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1 !!!!!!! bmcbupcjt!ufpsjb!

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3 ivae javaxisvilis saxelobis Tbilisis saxelmwifo uiversiteti fmj{cbs!obebsbjb-! sfwb{!bctbwb-!{fwjobs!gbdbdjb! bmcbupcjt!ufpsjb!

4 wigi warmoadges albatobis Teoriis saxelmzrvaelos, romelsic albatobis Teoriis masalis gadmocema eyrdoba olmogorovis aqsiomatiur midgomas. saxelmzrvaelos meore gamocema (pirveli gamoica 005 w.) gadamusavebulia, gasworebulia zogierti uzustoba, mravali oreqturuli Secdoma da gazrdilia savarjiso amocaata raodeoba. saxelmzrvaelo, romelsac Ta ertvis savarjiso amocaebi, gautvilia uiversitetebis matematiis mimartulebis studetebistvis. redaqtori: iv. javaxisvilis saxelobis Tbilisis saxelmwifo uiversitetis asoc. profesori, fizia-matematiis mecierebata doqtori b. dowviri recezetebi: iv. javaxisvilis saxelobis Tbilisis saxelmwifo uiversitetis asoc. profesori, fizia-matematiis mecierebata doqtori o. Rloti a. weretelis saxelobis qutaisis saxelmwifo uiversitetis sruli profesori, fizia-matematiis mecierebata doqtori g. soxaze Tbilisis uiversitetis gamomcemloba, 009 e. adaraia, r. absava, m. facacia ISBN

5 S e s a v a l i mecierebisa da teqiis TiTqmis yvela sferosi xsirad gvxvdeba SemTxvevebi, rodesac esa Tu is eqsperimeti (cda) a davirveba SeiZleba ertair pirobebsi mravaljer gameordes. c d a vuwodot pirobata raime gasazrvruli G ompleqsis (ertobliobis) gaxorcielebas. cdis Sedegi SeiZleba iyos ricxvi a raime abstraqtuli elemeti. Tu cdis Sedegi eutvis raime A simravles, masi ambobe, rom adgili aqvs A x d o m i l o b a s, Tu pirobata G ompleqsis gaxorcielebisas A xdomilobas adgili eqeba aucileblad, ada misi moxdea SeuZlebelia, masi, Sesabamisad, A xdomilobas uwodebe a u c i l e b e l s a S e u Z l e - b e l s. magalitad, G pirobata ompleqsi iyos ori satamaso amatlis gagoreba, masi xdomiloba A={i+j; i, j, 6} _ aucilebeli xdomilobaa, xolo B={i+j3} _ SeuZlebeli. xdomilobas, romelic pirobata garveuli ompleqsis gaxorcielebisas xa xdeba, xa ara, ewodeba S e m T x v e - v i T i. moviyvaot SemTxveviTi xdomilobis magalitebi:. G pirobata ompleqsi iyos moetis ertjer agdeba, xolo A _ gerbis mosvla.. G pirobata ompleqsi iyos raime fiziuri sididis gazomva, xolo A _ gazomvis Sedegi a,b. 3. G pirobata ompleqsi iyos q.tbilissi dabadebul bavsvta sqesis registracia, xolo A _ axalsobili vajia. 4. pirobata G ompleqsi iyos burtis amoreba yutida, romelsiac m TeTri da m witeli burtia, xolo A _ amorebuli burti witelia. moyvaili magalitebida Cas, rom pirobata garveuli ompleqsis gaxorcielebamde ar SegviZlia wiaswar vtqvat, cdis romel oretul Sedegs eqeba adgili, e.i. mati Sedegebi calsaxad ar gaisazrvreba pirobata ompleqsis gaxorcielebit. albatobis Teoria swored aset eqsperimetta matematiur modelebs Seiswavlis. caleuli cdis a 5

6 davirvebis SedegiT Zelia raime aozomierebis SemCeva, magram Tu gavixilavt cdata mimdiareobas mtliaobasi, masi SesaZlebelia garveuli aozomierebis SemCeva, romelic s t a t i s t i u r i m d g r a d o b i s, au fardobit sixsireta mdgradobis TvisebaSi vlideba. es Tviseba ariwereba Semdegairad: A xdomilobis fardobiti sixsire -jer Catarebul cdasi ewodeba W (A)= A / wilads, sadac A im cdata raodeobaa, romlis drosac adgili hqoda A xdomilobas. cxadia, rom 0W (A), xolo Tu A aucilebeli a SeuZlebeli xdomilobaa, masi, Sesabamisad, W (A)= da W (A)=0. fardobit sixsireta mdgradoba imasi mdgomareobs, rom Tu samarisad didia, fardobiti sixsire W (A) mcired irxeva (-is cvlilebisas) garveuli P(A) ricxvis gar- Semo, e.i. axlos iqeba P(A) ricxvta. P(A) ricxvs A xdomilobis a l b a T o b a s uwodebe. is aris A xdomilobis moxdeis SesaZleblobis raodeobrivi z o m a. atqvamis sailustraciod gavixilot magaliti. avagot grafii: abscista RerZze arvisot cdata ricxvi, xolo ordiatta RerZze fardobiti sixsire W (A)= /. SeviSavT, rom texili, romelic aertebs wertilebs (, W (A)) -is zrdasta ertad, Zalia swrafad uaxlovdeba wrfes W (A)=/. W (A) / 0 6

7 fardobit sixsireta mdgradobis es Tviseba SemCeuli iyo jer idev XVII sauuesi albatobis Teoriis s a w y i - s e b i s Semqmelebis mier. ase, magalitad, biufoma (XVII s.) moeta aagdo 4040-jer, mat Soris gerbi (A xdomiloba) movida A =048-jer da, masasadame, W (A)=0,508. pirsoma igive cda Caatara: =4000, A =0, e.i. W (A)=0,5005. xdomilobis albatobis zemot mocemuli gamarteba ar aris matematiurad macri. ragid didi ar uda iyos cda- Ta ricxvi, Cve ver vipovit P(A) albatobas zustad. P(A) ar warmoadges fardobit sixsireta mimdevrobis zrvars Cveulebrivi gagebit. martlac, fardobit sixsireta mimdevroba {W (A)} cdata erti seriisatvis gasxvavebuli iqeba, rogorc wesi, cdata meore seriis Sesabamis fardobit sixsireta mimdevrobisaga. garda amisa, siamdvilesi Cve gveqeba ara fardobit sixsireta usasrulo mimdevroba, aramed mxolod misi elemetebis sasruli raodeoba. amgvarad, rogorc Cas, albatoba uda gaisazrvros sxvaairad, magram ise, rom fardobiti sixsiris arisuli Tviseba SearCuebul iqas, e.i. raime azrit fardobiti sixsire cdata ricxvis zrdisas uda uaxlovdebodes Sesabamisi xdomilobis albatobas (ra azrit uda iyos miaxloeba, Cve amas SemdgomSi gavecobit!). 7

8 Tavi I elemetarul xdomilobata disretuli sivrce 8 $. albatobis gasazrvra da Tvisebebi albatobis Teoriis metodebit raime G cdasta daavsirebuli realuri amocais Seswavlisas, upirveles yovlisa, gamoyofe cdis SedegTa srulad armwer simravles, e.i. gamoyofe SemTxveviTi eqsperimetis yvela SesaZlo SedegTa simravles. am simravlis yovel elemets elemetaruli xdomiloba ewodeba, xolo TviT simravles _ elemetarul xdomilobata sivrce. sivrces martivi saxe aqvs im SemTxvevaSi, rodesac Sesabamisi eqsperimetis Sesa- Zlo SedegTa simravle sasruli a Tvladia. am TavSi mxolod aset s i v r c e e b s gavixilavt. SemoviRoT Semdegi: gasazrvra.. elemetarul xdomilobata disretuli sivrce ewodeba ebismier sasrul a Tvlad simravles ={,,...} magaliti.. vtqvat, moetas agdebe ertjer. am cdis armweri elemetarul xdomilobata sivrce Sedgeba ori elemetisaga ={, }, =s, =g. sadac s isavs safasurs, xolo g _ gerbs. magaliti.. vtqvat, moetas agdebe orjer. Sesabamisi elemetarul xdomilobata sivrce iqeba aseti ={,, 3, 4 }, sadac =gg, =gs, 3 =sg, 4 =ss. magaliti.3. ori da piri rigrigobit agdebs moetas. vtqvat, TamaSs iwyebs da igebs is, visac pirvelad mouva safasuri. am eqsperimetis Sesabamisi elemeta-

