Yiwuwar abu (Lissafi)

Yiwuwar abu (Lissafi)
probability measure (en)
Bayanai
Ƙaramin ɓangare na	mathematical object (en) , dimensionless quantity (en) da possibility (en)
Fuskar	number of outcomes (en)
Karatun ta	probability theory (en) da statistics (en)
ISQ dimension (en)	$1$
Image of function (en)	unit interval (en)
Quantity symbol (LaTeX) (en)	$p$ da $P$
Contributing factor of (en)	uncertainty (en)
Has characteristic (en)	central limit theorem (en) , law of large numbers (en) , probability measure (en) da probability distribution (en)
Notation (en)	P

 

Halin da za a iya juyawa lambobi da yawa ta amfani da dice biyu

Mai yiwuwa ya shafi abubuwan da suka faru da kuma bayanin lambobi na yadda za su faru. Halin yiwuwar wani abu shine lamba tsakanin 0 da 1; mafi girman yiwuwar, mai yiwuwa wani abu zai faru. [bayanin kula 1] Wannan lambar sau da yawa ana bayyana ta a matsayin kashi (%), daga 0% zuwa 100%. Misali mai sauƙi shine jefa tsabar kudi mai kyau (marasa son kai). Tunda tsabar kudi tana da kyau, sakamakon biyu ("kai" da "tsuntsaye") duka suna da yiwuwar daidai; yiwuwar "tsuntsayen" daidai yake da yiwuwar "saki"; kuma tunda babu wasu sakamako masu yiwuwa, yiwuwar ko dai "tsaki" ko "tsakiyoyi" shine 1/2 (wanda kuma za'a iya rubuta shi a matsayin 0.5 ko 50%).

Wadannan ra'ayoyin an ba su tsari na lissafi na axiomatic a cikin Ka'idar yiwuwar, wanda ake amfani dashi ko'ina a fannonin karatu kamar kididdiga, lissafi, kimiyya, kudi, caca, hankali na wucin gadi, ilmantarwa na inji, kimiyyar kwamfuta, ka'idar wasan, da falsafar, alal misali, zana ra'ayi game da yawan abubuwan da ake tsammani. Hakanan ana amfani da ka'idar yiwuwar don bayyana ainihin injiniyoyi da daidaito na tsarin rikitarwa.

Magana

Kalmar yiwuwar ya samo adalci ne daga Latin probabilitas, wanda kuma zai iya nufin "gaskiya", ma'auni na iko shaida a cikin shari'ar shari'a a Turai, kuma sau da yawa yana da alaƙa da daraja shaidar. A wata ma'ana, wannan ya bambanta da ma'anar zamani na yiwuwar, wanda ya bambanta shine ma'auni na nauyin shaidar kwarewa, kuma an kai shi daga tunani mai ma'ana da ƙididdigar ƙididdiga.

Fassara

Lokacin da ake hulɗa da gwaje-gwaje na bazuwar - watau, gwaje-gaje waɗanda ke bazuwar kuma an bayyana su sosai - a cikin saitin ka'ida (kamar jefa tsabar kudi), ana iya bayyana yiwuwar ta hanyar lambobi ta yawan sakamakon da ake so, an raba su da jimlar adadin duk sakamakon. Ana kiran wannan a matsayin yiwuwar ka'ida (ba kamar yiwuwar kwarewa ba, ma'amala da yiwuwar a cikin mahallin gwaje-gwaje na ainihi). Mai yiwuwa shine lamba tsakanin 0 da 1; mafi girman yiwuwar, mafi kusantar sakamakon da ake so zai faru. Misali, jefa tsabar kudi sau biyu zai samar da sakamakon "kai-kai", "kai-kibiya", "kai", da kuma "kibiya-kibiya". Halin yiwuwar samun sakamakon "kai-kai" shine 1 daga cikin sakamako 4, ko, a cikin lambobi, 1/4, 0.25 ko 25%. Halin samun sakamako na akalla kai ɗaya shine 3 daga 4, ko 0.75, kuma wannan taron zai fi faruwa. Koyaya, idan ya zo ga aikace-aikace mai amfani, akwai manyan nau'o'i biyu masu gasa na fassarorin yiwuwar, waɗanda mabiyan su ke da ra'ayoyi daban-daban game da ainihin yanayin yiwuwar:

