Basic ntsiab ntawm ntau yam kev tshawb xav. Cov kev cai ntawm qhov muaj tseeb kev tshawb xav

Muaj coob tus neeg, thaum ntsib nrog lub notion ntawm "probability kev tshawb xav", ntshai, xav tias nws yog ib yam dab tsi intolerable, nyuaj heev. Tab sis nws yog ua tau tsis yog li ntawd tragic. Niaj hnub no peb saib cov tswv yim ntawm ntau yam kev tshawb xav, kawm los daws teeb meem los ntawm qhob piv txwv.

science

Yuav ua li cas yog kev kawm ib ceg ntawm kev kawm txog zauv raws li ib tug "yuav tshwm kev tshawb xav"? Nws sau ntawv qauv ntawm random txheej xwm thiab zog. Rau cov thawj lub sij hawm qhov teeb meem ntawm kev txhawj xeeb Zaum nyob rau hauv lub xyoo pua, thaum kawm kev twv txiaj. Basic ntsiab ntawm ntau yam kev tshawb xav - kev tshwm sim. Nws yog tej qhov tseeb hais tias yog hais los ntawm kev los yog kev soj xyuas. Tab sis li cas yog kev? Lwm yooj yim lub tswvyim ntawm lub hom phiaj ntawm qhov muaj tseeb. Nws txhais tau tias qhov no yog ib feem ntawm qhov kev tshwm sim yog tsis ntawd tsim, thiab muaj ib lub hom phiaj. Nrog hais txog kev soj ntsuam, muaj cov neeg soj ntsuam nws tus kheej tsis koom nrog nyob rau hauv cov kev, tab sis tsuas yog ib tug tim khawv rau cov txheej xwm, nws twb tsis muaj ntxim rau dab tsi tshwm sim.

txheej xwm

Peb kawm tau hais tias qhov yooj yim lub tswvyim ntawm lub hom phiaj ntawm qhov muaj tseeb - cov kev tshwm sim, tab sis ho tsis xav txog cais. Tag nrho cov ntawm lawv yog muab faib mus rau hauv lub nram no:

• Txhim khu kev qha.
• Tsis yooj yim sua.
• Random.

Tsis muaj teeb meem dab tsi tshwm sim yog, uas yog raug saib los yog tsim nyob rau hauv lub chav kawm ntawm qhov xyaum ua tej no, lawv yuav cuam tshuam los ntawm no cais. Peb muaj txhua txhua hom ntawm ntsib nyias.

tej yam kev tshwm sim

Qhov no yog ib qhov tseeb uas yuav ua rau cov tsim nyog set ntawm cov kev ua ub. Nyob rau hauv thiaj li yuav zoo dua muab lub essence, nws yog zoo dua rau muab ib co piv txwv. Qhov no yog subordinate mus rau lub kev cai lij choj thiab physics, Science News for KIDS, economics, thiab lej uas siab dua. probability kev tshawb xav muaj xws li xws li ib qho tseem ceeb tswvyim raws li ib tug tseem ceeb tshwm sim. Ntawm no yog ib co piv txwv:

• Peb ua hauj lwm thiab tau txais remuneration nyob rau hauv daim ntawv ntawm cov nyiaj ua hauj lwm.
• Zoo dhau lub xeem, dhau ib tug kev sib tw rau nws tau txais remuneration nyob rau hauv daim ntawv ntawm cov kev nkag los rau ib tug tsev kawm ntawv.
• Peb tau nyiaj ua lag luam nyiaj nyob rau hauv lub bank, tau lawv rov qab, yog hais tias tsim nyog.

Tej yam zoo li no yeej muaj tseeb. Yog hais tias peb tau ua tiav raws li tag nrho cov tsim nyog tej yam kev mob, nco ntsoov mus nrhiav tau cov kev tsim tshwm sim.

tsis yooj yim sua tshwm sim

Tam sim no peb xav txog lub ntsiab ntawm lub hom phiaj ntawm qhov muaj tseeb. Peb muaj mus rau lub clarifications nyob rau hauv lub nram qab no ntawm cov txheej xwm - namely tus tsis yooj yim sua. Yuav pib stipulate lub feem ntau tseem ceeb txoj cai - qhov tshwm sim ntawm ib tug tsis yooj yim sua tshwm sim yog pes tsawg.