9 rul xdomilobata sivrce Sedgeba elemetta Tvladi raodeobisaga: ={,,...}, sadac =s, =gs,..., gg...gs,... elemetarul xdomilobata disretuli sivrcis ebismier A qvesimravles S e m T x v e v i T i x d o m i l o b a ewodeba. A-xdomilobis sawiaarmdego xdomiloba ariiseba A - Ti da is isavs A-s armoxdeas, e.i. A { : A}. Tu A xdomilobis yoveli elemeti eutvis B xdomilobas, masi were AB, (ABAB). A da B xdomilobata g a e r T i a e b a ewodeba yvela im elemetarul xdomilobata simravles, romlebic A da B xdomilobida erts maic eutvia da ariiseba AB a A+B simboloti. ori A da B xdomilobis T a a v e T a (amravli) ewodeba yvela im elemetarul xdomilobata simravles, romlebic eutvia rogorc A, ise B xdomilobas da ariiseba AB a AB simboloti. A da B xdomilobata s x v a o b a s uwodebe A xdomilobis yvela im elemetarul xdomilobata simravles, romlebic B-s ar eutvia da ariiseba A\B simboloti, a A-B. -s uwodebe a u c i l e b e l xdomilobas, xolo cariel simravles _ S e u Z l e b e l s. cxadia, rom A \ A, A \ B A B. xdomilobebisatvis adgili aqvs: a) A B A B ; b) A B A B. 9

10 gasazrvra.. A da B xdomilobebs u T a v s a d i ewodeba, Tu AB=. magaliti 4. vtqvat, cda mdgomareobs satamaso amatlis ertxel gagorebasi. Sesabamisi elemetarul xdomilobata sivrce iqeba = {,, 3, 4, 5, 6 }, sadac =, =,, 3,4,5,6. 0 vtqvat, A={ 6 }, B={ 3 }, C={, 4, 6 } da D={ 3, 6 }. advili misaxvedria, rom AC, AB=D, CD=A, AB=, {,, }. C 3 5 xdomilobebze zemot moyvaili operaciebi tradiciulad albatobis Teoriis termiebsi gamoitqmis Semdegairad: Tu A xdomilobis moxdea iwvevs B xdomilobas, masi vityvit, rom B moicavs A-s da es garemoeba ase Caiwereba AB. magalitad, A iyos xdomiloba _ SemTxveviT arceuli qali dedaa, xolo B _ SemTxveviT arceuli adamiai qalia. cxadia, AB. ori A da B xdomilobis g a e r T i a e b a AB (jami) isavs xdomilobas, romelic xdeba masi da mxolod ma- Si, rodesac A da B xdomilobebida erti maic xdeba. A da B xdomilobata TaaveTa AB (amravli) ewodeba xdomilobas, romelic xdeba masi da mxolod masi, rodesac A da B ertad xdeba. A da B xdomilobata A\B s x v a o b a ewodeba xdomilobas, romelic xdeba masi da mxolod masi, rodesac xdeba A da ar xdeba B. gasazrvra.3. ambobe, rom mocemulia elemetarul xdomilobata albatobebi, Tu -ze gasazrvrulia arauaryofiti ricxviti fuqcia P(), iseti, rom P ( ).

11 gasazrvra.4. A xdomilobis P(A) albatoba ewodeba A-Si Semavali elemetaruli xdomilobebis albatobata jams: P (A) P( ). A gasazrvra.5. sameuls (,,P), sadac elemetarul xdomilobata sivrcis yvela qvesimravleta lasia, ewodeba disretuli a l b a T u r i m o d e l i, au disretuli a l b a T u r i sivrce. SemTxveviTi eqsperimetis Sesabamisi albaturi modelis agebisas pirvel rigsi gamoyofe sivrces; Semdeg -is yovel elemets, raime mosazrebida gamomdiare, miawere albatobas. albatobis mocema ufro Zelia, vidre sivrcis ageba. Tu A xdomilobis moxdeis faqtze davirveba SesaZlebelia, aset davirvebas ver CavatarebT mis Sesabamis P(A) albatobaze. P(A) albatobis obieqtur arsebobaze migvititebs fardobiti sixsiris mdgradoba, romelic gasazrvruli gvqoda SesavalSi.

12 albatobis Teorias ar aiteresebs albaturi modeli (,,P) ramdead sworadaa agebuli. modelis sisworis saitxs, e.i. ramdead etaxmeba raime eqsperimetis Sedegs, romlistvisac iyo igi Sedgeili, ivlevs m a T e m a t i u - r i s t a t i s t i a. SeviSoT, rom yvela eqsperimeti ar SeiZleba ariweros disretul elemetarul xdomilobata sivrcit. ase, magalitad, Sesavlis meore magalitsi moyvaili eqsperimetis Sedegebi SeiZleba avsebde raime [a,b] itervals ricxvta RerZze. cxadia, am eqsperimetis Sesabamisi elemetarul xdomilobata sivrce =[a,b] ar aris disretuli. aseve, Tu G pirobata ompleqsi (cda) avadmyofisaga eleqtroardiogramis arebasi mdgomareobs, masi, cxadia, eqsperimetis Sedegi raime fuqcioaluri sivrcis elemets warmoadges. aseti eqsperimetebisatvis sawiroa aigos ufro zogadi albaturi modeli. zemot moyvaili gasazrvrebebida SegviZlia martivad davadgiot albatobis Semdegi Tvisebebi:. P()=, P()=0;. P(A B) P( ) P(A) P(B) P(A B) ; A B 3. (aditiurobis Tviseba) P(AB)=P(A)+P(B), Tu AB=; 4. P(A) P(A) ; 5. Tu A =,... da A A j =. j, masi P( A ) P(A ). (.) (.) toloba gamomdiareobs P( A ) P(A ) (.)

13 tolobida ((.) toloba Tavis mxriv miireba me-3 Tvisebida iduqciis wesit) da iqida, rom P( A ) 0, roca. j j (.)-s albatobis (sigma)_aditiurobas, au Tvladad aditiurobas uwodebe. 6. (axevrad aditiuroba). meore Tvisebida gamomdiareobs, rom P(A B) P(A) P(B). (.3) (.3) utolobas adgili aqvs xdomilobata Tvladi raodeobistvisac, au albatoba _axevrad aditiuria: P( A ) P(A ). j j j 7. Tu AB, masi P(B\A)=P(B)_P(A). aqeda gamomdiareobs albatobis m o o t o u r o b a : Tu AB, masi P(A)P(B). amgvarad, -7 Tviseba gvicveebs, albatoba ise iqa gasaz- Rvruli, rom mas SerCeoda SesavalSi gamartebuli xdomilobis fardobiti sixsiris advilad sacveebeli Tvisebebi: W ()=, W ()=0, j W ( A ) W (A ), Tu A A i j. i j $. lasiuri sqema vtqvat, elemetarul xdomilobata sivrce Sedgeba N elemetisga ={,,...,N N } da yvela elemetaruli xdomiloba tolalbaturia, e.i. P( )=P( )=...N=P( N ). gasazrvra.3-is ZaliT P( )=/N, =,,...,N. 3

14 Tu axla A xdomilobasi Semaval elemetarul xdomilobata raodeobas N(A) simboloti arvisavt,.4. gasazrvrida miireba gasazrvra.. Tu elemetarul xdomilobata sivrce Sedgeba N tolalbaturi elemetaruli xdomilobisaga, masi 4 N(A) P(A), (.) N sadac A ( lassi N elemetia). albatur models (,,P) uwodebe l a s i u r s. (.) tradiciulad iitxeba ase: lasiuri sqemis SemTxvevaSi A xdomilobis albatoba tolia xelsemwyob SemTxvevaTa N(A) ricxvi gayofili yvela SesaZlo SemTxvevaTa N=N() ricxvze. mas albatobis lasiur gasazrvrasac uwodebe. lasiuri sqema iseti eqsperimetis armweria, romelsac gaacia ertairad mosalodel SedegTa sasruli raodeoba. gavixilot z o g i e r T i d i s r e t u l i a l b a T u r i m o d e l i.. moetis ertjer agdebis SemTxvevaSi (igulisxmeba moeta sworia ) albatur models (.)-is ZaliT eqeba saxe: ={g,s}; P({g})=/, P({s})=/, N()=. moetis orjer agdebis SemTxvevaSi ={gg,gs,sg,ss}, P(gg)=P(sg)=P(gs)=P(ss)=/4, N N()=4. A xdomilobis _ ertxel maic dajdeba gerbi _ alba- Toba (.)-is ZaliT tolia 3/4, e.i. P(A)=3/4 da a.s. moetis -jer agdebis SemTxvevaSi i _ ={}, =(,,... ), sadac = g a s N()=, P()= -.