• Masu son kai suna ba da lambobi don bayyana wasu manufa ko yanayin jiki. Mafi mashahuriyar fasalin yiwuwar manufa ita ce yiwuwar maimaitawa, wanda ke da'awar cewa yiwuwar wani abu na bazuwar yana nuna yawan dangi na faruwar sakamakon gwaji lokacin da aka sake maimaita gwajin har abada. Wannan fassarar tana la'akari da yiwuwar zama dangi "a cikin dogon lokaci" na sakamakon. Canjin wannan shine yiwuwar yiwuwar, wanda ke fassara yiwuwar a matsayin yanayin wasu gwaje-gwaje don samar da wani sakamako, koda kuwa an yi shi sau ɗaya kawai.
• Masu ra'ayi suna ba da lambobi ta hanyar yiwuwar ra'ayi, wato, a matsayin matakin imani. An fassara matakin imani a matsayin "farashin da za ku saya ko siyar da fare wanda ke biyan 1 na amfani idan E, 0 idan ba E ba", kodayake wannan fassarar ba a yarda da ita ba a duniya. Mafi mashahuriyar fasalin yiwuwar ra'ayi shine yiwuwar Bayesian, wanda ya haɗa da ƙwararren ilimin da kuma bayanan gwaji don samar da yiwuwar. Ilimin kwararru yana wakiltar wasu (subjective) yiwuwar rarraba a baya. Wadannan bayanai an haɗa su a cikin aikin yiwuwar. Samfurin da ya gabata da yiwuwar, lokacin da aka daidaita, yana haifar da Rarraba yiwuwar baya wanda ya haɗa da duk bayanan da aka sani har zuwa yau. Ta hanyar yarjejeniyar yarjejeniyar Aumann, jami'an Bayesian waɗanda imanin su na baya yayi kama da juna zai ƙare da irin wannan imanin baya. Koyaya, isasshen abubuwan da suka dace na iya haifar da yanke shawara daban-daban, ba tare da la'akari da yawan bayanan da jami'an ke raba ba.

Tarihi

Nazarin kimiyya na yiwuwar ci gaba ne na zamani na lissafi. Yin caca ya nuna cewa akwai sha'awar ƙididdigar ra'ayoyin yiwuwar a cikin tarihi, amma ainihin bayanin lissafi ya tashi daga baya. Akwai dalilai na jinkirin ci gaban lissafi na yiwuwar. Ganin cewa wasannin sa'a sun ba da gudummawa ga nazarin lissafi na yiwuwar, batutuwan da suka dace [bayanin 2] har yanzu suna ɓoyewa ta hanyar camfi.

A cewar Richard Jeffrey, "Kafin tsakiyar karni na goma sha bakwai, kalmar 'mai yiwuwar' (Latin probabilis) tana nufin amincewa, kuma an yi amfani da ita a wannan ma'anar, ta hanyar da ta dace, ga ra'ayi da aiki. Wani mataki mai yiwuwa ko ra'ayi shine kamar yadda mutane masu hankali za su yi ko riƙe, a cikin yanayin. Koyaya, a cikin mahallin shari'a musamman, 'mai yiwuwa'. Hakanan yana iya amfani da shawarwari waɗanda akwai kyakkyawar shaida.

Gerolamo Cardano (karni na 16)

Christiaan Huygens ya buga ɗayan littattafai na farko game da yiwuwar (karni na 17).

Masanin ilimin Italiyanci na ƙarni na goma sha shida Gerolamo Cardano ya nuna ingancin bayyana Rashin daidaituwa a matsayin rabo na sakamako mai kyau ga sakamako mara kyau (wanda ke nuna cewa yiwuwar wani taron an ba shi ta hanyar rabo na sakamako masu kyau ga jimlar sakamakon da za a iya samu ). Baya ga aikin farko na Cardano, koyarwar yiwuwar ta kasance daga wasikar Pierre de Fermat da Blaise Pascal (1654). Christiaan Huygens (1657) ya ba da sanannen ilimin kimiyya na farko game da batun. Jakob Bernoulli's Ars Conjectandi (bayan mutuwarsa, 1713) da kuma Abraham de Moivre's Doctrine of Chances (1718) sun bi da batun a matsayin reshe na lissafi. Dubi Ian Hacking's The Emergence of Probability da James Franklin's The Science of Conjecture don tarihin farkon ci gaban ainihin manufar yiwuwar lissafi.