Los ntawm no formulation tsis tau derogated nyob rau hauv kev daws teeb meem. Muab ua piv txwv piv txwv ntawm tej txheej xwm:

• Dej yog khov ntawm ib tug kub ntawm plus kaum (nws yog tsis yooj yim sua).
• Qhov uas tsis muaj hluav taws xob tsis cuam tshuam rau ntau lawm (raws li tsis yooj yim sua li nyob rau hauv lub yav dhau los piv txwv li).

Ntau cov qauv muab yog tsis tsim nyog, raws li tau piav saum toj no kom meej meej heev cuam tshuam lub essence ntawm no. Tsis yooj yim sua tshwm sim yeej tsis tshwm sim thaum lub sij hawm xyaum ua tej yam nyob rau hauv txhua yam.

random txheej xwm

Los ntawm kev kawm cov ntsiab ntawm cov ntau yam kev tshawb xav, tshwj xeeb mloog yuav tsum tau them mus rau lub muab hom kev tshwm sim. Cov no yog cov sawv daws yuav kawm no science. Raws li ib tug tshwm sim ntawm qhov kev paub ntawm ib yam dab tsi yuav tshwm sim los yog tsis. Nyob rau hauv tas li ntawd, lub xeem ib tug unlimited tooj ntawm lub sij hawm yuav nqa tawm. Notable piv txwv muaj xws li:

• Pov lub txiaj - nws yog ib qho kev, los yog kev xeem, tsis yog ib tug dav dawb hau - qhov kev tshwm sim.
• Rub lub pob ntawm lub hnab blindly - xeem, twb ntes tau pob liab liab - qhov kev tshwm sim thiab thiaj li nyob.

Tej yam ntxwv uas yuav ua tau ib tug unlimited tooj, tab sis, nyob rau hauv Feem ntau, yog yuav tsum tau to taub. Yuav kom saib lub ntsab lug thiab systematize lub mas yuav kis tau paub txog tus txheej xwm ntawm ib lub rooj. probability kev tshawb xav kev tshawb fawb tsuas yog tus tom kawg zoo ntawm tag nrho cov hais.

kev cai

Muaj tseeb kev tshawb xav - qhov kev kawm uas kawm tau tsis muaj tej yam kev tshwm sim. Zoo li lwm tus, nws muaj ib co kev cai. Cov nram qab no cov kev cai ntawm qhov muaj tseeb kev tshawb xav:

• Lub convergence ntawm sequences ntawm random zog.
• Txoj cai ntawm cov coob.

Thaum xam tau ib tug complex yuav siv tau txoj kev yooj yim txheej xwm mus cuag tau yooj yim thiab ceev txoj kev. Nws yuav tsum tau muab sau tseg tias cov kev cai ntawm qhov muaj tseeb kev tshawb xav yuav muab tau yooj yim muaj pov thawj nrog rau cov kev pab los ntawm ib co ntawm cov theorems. Peb hais kom pib mus ntsib cov thawj txoj kev cai.

Lub convergence ntawm sequences ntawm random variables

Nco ntsoov tias lub convergence ntawm ob peb hom:

• Cov kab sab ntawm random variables convergence nyob rau hauv ntau yam.
• Yuav luag tsis yooj yim sua.
• Rms convergence.
• Convergence rau cov ceevxwm.

Yog li ntawd, nyob rau yoov, nws yog heev yooj yim kom to taub cov essence. Ntawm no yog cov ntsiab lus txhais hais tias yuav pab kom to taub lub npe. Yuav pib nrog tus thawj saib. Cov kab no hu ua convergence nyob rau hauv ntau yam, yog hais tias tus nram qab no mob: n txuas infinity, tus xov tooj nrhiav los ntawm ib theem zuj zus yog ntau tshaj pes tsawg thiab ze rau lub tsev.

Mus rau tom ntej no saib, luag yeej. Lawv hais tias lub ib theem zuj zus converges yuav luag muaj tseeb mus rau ib tug random nce mus nce los nrog n tsis dim mus infinity, thiab R, tsis dim mus rau ib tug nqi nyob ze rau kev sib sau.