15 . magaliti.3-is Sesabamis albatur models eqeba saxe: ={,,...}, P( )=/,... P( )= -... vipovot A xdomilobis _ piri moigebs _ albatoba. cxadia, A={, 3,... +,...}, da gasazrvra.4-is ZaliT gveqeba P(A) 0 P( ) / vtqvat, mocemulia raime elemetta sasruli simravleebi: E i (i) (i) { a,..., a }, i,,...,r. i E i saxis simravles vuwodot i moculobis g e e r a - l u r i ertoblioba. gavixilot Semdegi saxis G eqsperimeti: TiToeuli E i simravlida SemTxveviT virebt TiTo elemets. am eqsperimetis Sesabamisi albaturi modeli iqeba: () (r) { }, ( a,..., a ), j,,r, N N( ) j j 3 r. (daamticet!), P()=/N(). 4. vtqvat, mocemulia geeraluri ertoblioba E={a,...,a }. E geeraluri ertobliobida s moculobis SearCevi ewodeba d a l a g e b u l mimdevrobas ( a j,..., a ) j s. SerCeva SeiZleba orairad vawarmoot: vtqvat, Cvei G eqsperimeti mdgomareobs imasi, rom E simravlida SemTxveviT virebt romelirac a j elemets, Semdeg E \ { a j } simravlida romelime a j elemets da a.s. dasasrul, E \ { a,..., a } simravlida virebt romelime j j s a j (s ) elemets. aset process gaumeorebeli SerCeva S ewodeba, xolo TviT eqsperimetis Sedegs ( a j,..., a ) _ j S 5

16 s moculobis gaumeorebeli SemTxveviTi SearCevi. cxadia, gaumeorebel SerCevaTa raodeoba aris -elemetisaga s-elemetiai wyoba A s () S =(-)...(-s+). masasadame, gaxiluli eqsperimetis Sesabamis albatur models aqvs saxe: 6 ( a j,..., a ), N()=() S, P()=/() S. j S vipovot albatoba imisa, rom s moculobis gaumeorebel SearCevSi rigit pirveli da meore Sesabamisad a da a elemetia (xdomiloba A). xdomiloba A iseti elemetaruli xdomilobebisaga Sedgeba, romeltatvis a a a a : e.i. { : a a, a a }. j, j A j j Tu SerCevaSi ori elemeti fiqsirebulia, masi daar- Cei s- adgili SeiZleba daiavos geeraluri ertobliobis ebismierma - elemetma (-) s- -airad. amitom N(A)=(-) s-. ( ) s amrigad, P(A) / ( ). () s im SemTxvevaSi, rodesac SerCeva daulagebelia (SerCeva- Si Semavali elemetebis rigs ara aqvs misveloba), masi s sivrce Seicavs N( ) C elemets. martlac, yoveli aradaulagebeli SerCevida ( a j,..., a ) j s, romelic Seicavs sxvadasxva elemets, SeiZleba mivirot s! dalagebuli Ser- Ceva. s masasadame, s!n()=() s, e.i. N( ) () / s!. s C SeiSva. gaumeorebeli SerCeva TvalsaCiod SeiZleba warmovidgiot, rogorc yutsi motavsebuli gadaomrili burtis mimdevrobit amoreba. amorebuli burti yutsi ar dabrudeba. Tu burtebs gavuigivebt mat omrebs, masi omrebis mimdevroba (j,,j s ) izleva gaumeorebel SerCevas, amasta, j,,j s ricxvebi ertmaetisaga gasxvavebulia.

17 gavixilot Semdegi saxis eqsperimeti: E={a,...,a } geeraluri ertobliobida elemetebis amorceva SeiZleba gameorebitac, saxeldobr, CaviiSoT pirveli amorceuli elemetis omeri j ; davabruot igi geeralur ertobliobasi da gavimeorot amorceva. meore elemetis j omeric CaviiSoT da a.s. gavimeorebt am cdas s-jer (igulisxmeba eqsperimeti orgaizebulia ise, rom TiToeuli elemetis amorceva ertairad mosalodelia). aset process gameorebiti SerCeva ewodeba, xolo TviT eqsperimetis Sedegs =(j,j,.,j s ) s-moculobis gameorebiti SearCevi. cxadia, j,,j S ricxvebida SesaZloa ramdeime tolic iyos. gameorebit amoarcevta raodeoba S -is tolia. es gamomdiareobs me-3 modelida, Tu iq davusvebt E =E =...= E s =E. amrigad, gaxiluli eqsperimetis Sesabamisi albaturi modelia: =(j,j,.,j s ). N=N()= s, P()=/N(). vipovot albatoba imisa, rom s (s) moculobis gameorebit SerCevaSi yvela elemeti ertmaetisga gasxvavebuli iqeba (xdomiloba A). cxadia, xdomiloba A={:j j j s } imde elemetarul xdomilobebs Seicavs -da, ramdesac E-da gaumeorebeli SerCevis Sesabamisi elemetarul xdomilobata sivrce: N(A)=() S. amrigad, P(A)=() s / s. magaliti.. vtqvat, yutsi gadaomrili burtia, romeltaga m TeTria, daarcei -m burti i, vtqvat, Savi. vipovot albatoba imisa, rom s moculobis gameorebit SearCevSi zustad (s) burti TeTria (xdomiloba A ). cxadia, N=N()= s. vipovot A xdomilobasi Semavali elemetarul xdomilobata raodeoba N(A ). warmovidgiot, rom SerCevaSi ri- 7

18 git j -uri, j -uri,..., j -uri burtebi TeTria, daarcei i _ Savi. rigit j,...,j burtebi TeTri SeiZleba iyos m - airad, xolo daarce s- adgilebze Savi burtebis arceva SeiZleba (-m) s- -airad. masasadame, SerCevaTa raodeoba, e.i.,, s ) elemetarul xdomilobata raodeoba, romelsiac rigit j, j,,j burtebi TeTria, tolia m (m) s- -is. magram, -rigis arcevisatvis arsebobs C s variati ( C s aris s-elemetiai simravlis -elemetia qvesimravleta raodeoba). amitom N(A )= C s m (-m) s-. amrigad, N(A ) m m s P(A ) Cs ( ) ( ), N( ) s! C s.!(s )! ricxvta ertobliobas P(A ),, s ewodeba b i o m i - a l u r i gaawileba, romelic mirebulia lasiuri sqemis farglebsi. es biomialuri gaawilebis erzo SemTxvevaa. zogad sqemas Cve gavixilavt am Tavis $4-Si. magaliti.. vtqvat, yuti imave Semadgelobisaa, rac wia magalitsi. vipovot albatoba imisa, rom s moculobis g a u m e o r e b e l SerCevaSi zustad burti TeTri iqeba, s- i _ Savi (xdomiloba A). radgaac Cve gvaiteresebs SearCevis mxolod Semadgeloba da ara rigi, amitom am magalitis amosaxselad ar aris sawiro gavixilot gaumeorebeli SerCevis armwer elemetarul xdomilobata mtliai sivrce. samarisia avagot mxolod misi qvesivrce Semdegairad: Tu burtebs gavuigivebt mat omrebs, masi elemetarul xdomilobad (SerCevad) SegviZlia avirot {,,...,} simravlis s-elemetiai raime qvesimravle da, masasadame, 8

19 ={}, ( j,..., js ), j r, r, s, N()= C s. axla vipovot A xdomilobasi Semavali elemetarul xdomilobata raodeoba N(A ). TeTri burti m TeTri burtebida SeiZleba SevarCioT Cm -airad. daarcei s- s Savi burti -m Savi burtebida SeiZleba SevarCioT C - m airad. TeTri burtebis ebismier ertobliobas SevusabamoT s- Savi burtebis ebismieri ertoblioba. mivirebt amrigad, N(A P m, m s m ) C C. (, s) P(A CmC ) s C s m, 0, s. (.) ricxvta ertobliobas (P m, (0,s), P m, (,s),, P m, (s,s)) h i p e r g e o m e t r i u l i gaawileba ewodeba. (.) formulas SeiZleba mieces ufro martivi saxe: s!m!( m)!( s)! P m, (, s). (.3)!(s )!(m )![ m (s )]!! axla davadgiot P m, (,s)-is asimptoturi yofaqceva, rodesac. s mudmivia da m/=p=cost[0,] (magali- Tad, Tu = da m=, masi p=/). (.3)-is misveli da mricxveli gavyot S -ze, mivirebt: P m, s p(p )...(p )( p)...( p ) (, s) C s. s ( )...( ) / aqeda, roca gvaqvs * * a simbolo a b, sadac {a } da {b } ricxviti mimdevrobebia, arisavs lim. b 9

20 P m, s s (s, ) C p ( p), 0, s. (.4) (.4) Taafardobida SegviZlia davasvat, saxeldobr, Semdegi: roca geeraluri ertobliobis moculoba samarisad didia, masi praqtiulad gaumeorebeli SerCeva ar gasxvavdeba gameorebiti SerCevisaga. (.4) hipergeometriuli gaawilebisatvis cobilia z R v a r i T i Teoremebis saxelwodebit. vtqvat, xdeba lataria sportlotos gatamaseba. vipovot albatoba imisa, rom TamaSSi moawile gamoicobs sportis eqvsive saxeobas (xdomiloba A). cxadia, aq saqme gvaqvs hipergeometriul gaawilebasta. geeraluri ertobliobis moculobaa =49, xolo "TeTri" burtebis raodeoba m=6. amgvarad, P(A)=P 6,49 (6,6)7, $3. xdomilobata gaertiaebis albatoba' vtqvat, mocemulia disretuli albaturi sivrce (,,P). Teorema 3.. vtqvat, A ;,, masi, P( A ) P(A ) j P(A A ) j 0 P(A A A )... ( ) P( A ) (3.) i j i j damticeba. Teorema davamticot matematiuri iduqciis metodit. vtqvat, A da BB, masi (ix.$) P(AB)=P(A)+P(B)-P(AB). (3.) amgvarad, Teorema samartliaia, roca =. axla davusvat, rom Teorema samartliaia, roca =m da vacveot, rom igi samartliaia, roca =m+. i i