Ana iya gano ka'idar kurakurai zuwa Opera Miscellanea na Roger Cotes (bayan mutuwarsa, 1722), amma wani abin tunawa da Thomas Simpson ya shirya a 1755 (an buga 1756) ya fara amfani da ka'idar don tattaunawar kurakurai na lura. Sake bugawa (1757) na wannan tarihin ya sanya axioms cewa kuskuren da ba su da kyau suna da yiwuwar daidaito, kuma wasu iyakokin da za a iya ba da su sun bayyana kewayon duk kurakurai. Simpson ya kuma tattauna kurakurai masu ci gaba kuma ya bayyana yiwuwar yiwuwar.

Dokokin kuskure guda biyu na farko da aka gabatar sun samo asali ne daga Pierre-Simon Laplace . An buga dokar farko a cikin shekara ta 1774, kuma ta bayyana cewa ana iya bayyana mitar kuskure a matsayin aiki mai yawa na girman lambobi na kuskure - yin watsi da alamar. Dokar kuskure ta biyu ta gabatar a cikin 1778 ta Laplace, kuma ta bayyana cewa mitar kuskuren aiki ne mai yawa na murabba'in kuskuren. Dokar kuskure ta biyu ana kiranta rarraba ta al'ada ko dokar Gauss. "Yana da wahala a tarihi a danganta wannan dokar ga Gauss, wanda duk da sanannen sanannen saurinsa mai yiwuwa bai yi wannan binciken ba kafin ya kai shekaru biyu. " –

Daniel Bernoulli (1778) ya gabatar da ka'idar matsakaicin samfurin yiwuwar tsarin kurakurai na lokaci guda.

Carl Friedrich Gauss

Adrien-Marie Legendre (1805) ya haɓaka hanyar mafi ƙarancin murabba'i, kuma ya gabatar da shi a cikin Nouvelles méthodes pour la détermination des orbites des comètes (Sabon Hanyoyi don Tabbatar da Orbits of Comets). A cikin jahilci game da gudummawar Legendre, wani marubucin Irish-Amurka, Robert Adrain, editan "The Analyst" (1808), ya fara cire dokar kuskuren kuskure.

$$\phi (x)=ce^{-h^{2}x^{2}}$$

where $h$ is a constant depending on precision of observation, and $c$ is a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as John Herschel's (1850). Gauss gave the first proof that seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W.F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula  for r, the probable error of a single observation, is well known.

A cikin karni na sha tara, marubutan kan ka'idar gabaɗaya sun haɗa da Laplace, Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion da Karl Pearson. Augustus De Morgan da George Boole sun inganta bayanin ka'idar.

A cikin 1906, Andrey Markov ya gabatar da ra'ayin sarkar Markov, wanda ya taka muhimmiyar rawa a cikin ka'idar tsarin stochastic da aikace-aikacenta. Ka'idar yiwuwar zamani ta dogara da ka'idar ma'auni ta samo asali ne daga Andrey Kolmogorov a cikin 1931.

A gefen lissafi, masu ba da gudummawa ga The Educational Times sun haɗa da Miller, Crofton, McColl, Wolstenholme, Watson, da Artemas Martin. Dubi cikakkiyar lissafi don ƙarin bayani.

Ka'idar

Kamar sauran ra'ayoyin, ka'idar yiwuwar wakilci ne na ra'ayoyinta a cikin sharuddan tsari - wato, a cikin sharudda da za a iya la'akari da su daban daga ma'anar su. Wadannan ka'idoji da aka tsara ana sarrafa su ta hanyar ka'idoji na lissafi da tunani, kuma ana fassara duk wani sakamako ko fassara shi cikin yankin matsala. –

Akwai akalla yunkurin nasara guda biyu don tsara yiwuwar, wato tsarin Kolmogorov da tsarin Cox. A cikin tsarin Kolmogorov (duba kuma sararin yiwuwar), ana fassara saiti a matsayin abubuwan da suka faru da yiwuwar a matsayin ma'auni a kan aji na saiti. A cikin Ka'idar Cox, ana ɗaukar yiwuwar a matsayin na asali (watau, ba a ƙara nazarin shi ba), kuma an jaddada shi ne akan gina daidaitattun ƙimar yiwuwar ga shawarwari. A lokuta biyu, dokokin yiwuwar iri ɗaya ne, sai dai don cikakkun bayanai na fasaha.

Akwai wasu hanyoyin da za a iya tantance rashin tabbas, kamar Ka'idar Dempster-Shafer ko Ka'idar yiwuwar, amma waɗannan sun bambanta sosai kuma ba su dace da ka'idojin yiwuwar da aka fahimta ba.

