Chiphunzitso cha Mwina. Mwina chochitika, nthawi zina chochitika (Mwina chiphunzitso). Mwayekha atakwatirana zikuchitika chiphunzitso cha Mwina

Ndi chodziwikiratu kuti ambiri n'zotheka kuwerenga zochitika, omwe kumlingo mwangozi. Kuziyika izo mu mawu osavuta, n'zomveka kuti zimene mbali ya kyubu mu dayisi zidzagwa nthawi ina. Zinali funso kufunsa asayansi otchuka awiri, anaika maziko a sayansi, chiphunzitso cha Mwina, Mwina chochitika amene anaphunzira kwambiri zokwanira.

m'badwo

Mukayamba kufotokoza mfundo monga chiphunzitso cha Mwina, timachotsa zotsatirazi: ichi ndi chimodzi mwa nthambi za masamu kuti amaphunzira mosalekeza zochitika mwachisawawa. Mwachionekere, mfundo zimenezi silinena akamanena, kotero muyenera kuona mwatsatanetsatane.

Ndikufuna kuyamba ndi amene anayambitsa chiphunzitso. Monga tatchulazi, panali awiri, kuti Per Ferma ndi Blez Paskal. Iwo anali woyamba anayesa ntchito mitunduyi ndi masamu kuwerengera zotsatira za chochitika. Ambiri, zoyamba za sayansi izi ngakhale Ages Middle. Ngakhale anzeru osiyanasiyana ndiponso asayansi ayesetsa kudziwa kasino masewera monga roleti, craps, ndi zina zotero, potero kukhazikitsa chitsanzo, ndi kuchuluka imfa zingapo. maziko anali anaika mu m'ma chakhumi ndi chisanu ndi chiwiri anali akatswiri zatchulidwazi.

Poyamba, ntchito zawo sanathe zimachitika zofunika kwambiri mu munda, pambuyo pa zonse, kodi iwo anachita, iwo anali chabe mfundo wazotsatira ndi zatsopano anali bwino popanda kugwiritsa ntchito mitunduyi. M'kupita kwa nthawi, kunapezeka kuti iyende wamkulu, amene anaonekera chifukwa cha ananena osewera a mafupa. Iwo chida ichi wathandiza kubweretsa woyamba osiyana chilinganizo.

othandizana

Osanenapo munthu ngati Christiaan Huygens, m'kati kuphunzira phunziro limene lili ndi dzina la "chiphunzitso Mwina" (Mwina chochitika anafotokoza mu sayansi). Munthu uyu ndi chidwi kwambiri. Iye, komanso asayansi anapereka pamwamba kuyesera mu mawonekedwe a mitunduyi masamu kuti tikuona chitsanzo cha zochitika mwachisawawa. N'zochititsa chidwi kuti sanali kuuza Pascal ndi Fermat, kuti ntchito zake zonse zilibe alipo maganizo amenewo. Huygens anachokera malingaliro zoyambirira za chiphunzitso Mwina.

Choona chidwi ndi ntchito yake anabwera kale zotsatira za ntchito za apainiya, kukhala enieni, zaka makumi awiri kale. Pali pakati mfundo anazizindikira:

• monga mfundo mwayi Mwina makhalidwe;
• kuyembekezera kuti vuto apawokha;
• theorems wa Kuwonjezera ndi chulutsa ziyerekezo.

Komanso, munthu samaiwala Yakoba Bernulli, amene anathandiza kuti kuphunzira vuto. Kudzera lawo, ngakhalenso amene mayesero palokha, iye anatha kupereka umboni wa lamulo la ambirimbiri. Poyankha, asayansi Poisson ndi Laplace, amene ankagwira ntchito ku m'ma chakhumi ndi chisanu ndi chinayi, anatha kutsimikizira Lingaliro Lovomerezeka pachiyambi. Kuyambira nthawi imeneyo yosanthula zolakwa kuzipenya ife anayamba kugwiritsa ntchito chiphunzitso Mwina. Party padziko sayansi sakanakhoza ndi Russian asayansi, m'malo Markov, Chebyshev ndi Dyapunov. Iwo zochokera ntchito mwachita geniuses kwambiri, wotetezedwa nkhani monga nthambi ya masamu. Tinayesetsa Chiwerengerochi kumapeto kwa zaka za m'ma chakhumi ndi chisanu ndi chinayi, ndipo chifukwa chopereka lawo, zochitika kutsimikiziridwa monga:

• malamulo a ambiri;
• Chiphunzitso cha Markov unyolo;
• Chapakati malire Lingaliro Lovomerezeka.