21 arvisot dasvebis ZaliT P(B) m j m B A j, C j A j Am, j, m. j j P(A ) ijm P(A A )... ( ) i j m m P( j A j ), P( m j m C ) j j P(C ) j i jm P(C C )... ( ) i j m m P( j C j ). Tu am misvelobebs CavsvamT (3.)-da gamomdiare P( (m) A ) P(B A j j m ) P(B) P(A m m ) P( gamosaxulebasi, mivirebt Teoremis damticebas. * (3.) formulas b u l i s formula ewodeba. bulis formulis gamoyeebis sailustraciod gavixilot e.w. T a a m T x v e v i s a m o c a a : garveuli mimdevrobit galagebuli sagai SemTxveviT gadaaacvles. Tu romelime sagai erti da imave adgilas darca, masi vityvit, rom adgili aqvs TaamTxvevas. vipovot albatoba imisa, rom ertxel maic eqeba adgili TaamTxvevas. simartivisatvis gavuigivot sagebi mat omrebs, sawyisi dalageba iyos {,,...}. cxadia, am SemTxveviT eqsperimet- Ta, romelic mdgomareobs sagata gadaacvlebasi, daavsirebuli albaturi modeli iqeba j ={}, =(j,j,,j ), N=N()=!, P()=/N. vtqvat, A aris xdomiloba, romelic gvicveebs imas, rom TaamTxvevas adgili aqvs -ur omerze: A ={: j =}, saziebelia P( A ). amisatvis gamoviyeot bulis formula * simbolo isavs damticebis dasasruls. C j )

22 P( A ) = S -S + +(-) - S, sadac S P( A ). advili misaxvedria, rom N(A gvaqvs ) ( )!, N(A A ) ( )!,..., N( i... i i i i i t t A it ) ( )! ( )! P(Ai A... A ) i i.! xolo S =/! (viaida S jamsi C wevria, romlebic satitaod (-)!/! tolia). amrigad, e.i. P( )... ( )! 3! A j j! ( P(! 3!... ( ) A ) -e - 0,63. j j )! -e -,. am paragrafis dasasruls, bulis formulis gamoyeebis sailustraciod, gavixilot agretve g a l a g e b i s amocaa, romelsac gamoyeeba aqvs matematiur statistiasi a r a p a r a m e t r u l h i p o T e z a T a SemowmebisaTvis riteriumis asagebad. g a l a g e b i s a m o c a a : vtqvat, yutsi SemTxveviT vardeba s burti. davusvat, rom ebismieri fiqsirebuli bur- Tis Cavarda j-ur yutsi, je={,,...}, ertairad aris mosalodeli. vipovot albatoba imisa, rom. erti yuti maic darceba carieli (xdomiloba B).. yvela yuti daavebulia (xdomiloba B 0 ).

23 3. yutebis raodeoba darceba carieli (xdomiloba B ). burtebis yoveli fiqsirebuli galageba yutebsi SegviZlia warmovidgiot, rogorc s moculobis gameorebiti Sear- Cevi E={,,...} geeraluri ertobliobida. cxadia, rom ={}, j,..., j ), j,,, ( s N()= s, P()= s. vtqvat, A i xdomiloba isavs, rom i-uri yuti carielia. advili misaxvedria, rom da B B A, B 0 =\B (A A...A A... A ), (3.3) i... i i i sadac {j,j,,j - }=E\{i,i,,i }. radgaac N(A i j s i Ai... Ai ) ( ) j s s s da P(Ai... Ai ) ( ) / ( ),,. amitom bulis formulis ZaliT mivirebt: j P (B) ( ), (3.4) j j s sadac C ( j/ ). (3.4)-da gvaqvs j P 0 (s,)=p(b 0 )=-P(B)= 0 j s ( ) C ( / ). (3.5) vipovot axla P (s,)p(b ). vtqvat, i -uri, i -uri,...i -uri yutebi carielia; masi s burti darceil - yutsi SeiZleba galagdes ise, rom yvela - yuti daavebuli iyos, (-) s P 0 (s,-)-airad. masasadame, P(A i ( ) P0 (s, ) A i... A i A j... A j ). s s 3

24 4 Tu gamoviyeebt albatobis aditiurobis Tvisebas, gveqeba P (s, ) P(B s s ) ( ) P0 (s, ) i... i s s 0 s (3.6) 0 C()P(s,). C ( ) C ( /( )) gamotvlebis TvalsazrisiT saiteresoa SeviswavloT P (s,)-is asimptoturi yofaqceva, roca da s ise, rom s/=a+l, sadac a=cost. amisatvis jer davamticot ori martivi lema. lema. Tu t(0,), masi e t t t e damticeba. fuqcia f(t) l, t(0,) gavsalot t mwrivad t l t... (3.7) t axla, Tu t(0,), masi (3.7)-da gvaqvs da aqeda l t t l t t t t. t.... t t l( t) t, t 0 t. gavixilot sasrul jamta ori usasrulo ricxviti mimdevroba N() j ) j N() b j() j () () T a ( da T, =,,

25 sadac da N(), roca, a j ()>0, b j ()>0, j, N(). () lema. Tu {T } mimdevroba rebadia L sasruli ricxvisae lim T ( ) L da, garda amisa, yoveli ragid mcire dadebiti ricxvisatvis moizebeba iseti mteli dadebiti ricxvi (), rom a j (), b () j roca >(), Taabrad yvela j-satvis, jn()n, () masi {T } mimdevrobasac aqvs zrvari da igi L-is tolia: lim T ( ) L. damticeba. cxadia, aqeda T () T () () () () L (T T ) (T L). () () () L T T T L. (3.8) () radga {T } mimdevroba rebadia, amitom yoveli >0 ricxvisatvis arsebobs iseti mteli dadebiti ricxvi M(), rom adgili hqodes utolobas T ( ) L, roca >M(). (3.9) gvaqvs T a () N() N() () () j T ( a j() b j()) b j(). j j b j() 5

26 aqeda davusvat, T () N() () () j T b j(). j b j() a (3.0) =(/)/(L+/) da gamoviyeot lemis meore piroba. masi, rogoric ar uda iyos >(), gvaqvs a j() / b () L / j (3.) Taabrad yvela j-satvis, jn(). axla arvisot M'()-iT udidesi M() da ()-s Soris, e.i. M'()=max(M(), ()). masi (3.0) da (3.)-da davasvit, rom T () T () / L / N() j / b j() T L / (), roca >M'(), xolo (3.9)-da i da, masasadame, T ( ) L, roca >M'() () () T T, roca >M'(). (3.) (3.8), (3,9) da (3.) Taafardobida mivirebt utolobas 6 T ) ( L, roca >M'().

27 SeviswavloT asimptotia. viaida () C s ( ) (-) < () < da amitom, cxadia, gvaqvs ( ) s! Tu gamoviyeebt pirvel lemas -is ( )!C, s () ( ). t -satvis, mivirebt: s s e! () e. (3.3) dasvebis Taaxmad s/=a+l, amitom (3.3) miirebs saxes: e ( al ) sadac =e -s/ =e -(a+l) =e -a. ()!, (3.4)-da Cas, rom yoveli fiqsirebuli -satvis 0 ( ) da! gvaqvs 0 (3.4) 0 ( ), roca (3.5)! 0 () () P0 (s,m) ( ) T T R 0! () () e 0 T T R, (3.6) () sadac T ( ) (, N() [ ] ) 0 7

28 T () [ ] ( ) 0 0! R 8 R [ ] ( ) () [ ] ( ) 0! ([x] isavs x-is mtel awils) da 0<</. cxadia, [ ] () T da () [ ] 0! [ ] 0 0, roca. (3.7)! () T jamebi amayofilebe me- lemis pirobebs. martlac, pirveli piroba sruldeba trivialurad: [ ] () 0 0 lim T lim ( ) e, 0 0. (3.8)! Semdeg, Tu (3.4)-is marcxea mxaresi -s SevcvliT misi udidesi misvelobit, mivirebt l a ()! e. axla avirot ragid mcire dadebiti ricxvi. masi SegviZlia vipovot iseti dadebiti () ricxvi, rom! () 0 0, roca >(), Taabrad yvela -stvis, 0<<[ ]. lemis meore pirobac Sesrulda. amrigad, () 0 lim T e, (3.9) (3.6), (3.7), (3.8) da (3.9) Taafardobebida vrebulobt, rom P 0 (s, ) e 0, roca.