Aikace-aikacen

Ana amfani da ka'idar yiwuwar a rayuwar yau da kullun a cikin kimantawa da ƙira. Masana'antar inshora da kasuwanni suna amfani da kimiyyar lissafi don ƙayyade farashi da yanke shawara na kasuwanci. Gwamnatoci suna amfani da hanyoyin yiwuwar a cikin Tsarin muhalli, nazarin haƙƙin mallaka, da Tsarin kudi.

Misali na amfani da ka'idar yiwuwar a cikin cinikayya ta daidaito shine tasirin yiwuwar duk wani rikici na Gabas ta Tsakiya a kan farashin mai, wanda ke da tasiri a cikin tattalin arziki gaba ɗaya. Binciken da mai siyar da kayayyaki ya yi cewa yaƙi zai iya aika farashin kayan sama ko ƙasa, kuma yana nuna wasu 'yan kasuwa na wannan ra'ayi. Dangane da haka, ba a tantance yiwuwar da kanta ba ko kuma ba lallai ba ne a hankali. Ka'idar halayyar kuɗi ta fito ne don bayyana tasirin irin wannan tunanin rukuni akan farashi, akan manufofi, da kuma zaman lafiya da rikici.

Baya ga kimantawa na kudi, ana iya amfani da yiwuwar don nazarin abubuwan da ke faruwa a cikin ilmin halitta (misali, yaduwar cututtuka) da kuma ilimin muhalli (misali), murabba'in Punnett na halitta). Kamar yadda yake tare da kudi, ana iya amfani da kimanta haɗari azaman kayan aiki na kididdiga don lissafin yiwuwar abubuwan da ba a so su faru, kuma suna iya taimakawa tare da aiwatar da ladabi don kauce wa fuskantar irin waɗannan yanayi. Ana amfani da yiwuwar tsara wasannin sa'a don gidajen caca su iya samun tabbacin riba, duk da haka samar da biyan kuɗi ga 'yan wasan da ke da isasshen lokaci don ƙarfafa ci gaba da wasa.

Wani muhimmin aikace-aikacen ka'idar yiwuwar a rayuwar yau da kullun shine amintacce. Yawancin samfuran mabukaci, kamar motoci da kayan lantarki na mabukaci، suna amfani da ka'idar amintacce a cikin ƙirar samfur don rage yiwuwar gazawar. Halin gazawar na iya rinjayar yanke shawara na mai ƙerawa akan garantin samfurin.

Misali na harshe na cache da sauran samfuran harshe na kididdiga waɗanda ake amfani da su a cikin sarrafa harshe na halitta suma misalai ne na aikace-aikacen ka'idar yiwuwar.

Magani na lissafi

Ƙididdigar yiwuwar (hadari) vs odds

Ka yi la'akari da gwaji wanda zai iya samar da sakamako da yawa. Tarin duk sakamakon da zai yiwu ana kiransa samfurin samfurin gwajin, wani lokacin ana nuna shi kamar haka

Ω

{\displaystyle \Omega }

. An kafa saitin wutar lantarki na sararin samfurin ta hanyar la'akari da duk tarin daban-daban na sakamakon da zai yiwu. Misali, mirgina die na iya samar da sakamako shida. Ɗaya daga cikin sakamakon da zai yiwu yana ba da lambar da ba ta dace ba a kan mutuwar. Don haka, ɓangaren {1,3,5} wani bangare ne na saitin wutar lantarki na samfurin sarari dice. Wadannan tarin ana kiransu "al'amuran". A wannan yanayin, {1,3,5} shine abin da ya faru cewa mutuwar ta fadi a kan wani lamari mai ban mamaki. Idan sakamakon da ya faru ya fadi a cikin wani taron, an ce taron ya faru.

Wata yiwuwar wata hanya ce ta ba da kowane taron darajar tsakanin sifili da ɗaya, tare da buƙatar cewa taron ya ƙunshi duk sakamakon da zai yiwu (a cikin misalinmu, an ba da taron {1,2,3,4,5,6}) darajar ɗaya. Don cancanta a matsayin yiwuwar, sanya dabi'u dole ne ya gamsar da abin da ake buƙata cewa don kowane tarin abubuwan da suka faru na juna (al'amuran da ba su da sakamako na kowa, kamar abubuwan da suka gabata {1,6}, {3}, da {2,4}), yiwuwar cewa aƙalla ɗaya daga cikin abubuwan da suka faruwa zai faru an ba shi ta jimlar yiwuwar duk abubuwan da suka biyo baya.