Tus tom ntej hom - ib tug convergence ntawm rms. Thaum uas siv cov SC-kawm convergence ntawm vector random dab thiaj li rau txoj kev tshawb ntawm random tswj dab.

Yog kawg ntaus, cia saib luv luv thiab mus ncaj qha mus rau hauv kev daws ntawm teeb meem. Convergence rau cov ceevxwm muaj lwm lub npe - "zog", ces piav vim li cas. Tsis muaj zog convergence - yog lub convergence ntawm lub tsev zog ntawm tag nrho cov ntsiab lus ntawm cov rooj saab laaj ntawm cov kev txwv faib nuj nqi.

Nco ntsoov kom cov lus cog tseg: tsis muaj zog convergence yog txawv los ntawm tag nrho cov saum toj no hais tias tus random nce mus nce los yog tsis txhais rau qhov yuav tshwm qhov chaw. Qhov no yog tau vim tias tus mob yog tsim heev dua lwm yam siv faib zog.

Txoj cai ntawm cov coob

Great pab nyob rau hauv lub pov thawj ntawm cov kev cai lij choj yuav tsum theorems ntawm ntau yam kev tshawb xav, xws li:

• Chebyshev inequality.
• Chebyshev lub theorem.
• Generalized Chebyshev theorem.
• Markov theorem.

Yog hais tias peb xav txog tag nrho cov theorems, ces qhov teeb meem yuav siv li ob peb kaum tawm los xij. Peb muaj lub ntsiab hauj lwm - yog daim ntawv thov ntawm ntau yam kev tshawb xav nyob rau hauv xyaum. Peb muaj koj txoj cai tam sim no thiab ua nws. Tab sis ua ntej peb xav txog lub axioms ntawm ntau yam kev tshawb xav, lawv yog qhov tseem ceeb koom tes nyob rau hauv kev daws teeb meem.

axioms

Los ntawm cov thawj, peb twb pom, thaum sib tham txog cov tsis yooj yim sua tshwm sim. Wb nco ntsoov: qhov tshwm sim rau ib tug tsis yooj yim sua tshwm sim yog pes tsawg. Piv txwv li peb tau muab ib tug heev tiag tiag li thiab nco: cov daus poob nyob rau ntawm ib tug huab cua kub peb caug degrees Celsius.

Qhov thib ob yog raws li nram no: ib tug tej yam kev tshwm sim muaj tshwm sim nrog ntau yam kev sib sau. Tam sim no peb yuav qhia li cas nws yog sau ntawv nrog kev pab los ntawm lej lus: P (B) = 1.

Peb: A random tshwm sim tej zaum yuav tshwm sim los yog tsis, tab sis lub tseem yog ib txwm sib txawv ntawm xoom mus rau ib. Cov los ze zog nws yog kev sib sau, cov ntau muaj feem; yog hais tias tus nqi yog nyob ze rau zero, qhov yuav tshwm yog muaj tsawg kawg nkaus. Peb sau tsab ntawv no nyob rau hauv lej lus: 0

Xav txog lub xeem, plaub axiom, uas yog: lub sum ntawm qhov yuav tshwm ntawm ob txheej xwm yog sib npaug zos rau cov sum ntawm lawv probabilities. Sau zauv cov nqe lus: P (A + B) = P (A) + P (B).

Lub axioms ntawm ntau yam kev tshawb xav - nws yog ib tug yooj yim txoj cai uas yuav tsis tau yooj yim yuav tsum nco ntsoov. Wb sim daws tau ib co teeb meem, raws li nyob rau hauv twb nrhiav kev txawj ntse.

lottery ticket

Ua ntej, xav txog qhov nyuaj piv txwv - ib tug lottery. Xav txog tej yam uas koj yuav ib tug lottery ticket rau hmoov zoo. Yuav ua li cas yog qhov yuav tshwm hais tias koj yuav yeej tsawg kawg yog nees nkaum rubles? Tag nrho cov kev yog muab kev koom tes nyob rau hauv ib txhiab daim pib, ib tug ntawm cov uas muaj ib tug nqi zog ntawm tsib puas rubles, kaum puas rubles, nees nkaum thiab tsib caug rubles, thiab ib puas - tsib. Cov hauj lwm ntawm cov kev tshawb xav ntawm ntau yam raws li nyob rau hauv yuav ua li cas nrhiav tau ib txoj kev muaj hmoov. Tam sim no ua ke peb tsom xam qhov kev txiav txim saum toj no lub paub tab saib.

