- Ama-axiom ka-Kolmogorov achaza ngokusemthethweni amathuba njengesilinganiso esingelona-negative, esijwayelekile, kanye nesengezo esingu-σ.
- Kusukela kulawa ma-axiom, izakhiwo ezifana no-P(∅)=0, 0≤P(A)≤1, imithetho yokwengeza, nobudlelwane nokuhambisanayo kutholwa.
- Izakhiwo ezifana nesikhala samathuba (Ω, F, P), amathuba anemibandela, nokuzimela kuvela ngokuqondile kulolu hlaka lwe-axiomatic.
Umbuzo othi "yini ama-axioms of probability?" kubonakala kulula, kodwa impendulo iholela ekwakhiweni kwezibalo okuqinile kakhulu., eyaqala ukuhlelwa ngokuqinile ngekhulu lama-20 nomsebenzi ka-Andrey Kolmogorov. Lawa ma-axiom ayisisekelo sayo yonke ithiyori yamathuba esimanjemanje, kusukela ocwaningweni lwemidlalo yenhlanhla ukuya kumamodeli ezibalo ayinkimbinkimbi asetshenziswa kusayensi yedatha, ezezimali, nobunjiniyela.
Ngaphambi kwe Kolmogorov ngokusemthethweniNgaleso sikhathi, amathuba ayeqondwa ngendlela enembile, exhunywe nomqondo wokuphindaphinda noma ithuba.Futhi izazi zezibalo ezihlukahlukene zasebenzisa izincazelo ezihlukahlukene. Namuhla, uma sikhuluma ngama-axiom of probability, sibhekisela kusethi encane yemithetho okumele itholwe noma yimuphi umsebenzi wamathuba ukuze sikwazi ukwenza izibalo ezihambisanayo, sigweme ukungqubuzana, futhi sakhe amathiyori anamandla.
I-intuition eyisisekelo: okuhlangenwe nakho okungahleliwe nemicimbi
Ukuze uqonde ama-axiom of probability, isinyathelo sokuqala ukwazi ukuthi kuyini ukuhlola okungahleliwe nokuthi yini esiyibiza ngokuthi umcimbi.Ukuhlolwa okungahleliwe yinoma iyiphi inqubo umphumela wayo ongenakubikezelwa ngokuqiniseka, nakuba sazi yonke imiphumela engaba khona; izibonelo zakudala ukupheqa uhlamvu lwemali noma ukugingqa idayizi.
Sibiza isikhala sesampula, ngokuvamile esichazwa ngu-Ω, isethi yayo yonke imiphumela engaba khona yalokhu kuhlolwa.Uma sijikijela uhlamvu lwemali, isibonelo, isikhala sesampula singabhalwa ngokuthi Ω = {H, T}, lapho u-H emele "amakhanda" futhi u-T amele "imisila". Ingxenye ngayinye ka-Ω ibizwa ngokuthi umphumela wokuqala.
Umcimbi unoma iyiphi isethi engaphansi ye-Ω esingathanda ukuyibuka.Ngakho-ke, uma ukuhlolwa kuwukuphonswa kohlamvu lwemali, isethi {H} umcimbi "amakhanda akhuphuka", isethi {T} umcimbi "imisila ekhuphuka", futhi Ω ngokwayo umcimbi "amakhanda noma imisila ekhuphuka", okungukuthi, umcimbi othile.
Eminye imicimbi ibaluleke kakhulu: isenzakalo esingenakwenzeka, isenzakalo sokuqala, kanye nesenzakalo esithile.Isethi engenalutho ∅ imele umcimbi ongenakwenzeka, njengoba ingenawo umphumela; isethi enento eyodwa {ω}, eno-ω kokuthi Ω, imele umcimbi wokuqala; futhi u-Ω ngokwawo umcimbi othile, lowo ohlale uvela lapho ukuhlola kwenziwa.