A rubuta yiwuwar wani abu A kamar haka

P (A)

{\displaystyle P (A) }

,

p (A)

{\displaystyle p (A) }

, ko kuma

Pr

(A)

{\displaystyle {\text{Pr}} (A) }

. Wannan ma'anar lissafi na yiwuwar na iya fadada zuwa sararin samfurori marasa iyaka, har ma da sararin samfar da ba a iya lissafawa ba, ta amfani da manufar ma'auni.

A">A hA ko kari na wani taron A shine taron [ba A] (wato, taron A ba ya faruwa), sau da yawa ana nuna shi kamar yadda take A ′

,

A

c

{\displaystyle A',A^{c}}

,

A ̄

,

A

Sanya

, ¬ A

{\displaystyle {\overline {A}},A^{\complement },\neg A}

, ko kuma

∼

A

{\displaystyle {\sim }A}

; yiwuwar ta ta ba da P (ba A) = 1 − Samfuri:Sfrac = Samfuri:Sfrac. P (A). A matsayin misali, damar da ba za a mirgine shida a kan gefen gefen shida ba shine 1 - (hanzarin mirgine shida) = 1 − ⁠1/6⁠ = ⁠5/6⁠ . Don ƙarin bayani, duba Complementary event.

IdA AB' biyu A da B sun faru a kan aikin gwaji guda ɗaya, ana kiran wannan haɗuwa ko yiwuwar hadin gwiwa na A da B, wanda aka nuna kamar haka

P (A cikin B).

{\displaystyle P (A\cap B). }

$P(A\cap B).$

Abubuwan da suka faru masu zaman kansu

Abubuwan da suka faru A da B da aka nuna a matsayin masu banbanci da ba masu banbancin juna ba a sararin samaniya Ω

IdA aB biyu sun faru, A da B sun kasance masu zaman kansu to yiwuwar hadin gwiwa shine

P (A)

da kuma

B) = P (A) = P.

{\displaystyle P (A{\mbox{ da }}B) =P (A\cap B) =P, A) P (B). }

Misali, idan an juya tsabar kudi biyu, to damar zama kawuna biyu shine

1 2

×

1 2

=

1 4

.

{\displaystyle {\tfrac {1}{2}}\times {\tfraz {1} {2}}={\tfrac{1}{4}} . }

Abubuwan da suka faru na juna

IdA ko dai aB da ya faru A ko abin da ya gabata B zai iya faruwa amma ba duka biyu ba a lokaci guda, to ana kiran su abubuwan da suka faru na juna.

Idan abubuwa biyu sun bambanta da juna, to yiwuwar duka biyun suna faruwa kamar haka

P (A cikin B)

{\displaystyle P (A\cap B) }

da kuma

P (A)

da kuma

B) = P (Abin B) = 0

{\displaystyle P (A{\mbox{ da }}B) =P (A\cap B) =0}

Idan abubuwa biyu sun bambanta da juna, to yiwuwar ko dai ya faru kamar haka

P (A cikin B)

{\displaystyle P (A\cup B) }

da kuma

P (A)

ko kuma

B) = P (A) + P (B) − P (A" + P (A, 0 = P (B")

{\displaystyle P (A{\mbox{ ko }}B) =P (A\cup B) =P, +P (B) -P (A \cap B) = P (A) +P (P) -0=P (A) +P (B).

$$P(A{\mbox{ or }}B)=P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-0=P(A)+P(B)$$

Misali, damar da za a yi amfani da 1 ko 2 a kan gefen gefen shida shine

P (1

ko kuma

2) = P (1) + P (2) =

1 6

+

1 6

=

1 3

.

{\displaystyle P (1{\mbox{ ko }}2) =P (1) +P (2) ={\tfrac {1}{6}}+{\tfrak {1} {6}}={\tflak {1}. }

$P(1{\mbox{ or }}2)=P(1)+P(2)={\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{3}}.$

Ba (ya zama dole) abubuwan da suka faru na juna ba

Idan abubuwan da suka faru ba su da alaƙa da juna to

P

(Abin da ya fi dacewa da shi)

A

ko kuma

B

)

= P (A B) = P

(A)

+ P

(B)

- P

(Abin da ya fi dacewa da shi)

A

da kuma

B

)

.