Choncho, mbiri ya kubadwa kwa sayansi ndi umunthu chachikulu chimene chinathandiza izo, chirichonse kwambiri kapena zochepa bwino. Tsopano ndi nthawi kuti tizindikire zoona zonse.

mfundo zofunika

Musanayambe kukhudza malamulo ndi theorems ayenera kuphunzira mfundo yaikulu ya chiphunzitso Mwina. Chochitika ichi ali ndi mbali lalikulu. Nkhani imeneyi ndi m'malo ambiri, koma sadzakhoza kumvetsa onse popanda izo.

Chochitika chiphunzitso Mwina - izo akonzedwa aliyense wa zakambidwa zimenezo. Zikhulupiriro za zodabwitsazi apo si kokwanira. Choncho, Lotman wasayansi ntchito m'dera lino, wachita posonyeza kuti mu nkhani iyi tikulankhula zimene "chinachitika, ngakhale sanathe kuchitika."

zochitika mwachisawawa (Mwina chiphunzitso amasamalirira wapadera kwa iwo) - ndi mfundo zimene zimaphatikizapo mwamtheradi chodabwitsa chilichonse ndi mwayi kulankhula. Kapena M'malo mwake, sewero izi sizingatheke mu ntchito zosiyanasiyana zinthu. Ndi ofunika podziwa kuti kutenga buku lonse la chododometsa zikuchitika zochitika chabe mwachisawawa. Mwina chiphunzitso zikusonyeza kuti zinthu zonse Mungathe kubwereza zonse. Ndi khalidwe lawo umatchedwa "ndi" kapena "mayeso."

chochitika chidwi - ichi ndi chodabwitsa kuti ndi zana pa zana mu mayeso zichitike. Choncho, mwambo zosatheka - ichi ndi chinachake kuti sizichitika.

Yonseyi awiriawiri Action (conventionally nkhani A ndi mlandu B) ndi chodabwitsa umene umachitika nthawi imodzi. Iwo amatchedwa AB.

Kuchuluka kwa awiriawiri zochitika A ndi B - C ndi, mwa kulankhula kwina, ngati mmodzi wa iwo adzaposa (A kapena B), inu mumatenga C. The chilinganizo anafotokoza chodabwitsa kwalembedwa monga C = A + B.

Zochitika atakwatirana mchiphunzitso cha Mwina zikutanthauza kuti milandu iwiri ndi matanthauzo ofananafanana. Pa nthawi yomweyo ali Mulimonsemo sangathe zimachitika. zochitika olowa chiphunzitso Mwina - ndi antipode awo. Zimenezi zikutanthauza kuti ngati A chinachitika, zilibe akuletsa C.

Kutsutsana chochitika (Mwina chiphunzitso asamalira iwo mwatsatanetsatane), ndi losavuta kumvetsa. Ndi bwino kuthana nawo poyerekeza. Iwo ali pafupi ofanana zosagwirizana zikuchitika chiphunzitso cha Mwina. Komabe, kusiyana awo kuti munthu wa kuchuluka kwa zochitika Mulimonsemo muzichita.