29 aseve advili misaxvedria, rom yoveli fiqsirebuli - satvis () 0 P 0 (s, ) e, roca. garda amisa, P 0 (s, ) -s mamravli (3.6)-Si SegviZlia gadavwerot rogorc (), magram (3.5)-is ZaliT 0 (), roca.! amrigad, damticda Semdegi: Teorema 3.. Tu s/=a+l, sadac a mudmivi ricxvia, masi yoveli fiqsirebuli -satvis 0 0 P (s, ) e,! roca. (3.0) SemoviRoT arisva e (, ),! 0,,,... ricxvta am ertobliobas p u a s o i s g a a w i l e b a ewodeba. advili misaxvedria, rom 0 (, ). amgvarad, P (s,) _ albatoba imisa, rom yutida carieli armocdeba yuti, asimptoturad (roca ) uaxlovdeba puasois gaawilebas (, 0 ), =,,.... $4. biomialuri gaawileba vtqvat, vawarmoebt orelemetiai geeraluri ertobliobida -jer gameorebit SerCevas. ert-ert elemets davarqvat warmateba da arvisot -it, xolo meore elemets _ marcxi 'da arvisot igi 0-iT. am eqsperimetis Sesabamisi elemetarul xdomilobata sivrce iqeba Semdegi struqturis: 9

30 { : ( a,..., a ), ai 0,}, N( ) mivuwerot yovel elemetarul =(a,, a ) xdomilobas albatoba P( ) p a a sadac p da q iseti arauaryofiti ricxvebia, rom p+q=. imisatvis, rom davrwmudet P(), amayofilebe gasazrvra.3 pirobebs, uda davamticot P( ) tolobis samartliaoba. martlac, gavixilot -da elemetarul =(a,, a ) xdomilobata iseti simravle, romeltatvisac i i q a, 0,. i cxadia, es simravle Seicavs elemetta amitom P( ) P( ) i C 0 i p i, q C raodeobas. (p q). amrigad, sivrce yvela misi qvesimravleta sistemit da P(A) P( ), A albatobit, gasazrvravs raime A albatur models, romelic Seesabameba orelemetiai geeraluri ertobliobida -jer gameorebit SerCevas. vtqvat, =, masi sivrce Seicavs or wertils: = ( warmateba ) da =0 ( marcxi ). albatobas P({})=p vuwodot warmatebis albatoba. Cve vaxavt ($3, Tavi II), rom gaxiluli albaturi modeli, romelic arwers orelemetiai geeraluri ertobliobida -jer gameorebit SerCevas, SeiZleba mirebul iqes rogorc damouidebel cdata Sedegi warmatebis. 30

31 p albatobit, romelic cdida cdamde ucvlelia, e.i. warmatebis () mosvlis albatoba SearCevis yovel fiqsirebul adgilze p-s tolia. axla vipovot albatoba imisa, rom SerCevaSi iqeba zustad ertiai (xdomiloba A ), e.i. albatoba imisa, rom zustad eqeba adgili warmatebas. cxadia, rom P(A ) Cp q, 0,,,...,. viaida iseti -ebis raodeoba, romlebic zustad er- Tiaebs Seicavs, emtxveva adgilida adgilis arcevata raodeobas, rac C -is tolia. ricxvta ertobliobas b(,, p) P(A ) ewodeba biomialuri gaawileba. 0, SeiSvebi:. biomialuri gaawilebis erzo saxe Cve miviret $-Si (magaliti ) lasiuri sqemis sasualebit. martlac, Tui im magalitsi TeTri burtis gamoceas warmatebit () arvisavt, Savi burtis gamoceas i _ marcxit (0), TiToeuli cdisatvis gveqeboda p=m/ da, Sesabamisad, q=-m/ albatobebis mqoe ori elemetaruli xdomilobisaga Semdgari sivrce.. imave paragrafis meore magalitsi Cve SemovitaeT hipergeometriuli gaawileba P,m (,s), =0,,,s. vtqvat,, m ise, rom m/ p [0,], masi s P m, (,s) Cs p ( p) (4.) (4.) mticdeba (.4)-is aalogiurad. 3

32 3 Tavi II elemetarul xdomilobata ebismieri sivrce $. olmogorovis aqsiomatia pirvel TavSi Cve gavixilet iseti SemTxveviTi eqsperimeti, romelta SesaZlo SedegTa simravle sasruli a Tvladi iyo. A xdomilobis P(A) albatoba gavsazrvret elemetarul xdomilobata P() albatobit da sameuls (,,P), sadac elemetarul xdomilobata sivrcis yvela qvesimravleta simravlea, vuwodet disretuli albaturi modeli, au d i s r e t u l i a l b a T u r i s i v r c e. magram, rogorc arisuli iyo imave TavSi, yvela eqsperimeti ar SeiZleba ariweros elemetarul xdomilobata disretuli sivrcit. magalitad, gavixilot eqsperimeti, romelic mdgomareobs simetriuli moetis usasrulod agdebasi. cxadia, am eqsperimetis armweri elemetarul xdomilobata sivrcea { : ( a,..., a,...), ai 0,}, e.i. yvela (a,a,...) mimdevrobata ertoblioba, romelta elemetebi Rebulobe misvelobebs 0 a. cobilia, rom yoveli a[0,] ricxvi SeiZleba calsaxad gavsalot usasrulo orwiladad a a a... ( ai 0,). aqeda cxadia, rom -s wertilebsa da [0,) itervalis wertilebs Soris arsebobs urtiertcalsaxa Taadoba. amgvarad, simravles aqvs otiuumis simzlavre. cxadia, gaxiluli eqsperimeti evivaleturia eqsperimetisa, romelic mdgomareobs [0,] itervalida wertilis SemTxveviT arcevasi. simetriulobis mosazrebida ga-

33 momdiare cxadia, rom eqsperimetis yvela Sedegi uda iyos tolalbaturi, magram [0,] simravle aratvladia da, Tu CavTvliT, rom misi albatoba -is tolia, masi yoveli [0,) Sedegis P() albatoba uewvelad 0-is toli uda iyos. magram, ase albatobis mocema (P()=0, [0,) arafers gvazlevs. saqme is aris, rom Cve daiteresebuli vart ara imit, ra albatobit moxdeba esa Tu is Sedegi, aramed imit, ras udris albatoba imisa, rom eqsperimetis Sedegi mieutveba raime mocemul A simravles [0,)-da. disretuli sivrcis SemTxvevaSi P() albatobebit Cve gavsaz- RvreT A xdomilobis P(A) albatoba: P (A) P( ). A magram gasaxilvel SemTxvevaSi P()=0, [0,) tolobida Cve ar SegviZlia gavsazrvrot, magalitad, albatoba imisa, rom SemTxveviT arceuli wertili eutvis [0,/) simravles. amave dros ituiciurad cxadia, rom es alba- Toba /-is tolia. es SeiSvebi migvaisebs, rom albaturi modelis agebis dros, im SemTxvevaSi, rodesac aratvladia, albatoba uda iyos mocemuli ara caleuli elemetaruli xdomilobebisatvis, aramed garveul lassi Semavali simravleebisatvis. aseti simravleta lasi uda Seadgio davirvebadma xdomilobebma da es lasi uda iyos Caetili gaertiaebis, TaaveTisa da damatebis operaciebis mimart, rogorc es iyo disretuli sivrcis SemTxvevaSi. am mizit sawiroa SemovitaoT gasazrvra.. vtqvat, elemetarul xdomilobata sivrce ebismieri simravlea. sivrcis raime qvesimravleta A lass a l g e b r a ewodeba, Tu Sesrulebulia pirobebi:. A,. Tu E A, E A, masi E E A, E E A, 3. Tu EA, masi E A. 33

34 advili SesamCevia, rom Tu pirobasi movitxovt mxolod ert-erti Taafardobis Sesrulebas, masi meorec Sesruldeba 3-is ZaliT. magaliti. vtqvat, =[a,b), -a, b<. A [a,b] iyos iset qvesimravleta lasi [a,b]-da, romeltaga TiToeuli Sedgeba [c,c ), [c,c ], (c,c ], (c,c ) tipis itervalta sasruli gaertiaebisga. advili Sesamowmebelia, rom A [a,b] algebraa. A [a,b] algebras uwodebe b o r e l i s a l g e b r a s [a,b] itervalsi. erzod, A () = A (-,) uwodebe borelis algebras amdvil ricxvta R () RerZze. 34 magaliti. vtqvat, () I[ a, b) [ aj,bj), a=(a,a,,a ), b=(b,b, b ), -a i, b i, i=,,, -gazomilebiai itervalia evlides R () sivrcesi. A ( ) iyos iset qvesimravleta () lasi I[ a,b) -da, romeltaga TiToeuli Sedgeba [c,d ) [c,d ), [c,d ] [c,d ], (c,d ) (c,d ), (c,d ) (c,d ) tipis itervalta sasruli gaertiaebisaga. A ( ) [a, b) lasi () algebraa; mas ewodeba borelis algebra I[a, b) -Si. erzod, A () (-, ) -s uwodebe borelis algebras evlides gazomilebida R () -sivrcesi. gasazrvra.. -s qvesimravleta lass ewodeba -algebra, Tu is algebraa da, garda amisa, Sesrulebulia piroba: j Tu A,,A,, masi A da A. SeiSva. pirobasi ori Taafardobida ert-ertis motxova savsebit samarisia, viaida A = A da A = = A. Tu A raime simravleta lasia -da, masi arsebobs -algebrebi, romlebic mas Seicave. -algebrata aseti