{\displaystyle P\left (A{\hbox{ ko }}B\right) =P (A\cup B) =P\left, A\right) +P\lept (B\right). }

An sake rubuta shi,

P

(Abin da ya fi dacewa da shi)

B B B B

)

= P

(A)

+ P

(B)

- P

(Abin da ya fi dacewa da shi)

B B B B

)

{\displaystyle P\left (A\cup B\right) =P\left, A\right) +P\lett (B\right) -P\lept (A\cap B\right).

$$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$$

Misali, lokacin da ka zana katin daga katunan katunan katako, damar samun zuciya ko katin fuska (J, Q, K) (ko duka biyun) shine

13 52

+

12 52

-

3 52

=

11 26

,

{\displaystyle {\tfrac {13}{52}}+{\tfrac{12}{52}},-{\tfrak {3}{52}={\tfranc {11}{26}},}

tunda daga cikin katunan 52 na bene, 13 sune zukata, 12 sune katunan fuska, kuma 3 duka biyu ne: a nan yiwuwar da aka haɗa a cikin "3 da suke duka biyu" an haɗa su a cikin kowane ɗayan "zuciya 13" da "katunan fuska 12", amma ya kamata a ƙidaya sau ɗaya kawai.

Ana iya fadada wannan don abubuwa da yawa da ba su da mahimmanci ga juna. Ga abubuwan da suka faru guda uku, wannan ya ci gaba kamar haka:

P

(Abin da ya fi dacewa da shi)

Sanya B C

)

=

P

(Abin da ya fi dacewa da shi)

(Abin da ya fi dacewa da shi)

B B B B

)

C.E.

)

=

P

(Abin da ya fi dacewa da shi)

B B B B

)

+ P

(C)

- P

(Abin da ya fi dacewa da shi)

(Abin da ya fi dacewa da shi)

B B B B

)

C.E.

)

=

P

(A)

+ P

(B)

- P

(Abin da ya fi dacewa da shi)

B B B B

)

+ P

(C)

- P

(Abin da ya fi dacewa da shi)

(Abin da ya fi dacewa da shi)

Sashin C

)

Sanya

(Abin da ya fi dacewa da shi)

B Sanya C

)

)

=

P

(A)

+ P

(B)

+ P

(C)

- P

(Abin da ya fi dacewa da shi)

B B B B

)

-

(Abin da ya fi dacewa da shi)

P

(Abin da ya fi dacewa da shi)

Sashin C

)

+ P

(Abin da ya fi dacewa da shi)

B Sanya C

)

- P

(Abin da ya fi dacewa da shi)

(Abin da ya fi dacewa da shi)

Sashin C

)

Sanya

(Abin da ya fi dacewa da shi)

B Sanya C

)

)

)

P

(Abin da ya fi dacewa da shi)

Sanya B C

)

=

P

(A)

+ P

(B)

+ P

(C)

- P

(Abin da ya fi dacewa da shi)

B B B B

)

- P

(Abin da ya fi dacewa da shi)

Sashin C

)

- P

(Abin da ya fi dacewa da shi)

B Sanya C

)

+ P

(Abin da ya fi dacewa da shi)

Sanya B C

)

{\displaystyle {\begin{aiki}P\hagu (A\cup B\dama) =&P\hagu) \cap&P\ hagu (\hagu) \cap\hagu) (\hagu\dama) \cap\dama (\dama) (\ga dama\\ga dama) \cap/ga dama\C\ga dama (\ga) \capcap&P/ga dama) \ga\ga dama

Ana iya ganin, to, cewa ana iya maimaita wannan tsari don kowane adadi na abubuwan da suka faru.

Halin yiwuwar

A"><A class="cx-link" data-linkid="473" href="./Conditional_pro'B'" id="mwAcI" rel="mw:WikiLink" title="Conditional probability">Halin yiwuwar yanayi shine yiwuwar wani abu A, idan aka ba da faruwar wani abu B. An rubuta yiwuwar yanayi

P (A cikin B)

{\displaystyle P (A\mid B) }

, kuma ana karantawa "hawancin A, an ba da B". An bayyana shi ta hanyar

$$P(A\mid B)={\frac {P(A\cap B)}{P(B)}}\,$$

Idan

P (B) = 0

{\displaystyle P (B) =0}

sa'an nan

P (A cikin B)

{\displaystyle P (A\mid B) }

Ba a bayyana shi ba ta wannan magana. A wannan yanayin

A

{\displaystyle A}

da kuma

B

{\displaystyle B}

suna da 'yanci, tun lokacin da

P (A) = P (A), P (B) = 0.