Mofanana ayenera zochitika - zochita amenewo, n'zotheka kubwereza ndi wofanana. Kuti bwino, inu mukhoza kulingalira ikukankhidwa ndalamayo: imfa ya munthu wa m'mbali mwake mofanana Nkosatheka imfa ena.

n'kosavuta tione chitsanzo cha akukondera chochitika. Tiyerekeze pali Zomwe zinachitika mu yanthaŵi A. The choyamba - mpukutu wa kufa mkubwela kwa nambala wosamvetseka, ndipo chachiwiri - kuoneka chiwerengero asanu pa dayisi lapansi. Ndiye likukhalira kuti ndi wapadera V.

zochitika Independent chiphunzitso Mwina muno yekha kawiri kapena zambiri ndipo kumafuna wosadalira kanthu kwa mzake. Mwachitsanzo, A - pa imfa michira ndalama mafunde ache, ndi B - dostavanie Jack kwa sitimayo. Iwo ndi zochitika wodzilamulira mu chiphunzitso Mwina. Kuyambira zinadziwika.

zochitika yozungulira chiphunzitso Mwina ndi chovomerezeka okha zawo. Akusonyeza kudalira wa munthu pa ena, ndiye chodabwitsa angayambe mu Koma ngati pamene A kunachitika kale kapena M'malo mwake, sizinachitike pamene - chikhalidwe chachikulu B.

Zotsatira za kafukufuku mwachisawawa ongokhala chigawo limodzi - ndi zochitika pulayimale. Mwina chiphunzitso limanena kuti chinthu chimene anachita kamodzi kokha.

chilinganizo zoyambirira

Choncho, pamwamba ankatengedwa maganizo "chochitika", "Mwina chiphunzitso" matanthauzo a mawu ofunika a sayansi anapatsidwanso. Tsopano ndi nthawi dziŵani yokha ndi mitunduyi zofunika. mawu awa mwamasamu anatsimikizira mfundo zonse waukulu chonchi nkhani zovuta monga chiphunzitso cha Mwina. Mwina chochitika ndipo chimagwira ntchito yaikulu.

Ndikwabwino kuyambitsa ndi mitunduyi zofunika combinatorics. Ndipo musanayambe iwo, ndi bwino kuganizira icho chiri.

Combinatorics - makamaka nthambi ya masamu, iye wakhala akuphunzira nambala yaikulu integers, ndi permutations zosiyanasiyana za manambala ndi zinthu zawo, deta zosiyanasiyana, etc., zikubweretsa angapo ma ... Kuwonjezera pa chiphunzitso cha Mwina, ntchito imeneyi n'kofunika kuti ziwerengero sayansi ya kompyuta ndi cryptography.

Kotero tsopano inu mukhoza kupita mu ulaliki okha ndi tanthauzo mitunduyi awo.

Choyamba mwa zinthuzi ndi mawu akuti chiwerengero cha permutations, ndi motere:

P_n = N ⋅ (n - 1) ⋅ (n - 2) ... 3 2 ⋅ ⋅ 1 = N!

Aone kuti pali likukhudza okha ngati zinthu zosiyanasiyana yekha mu dongosolo la dongosolo.

Tsopano Kusinthaku chilinganizo, izo zikuwoneka ngati izi tidzakambirana:

A_n ^ mamita = N ⋅ (n - 1) ⋅ (n-2) ⋅ ... ⋅ (n - mamita + 1) = N! (N - m)!

Mawu amenewa amagwira ntchito osati chinthu chokha dongosolo Kusinthaku, komanso kapangidwe kake.

The aone kuti pali magawo atatu a combinatorics, ndipo otsiriza, wotchedwa chilinganizo chiwerengero cha ma:

C_n ^ mamita = N! : ((N - m))! : M!

Kuphatikiza otchedwa zitsanzo zosankhidwazi, zimene sizinalembedwa analamula, motero, ndi ntchito lamulo limeneli.

Ndi mitunduyi ya combinatorics anamvetsa mosavuta, mukhoza tsopano kupita tanthauzo chakale cha Mwina. Izo zikuwoneka ngati mawu amenewa motere:

P (A) = mamita: N.

Mu dongosolo ili, mamita - ndi chiwerengero cha zinthu abwino chochitika A, ndipo N - chiwerengero cha mofanana kwathunthu zonse zochitika pulayimale.

