A ka'idar yiwuwa. Yiwuwar samun wani taron, riqo na wuccin aukuwa (yiwuwa ka'idar). Independent da kuma m aukuwa a ka'idar yiwuwa

Abu ne mai wuya cewa mutane da yawa zaton yana yiwuwa ya ƙidaya events, wanda wasu har mai haɗari. Don saka shi a cikin sauki kalmomi, shi ne shi idon basira zuwa ga sani da gefen shigen sukari a cikin dan lido zai fada na gaba lokaci. Shi wannan tambaya biyu mai girma masana kimiyya, aza harsashin ginin wannan kimiyya, ka'idar da Yiwuwar, yiwuwar taron a cikin abin da karatu a baje isa.

ƙarni

Idan ka yi kokarin ayyana irin wannan ra'ayi kamar yadda ka'idar yiwuwa, mu samu wadannan: wannan shi ne daya daga cikin rassan lissafi da cewa karatu da haƙuri da bazuwar abubuwan. A bayyane yake, wannan ra'ayi da gaske ba Ya bayyana ainihi, saboda haka kana bukatar ka yi la'akari da shi a cikin mafi daki-daki.

Ina son fara tare da kafa ka'idar. Kamar yadda aka ambata a sama, akwai biyu, cewa Per Ferma da Blez Paskal. Sun kasance sũ ne na farko yunkurin yin amfani da dabarbari da kuma ilmin lissafi lissafin yin lissafi da sakamako daga wani taron. A general, da rudiments na wannan kimiyya ne ko da a tsakiyar zamanai. Duk da yake daban-daban gabascin da masana kimiyya sun yi kokari don nazarin gidan caca da wasannin irin su caca, craps, da sauransu, game da shi, to kafa juna, da kuma yawan asarar da dama. Kafuwar aka kuma aza a cikin goma sha bakwai karni shi ya waccan mujalla da muka malamai.

Da farko, su aiki ba za a iya dangana ga mai girma nasarori a wannan filin, bayan duk, abin da suka aikata, suka kasance kawai empirical facts da gwaje-gwajen sun fili ba tare da yin amfani da dabarbari. A tsawon lokaci, ya juya ya cimma babban sakamako, wanda ya bayyana a matsayin wani sakamakon kallo na simintin gyaran daga cikin ƙasusuwansa. An wannan kayan aiki ya taimaka wajen kawo farko jinsin dabara.

magoya bayan

Ba a ma maganar irin wannan mutum a matsayin Christiaan Huygens, a kan aiwatar da nazarin batun da ake kira da sunan "Yiwuwar ka'idar" (Yiwuwar taron Highlights da shi a cikin wannan kimiyya). Wannan mutum ne da ban sha'awa sosai. Ya, kazalika da masana kimiyya ya gabatar a sama ana kokarin a cikin hanyar ilmin lissafi dabarbari zuwa kacokan a juna na bazuwar events. Abin lura shi ne cewa ya ba raba shi da Pascal da Fermat, cewa shi ne dukan aikinsa ba zoba da waɗanda hankalinsu. Huygens samu asali Concepts na yiwuwa ka'idar.

An ban sha'awa Gaskiyar ita ce aikinsa zo tun kafin sakamakon ayyukan majagaba, su zama daidai, ashirin da shekaru a baya. Akwai kawai daga cikin matsalolin gano su ne:

• kamar yadda manufar yiwuwa dabi'u damar.
• fata ga mai hankali hali.
• theorems na Bugu da kari kuma multiplication na yiwuwa.

Har ila yau, wanda ba zai iya manta da Yakoba Bernulli, wanda kuma gudummawar da nazarin matsalar. Ta hanyar nasu ba, ba wanda masu zaman kansu da gwaje-gwaje, ya ya iya samar da hujja na dokar manyan lambobi. Bi da bi, masana kimiyya Poisson da Laplace, wanda ya yi aiki a cikin farkon karni na sha tara, sun iya tabbatar da ainihin Theorem. Daga wannan lokacin don nazarin kurakurai a cikin lura da muka fara yin amfani da yiwuwar ka'idar. Jam'iyyar kusa da wannan kimiyya iya ba da kuma Rasha masana kimiyya, wajen Markov, Chebyshev da Dyapunov. Suna dogara ne a kan aikin yi girma 'yan baiwa, kulla da batun kamar yadda wani reshe na lissafi. Mun yi aiki da wadannan Figures a karshen karni na sha tara, da kuma godiya ga taimako, da aka tabbatar da mamaki kamar:

• Shari'ar manyan lambobi.
• Ka'idar Markov sarƙoƙi.
• A tsakiyar iyaka Theorem.