{\displaystyle P (A\cap B) =P (A) P (B) =0. }

Koyaya, yana yiwuwa a ayyana yiwuwar yanayi don wasu abubuwan da suka faru, misali ta hanyar amfani da σ-algebra na irin waɗannan abubuwan (kamar waɗanda ke tasowa daga ci gaba mai canzawa).

Misali, a cikin jaka na 2 jan balls da 2 blue balls (4 balls a duka), yiwuwar ɗaukar jan ball ne

1

/

2 ;

{\displaystyle 1/2;}

Duk da haka, lokacin da ake ɗaukar kwallon na biyu, yiwuwar zama ko dai jan ko blue ball ya dogara da kwallon da aka ɗauka a baya. Misali, idan an dauki jan ball, to yiwuwar sake karbar jan ball zai kasance

1

/

3 ,

{\displaystyle 1/3,}

tunda kawai 1 ja da 2 blue balls zasu kasance sun rage. Kuma idan an dauki kwallon shuɗi a baya, yiwuwar daukar jan kwallon zai kasance

2

/

3.

{\displaystyle 2/3.}

$2/3.$

Da yiwuwar baya

A cikin Ka'idar yiwuwar da aikace-aikace, dokar Bayes ta danganta rashin daidaito na taron

A

1

{\displaystyle A_{1}}

zuwa taron

A

2

,

{\displaystyle A_{2},}

kafin (kafin) da kuma bayan (bayan) yanayin a wani taron

B.

{\displaystyle B.}

Rashin daidaituwa

A

1

{\displaystyle A_{1}}

zuwa taron

A

2

{\displaystyle A_{2}}

shine kawai rabo na yiwuwar abubuwan da suka faru guda biyu. Lokacin da abubuwa da yawa suka faru

A

{\displaystyle A}

suna da ban sha'awa, ba kawai biyu ba, ana iya sake fasalin dokar kamar yadda baya ya dace da lokutan da suka gabata.

P (A)

Sanyawa

B) Sanya P (B)

Sanyawa

A)

{\displaystyle P (A negB) \propto P (A) P (B negA) }

inda alamar daidaito tana nufin cewa gefen hagu daidai yake da (watau, daidai yake da lokaci) gefen dama kamar yadda

A

{\displaystyle A}

ya bambanta, don daidaitawa ko bayarwa

B

{\displaystyle B}

(Lee, 2012; Bertsch McGrayne, 2012). A cikin wannan nau'in ya koma Laplace (1774) da Cournot (1843); duba Fienberg (2005).

Taƙaitaccen yiwuwar

Taƙaitaccen yiwuwar

Abin da ya faru	Mai yiwuwa
A	$P(A)\in [0,1]$
ba A ba	$P(A^{\complement })=1-P(A)\,$
A ko B	${\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\P(A\cup B)&=P(A)+P(B)\qquad {\mbox{if A and B are mutually exclusive}}\\\end{aligned}}$
A da B	${\begin{aligned}P(A\cap B)&=P(A|B)P(B)=P(B|A)P(A)\\P(A\cap B)&=P(A)P(B)\qquad {\mbox{if A and B are independent}}\\\end{aligned}}$
An ba da B	$P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {P(B|A)P(A)}{P(B)}}\,$

Dangantaka da bazuwar da yiwuwar a cikin injiniyan lissafi

A cikin sararin samaniya mai mahimmanci, bisa ga ra'ayoyin Newtonian, ba za a sami yiwuwar ba idan an san dukkan yanayi (Aljanin Laplace) (amma akwai yanayi inda hankali ga yanayin farko ya wuce ikonmu na auna su, watau sanin su). A cikin yanayin motar roulette, idan an san ƙarfin hannu da lokacin wannan ƙarfin, lambar da kwallon za ta tsaya za ta zama tabbatacciya (ko da yake a matsayin al'amari mai amfani, wannan zai iya zama gaskiya ne kawai game da motar roulette wanda ba a daidaita shi daidai ba - kamar yadda Thomas A. Bass' Newtonian Casino ya bayyana). Wannan kuma yana ɗaukar ilimin inertia da friction na motar, nauyi, santsi, da kuma zagaye na kwallon, bambance-bambance a cikin saurin hannu yayin juyawa, da sauransu. Bayani mai yiwuwa zai iya zama da amfani fiye da injiniyoyin Newtonian don nazarin tsarin sakamakon maimaitawa na motar roulette. Masana kimiyyar lissafi suna fuskantar irin wannan halin da ake ciki a cikin ka'ida kinetic na iskar gas, inda tsarin, yayin da yake da ƙayyadewa a ka'idar, yana da rikitarwa sosai (tare da yawan kwayoyin yawanci tsari na girman Avogadro mai ɗorewa 6.02) cewa kawai bayanin ƙididdigar dukiyarsa zai yiwu.