Yog hais tias peb txhais los ntawm ib nqi zog ntawm tsib puas rubles, ces qhov tshwm sim rau ib tug yog sib npaug zos rau 0,001. Yuav ua li cas peb tau txais? Cia li xav tau tus xov tooj ntawm "muaj hmoo" daim pib muab faib los ntawm tag nrho cov (nyob rau hauv cov ntaub ntawv no: 1/1000).

Nyob rau hauv - ib tug nce ntawm ib puas rubles, qhov yuav tshwm yuav tsum yog sib npaug zos rau 0.01. Tam sim no peb tau ua nyob rau hauv tib txoj kev raws li lub xeem txiav txim (10/1000)

C - payoff yog nees nkaum rubles. Nrhiav qhov kwv yees, nws yog sib npaug zos rau 0.05.

Tus so ntawm lub daim pib peb tsis xav, raws li lawv cov nqi zog nyiaj yog tsawg tshaj li teev nyob rau hauv lub mob. Thov plaub axiom: Qhov ntau yam ntawm winning tsawg kawg yog nees nkaum rubles yog P (A) + P (B) + P (C). Cov tsab ntawv P qhia qhov tshwm sim ntawm keeb kwm ntawm cov kev tshwm sim, peb nyob rau hauv lub kauj ruam dhau twb pom lawv. Nws tseem raug nyob tsuas mus rau nteg cia tus tsim nyog cov ntaub ntawv, cov lus teb uas peb tau txais 0,061. Qhov no muaj pes tsawg tus yuav tau cov lus teb rau lo lus nug ntawm cov hauj lwm.

lawj ntawm phaib

Teeb meem rau ntau yam kev tshawb xav, tseem muaj ntau, piv txwv li, siv sij hawm tom ntej no txoj hauj lwm. Ua ntej koj yuav lawj ntawm peb caug-rau phaib. Koj ua hauj lwm - mus kos ob cards nyob rau hauv ib tug kab, tsis muaj nws pawg, tus thawj thiab thib ob cards yuav tsum ARRANGEMENT, suits tsis os.

Yuav pib, nrhiav qhov kwv yees hais tias tus thawj card yog ib tug ace, qhov no faib los ntawm plaub thiab peb caug-rau. Teem caij rau ib sab. Peb tau txais ib tug thib ob daim ntawv yog ib qho ace nrog qhov tshwm sim ntawm peb puas thiab peb caug tsib. Qhov ntau yam ntawm lub thib ob kev tshwm sim nyob rau hauv uas card peb rub ib tug thawj, peb xav nyob rau hauv, nws yog ib tug ace los yog tsis. Los ntawm no nws hais ntxiv hais tias nyob rau hauv cov kev tshwm sim nyob rau cov kev tshwm sim A.

Cov kauj ruam tom ntej peb nrhiav qhov kwv yees ntawm simultaneous siv, piv txwv li, muab A thiab B. Lawv ua hauj lwm yog raws li nram no: qhov tshwm sim rau ib tug kev tshwm sim khoo los ntawm cov zwj ceeb tshwm sim rau lwm tus, peb xam, piv txwv tias yog tias tus thawj tshwm sim muaj tshwm sim, piv txwv li, tus thawj card peb rub ib tug ace.

Nyob rau hauv thiaj li yuav ua tag nrho cov no yog meej, muab lub los xws caij li lub zwj ceeb tshwm sim rau cov kev tshwm sim. Nws yog xam los ntawm piv txwv tias yog hais tias kev tshwm sim A tshwm sim. Nws yog xam raws li nram no: P (B / A).