Ulimi lwethiyori emisiwe lusiza kakhulu ocwaningweni lwamathuba.Uma u-A no-B kuyizehlakalo, khona-ke u-A ∩ B umele ukwenzeka ngesikhathi esisodwa kuka-A no-B, u-A ∪ B umele ukwenzeka kwesinye sazo, futhi umphelelisi ka-A, evame ukubhalwa ngokuthi ̄A noma Ω \ A, imele “ukungenzeki kuka-A”. Lokhu notation kanye nezakhiwo zamasethi kuzosetshenziswa ngokuqondile ekwakhiweni kwama-axiom.
Izincazelo zomqondo wamathuba
Nakuba ama-axiom kaKolmogorov enikeza isisekelo sezibalo sokungenzeka, igama elithi "amathuba" ngokwalo lingahunyushwa ngezindlela ezihlukahlukene.Ngokomlando, kuye kwavela ukuhunyushwa okuhlukene mayelana nokuthi kusho ukuthini ukwabela inombolo P(A) emcimbini A.
Encazelweni yakudala ye-Laplace, evumeleke ezikhaleni ezinomkhawulo ezinemiphumela elinganayo, amathuba okuthi A yinani eliphakathi kwenani lezigameko ezivumayo kanye nenani lezigameko ezingaba khona.Uma isikhala sesampula sinemiphumela engenzeka ngokulinganayo (okungukuthi, #Ω = n) futhi umcimbi A uqukethe i-n_A yale miphumela (#A = n_A), khona-ke amathuba anikezwa ngu-P(A) = n_A / n. Le fomula inembile uma yonke imiphumela inethuba elifanayo lokwenzeka.
Mina incazelo ye- frequentist Ixhumanisa amathuba nemvamisa ehlobene ebonwa ekuphindaphindweni kokuhlolwa.Kuleli phuzu lokubuka, siphinda ukuhlola okungahleliwe izikhathi ezingu-n futhi sibale ukuthi isigameko A senzeka kangaki, sibiza le nombolo ngokuthi n_A; bese sibheka umkhawulo, njengoba n ikhula, ingxenye n_A / n. Amathuba ka-A angaba ngu-P(A) = lim_{n→∞} (n_A / n), inqobo nje uma lo mkhawulo ukhona.
Kukhona futhi ukuhumusha okuzimele, okusetshenziswa kakhulu kwizibalo ze-Bayesia, lapho amathuba ahlotshaniswa nezinga lenkolelo yesihloko esinengqondo.Ngale ndlela, u-P(A) ulinganisa ukuthi umuntu uqiniseka kangakanani ngokwenzeka kuka-A, kucatshangelwa ulwazi olukhona. Akukona ulwazi "oluphethe" amathuba, kodwa isihloko esihlola ukungaqiniseki ngokuhambisana.
Naphezu kwalezi zincazelo ezahlukene, zonke zingaba khona ngaphakathi kohlaka olufanayo lwe-axiomatic ye-Kolmogorov.Ngamanye amazwi, kungakhathaliseki ukuthi ukhetha umbono we-classical, frequentist, noma subjective, ekugcineni amathuba azomodela ngokwezibalo umsebenzi ongu-P othobela isethi encane yama-axiom mayelana nesikhala somcimbi.
Ukwakhiwa okusemthethweni: izikhala zamathuba kanye nama-σ-algebra
U-Kolmogorov wachaza amathuba ngokukathathu (Ω, F, P), okubizwa ngokuthi indawo yamathuba.Kulokhu kathathu, u-Ω uyisampula yesikhala, u-F uyisethi yemicimbi engenzeka (ngokobuchwepheshe, i-σ-algebra yamasethi angaphansi ka-Ω), futhi u-P umsebenzi wamathuba.
I-σ-algebra F iqoqo elikhethekile lamasethi angaphansi ka-Ω anelisa izici ezithile.Ngokuvamile, u-F udinga ukuqukatha isethi engenalutho, ivalwe ngaphansi kokupheleliswa (uma u-A eku-F, khona-ke umphelelisi wayo futhi ungu-F), futhi uvalwe ngaphansi kwezinyunyana ezingabaleki (uma u-A₁, A₂, ... ziku-F, khona-ke inyunyana yazo zonke iku-F). Lesi sakhiwo siqinisekisa ukuthi singasebenza ngemisebenzi emisiwe ngaphandle kokushiya indawo yonke yezehlakalo ezinamathuba achazwe kahle.