Ana buƙatar Ka'idar yiwuwar don bayyana abubuwan da suka faru. Binciken juyin juya hali na farkon karni na 20 shine halin bazuwar duk matakai na zahiri da ke faruwa a ma'auni na atom kuma ana sarrafa su ta hanyar dokokin injiniya. Ayyukan raƙuman ruwa suna canzawa ta hanyar ƙaddara amma, bisa ga Fassarar Copenhagen, yana hulɗa da yiwuwar lura, sakamakon da aka bayyana ta hanyar rushewar aikin raƙuman raƙuman lokacin da aka yi kallo. Koyaya, asarar ƙaddara don kare kayan aiki bai sadu da amincewar duniya ba. Albert Einstein ya yi magana a cikin wata wasika ga Max Born: "Na gamsu cewa Allah ba ya wasa da dice". Kamar Einstein, Erwin Schrödinger, wanda ya gano aikin raƙuman ruwa, ya yi imanin cewa quantum mechanics kusanci ne na kididdiga ƙididdiga na ainihin Gaskiya. A wasu fassarori na zamani na ƙididdigar ƙididdiga na ma'auni, ana kiran ƙididdigat ƙididdigal don lissafin bayyanar sakamakon gwaji na ra'ayi.

Dubi kuma

 

• Ba zato ba tsammani
• Daidaitawa daidai
• Rashin fahimta
• Heuristic (halayyar mutum)

 

Bayanan littattafai

• Kallenberg, O. (2005) Mai yiwuwa Symmetries da Invariance Principles. Springer-Verlag, New York. 510 shafi na  ISBN 0-387-25115-4
• Kallenberg, O. (2002) Tushen yiwuwar zamani, 2nd ed. Springer Series a cikin Kididdiga. 650 shafi na  ISBN 0-387-95313-2
• Olofsson, Peter (2005) Mai yiwuwa, Kididdiga, da Tsarin Stochastic, Wiley-Interscience. 504 shafi na .ISBN 0-471-67969-0

Haɗin waje

• Gidan gwaje-gwaje na kama-da-wane a cikin yiwuwar da kididdiga (Univ. na Ala.-Huntsville)
•  
• Probability a kanA Zamaninmua cikinBBC
• Mai yiwuwa da Kididdiga EBook Archived 2008-12-16 at the Wayback Machine
• Edwin Thompson Jaynes. Ka'idar yiwuwar: Ma'anar Kimiyya . Preprint: Jami'ar Washington, (1996). - Lissafin HTML tare da hanyoyin haɗi zuwa fayilolin PostScript da PDF (surori uku na farko)
• Mutanen daga Tarihin yiwuwar da Kididdiga (Univ. na Southampton)
• Mai yiwuwa da Kididdiga a kan Shafukan Amfani da Farko (Univ. na Southampton)
• Amfani da Alamomi na Farko a cikin Mai yiwuwa da Kididdiga Archived 2017-06-30 at the Wayback Machine akan Amfani da alamomi daban-daban na lissafiAmfani da Farko na Alamomin Lissafi daban-daban Archived 2016-02-20 at the Wayback Machine
• Koyarwa game da yiwuwar da Bayes' theorem da aka tsara don ɗaliban Jami'ar Oxford na shekara ta farko
• Gabatarwa ga Mai yiwuwa - eBook Archived 2011-07-27 at the Wayback Machine An adana 27 Yuli 2011 a , na Charles Grinstead, Laurie Snell Tushen Archived 2012-03-25 at the Wayback Machine An adana Wayback Engine (GNU Free Documentation License)
• (a cikin Turanci da Italiyanci) Bruno de Finetti, Probabilità da induzione Archived 2013-04-08 at the Wayback Machine, Bologna, CLUEB, 1993. (digital version) ISBN 88-8091-176-7
• Richard Feynman's Lecture on probability.