35 lasi aracarielia, viaida -s yvela qvesimravleta simravle, romelic -algebraa, Seicavs A -s. axla gavixilot = * cxadia, rom 0. algebraa, 0. A, 3 0. yvela -stvis, e.i. -ze ufro mcire 'algebra, romelic Seicavs A -s, ar arsebobs, viaida ert-erti -s tolia Tviseba gvazlevs safuzvels -s vuwodot A-s Semcveli u m c i r e s i -algebra da mas arvisavt ase: A ). gasazrvra.3. borelis A [a,b] algebris Semcvel umcires -algebras [a,b] =( A [a,b] ) ewodeba b o r e l i s -algebra, au borelis qvesimravleta -algebra [a,b] itervalsi. erzod, umcires -algebras (), romelic Seicavs A (-,) algebras, ewodeba borelis -algebra R () =(-,)-Si. axla vaxot, ramdead mdidaria [a,b] borelis qvesimravleta -algebra, CveTvis cobili ra simravleebi Sedis am sigma algebrasi? magalitad:. caleuli wertili borelis qvesimravlea. martlac, Tu c[a,b], masi moizebeba iseti N 0 ricxvi, rom ( c / N,c / N )[a,b], NN 0, amitom { c} (c / N,c / N) [a,b]. NN 0 * simravleta lasebsi CarTvis operacia, TaaveTa da gaertiaeba Cveulebrivad gaigeba. 35

36 . racioalur wertilta simravle borelis simravlea, rogorc caleul wertilta Tvladi gaertiaeba. 3. iracioalur wertilta simravle borelis simravlea, e.i. rogorc racioalur wertilta simravlis damateba. 4. ebismieri Ria simravle eutvis [a,b] -s, viaida Ria simravle TaauveT itervalta sasruli a Tvladi raodeobis gaertiaebaa. 5. ebismieri Caetili simravle borelis simravlea [a,b]-si, rogorc Ria simravlis damateba. 6. Tu f() aris uwyveti fuqcia [a,b]-ze, masi ebismieri amdvili x-satvis {: f()x} [a,b], radga {: f()x} Caetili simravlea (daamticet). samaod Zelia moviyvaot iseti simravlis magaliti [a,b]-da, romelic [a,b] -lass ar eutvis. amgvarad, [a,b] lasi imdead mdidaria, rom is uewvelad samarisia praqtiuli mizebisatvis..3. gasazrvra gavazogadot: gasazrvra.4. borelis A () ( a, b) algebris Semcvel umcires -algebras ( ) [a, b) = A () ( a, b) ewodeba borelis -algebra, au borelis qvesimravleta -algebra I ( ) [a, b) (ix. magaliti ) -gazomilebia itervalsi, erzod, ( ) = () 36 (, ) umcires -algebras, romelic A () (, ) -s Seicavs, ewodeba borelis -algebra evlides gazomilebida R () sivrcesi. SeiSva. Tu sivrce Tvladia, masi umciresi -algebra, romelic Seicavs -s caleuli wertilebisga Sedgeil lass, emtxveva -s yvela qvesimravleta simravles. SeiSva 3. vtqvat, A iyos iset qvesimravleta lasi (-,)-da, romeltaga TiToeuli Sedgeba [a,b) tipis TaauveT itervalta sasruli gaertiaebisaga. A algebraa da A. SeiSva 4. () wyvils, sadac -algebraa a -algebra, ewodeba z o m a d i sivrce. magalitad, (R (), () ) zomadi sivrcea.

37 vtqvat, G raime SemTxveviTi eqsperimetia. am eqsperimetebta daavsirebuli ama Tu im albaturi amocais formalizebisatvis sawiroa G-s SevusabamoT () zomadi sivrce. arisavs eqsperimetis SedegTa simravles. simravleta algebris a -algebris gamoyofa -da gapirobebulia, erti mxriv, gasaxilavi amocais arsit, meore mxriv, simravlis buebit. iseve rogorc I TavSi, -Si Semaval simravleebs xdomilobebs vuwodebt; TviT simravles aucilebel xdomilobas. -is gasazrvrida cxadia, rom carieli simravle ; mas SeuZlebel xdomilobas uwodebe. A -s ewodeba A xdomilobis sawiaarmdego xdomiloba. Tu A, B da A B =, masi A da B xdomilobebs utavsebadi ewodeba, e.i. mati ertad moxdea SeuZlebelia. axla SegviZlia gadavidet albatobis gamsazrvravi aqsiomebis Camoyalibebaze. am mizit gavixilot (A zomadi sivrce, sadac A algebraa. gasazrvra.5. A algebraze gasazrvrul P(. ) fuqcias ewodeba albatoba (A ) zomad sivrceze, Tu is amayofilebs Semdeg aqsiomebs:. A A P(A)0; (P-s arauaryofitobis aqsioma);. P()=, (ormirebis aqsioma); 3. Tu {A } xdomilobata mimdevroba A -da isetia, rom masi i A A, A A j =, j, i A j ) P(A j). j j P ( (.) mesame aqsiomis evivaleturia (.)-is Sesruleba xdomilobata sasruli ricxvisatvis da Semdegi uwyvetobis aqsioma: 37

38 3. vtqvat, {B } xdomilobata mimdevroba isetia, rom masi B + B da B B A, P(B )P(B), roca. evivaleturobis damticeba. vtqvat, Sesrulebulia me-3 aqsioma da B + B, B B, masi B, C B B xdomilobata mimdevroba wyvilwyvilad utavsebadia da B B ( C ), =,,.... me-3 aqsiomis ZaliT mivirebt, rom mwrivi P(B ) = P(B) + P(C ) rebadia. es i isavs, rom, roca j P(B ) = P(B) + P(C ) P(B). amgvarad, 3 aqsioma Sesrulebulia. SebruebiT, Tu {A } utavsebadi xdomilobata mimdevrobaa, masi P( A ) P( A ) P( A ) j j j j j j da 3 aqsiomis ZaliT adgili aqvs tolobas j P(A ) lim P(A ) lim P( A ) j limp( A j ) P( j j j j A j ) P( A j ). j j 38

39 (,A,P)-sameuls ewodeba albaturi modeli farto azrit, au a l b a T u r i s i v r c e f a r T o a z r i T. Tu algebra aris -algebra, (=()), masi me-3 aqsiomasi j A j piroba ((,)-ze gasazrvruli albatobisatvis) Sesruldeba avtomaturad. (,,P)-sameuls, sadac -algebraa, ewodeba ubralod albaturi modeli, au a l b a T u r i s i v r c e. amgvarad, albaturi sivrcis ageba isavs (,) zomad sivrceze iseti arauaryofiti Tvladad-aditiuri P() zomis mocemas, romlistvisac P()=. albatobis Teoriis aqsiomatia am saxit iqa Camoyalibebuli aademios a. o l - m o g o r o v i s mier. axla davubrudet gasazrvra.5-s. vtqvat, (,A,P) albaturi sivrcea (A-algebraa). rogorc vaxet, yovel algebras SeiZleba davuavsirot A-s momcveli =( A) umciresi -algebra. buebrivad ismeba itxva: A -ze mocemuli albaturi P zoma gasazrvravs Tu ara zomas =(A )-ze da es gasazrvra calsaxaa? sxva sityvebit rom vtqvat, (,,P) albaturi sivrcis agebisatvis samarisia Tu ara P-s mocema mxolod romelime A algebraze, romlistvisac =(A). pasuxs am itxvaze izleva a r a T e o d o r i s Teorema, romelsac Cve damticebis garese moviyvat. arateodoris Teorema. vtqvat, (,A,P) albaturi sivrcea farto gagebit. masi arsebobs =(A)-ze gasazrvruli iseti ertaderti albaturi Q zoma, rom Q(A)=P(A ), roca A A. amgvarad, yoveli (,A,P) albaturi sivrce farto azrit gasazrvravs ertadert (,,P) albatur sivrces, sadac =(A). sxva sityvebit rom vtqvat, Tu Cve gvaqvs agebuli 39