Ngokusemthethweni, u-F uyi-σ-algebra ngaphezu kuka-Ω lapho: Isethi engenalutho ethi ∅ eka-F; uma u-A eku-F, khona-ke ukugcwaliseka kuka-A kokuthi Ω nakho kungokuka-F; futhi uma u-A₁, A₂, … ingukulandelana (okulinganiselwe noma okungenamkhawulo) kwezakhi zika-F, khona-ke inyunyana ethi A₁ ∪ A₂ ∪ … iku-F. Ezimweni eziningi, u-F ubizwa nangokuthi inkundla ye-Borel noma u-σ-field.
Umsebenzi wamathuba okuthi P uchazwe kokuthi F futhi yabela umcimbi ngamunye E ngo-F inombolo yangempela engeyona inegethivu.Sibe sesisho ukuthi u-P(E) uku-ℝ kanye no-P(E) ≥0 kuwo wonke u-E kokuthi F. Ngokujwayelekile ithiyori yokulinganisa, izinyathelo zingathatha amanani angenamkhawulo, kodwa kumbono wamathuba ajwayelekile, u-P(E) uhlala enomkhawuko, okuletha umehluko othile maqondana nezinyathelo ezijwayelekile.
Lesi sakhiwo (Ω, F, P) esino-P(Ω) = 1 yilokho esikubiza ngokuthi isikhala esingaba khona.Isimo P(Ω) = 1 sibalulekile ngoba simele umqondo wokuthi, lapho wenza ukuhlola, umphumela othile kokuthi Ω uyenzeka ngempela; ayikho "imiphumela efihliwe" ngaphandle kwesikhala sesampula.
Ama-axiom amathathu kaKolmogorov
Ithiyori ye-axiomatic ka-Kolmogorov isekelwe kuma-axiom amathathu ayisisekelo okumele anelise noma yimuphi umsebenzi wamathuba.Alula ukuwasho, kodwa anamandla ngokwedlulele, ngoba cishe zonke izici ezivamile zamathuba atholakala kuzo.
I-axiom yokuqala - Non-negativity: Kunoma yimuphi umcimbi A we-σ-algebra F, sine-P(A) ≥ 0. Okusho ukuthi, amathuba awalokothi abe negethivu. Kwezinye izinkolelo-mbono ezingavamile, kukhulunywa "ngamathuba angemahle," kodwa le mibono iyaphambuka kuhlaka luka-Kolmogorov lwakudala.
I-axiom yesibili - Ukujwayela: Amathuba okuba kwenzeke isenzakalo esithile alingana no-1, okungukuthi, P(Ω) = 1. Le-axiom isungula isivumelwano sokuthi u-1 uhambisana nokuqiniseka okungu-100%, futhi u-0 uhambisana nokungenzeki. Ezinguqulweni eziningi eziyisisekelo, le axiom ingaqondwa nangokuthi ithi isamba samathuba ayo yonke imiphumela eyisisekelo ka-Ω ilingana no-1.
I-axiom yesithathu - σ-additivity: Uma u-A₁, A₂, … kuwukulandelana kwezehlakalo ezingahlangani ngambili (ephinde zibizwe ngemicimbi ekhethekile), bese u-P(∪ᵢ Aᵢ) = Σᵢ P(Aᵢ). Lokhu kuyiqiniso kukho kokubili iqoqo lemicimbi elinesiphelo nelingapheli. Lesi sakhiwo esingenakubalwa siwumehluko omkhulu uma kuqhathaniswa nokunezelwa okulinganiselwe.
Ezimweni ezilula, abanye ababhali basebenza kuphela ngokungeza okunomkhawulo., edinga ukuthi u-P(A ∪ B) = P(A) + P(B) wemicimbi engahlangani A no-B, nokuthi lokhu kudlulela enanini elilinganiselwe lamasethi. Kulokhu, kwanele ukusebenza ngesethi ye-algebra, hhayi ngempela i-σ-algebra, kodwa indlela evamile emathubeni esimanje iwukudinga ukungezwani kuka-σ.