Saboda haka, tarihin haihuwar kimiyya da tare da manyan mutane da cewa gudummawar da shi, duk abin da yake fiye ko žasa bayyananne. Yanzu yana da lokaci zuwa nama daga duk hujjojin.

asali Concepts

A gabãnin ku shãfe dokoki da theorems kamata koyi ainihin Concepts na yiwuwa ka'idar. Event ya mamaye wani gagarumar rawar. Wannan topic ne wajen m, amma ba za su iya fahimta duk sauran ba tare da shi.

Taron a Yiwuwar ka'idar - shi Wani sa na sakamakon da gwajin. Concepts wannan sabon abu akwai bai isa ba. Saboda haka, Lotman masanin kimiyya aiki a cikin wannan wuri, ya bayyana cewa, a cikin wannan harka muna magana ne game da abin da "ya faru, ko da yake da shi ba zai faru."

Random events (yiwuwa ka'idar biya musamman da hankali ga su) - shi ne ra'ayi cewa ya shafi cikakken wani sabon abu da ciwon da yiwuwar faruwa. Ko, a maimakon haka, wannan labari ba zai iya faruwa a wasan kwaikwayon na dama yanayi. Shi ne ma daraja da sanin cewa zauna cikin duka girma daga cikin mamaki da ke faruwa kawai bazuwar events. Yiwuwar ka'idar shawara cewa dukkan yanayi za a iya maimaita kullum. Shi ne mai raRaɗi da aka kira "kwarewa" ko "gwajin."

Muhimmanci taron - wannan shi ne wani sabon abu da yake xari bisa dari a cikin wannan gwajin da ya faru. Haka kuma, ba zai yiwu ba taron - wannan shi ne wani abu da ba ya faru.

Hada da nau'i-nau'i Action (conventionally al'amarin A da hali B) shi ne sabon abu wanda ya auku a lokaci guda. Suna kira a matsayin AB.

The adadin nau'i-nau'i daga events A da B - C ne, a cikin wasu kalmomi, idan akalla daya daga cikinsu zai (A ko B), ka samu wani C. The dabara aka bayyana sabon abu ne da aka rubuta a matsayin C = A + B.

M aukuwa a ka'idar yiwuwa ya nuna cewa biyu lokuta ne iri dabam dabam. A lokaci guda su ne a wani hali ba zai iya faruwa. Hadin gwiwa events in yiwuwa ka'idar - shi ne su antipode. Manufarta shi ne cewa idan A faru, shi ba ya hana C.

Hamayya taron (yiwuwa ka'idar gan su a babban daki-daki), su ne sauki fahimta. Shi ne mafi kyau ga magance su a kwatanta. Su ne kusan guda a matsayin m aukuwa a ka'idar yiwuwa. Duk da haka, su bambancin shi ne cewa daya daga wani jam'i na mamaki a wani hali ya kamata faruwa.

Daidai da m events - wadanda ayyuka, da yiwuwar maimaitawa ne daidai. Don bayyana shi, za ka iya kwatanta tossing tsabar kudin: asarar daya daga sãsanninta ne daidai da m asarar wasu.

yana da sauki ga la'akari da misalin fi son taron. Misali akwai wani episode a cikin episode A. The farko - a yi na wata mutu tare da zuwan wani m yawan, da kuma na biyu - da bayyanar da yawan biyar a kan dan lido. To, shi dai fitar da cewa A ne madaukakan V.

Independent events in yiwuwa ka'idar an kimanta kawai a kan biyu ko fiye da lokatai da ya unsa mai zaman kanta da wani mataki daga wasu. Alal misali, A - a asara wutsiyoyi tsabar kudin tossing, da kuma B - dostavanie jack daga cikin bene. Bã su da m events in yiwuwa ka'idar. Daga wannan lokacin da ya bayyana.

Dogara da abubuwan da suka faru a cikin yiwuwa ka'idar ne ma ya halatta ne kawai don su sa. Su nufa dogara da daya a kan wasu, cewa shi ne, sabon abu zai iya faruwa a kawai a cikin yanayin lokacin A ya riga ya faru, ko kuma, a akasin haka, bai faru ba a lokacin da shi ne - da babban sharadin B.

