 ) ( = B P , maSin gvaqvs Semdegiganmarteba 1. A xdomilebas ewodeba B xdomilebisagandamou-kidebeli, Tu P ) ( ) | ( A P B A P = , (1)xolo Tu ) ( ) | ( A P B A P = , maSin gvaqvsdamokidebuli xdomilebebi.Tu gavixsenebT pirobiTi albaTobis ganmartebas) ( / ) ( ) | ( B P B A P B A P = ,maSin (1) Tanafardobidan miviRebT, rom) ( ) ( ) ( B P A P B A P = . (2)piriqiT, im SemTxvevaSi, roca 0 ) ( = B P , (2) Tanafardobidan mii-Reba (1) Tanafardoba.SeniSvna. zogierT saxelmZRvaneloSi xdomilebaTa damoukide-bloba ganimarteba (2) TanafardobiT ((1) da (2) eqvivalenturia, Tu0 ) ( = B P ). mas aqvs is upiratesoba, rom misi gamoyeneba SesaZlebeliamaSinac, roca xdomilebebis albaTobebi nulovania (magaliTad, Se-uZlebelixdomilebadamoukidebelianebismierixdomilebisagan A P( )= ( P )=0= ( ) ( P A P )) dagardaamisa,igisimetriulia A daB -s mimarT (aseT SemTxvevaSi, Tu A damoukidebelia B -sagan, maSinBdamoukidebelia A-sagan), magram (1) Tanafardobis upiratesoba isaris, rom iqidan Cans aRniSnuli ganmartebis Sinaarsi. cxadia, rom aucilebeli xdomileba damoukidebelia nebismie-ri xdomilebisagan:P ) (1) () () () | ( A PA PPA PA P = =OO = O ,rac bunebrivia imis gamo, rom am SemTxvevaSi piroba ar warmoadg-ens damatebiT informacias (Cven isedac vicodiT, rom aucilebelixdomileba es is xdomilebaa, romelic yovelTvis xdeba) da amitomalbaToba arc unda Secvliliyo.vinaidan A da Bxdomilebebis damoukidebloba niSnavs, rominformacia A -s moxdenis Sesaxeb ar cvlis B -s albaTobas, bunebr-ivia, rom informaciam A -s ar moxdenis Sesaxeb agreTve ar unda Se-cvalos B -s moxdenis albaToba. marTlac samarTliania SemdegiTeorema 1. Tu A da B xdomilebebidamoukideblia,maSinxdomilebebi A da BagreTve damoukidebelia.damtkiceba. advilidasanaxia,rom ) ( \ B A B B A = (vinaidan) ( \ B A B B A = ), amitom sxvaobis albaTobis formulis gamoyenebiTTeoremis pirobebSi vRebulobT, rom49) ( ) ( ) ( )) ( 1 ( ) ( ) ( ) ( ) ( ) ( ) () 2 (B P A P B P A P B P A P B P B A P B P B A P = = = = .Sedegi. Tu A da Bxdomilebebi damoukideblia, maSin damou-kidebelia A da B .magaliTi 1. kartebis nakrebidan (romelSic 36 kartia) SemTxv-eviT iReben erT karts. ganvixiloT xdomilebebi: A iyos xdomile-ba, rom amoRebuli karti agurisa, xolo Biyos xdomileba, romamoRebuli karti mefea. arian Tu ara es xdomilebebi damoukidebe-li?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 2. davuSvaT, vagorebT or saTamaSo kamaTels. ganvix-iloT xdomilebebi: A -- pirvel kamaTelze movida kenti qula, B --meore kamaTelze movida kenti qula, C -- orive kamaTelze mosulqulaTa jami kentia. gavarkvioT am xdomilebebis damoukideblobissakiTxi.cxadia, rom 2 / 1 6 / 3 ) ( ) ( = = = B P A P , xolo 4 / 1 36 / 3 3 ) ( = = B A P .amitom A da Bxdomilebebi damoukideblia.davaleba 1. SeamowmeT, rom 2 / 1 ) ( = C P .SevniSnoT, rom A da Bxdomilebidan erT-erTis moxdenis pi-robaSi Cxdomileba xdeba maSin da mxolod maSin, roca Sesabamis-ad, an pirvel an meore kamaTelze movida luwi qula, anu gvaqvs Ta-nafardobebi:B A C A = da B A C B = .vinaidan, Teorema 1-is ZaliT, xdomilebebi A da Bda A daBagreTve damoukideblebia, amitom4 / 1 ) ( ) ( ) ( = = B P A P B A Pda 4 / 1 ) ( = B A P .Sesabamisad,4 / 1 ) ( ) ( = = B A P C A Pda 4 / 1 ) ( ) ( = = B A P C B P .esTanafardobebiki, 2 / 1 ) ( = C P albaTobisgaTvaliswinebiT,niSnavs,romdamoukideblebia A da C da B da C xdomilebaTawyvilebic.ganmarteba2. xdomilebaTaerTobliobasnA A A ,..., ,2 1ewodebawyvil-wyvilad damoukidebeli Tu nebismieri ori xdomileba am er-Tobliobidan damoukidebelia, anuj i A P A P A A Pj i j i= = ), ( ) ( ) ( .winamagaliTSiCvenvnaxeT,romxdomilebebi A, B da Cwyvil-wyvilad damoukidebelia.ganmarteba3. xdomilebaTaerTobliobasnA A A ,..., ,2 1ewodebaerToblivaddamoukidebeli Tunebismieri n k s raodenobisaTvis50da erTmaneTisagan gansxvavebuliki i i ,..., ,2 1indeqsebisaTvis, romelTa-gan TiToeuli icvleba erTidan n -mde:) ( ) ( ) ( ) (2 1 2 1 k ki i i i i iA P A P A P A A A P = .cxadia, rom Tu xdomilebebi erToblivad damoukidebelia, ma-Sin isini iqnebian wyvil-wyvilad damoukidebeli. piriqiT, ki sazog-adod swori ar aris. amis magaliTad gamodgeba magaliTi 2.davaleba 2. SeamowmeT, rom xdomilebebi magaliTi 2-dan ar ar-ian erToblivad damoukidebeli.magaliTi 3. davuSvaT, vagdebT sam monetas. SemoviRoT xdomi-lebebi: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 xdomilebadamoukidebelia,xolosamiveerTaddamokidebulia,vinaidanTuCven gvecodineba rom magaliTad,1A da2A moxda, maSin Cven zustadviciT, rom3A agreTve moxda.davaleba 3. SeamowmeT, rom xdomilebebi1A ,2A da3A wyvil-wyvilad damoukidebelia.Teorema 2. TunA A A ,..., ,2 1 xdomilebebi erToblivad damoukide-belia, maSin xdomilebebinA A A ,..., ,2 1 agreTve erToblivad damoukid-eblebia.davaleba 4. daamtkiceT Teorema 2.amocana 1 (dabadebis dReebze). vipovoT albaToba imisa, konkr-etuli skolis rom 150 moswavlidan erTi mainc dabadebulia mocem-ul fiqsirebul 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, romiAxdomilebebi erToblivad damoukidebelia da365 / 1 ) ( =iA P .gardaamisa,iiA A1501 =. = damaSasadame,sapovneliadamoukidebel xdomilebaTa gaerTianebis albaToba. gadavideT sawi-naaRmdego xdomilebaze da visargebloT de-morganis kanoniT. maSingvaqvs:) ( ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) (115011501n iiiiA P A P A P A P A P A P = = . = == =.ramdenadac 365 / 1 1 ) ( =iA P , Tu visargeblebT niutonis binomisformuliT== +njj jnnx C x1) 1 ( , vRebulobT51 + + =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 < ~ < ,amitom mwkrivis niSancvladobis gamo, Tu gadavagdebT mwkrivis wev-rebs dawyebuli me-5 wevridan (mwkrivebis zogadi Teoriidan gamomd-inare), SesaZlebelia vamtkicoT, rom2 3(0.41) (0.41)( ) 0.41 0.41 0.08 0.01 0.34.2 6PA ~ + ~ + =52$12. sruli albaTobis formulaxdomilebaTaerTobliobasnA A A ,..., ,2 1ewodeba xdomilebaTasrulisistema, Tuesxdomilebebiwyvil-wyviladuTavsebadiadamaTi gaerTianeba emTxveva aucilebel xdomilebas: = j iA A , rocaj i =da O = =iniA1. sxva sityvebiT, elementarul xdomilebaTa sivrcedayofilia (daxleCilia) TanaukveT nawilebad. qvemoT moyvanil nax-azze elementarul xdomilebaTa sivrce warmodgenila marTkuTxed-is saxiT da xdomilebaTa sruli sistema Sedgeba xuTi TanaukveTiE D C B A , , , ,xdomilebisagan.xdomilebaTa sruli sistemaa nebismieri A xdomileba da misisawinaaRmdego A xdomileba, vinaidan A A = da O =A A .qvemoT Cven moviyvanT formulas, romelsac sruli albaTobisformula ewodebadaromelicwarmoadgensZiriTadsaSualebasrTuli xdomilebebis albaTobebis gamosaTvlelad pirobiTi alba-Tobebis saSualebiT.Teorema 1.TunA A A ,..., ,2 1 xdomilebaTa sruli sistemaa iseTi,rom misi TiToeuli xdomilebis albaToba aranulovania ( ( ) 0,iPA >1, 2,..., i n = ),maSinnebismieri B xdomilebisalbaTobagamoiTvlebaformuliT) | ( ) ( ) (1iniiA B P A P B P== , (1)romelsac sruli albaTobis formula ewodeba.damtkic