Kusuka kulesi sihloko sesithathu lapho kuvela khona imiphumela embalwa ebalulekile, njengokulingana, ukungalingani, kanye nemithetho yamathuba.Futhi usenhliziyweni yesixhumanisi phakathi kwamathuba kanye nethiyori yokulinganisa, ehlola izilinganiso ngamasethi ngendlela evamile.
Izakhiwo ezithathwe kuma-axiom
Ngokusekelwe kuma-axiom amathathu ka-Kolmogorov, sikwazile ukufakazela izakhiwo ezimbalwa eziyisisekelo neziwusizo kakhulu.Lezi zakhiwo azicatshangwa kusengaphambili: ziyimiphumela enengqondo yama-axioms.
Enye yezakhiwo zokuqala yi-monotonicity of probability.Uma u-A no-B kuyizehlakalo kokuthi F futhi A aqukethwe kokuthi B (A ⊆ B), bese kuba ngu-P(A) ≤ P(B). Umbono unengqondo: uma u-B ehlanganisa yonke into engenzeka ku-A, futhi mhlawumbe nangaphezulu, khona-ke u-B akanakuba namathuba aphansi kuno-A.
Enye impahla eyisisekelo ukuthi amathuba okuba kwenzeke isenzakalo esingenakwenzeka nguziro.Kusukela endaweni yokubuka ehlelekile, kusetshenziswa u-σ-additivity, sicabangela ukulandelana lapho u-E₁ = A, E₂ = B \ A no-Eᵢ = ∅ ku-i ≥ 3, esimweni lapho u-A ⊆ B. Njengoba i-Eᵢ ingahlangani futhi inyunyana yabo ingu-B, isamba samathuba okuba khona kwe-conver (B). Uma sicabanga ukuthi u-P(∅) = a > 0, khona-ke isamba sika-P(∅) izikhathi eziningi ngokungenamkhawulo sizoqhuma sibe okungapheli, okungahambisani no-P(B) ophelele. Ngakho siphetha ngokuthi P(∅) = 0.
Ngakho-ke, singasho ukungalingani 0 ≤ P(E) ≤ 1 kunoma yimuphi umcimbi E ngo-F.Besivele sazi ukuthi i-P(E) ≥0 kusuka ku-axiom yokuqala. Ukwazi ukuthi P(Ω) = 1 nokusebenzisa i-monotonicity ngo-E ⊆ Ω, kulandela ukuthi P(E) ≤ P(Ω) = 1. Ngakho, wonke amathuba ahlala ephakathi kuka-0 no-1, kuhlanganisa.
Ubunikazi obuvame ukusetshenziswa yilokho okubizwa ngokuthi umthetho ongeziwe wanoma yiziphi izehlakalo ezimbili.Emicimbini engu-A no-B ku-F, ibamba ukuthi P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Le fomula ilungisa “ukubala okuphindwe kabili” komcimbi ovamile A ∩ B, owengezwa kabili uma sihlanganisa u-P(A) no-P(B) ngaphandle kokulungiswa.
Omunye umphumela obalulekile ubuhlobo phakathi komcimbi kanye nomphelelisi waso.Uma sichaza ukugcwaliseka kuka-A ngo-̄A, bese kuba ngu-P(̄A) = 1 − P(A). Lokhu kulingana kudlulisa umqondo wokuthi "u-A uyenzeka noma u-A angenzeki," futhi ayikho enye into engenzeka ngaphakathi kuka-Ω.
Kulokhu, kuyacaca futhi ukuthi u-P(A) = 0 akusho ukuthi u-A umcimbi ongenakwenzeka.Ngokwezibalo, kungenzeka ukuthi umcimbi ube namathuba anguziro ngaphandle kokuba isethi engenalutho (lokhu kuvela, isibonelo, ezikhaleni eziqhubekayo), kodwa ezingeni eliphansi kakhulu, u-P(A) = 0 ngokuvamile uhlotshaniswa nezenzakalo cishe ezingenakwenzeka.