A sakamakon bazuwar gwajin ya kunshi guda bangaren - yana da firamare events. Yiwuwar ka'idar ya ce shi ne wani sabon abu da aka yi sau ɗaya kawai.

asali dabara

Saboda haka, sama an dauke da manufar "aukuwa", "Yiwuwar ka'idar", ma'anar key sharuddan da wannan kimiyya da aka ma ba. Yanzu yana da lokaci zuwa familiarize da kanta tare da muhimmanci dabarbari. Wadannan maganganu suna da shifran tabbatar duk manyan mahanga a cikin irin wannan da wuya batu kamar yadda ka'idar yiwuwa. Yiwuwar samun wani taron da kuma taka wata babbar rawar.

Better to fara da asali dabarbari na combinatorics. Kuma kafin ka fara su, yana da daraja la'akari da abin da shi ne.

Combinatorics - shi ne da farko wani reshe na lissafi, ya ya aka karatu a babbar dama integers, da kuma daban-daban permutations na biyu da lambobi da su abubuwa, daban-daban data, da dai sauransu, abu zuwa yawan haduwa ... Bugu da kari ga ka'idar yiwuwa, wannan masana'antu da muhimmanci ga statistics, kimiyyar kwamfuta da cryptography.

Saboda haka yanzu za ka iya matsa wa gabatar da kansu da kuma definition dabarbari.

A farko daga cikin wadannan shi ne magana ga yawan permutations, shi ne kamar haka:

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

Jadawalin shafi kawai a yanayin saukan idan abubuwa bambanta kawai a cikin tsari na tsari.

Yanzu jeri dabara, shi kama da wannan za a yi la'akari:

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

Wannan magana ne m ba kawai ga kawai kashi na domin jeri, amma kuma ga abun da ke ciki.

The uku Jadawalin combinatorics, da kuma shi ne karshen, kira da dabara ga yawan haduwa:

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

Hade kira Samfur, wanda aka ba da umarnin, bi da bi, su da kuma amfani da wannan mulkin.

Tare da dabarbari na combinatorics zo domin ya fahimci sauƙi, za ka iya yanzu je na gargajiya definition na yiwuwa. Yana kama da wannan magana kamar haka:

P (A) = m: n.

A wannan dabara, m - shi ne yawan yanayi moriya ga taron A, da kuma n - yawan daidai da kuma gaba daya duk na farko events.

Akwai da yawa maganganu a cikin labarin ba zai iya daukan wani abu amma ya shafa za su zama mafiya muhimmanci kamar, misali, yiwuwar events nawa:

P (A + B) = P (A) + P (B) - wannan Theorem domin ƙara kawai iri dabam dabam events.

P (A + B) = P (A) + P (B) - P (AB) - amma wannan ne kawai domin kara jituwa.

A yiwuwa na taron ayyukan:

P (A ⋅ B) = P (A) ⋅ P (B) - wannan Theorem ga m events.

(P (A ⋅ B) = P (A) ⋅ P (B | A); P (A ⋅ B) = P (A) ⋅ P (A | B)) - kuma wannan ga dogara.

Ƙare jerin events dabara. A ka'idar yiwuwa ya gaya mana Theorem Bayes, wanda yai kama da wannan:

P (H_m | A) = (P (H_m) P (A | H_m)): (Σ_ (k = 1) ^ n P (H_k) P (A | H_k)), m = 1, ..., n

A wannan dabara, H _1, H _2, ..., H _n - shi ne mai cikakken sa na shiriritar.

A wannan tasha, samfurori dabarbari aikace-aikace zai yanzu a dauke for takamaiman ayyuka daga yi.

misalai

Idan ka bincika a hankali wani reshe na lissafi, shi ne ba tare da darussan da samfurin mafita. Kuma ka'idar yiwuwa: events, misalai nan ne wani na game bangaren na mai gaskatãwa kimiyya lissafin.

Da dabara ga yawan permutations

Alal misali, a cikin wani katin bene da talatin cards, da suka fara da maras muhimmanci daya. Next tambaya. Ta yaya da yawa hanyoyin da za a ninka bene haka cewa katunan tare da wata fuska darajar daya da biyu da aka ba located gaba?

The aiki da aka saita, yanzu bari mu matsa zuwa magance shi. Da farko kana bukatar ka ƙayyade yawan permutations talatin abubuwa, don wannan dalili muka dauki sama dabara, shi dai P_30 = 30!.