40 (,A,P) albaturi sivrce farto azrit, SegviZlia masive vigulisxmot, rom P zoma mocemulia ara marto A-ze, aramed =(A )-zedac. axla davubrudet zemot gaxilul eqsperimets, romelic 0 moavetze wertilis SemTxveviT arcevasi mdgomareobs. arvwerot am eqsperimetis Sesabamisi albaturi sivrce. cxadia, rom =[0,]. cxadia agretve, rom xdomilobebad uda CaiTvalos elemetarul xdomilobata is simravleebi, romlebic buebrivad davirvebadia eqsperimetis dros. ase, magalitad, A [0,] algebris simravleebi davirvebad xdomilobebad uda CaiTvalos. amgvarad, am eqsperimetis Sesabamisi zomadi sivrcea (, [0,] ). ituiciurad cxadia, magalitad, albatoba imisa, rom Sem- TxveviT arceuli wertili moxvdeba [a,b), (a,b], [a,b] da (a,b) tipis romelime <a,b> itervalsi, b-a-s tolia. zustad aseve, albatoba imisa, rom wertili moxvdeba A [0,] algebris romelime A a j, b j simravleze, ( b j a j )- j j is tolia. aqeda gamomdiare, A [0,] algebraze, buebrivia, P albatoba gaisazrvros ase: P(A)= ( b j a j ), A a j, b j A [0,]. j j P-zoma Tvladad aditiuria (amis dasamticeblad III Tavis Teorema 3.-Si uda davusvat, rom F(x)=x, x[0,]). arateodoris Teoremis ZaliT P albaturi zoma [0,] -zedac iqeba gasazrvruli. [0,] -ze asetairad gasazrvrul zomas uwodebe l e b e g i s zomas. masasadame, gaxiluli eqsperimetis Sesabamisi albaturi sivrcea ([0,], [0,], ). zemot arviset, rom =[0,]-ze -is gasazrvra, disretuli modelis aalogiurad (-yvela qvesimravleta lasia [0,]-da), garveul sizeleebs iwvevs. martlac, -ze 40

41 zomis mocema ise, rom <a,b> itervalisatvis mis sigrzes emtxveodes, e.i. (<a,b>)=b-a, SeuZlebelia, viaida -Si arsebobs iseti simravleebi, romlebic zomadi ar aria, e.i. gasazrvruli ar aris aset simravleebze (ix... -,, 974,.80). cxadia, ({})=0, [0,]. albatoba imisa, rom eqsperimetis Sedegi racioaluri ricxvia, ulis tolia. martlac, vtqvat, A {r } ra- j cioalur wertilta simravlea [0,]-da. rogorc vicit, A B [0,], amitom me-3 aqsiomis ZaliT (A) j ({r }) 0. gasazrvra.6. ambobe, rom gvaqvs amocaa geometriuli albatobis Sesaxeb, Tu elemetarul xdomilobata sivrcea evlides R () sivrcis borelis qvesimravle (), romelsac sasruli lebegis zoma gaacia, e.i. 0< () ()< () lebegis gazomilebiai zomaa, romelic paralelepipedze mis moculobas emtxveva), xolo misi ebismieri A qvesimravlis (A () ) albatoba () (A) P (A) () ( ) formulit moicema. magaliti 3. ( Sexvedris amocaa ) ori da moqalaqe SeTaxmda garveul adgilas Sexvde ertmaets, sa- Ramos 8-da 9 saatamde. TiToeuli matgai midis am adgilas ertimeorisaga damouideblad. is moqalaqe, romelic mividoda daisul adgilas, 0 wutis (/3) saatis) gamavlobasi ucdis meores da mere midis. vipovot da moqalaqeta Sexvedris albatoba. j j 4

42 cxadia, am eqsperimetis SesaZlo Sedegia =[0,][0,] vadratis yoveli (x,y) wertili, sadac x da y arisavs Sesabamisad -isa da -is mosvlis momets, xolo Sesabamisi albaturi sivrcea (, ( ) ), P), sadac ( arisavs vadratis yvela borelis qvesimravleta lass (ix. gasaz- Rvra.4) da () (A) P (A). () ( ) cxadia, CveTvis saitereso xdomilobaa A (x, y) : x y ) 3 xolo misi albatoba i P(A)=5/9. y /3 A 0 /3 x (, 4 $. albatobis Tvisebebi. P()=0. es gamomdiareobs += tolobida da me- da me-3 aqsiomida.. P( A )=-P(A), radga A A da A A. 3. Tu AB, masi P(A) P(B). es gamomdiareobs iqida, rom P(A)+P( A B )=P(B).

43 4. P(A), viaida A. 5. P( A B)=P(A)+P(B)-P( A B), radga A B A (B \ A B) da P(B \ AB) P(B) P(A B). 6. P( A B) P(A) + P(B). 7. ( A ) S - S (-) - S, P j j sadac S P(A A... A ). j i... i j i i es formula Cve davamticet (ix. Teorema 3.., Tavi ) disretuli sivrcis SemTxvevaSi. igi aalogiurad damticdeba ebismieri sivrcis SemTxvevaSic. 8. (-axevrad aditiuroba) P( A j ) P(A j). j damticeba. SemoviRoT arisvebi: B =A, B =A \A, B 3 =A 3 \( A A),..., B j =A j \( j i j j A ),..., j. cxadia, rom A B da B, j. j j j j B j amrigad, P( A j j ) P(B j) P(A j) j j, viaida A j B j. 9. Tu {A } simravleta mootourad zrdadi mimdevrobaa, e.i. A A + da A j j A, masi lim P(A ) P(A). damticeba. gavixilot {B =A\A } simravleta mimdevroba. cxadia, rom B + B da B j j. uwyvetobis aqsiomis ZaliT mivirebt P(A\A )=P(A)-P(A ) 0, roca. 43

44 44 $3. pirobiti albatoba. xdomilobata damouidebloba vtqvat, raime cdis armweri sivrce (,,P) disretulia, xolo A da B am cdasta daavsirebuli raime xdomilobebia, e.i. A, B. davusvat agretve, rom B xdomilobas hqoda adgili am eqsperimetis dros. ra SeiZleba itqvas amis Semdeg A xdomilobis albatobaze? mas Cve arvisavt P(A/B) a p B (A) simboloti (iitxeba: A xdomilobis pirobiti albatoba im pirobit, rom xdomiloba B moxda ). gvaqvs ra iformacia imis Sesaxeb, rom gaxorcielda B xdomiloba, gavixilot axla ara yvela elemetarul xdomilobata simravle, aramed mxolod yvela elemetaruli xdomilobis ertoblioba B-da da yovel j B elemetarul xdomilobas SevusabamoT raime arauaryofiti P( j /B) ricxvi ( j -is albatoba B pirobit) ise, rom adgili hqodes tolobas P( / B). (3.) jb j pirobiti albatobebi P( j /B), j=,,..., romlebic amayofilebe (3.) motxovas, SeiZleba mivirot, magalitad, P( j )-is P(B)-ze gayofit, e.i. P( j) P( j / B), j =,,.... P(B) cxadia, rom asetairad gasazrvruli P( j /B) albatobebi (3.) motxovas amayofilebs. amrigad, imisatvis, rom gamovtvalot P(A/B) pirobiti albatoba, sawiroa avjamot (ix. gasazrvra.4, Tavi I) pirobiti P( j /B) albatobebi yvela im j elemetaruli xdomilobebisatvis, romlebic mieutvebia A da B-s ertdroulad a, rac igivea A B xdomilobas. aqeda mivirebt P(A / B) jab P( j / B) P(B) jab P(A B) P( j). P(B)

45 lasiuri albaturi sivrcis SemTxvevaSi gveqeba: P(A / B) N(A B). N(B) magaliti. vtqvat, yutsi motavsebulia N burti, romeltaga N TeTria, xolo N-N Savi. ras udris P(A/B) albatoba imisa, rom meored amorebuli burti TeTria (A xdomiloba), im pirobit, rom pirvelad amorebuli burti TeTria (B-xdomiloba), cxadia, rom N( )=N(N-), N( A B)=N (N -) da N(B)=N (N-). amitom N P(A / B). N magaliti. vtqvat, 3-jer vagdebt simetriul moetas. masi ={}, =(,, 3 ), N()=8, sadac i = im SemTxvevaSi, rodesac adgili eqeba warmatebas (gerbi), i =0 marcxis (safasuris) SemTxvevaSi. A iyos xdomiloba imisa, rom zustad ertxel eqeba adgili warmatebas, e.i. A={(,0,0), (0,,0), (0,0,)}, xolo xdomiloba B etricxvjer eqeba adgili warmatebas, e.i. B={(,0,0), (0,,0), (0,0,), (,,)}. cxadia, rom P(A/B)=3/4. Cve SegviZlia axla gadavidet P(A/B)-s zogad gasaz- Rvraze. gasazrvra 3.. vtqvat, mocemulia (,,P) albaturi sivrce da A da B ebismieri xdomilobebia -da. Tu P(B)>0, masi A xdomilobis pirobiti albatoba B xdomilobis moxdeis pirobit (B pirobit) ewodeba sidides P(A B) P(A / B). (3.) P(B) 45

46 am gasazrvrida usualod gamomdiareobs pirobiti albatobis Semdegi Tvisebebi: 46 P(B/B)=, P(/B)=0, P(A/B)=, BA, P(A A /B)=P(A /B)+P(A /B), Tu A A. (3.)-da, mivirebt agretve, rom ori xdomilobis ertad moxdeis albatoba tolia ert-erti matgais albatobis amravlisa meoris pirobit albatobaze pirvelis pirobit, e.i. P(AB) = P(B)P(A / B) = P(A)P(B / A). (3.3) gasazrvra 3.. A da B xdomilobebs damouidebeli ewodeba, Tu sruldeba P(A B) P(A)P(B) (3.4) toloba. moviyvaot damouidebel xdomilobata zogierti Tviseba.. Tu P(B)>0, masi A da B xdomilobebis damouidebloba evivaleturia P(A/B)=P(A) tolobis. damticeba cxadia. xdomilobata damouideblobis azri imasi mdgomareobs, rom damouidebel xdomilobataga ert-ertis moxdea aravitar gavleas ar axdes meore xdomilobis albatobaze.. Tu A da B damouidebeli xdomilobebia, masi damouidebelia A da B xdomilobebi. martlac, P( A B) = P(B \ AB) = P(B) - P(AB) = = P(B)( - P(A)) = P(B)P( A ). Sedegi. Tu A da B xdomilobebi damouidebelia, masi damouidebelia A da B xdomilobebic. masasadame, SegviZlia davasvat: Tu A da B damouidebeli xdomilobebia, masi damouidebelia agretve yoveli ori xdomiloba ( A, B), (A, B ), ( A,B ).