Isibonelo esisebenzayo: ukuphenyisisa uhlamvu lwemali
Isibonelo sakudala nesisebenzayo sokubuka ama-axiom ka-Kolmogorov ukuphonswa kohlamvu lwemali.Ake sicabange, okokuqala, ukuthi uhlamvu lwemali lungahlala kuphela "emakhanda" (H) noma "emisileni" (T), nokuthi lena ukuphela kwemiphumela engenzeka.
Bese sichaza isikhala sesampula ngokuthi Ω = {H, T}Izehlakalo ezingaba khona zakha i-σ-algebra F ehlanganiswe {∅, {H}, {T}, {H, T}}. Kulo mongo, isenzakalo esingenakwenzeka ngu-∅, imicimbi yokuqala ithi {H} kanye no-{T}, kanti isenzakalo esithile sithi {H, T}.
Kusuka ku-axiom ka-Kolmogorov, siyazi ukuthi P (∅) = 0 kanye no-P (Ω) = 1Uma sicabanga ukuthi uhlamvu lwemali lulungile, okungukuthi, alukhethi noma yiluphi uhlangothi, khona-ke ukulinganisa kuphakamisa ukuthi P({H}) = P({T}). Njengoba isamba esingu-P({H}) + P({T}) kufanele silingane no-1, siphetha ngokuthi kokubili kufaneleka u-1/2.
Ngakho-ke, amathuba okuthola "amakhanda noma imisila" ngu-P({H, T}) = 1Amathuba okuthola "amakhanda" ngu-P({H}) = 1/2 futhi amathuba okuthola "imisila" ngu-P({T}) = 1/2. Isamba samathuba emicimbi yokuqala siqeda ingqikithi yamathuba esikhala.
Lo modeli, nakuba ulula, ubonisa indlela ama-axiom aziphatha ngayo ekusebenzeni nokuthi akuvimbela kanjani ukungqubuzana ekubaleni okungenzeka.Uma singachazi ngokucophelela indawo yesampula, singenza amaphutha amakhulu, ngoba noma imuphi umcimbi uhlala uyingxenye ka-Ω; uma isethi engaphansi ingangeni ku-Ω, amathuba ayo awachazwanga.
Amathuba ezikhaleni ezinomkhawulo nezibalekayo
Uma isikhala sesampula sinomkhawulo noma sibalwa, amathuba angachazwa ngendlela ephathekayo kakhulu.Ake sithi Ω = {ω₁, ω₂, …} iyisethi elinganiselwe noma ebalekayo yemiphumela engaba khona.
Uma u-A kuwumcimbi oqukethe eminye yale miphumela, njengokuthi A = {ω₁*, …, ω_{k*}, …}Ngakho-ke, amathuba okuthi A angabonwa njengesamba samathuba ezenzakalo eziyisisekelo ezihambisanayo: P(A) = P(∪ᵢ {ω_{i*}}) = Σᵢ P({ω_{i*}}). Lolu uhlelo lokusebenza oluqondile lokuhlanganisa (noma u-σ-additivity) kumasethi ama-disjoint.
Esimeni esithile lapho isikhala sesampula sinomkhawulo, nge-#Ω = n, futhi yonke imiphumela ingalingana.Sino-P({ωᵢ}) = 1/n ku-i ngayinye. Uma u-A equkethe u-k imiphumela ehlukene ku-Ω, bese u-P(A) = Σ_{i=1}^k P({ω_{i*}}) = k/n = (#A)/(#Ω). Lena ncamashi ifomula yakudala ye-Laplace ehunyushwe kabusha ngaphakathi kohlaka lwesimanje lwe-axiomatic.
Uma isikhala sesampula singenakubalwa, isamba samathuba emicimbi yokuqala sisadinga ukuhlangana sibe ngu-1.Okusho ukuthi, Σᵢ P({ωᵢ}) = 1. Yilapho-i-σ-additivity ibonisa amandla ayo, okusivumela ukuthi sibhekane nezibalo ezilinganiselwe kuphela, kodwa futhi nochungechunge olungapheli lwezenzakalo.
Amathuba anemibandela kanye nendima yama-axiom.