Pali mawu ambiri m'nkhani sati ankaona chilichonse koma anakhudzidwa adzakhala ofunika kwambiri monga Mwachitsanzo, Mwina zochitika chimachititsa:

P (A + B) = P (A) + P (B) - Lingaliro Lovomerezeka izi olembapo yekha zochitika zosiyana kwenikweni;

P (A + B) = P (A) + P (B) - P (AB) - koma izi zokha olembapo n'zogwirizana.

Mwina ntchito chochitika:

P (A ⋅ B) = P (A) ⋅ P (B) - Lingaliro Lovomerezeka izi zochitika palokha;

(P (A ⋅ B) = P (A) ⋅ P (B | A); P (A ⋅ B) = P (A) ⋅ P (A | B)) - ndipo ichi kwa yozungulira.

Yatha mndandanda wa zinthu chilinganizo. Chiphunzitso cha Mwina limatiuza Lingaliro Lovomerezeka Bayes, zomwe zikuwoneka ngati izi:

P (H_m | A) = (P (H_m) P (A | H_m)): (Σ_ (k = 1) ^ N P (H_k) P (A | H_k)), mamita = 1, ..., N

Mu dongosolo ili, H _1, H _2, ..., H _N - ndi lonse la malingaliro.

Pa amasiya izi, zitsanzo mitunduyi ntchito tikambirana pochita yeniyeni ndi mwambo.

zitsanzo

Ngati inu mosamala nthambi iliyonse ya masamu, si popanda zokambirana komanso njira nyemba. Ndipo chiphunzitso cha Mwina: zochitika, zitsanzo pano ali ndi yofunika zikuluzikulu za kutsimikizira kuwerengetsera sayansi.

Chilinganizo chiwerengero cha permutations

Mwachitsanzo, mu khadi sitimayo makadi makumi atatu, kuyambira ndi munthu mwadzina. funso lotsatira. Kodi njira pindani sitimayo kuti makadi ndi phindu nkhope ya munthu ndi awiri sanali ili pafupi ambiri?

Ntchito wakhazikika, tsopano tiyeni tisunthire kuthana nazo. Choyamba muyenera kudziwa chiwerengero cha permutations wa zinthu makumi atatu, chifukwa cha ichi tiyenera kutenga chilinganizo pamwamba, likukhalira P_30 = 30.

Kuchokera pa ulamuliro uwu, tidziwa bwanji njira zambiri pali n'kutsamira sitimayo m'njira zambiri, koma tiyenera deducted kwa iwo ndi anthu amene khadi loyamba ndi lachiwiri adzakhala wotsatira. Kuti tichite zimenezi, kuyamba ndi zosinthika, pamene woyamba ili pa lachiwiri. Iwo likukhalira kuti mapu woyamba kutenga malo makumi awiri ndi anayi - kuchokera loyamba mpaka makumi awiri naini, ndi khadi wachiwiri kwa wachiwiri kwa makumi atatu, akutembenukira mipando makumi awiri zisanu ndi zinayi awiriawiri makadi. Zikatero, ena akhoza kutenga mipando twente eyiti, ndi kuti chilichonse. Ndiko, kwa rearrangement wa makadi twente eyiti kuti makumi asanu ndi atatu options P_28 = 28

Zotsatira zake n'zakuti ngati tilingalira za nkhaniyi, pamene khadi woyamba ndi mwayi wachiwiri owonjezera kuti 29 ⋅ 28 = 29!

Kugwiritsa ntchito njira yofanana, muyenera awerengere nambala ya options kuthetsa kwa choncho pamene khadi loyamba lili pansi wachiwiri. Analandiranso 29 ⋅ 28 = 29!

Kuchokera pamenepa, ndizoonekeratu kuti mungachite owonjezera 2 ⋅ 29! Pamene njira kofunika kusonkhanitsa sitimayo 30! - 2 ⋅ 29. Imangokhala yokha kuwerengera.