Bisa wannan doka, da muka sani da yawa zabin akwai to kwanta bene a hanyoyi masu yawa, amma dole ne mu za a deducted daga cikinsu akwai waɗanda a cikin abin da na farko da na biyu katin zai zama na gaba. Don yin wannan, a fara da wani bambanci, a lokacin da na farko da aka located a kan na biyu. Sai dai itace cewa da farko map iya daukar ashirin da tara wuraren - daga na farko zuwa na ashirin da tara, kuma na biyu katin daga biyu zuwa talatin, ya jũya ashirin da tara kujeru ga nau'i-nau'i daga cards. Bi da bi, da sauransu iya dauka da ashirin da takwas kujeru, da kuma a cikin wani tsari. Wannan shi ne, domin rearrangement na ashirin da takwas cards sun ashirin da takwas zabin P_28 = 28!

A sakamakon haka ne cewa idan mun yi la'akari da yanke shawara, idan na farko da katin ne a kan na biyu karin damar samun 29 ⋅ 28! = 29!

Amfani da wannan hanya, kana bukatar yin lissafi da yawan m zaɓuɓɓuka saboda haka al'amarin a lokacin da na farko da katin da aka located a karkashin biyu. Har ila yau samu 29 ⋅ 28! = 29!

Daga wannan shi ya bi cewa karin zažužžukan 2 ⋅ 29!, Yayin da zama dole wajen tattara bene 30! - 2 ⋅ 29!. Ya zauna kawai yin lissafi.

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

Yanzu muna bukatar ninka tare duk da lambobi daga daya zuwa ashirin da tara, sa'an nan kuma a karshen duk ta tara da 28. Amsar samu 2,4757335 ⋅ 〖〗 10 ^ 32

Misalan mafita. Da dabara ga yawan masauki

A wannan matsala, kana bukatar ka gano yadda da yawa akwai hanyoyin da za a sa da goma sha biyar kundin a kan wani shiryayye, amma a karkashin yanayin da cewa kawai talatin kundin.

A wannan aiki, da yanke shawara kadan sauki fiye da baya. Amfani da aka riga aka sani dabara, shi wajibi ne yin lissafi da duka yawan talatin wurare goma sha biyar kundin.

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

Martani, bi da bi, za su zama daidai ga 202 843 204 931 727 360 000.

Yanzu kai da aiki dan kadan mafi wuya. Kana bukatar ka sani da yawa akwai hanyoyin da za a gabatar da talatin da biyu littattafai a kan shelves, tare da proviso cewa kawai goma sha biyar kundin iya kasance a kan wannan shiryayye.

Kafin farkon na yanke shawara son bayyana cewa wasu daga cikin matsalolin da za a iya warware a hanyoyi da dama, da kuma a cikin wannan akwai hanyoyi biyu, amma a biyu wanda kuma wannan dabara ne amfani.

A wannan aiki, za ka iya yi amsar daga baya daya, saboda akwai mun lasafta yawan lokuta za ka iya cika fitar da shiryayye goma sha biyar littattafai a cikin hanyoyi daban-daban. Sai ya juya A_30 ^ 15 = 30 ⋅ 29 ⋅ 28 ⋅ ... ⋅ (30 - 15 + 1) = 30 ⋅ 29 ⋅ 28 ⋅ ... ⋅ 16.

Na biyu rajimanti lasafta ta da dabara garambawul, saboda an sanya goma sha biyar littattafai, yayin da saura na da goma sha biyar. Mun yi amfani da dabara P_15 = 15!.

Sai dai itace cewa da Miliyan Xari so A_30 ^ 15 ⋅ P_15 hanyoyi, amma, a cikin Bugu da kari, da samfurin na duk lambobi daga talatin zuwa goma sha shida za su yawaita da samfurin na lambobin daga daya zuwa goma sha biyar, a karshen fitar da samfurin na duk lambobi daga daya zuwa talatin, cewa shi ne amsar ne 30!

Amma wannan matsala za a iya warware a wani daban-daban hanya - sauki. Don yin wannan, za ka iya tunanin cewa akwai daya shiryayye talatin littattafai. Dukan su suna sanya a kan wannan jirgin sama, amma saboda yanayin na bukatar cewa akwai biyu shelves, daya dade muna sawing a rabin, biyu jũya goma sha biyar. Daga wannan itace cewa wannan tsari zai iya zama P_30 = 30!.