Isici esiyinhloko salo mbono ukuqonda ukuthi amathuba ashintsha kanjani uma sazi ukuthi isehlakalo esithile sesenzekile kakade.Lapha yilapho kungena khona amathuba anemibandela, ngokuvamile abhalwa ngokuthi P(A | B), okusho ukuthi "amathuba okuthi A anikezwe ukuthi u-B wenzekile".
Ifomula eyisisekelo yamathuba anemibandela ithi P(A | B) = P(A ∩ B) / P(B), inqobo nje uma P(B) > 0Le ncazelo ihambisana nama-axiom ka-Kolmogorov, futhi empeleni, ku-B ngamunye ono-P(B) > 0, umsebenzi othi A ↦ P(A | B) uphinde wenelise ama-axiom amathathu lapho sikhawulela isikhala somcimbi ku-B.
Lokhu kusho ukuthi i-P(· | B) yona ngokwayo ingumsebenzi wamathuba phezu kwesampula "entsha" yesikhala B.Ngenxa yalokho, zonke izakhiwo eziyisisekelo zibambela amathuba anemibandela: P(̄A | B) = 1 − P(A | B), P(∅ | B) = 0, i-monotonicity enemibandela (uma A₁ ⊆ A₂, bese u-P(A₁ | B) ≤ P(A₂ | B₪) kanye nefomula B(A₂ | B))) P(A₁ | B) + P(A₂ | B) − P(A₁ ∩ A₂ | B).
Incazelo yamathuba anemibandela iphinde iveze ubudlelwano obubalulekile P(A ∩ B) = P(A) P(B | A), lapho u-P(A) > 0.Ngokuvumelana, singabhala u-P(A ∩ B) = P(B) P(A | B), inqobo nje uma u-P(B) > 0. Lokhu kulingana kusiza ukubola amathuba ahlangene futhi kuyisisekelo semiphumela eminingana, njenge-Bayes' Theorem.
Kuyathakazelisa ukuqaphela ukuthi amathuba "angenamibandela" angabonwa njengendaba ethile yamathuba anemibandela.Ngempela, singabhala u-P(A) = P(A ∩ Ω) / P(Ω) = P(A | Ω), kusukela ku-P(Ω) = 1. Lokhu kuqinisa umqondo wokuthi, ngokomqondo, wonke amathuba anemibandela kolunye ulwazi lwangemuva, ngisho noma kuwulwazi kuphela esilusebenza ngaphakathi kuka-Ω.
Ukuzimela kwemicimbi
Omunye umqondo oyinhloko oncike kuma-axiom yilowo wokuzimela phakathi kwezenzakalo.Izehlakalo ezimbili u-A no-B zizimele uma ukwenzeka kwesinye kungaguquli amathuba kwesinye.
Olimini oluhlelekile, u-A no-B bazimele uma u-P(A ∩ B) = P(A) P(B)Ngokwemibandela yamathuba anemibandela, lokhu kusho ukuthi uma u-P(B) > 0, bese u-P(A | B) = P(A), futhi uma u-P(A)> 0, bese kuba ngu-P(B | A) = P(B). Okungukuthi, ukwazi ukuthi u-B wenzeka akushintshi amathuba ka-A, futhi ngokuphambene nalokho.
Wonke umcimbi uzimele kumcimbi ongenakwenzeka ∅ kanye nomcimbi othile Ω.Kusethi engenalutho, P(A ∩ ∅) = 0 kanye no-P(∅) = 0, ngakho ubudlelwano bubambe kancane. Ngomcimbi othile, u-P(A ∩ Ω) = P(A) no-P(Ω) = 1, ngakho-ke u-P(A ∩ Ω) = P(A) P(Ω) = P(A).
Umbuzo ojwayelekile ukuthi izehlakalo ezimbili ezihlukene zingazimela yini.Ngokuvamile, uma u-A no-B behlukene futhi kokubili kunamathuba amahle, khona-ke u-P(A ∩ B) = 0, kodwa u-P(A) P(B) > 0, okwephula incazelo yokuzimela. Ngakho-ke, ezimweni eziningi, izehlakalo ezimbili ezingahlangani ezinamathuba angewona uziro azizimele, njengoba ukwenzeka kwesinye akuhlanganisi ukuba nokwenzeka kwesinye.