30! = 29! ⋅ 30; 30 - 2 ⋅ 29! = 29! ⋅ (30 - 2) = 29! ⋅ 28

Tsopano tiyenera kuchulukitsa pamodzi onse manambala kuyambira wani mpaka twente-naini, ndiyeno pa mapeto a zonse kuchulukitsa ndi 28. Yankho analandira 2,4757335 ⋅ 〖〗 10 ^ 32

Zitsanzo njira. Chilinganizo chiwerengero cha malawi

Mu vuto limeneli, muyenera kupeza angati pali njira kuika mabuku khumi ndi zisanu shelefu, koma pansi chikhalidwe chimene makumi atatu okha mabuku.

Mu ntchito iyi, chosankha zosavuta kuposa m'mbuyomu. Kugwiritsa chilinganizo kale, m'pofunika kuwerengetsa chiwerengero cha malo makumi khumi mabuku.

A_30 ^ 15 = 30 ⋅ 29 ⋅ ... ⋅ 28⋅ (30 - 15 + 1) = 30 ⋅ 29 ⋅ 28 ⋅ ... ⋅ 16 = 202 843 204 931 727 360 000

Anayankha, motero, adzakhala wofanana 202 843 204 931 727 360 000.

Tsopano tengani ntchito pang'ono kovuta. Muyenera kudziwa angati pali njira kukonza mabuku makumi awiri pa maalumali, ndi proviso kuti mabuku khumi okha adzakhala pa alumali chomwecho.

Pamaso pa chiyambi cha zochita akufuna kufotokozera kuti mavuto ena zingathe kuthetsedwa m'njira zingapo, ndipo iyi pali njira ziwiri, koma onse limodzi ndi chilinganizo amagwiritsa ntchito mawu omwewa.

Mu ntchito iyi, inu mukhoza kutenga yankho limodzi yapita, chifukwa pali ife masamu chiwerengero cha nthawi inu mukhoza kulemba shelufu mabuku khumi m'njira zosiyanasiyana. Anamuuza A_30 ^ 15 = 30 ⋅ 29 ⋅ 28 ⋅ ... ⋅ (30 - 15 + 1) = 30 ⋅ 29 ⋅ 28 ⋅ ... ⋅ 16.

The Regiment chachiwiri masamu ndi chilinganizo reshuffle, chifukwa aikidwa mabuku fifitini, pamene yotsala fifitini. Ife ntchito chilinganizo P_15 = 15.

Iwo likukhalira kuti ndalama adzatero A_30 ^ 15 ⋅ P_15 njira, koma, kuwonjezera, mankhwala a manambala onse pa makumi kwa sikisitini kuti kuchulukitsa ndi mankhwala a manambala kuyambira munthu mpaka fifitini, pamapeto kuzimitsa mankhwala a manambala onse kwa wina ndi makumi atatu, ndiye yankho ndi 30!

Koma vuto izi zikhoza kuthetsedwa m'njira yosiyana - mosavuta. Kuchita izi, inu mukhoza kulingalira kuti pali alumali wina mabuku makumi atatu. Onse a iwo anayikidwa pa ndege izi, koma chifukwa chikhalidwe amafuna kuti panali maalumali awiri, wina yaitali ife kumaliza mu theka, ankasinthana awiri khumi. Kuyambira lero likukhalira kuti makonzedwe zingakhale P_30 = 30.

Zitsanzo njira. Chilinganizo chiwerengero cha osakaniza

Amene amaonedwa ndi zosinthika cha vuto lachitatu la combinatorics. Muyenera kudziwa mmene njira zambiri pali kukonza mabuku khumi ndi zisanu chikhalidwe chimene muyenera kusankha pa makumi chimodzimodzi.

Chigamulo, ndithudi, ntchito njira chiwerengero cha ma. Ku chikhalidwe chimene zimaonekeratu kuti dongosolo la mabuku amodzi khumi si zofunika. Moti poyamba muyenera kupeza chiwerengero cha ma makumi khumi mabuku.

C_30 ^ 15 = 30! : ((30-15))! : 15! = 155117520

Ndizo zonse. Kugwiritsa chilinganizo uno, mu nthawi yaifupi zotheka kuthetsa vuto limeneli, yankho, motero, wofanana 155.117.520.