Misalan mafita. Da dabara ga yawan haduwa da

Wanda yana dauke da wani bambance-bambancen da uku matsalar da combinatorics. Kana bukatar ka sani da yawa hanyoyi akwai shirya da goma sha biyar littattafai a kan yanayin da cewa dole ne ka zabi daga talatin daidai da wannan.

Domin yanke shawara za, ba shakka, amfani da dabara domin yawan haduwa. Daga cikin yanayin da ya bayyana cewa, domin na wannan goma sha biyar littattafai ne ba muhimmanci. Saboda haka da farko kana bukatar ka sami fita da jimlar yawan haduwa da talatin da goma sha biyar littattafai.

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

Shi ke nan. Amfani da wannan dabara, a cikin guntu lokaci zai yiwu a warware irin wannan matsala, amsar, bi da bi, daidai to 155.117.520.

Misalan mafita. The classic definition na yiwuwa

Amfani da dabara ba sama, wanda zai iya samun amsar a cikin sauki aiki. Amma shi zai gani a sarari kuma bi hanya na aiki.

The aiki ba cewa a cikin wani akwatin jefa kuri'a akwai goma gaba daya m bukukuwa. Daga cikin wadannan, hudu rawaya da shida blue. Dauka daga akwatin jefa kuri'a daya ball. Wajibi ne a san yiwuwa dostavaniya blue.

Don warware matsalar ya zama dole to designate dostavanie blue ball taron A. Wannan kwarewa iya samun goma sakamakon, wanda, bi da bi, na farko da kuma daidai da m. A daidai wannan lokaci, shida daga cikin goma ne m zuwa ga taron A. Warware da wadannan dabara:

P (A) = 6: 10 = 0.6

Bin wannan dabara, mun koyi cewa da yiwuwar dostavaniya blue ball ne 0.6.

Misalan mafita. Yiwuwar events adadin

Wa zai zama wani bambanci da aka warware ta yin amfani da dabara na Yiwuwar events adadin. Saboda haka, ba sharadin cewa akwai biyu lokuta, da farko daya ne m da biyar fari bukukuwa, yayin da na biyu - takwas launin toka da hudu fari bukukuwa. A sakamakon haka, na farko da na biyu kwalaye sun dauka a kan daya daga cikinsu. Wajibi ne a gano abin da su ne chances cewa rasa da bukukuwa ne m da fari.

Don warware wannan matsala, shi wajibi ne don gane da taron.

• Kamar wancan ne, A - muna da wani m ball na farko akwatin: P (A) = 1/6.
• A '- fari kwan fitila ma dauka daga cikin na farko akwatin: P (A') = 5/6.
• The - riga cirewa m ball na biyu wuriyar: P (B) = 2/3.
• B '- dauki wani m ball na biyu aljihun tebur: P (B') = 1/3.

Bisa ga matsalar shi ne cewa wajibi ne daya daga cikin mamaki ya faru: AB 'ko' B. Amfani da dabara, mun samu: P (AB ') = 1/18, P (A'B) = 10/18.

Yanzu da dabara na halitta yiwuwar aka yi amfani. Next, to gano amsar, kana bukatar ka nema su lissafi kara:

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

Wannan yadda, ta amfani da dabara, ba za ka iya warware irin wannan matsaloli.

sakamakon

A takardar da aka gabatar da bayanai a kan "yiwuwa ka'idar", yiwuwar events cewa taka muhimmiyar rawa. Hakika, ba duk abin da aka gani, amma a kan tushen da rubutu gabatar, za ka iya rubuce samun matsahi na saba da wannan reshe na lissafi. Dauke kimiyya iya zama da amfani ba kawai a cikin sana'a kasuwanci, amma kuma a cikin rayuwar yau da kullum. Za ka iya amfani da shi zuwa lissafi wani yiwuwar wani taron.

The rubutu da aka ma ta shafa gagarumin kwanakin a tarihin ci gaban yiwuwa ka'idar a matsayin kimiyya, da kuma sunayen mutanen da waɗanda ayyukansu aka sa a cikin ta. Wannan yadda mutum son sani ya kai ga cewa mutane sun koyi su ƙidaya, ko da bazuwar abubuwan. Da zarar sun kasance kamar sha'awar a cikin wannan, amma a yau an riga an sanar wa dukan. Kuma babu wanda zai iya ce abin da zai faru da mu a nan gaba, abin da wasu m binciken alaka da ka'idar karkashin shawara, za a aikata. Amma abu daya ne don tabbatar da - da binciken har yanzu ba shi daraja!