47 SeiSva. xsirad ertmaetsi ureve xdomilobata damouideblobisa da utavsebadobis albatur azrs; SeiZleba es gamowveuli iyos damouideblobisa da utavsebadobis termiis foetiurad garveuli siaxlovit. davusvat, A da B iseti xdomilobebia, rom P(A)>0, P(B)>0. Tu A da B xdomilobebi utavsebadia, masi A B= da amitom P( A B)=0. magram Tu P( A B) = 0, masi (3.4)-s ar eqeba adgili da, masasadame, A da B xdomilobebi ar iqebia damouidebeli. SebruebiT, Tu A da B xdomilobebs aqvt dadebiti albatoba da damouidebelia, e.i. sruldeba (3.4) toloba, masi P( A B) > 0 da amitom A da B xdomilobebi ar iqebia utavsebadi. magaliti 3. vtqvat, A xdomiloba arisavdes simetriuli moetis orjer zedized agdebisas (N()=4) pirvelad warmatebis mosvlas (ix. magaliti ), xolo B meored marcxis mosvlas. cxadia, amitom N(A)=, N(B)= da N(A B). P(A B) P(A)P(B). 4 amrigad, A da B xdomilobebi damouidebelia. magaliti 4. vtqvat, eqsperimeti mdgomareobs ori moetis usasrulo raodeobit agdebasi a, rac igivea, erteulovai vadratida wertilis SemTxveviT arcevasi (ix. $, Tavi ). cxadia, am eqsperimetis armweri albaturi sivrcea (,,P), sadac =[0,][0,]. -borelis simravleta lasia -da, xolo P= () orgazomilebiai l e b e g i s zomaa. vtqvat, a, b[0,] da gavixilot xdomilobebi: A={(x,y):xa, (x,y) }, B={(x,y):yb, (x,y) }. cxadia, rom P(AB) = () (AB) = (-a)(-b) = P(A)P(B). 47

48 masasadame, A da B xdomilobebi damouidebelia. gasazrvra 3.3. A i, i=,,... xdomilobebs ewodeba e r T o b l i v a d d a m o u i d e b e l i, Tu mat Soris ebismieri m(m<) xdomilobebisatvis sruldeba 48 m P( A ) P(A Taafardoba. j m i i ) j j j SeviSoT, rom xdomilobata wyvil-wyvilad damouidebloba ar isavs ertoblivad damouideblobas. martlac, vtqvat, =(,, 3, 4 ) da TiToeuli i, i =, 4, tolad SesaZlebelia, e.i. masi xdomilobebi P( i )=/4, i =, 4, A ={, }, A ={, 3 } da A 3 ={, 4 } wyvil-wyvilad damouidebelia, magram P(A A A3 ) P(A)P(A )P(A3 ). 4 ertoblivad damouidebel xdomilobata mimdevrobis magalitad SeiZleba davasaxelot -jer Catarebuli caleuli cdis SedegTa mimdevroba, romelic gaxiluli iyo I Tavis $4-Si. amasi rom davrwmudet, sawiroa SemoviyvaoT cdata damouideblobis ceba. gavixilot (,,P ),..., ( m, m,p m ) albaturi sivrceebi, romlebic arwere Sesabamisad G,G,...,G m eqsperimetebs. gavixilot agretve rtuli G eqsperimeti, au sxvaairad Sedgeili eqsperimeti, romlis armweri albaturi sivrce iyos (,,P), sadac =... m aris,..., m sivrceta pirdapiri amravli, au =(,..., m ), =, =, m, elemetta simravle, 3

49 xolo aris umciresi -algebra, warmoqmili A={:=(,..., m ), A,..., m A m }=A...A m, A, =, m saxis martutxedebisaga. Cve vityvit, rom G, G,..., G m cdebi damouidebelia, Tu ebismieri A=A...A m, A, =, m, martutxedebisatvis sruldeba toloba P(A) = P (A )... P m (A m ). (3.5) xdomiloba, romelic i-ur cdas uavsirdeba, SeiZleba ariweros ara marto rogorc i lasis simravle, aramed rogorc lasis simravlec. amisatvis moviqcet Semdegairad: vtqvat, A i i, i=, m da gavixilot -da simravleebi: B j =... j- A j j+... m, j=, m. B j -is saxis simravleebs uwodebe cilidruls A j fu- ZiT. (3.5)-da cxadia, rom P(B i )=P i (A i ), i=, m da xdomilobebi B,B,...,B m ertoblivad damouidebelia. martlac, vtqvat, m. viaida B... B =A... A +... m (3.5)-da gvaqvs P(B... B ) =P(B )...P(B ). amrigad, caleul cdebta daavsirebuli xdomilobebi, romlebic ariwereba erti (,,P) albaturi sivrcis xdomilobebis sasualebit, damouidebeli armocda. axla davubrudet I Tavis $4-Si gaxilul berulis sqemas. am paragrafsi gaxiluli albaturi sivrce (,,P), sadac ={: =(a,...,a )}, a j = 0,, ={A:A} da P( ) p a a q, 49

50 romelic arwers orelemetiai geeraluri ertobliobida -jer gameorebit amorcevas, warmoadges rtuli G eqsperimetis Sedegs. martlac, aq gaxilul yovel caleul cdas ori Sedegi aqvs warmateba () da marcxi (0). warmatebis albatobaa p, marcxisa i q=-p. amrigad, rigit i-uri cda G i ariwereba ( i, i,p i ) albaturi sivrcit, sadac 50 i ={0,}, i ={{0}, {},, i }, P i ({})=p, P i ({0})=q=-p. rtuli G eqsperimetis armweri (,,P) albaturi sivrcis ostruqciida Cas, rom is warmoadges G i eqsperimetis armweri ( i, i, P i ) albatur sivrceta pirdapir amravls: sadac (=..., P(A) ={A:A}, P), ai ai p q { ( a,..., a ) A}, A, a i = 0,. cdata G, G,..., G mimdevroba damouidebelia, viaida asetairad agebuli P albaturi zoma amayofilebs (3.5) motxovas. vacveot, rom warmatebis () mosvlis albatoba cdis yovel -ur fiqsirebul adgilze p-s tolia, e.i. vacveot, rom ai ai p q { : a } a,..., a a,..., a ( a,..., a a,..., a ) q ( a,..., a a,..., a ) P{: a =} = p, martlac, P{: a =}= = p p = j j () j p Cp q p. j0 aseve miireba, rom P{: a =0} =q=-p. amrigad, gaxilul cdata mimdevroba G,G,...G, romlebsac or-ori Sedegi aqvs _ warmateba da marcxi, damoui-

51 debelia, xolo warmatebis albatoba cdida cdamde ucvlelia. aset cdebs berulis damouidebel cdata sqemas uwodebe a, ubralod, berulis sqemas. i. b e r u l i iyo pirveli, romelmac Seiswavla xseebuli albaturi modeli da daamtica mistvis did ricxvta aois samartliaoba. $4. jamis albatobis gamotvla urtiertdamouidebeli xdomilobebisatvis vtqvat, A,A,...,A damouidebeli xdomilobebia. gamartebis Taaxmad, A isavs xdomilobas, romelic xdeba masi da mxolod masi, rodesac xdeba erti maic A,,, xdomilobatagai. am xdomilobis sawiaarmdego xdomiloba iqeba A, viaida amitom vwert aqeda A =\ P( A A A )=.. P P( A )=-PP( A ). pirobis Taaxmad, A,A,...,A xdomilobai damouidebeli aria, amitom damouidebeli iqebia agretve mati sawiaarmdego xdomilobai A,A,..., A. vicit, rom damouidebel xdomilobata amravlis albatoba TaamamravlTa albatobebis amravlis tolia da amitom uaaseli toloba Semdegairad Caiwereba: P( A ) P(A )P(A )...P(A ). j j 5