Zitsanzo njira. The tanthauzo tingachipeze powerenga za Mwina

Kugwiritsa chilinganizo anapatsidwa pamwamba, munthu akhoza kupeza yankho ntchito yosavuta. Koma izo kuona ndi kutsatira zochita.

Ntchito linaperekedwa kuti urn ndi pali khumi mipira kwathunthu zofanana. Awa, anayi chikasu ndi asanu buluu. Anatengedwa kuchokera urn mpira umodzi. M'pofunika kudziwa Mwina dostavaniya buluu.

Pofuna kuthetsa m'pofunika tisonyezeni dostavanie buluu mpira chochitika A. Zimenezi zikhoza kukhala zotsatira zake khumi, amene nawonso, pulayimale ndipo mofanana mwina. Pa nthawi yomweyo, zisanu ndi khumi ndi abwino chochitika A. kuthetsa chilinganizo zotsatirazi:

P (A) = 6: 10 = 0.6

Kugwiritsa chilinganizo ichi, taphunzira kuti n'zotheka dostavaniya buluu mpira ndi 0,6.

Zitsanzo njira. Mwina zochitika kuchuluka

Ndani zosinthika amene anathana ndi ntchito chilinganizo cha Mwina zochitika kuchuluka. Choncho, amapatsidwa chikhalidwe chimenecho pali milandu iwiri, woyamba ndi imvi ndi mipira asanu woyera, pamene lachiwiri - eyiti imvi ndi anayi woyera mipira. Chifukwa, mabokosi yoyamba ndi yachiwiri amayamba mmodzi wa iwo. M'pofunika kupeza chomwe ndi mwayi munalibe mipira ndi imvi ndi woyera.

Kuti athetse vutoli, m'pofunika kudziwa chochitika.

• Choncho, A - tili ndi imvi mpira wa bokosi loyamba: P (A) = 1/6.
• A '- woyera babu naonso anatengedwa kuchokera bokosi loyamba: P (A) = 5/6.
• The - kale yotengedwa imvi mpira wa ngalande chachiwiri: P (B) = 2/3.
• B '- anatenga imvi mpira wa kabati chachiwiri: P (B) = 1/3.

Malinga ndi vuto m'pofunika kuti mmodzi wa chododometsa chinachitika: AB 'kapena' B. Kugwiritsa ntchito njira timapeza: P (AB ') = 1/18, P (A'B) = 10/18.

Tsopano chilinganizo cha kuchulukana Mwina ankagwiritsa ntchito. Kenako, kuti tipeze yankho, muyenera kutsatira aone kuti pali awo kuwonjezera:

P = P (AB '+ A'B) = P (AB') + P (A'B) = 11/18.

Ndi momwe, pogwiritsa ntchito njira, mukhoza kuthetsa mavuto amenewa.

chifukwa

pepala unaperekedwa kwa zinthuzo "Mwina chiphunzitso" Mwina zochitika kuti mbali yofunika kwambiri. N'zoona kuti zonse kale, koma pamaziko a m'lemba m'nkhaniyo, mukhoza theoretically Dziŵani nthambi izi wa masamu. Akuti sayansi kungakhale kothandiza osati malonda a akatswiri, komanso moyo wa tsiku ndi tsiku. Mukhoza ntchito kuwerengera mwawi chochitika.

Malemba, anakhudzanso ndi zibwenzi kwambiri mbiri ya chitukuko cha chiphunzitso Mwina sayansi, ndi mayina a anthu amene ntchito achita mu izo. Umo ndi momwe anthu chidwi kwachititsa kuti anthu aphunzira kuwerenga, ngakhale zinthu mwachisawawa. Akayamba kungofuna izi, koma lero kale kudziwika kwa onse. Ndipo palibe wina akhoza kunena chimene chidzachitikira m'tsogolo, chimene ena atulukira wanzeru zokhudzana ndi chiphunzitso pansi kulingalira, kuti muzipereke. Koma chinthu chimodzi motsimikiza - phunziro akadali sikuthandiza!