Lapho kukhulunywa ngemicimbi engaphezu kwemibili, kuvela imibono eminingana yokuzimela.Singaba nokuzimela ngokubili, ukuzimela ngokuhlanganyela, nezinye izinhlobo. Nokho, kuzo zonke lezi zimo, isiqalo sihlala sihlobene P (A ∩ B) = P (A) P (B), ngokusekelwe kuma-axiom ka-Kolmogorov kanye nencazelo yamathuba anemibandela.
Imithetho esebenzayo nemithetho yakudala yamathuba
Ngaphandle kwezakhiwo zabo ezisemthethweni, ama-axiom avumela ukwakhiwa kwemithetho eminingi yokusebenza, ewusizo emsebenzini wansuku zonke walabo abenza izibalo zokungenzeka.Omunye wabo yilokho okubizwa ngokuthi umthetho wokwengeza, osekushiwo kakade efomini P (A ∪ B) = P (A) + P (B) − P (A ∩ B), enganwetshwa kwinani elikhulu lemicimbi ngokusebenzisa isimiso sokufakwa-ukukhishwa.
Omunye umthetho osetshenziswa kakhulu ubudlelwano phakathi komcimbi kanye nengxenye "yangaphandle" yomunye umcimbi.Ku-A no-B ku-F, okulandelayo kubamba: P(A ∩ ̄B) = P(A) − P(A ∩ B). Lokhu kumane kuhlukanise u-A kube izingxenye ezimbili: ingxenye eyenzeka kanye no-B (A ∩ B) kanye nengxenye eyenzeka ngaphandle kuka-B (A ∩ ̄B). Lezi zingxenye ezimbili azihlangani, futhi inyunyana yazo ingu-A, okuholela ekulinganeni kwangaphambilini ngokuhlanganisa.
Umthetho wamathuba aphelele kanye ne-Bayes' Theorem, nakuba ingacacisiwe ngokugcwele lapha, futhi incike ngokuqondile kuma-axiom.Umthetho wamathuba aphelele uhlanganisa amathuba anemibandela abe ingxenye yesikhala sesampula, kuyilapho i-Bayes' Theorem "iguqula" izimo, okuvumela amathuba ukuthi abuyekezwe ngokusekelwe ebufakazini obusha.
Ezinguqulweni ezengeziwe ze-didactic, amanye "ama-axiom asebenzayo" okulula ukuwakhumbula nawo afakwe ohlwini.Isibonelo: amathuba aphezulu ngu-1 (100%); isamba samathuba azo zonke izici esikhaleni sesampula silingana no-1; kanye namathuba omcimbi X owengezwe emathubeni okuthi "hhayi u-X" ahlala engu-1. Lezi zitatimende ziwukubonakaliswa okuqondile kwama-axiom asemthethweni.
Ngale sethi yemithetho, kuyenzeka ukuthi kuxazululwe izinkinga kusukela kumageyimu alula ukuya kumamodeli ayinkimbinkimbi anezinhlobonhlobo eziningi.Inzuzo enkulu ukuthi, ngemuva kwawo wonke amafomula namaqhinga okubala, ukusekelwa okunengqondo kuhlala kuyi-tripod ye-axiomatic efanayo.
Ama-axioms ka-Kolmogorov wokungenzeka ahlinzeka ngesisekelo esiqinile kodwa esiguquguqukayo sokubhekana nokungaqiniseki.Ngokusekelwe ezimisweni ezintathu ezilula—ukungabi negativity, ukujwayela, kanye no-σ-additivity—ithiyori yonke enothile yakhiwe, ekwazi ukuhlanganisa ukuhumusha kwakudala, kaningi, nokucabangelayo, ukuphatha izikhala ezinomkhawulo noma ezingapheli, echaza amathuba anemibandela nokuzimela, kanye nokusekela izinhlelo zokusebenza cishe kuyo yonke imikhakha yesayensi nezobuchwepheshe.