Peb cuag cov tshuaj rau peb teeb meem: P (A _* B) = P (A) _* P (B / A) los yog P (A _* B) = P (B) _* P (A / B). Qhov ntau yog _{(4/36) * ((3/35)} / (4/36) yog xam los ntawm pub muab kwv yees mus rau qhov ze puas Peb muaj: .. _{0,11 * (0,09 / 0,11) =} 0,11 * 0 , 82 = 0.09. qhov ntau hais tias peb kos tawm ob ARRANGEMENT nyob rau hauv ib tug kab yog sib npaug zos rau cuaj hundredths. tus nqi yog heev me me, nws hais ntxiv hais tias qhov tshwm sim ntawm cov kev tshwm sim tshwm sim yog tsis tshua muaj tsis tshua muaj.

tsis nco qab lawm chav tsev

Peb muaj ua tawm ib co ntau txoj kev xaiv ntawm cov hauj lwm uas kawm cov kev tshawb xav ntawm ntau yam. Piv txwv ntawm cov kev daws ntawm ib co ntawm cov uas koj twb pom nyob rau hauv no tsab xov xwm, sim kom daws tau qhov nram qab no teeb meem: Tus me nyuam tub tsis nco qab tus xov tooj rau lub xeem lej ntawm nws cov phooj ywg, tab sis txij li thaum hu yog qho tseem ceeb heev, ces pib tuaj tos txhua tus nyob rau tig. Peb yuav tsum xam qhov tshwm sim hais tias nws yuav hu rau tsis muaj ntau tshaj peb lub sij hawm. qhov nyuaj tov ntawm qhov teeb meem, Yog hais tias koj paub cov kev cai, cov cai thiab axioms ntawm ntau yam kev tshawb xav.

Ua ntej koj pom ib tug daws, sim los daws kom tau lawv tus kheej. Peb paub hais tias cov tom kawg daim duab tej zaum yuav tsum yog los ntawm xoom cuaj, rau ib tug tag nrho ntawm kaum qhov tseem ceeb. Muaj tseeb tau yuav tsum tau yog 1/10.

Tom ntej no peb yuav tsum xav txog kev xaiv rau lub keeb kwm ntawm cov txheej xwm, qhia rau peb xav hais tias cov me nyuam tub guessed txoj cai thiab yeej muaj txoj cai, qhov yuav tshwm ntawm tej txheej xwm no yog sib npaug zos rau 1/10. Qhov thib ob xaiv: tus thawj hu ris hauv qab, thiab lub thib ob lub hom phiaj. Peb xam qhov tshwm sim ntawm tej txheej xwm: 9/10 multiplied by 1/9 nyob rau hauv lub kawg peb tau raws li 1/10. Cov kev xaiv thib peb: tus thawj thiab thib ob hu muab tau cov tsis ncaj ncees lawm qhov chaw nyob, tsuas yog tus thib peb me nyuam tub yog qhov uas nws xav. Xam qhov tshwm sim rau tej txheej xwm: 9/10 multiplied by 8/9 thiab 1/8, peb tau raws li ib tug tshwm sim ntawm 1/10. Lwm yam kev xaiv rau tus mob ntawm qhov teeb meem peb tsis xav, qhov no tseem rau peb mus rau nteg cia cov tau, nyob rau hauv lub kawg peb tau muaj ib tug 3/10. Teb: Qhov ntau hais tias ib tug me nyuam tub yuav hu xov tooj tsis muaj ntau tshaj peb lub sij hawm, sib npaug zos rau 0.3.

Phaib nrog cov zauv

Ua ntej koj yuav cuaj phaib, txhua tus uas yog sau ib tug xov tooj ntawm ib tug mus cuaj, cov zauv tsis rov qab. Lawv muab tso rau hauv ib lub thawv thiab sib tov meej. Koj yuav tsum xam qhov tshwm sim hais tias tus

• dov ib tug txawm tooj;
• ib tug ob tug.

Ua ntej cov tswv yim rau qhov kev txiav txim stipulate uas m - yog tus naj npawb ntawm kev vam meej tus neeg mob, thiab n - yog tag nrho cov xov tooj ntawm cov kev xaiv. Cia peb nrhiav qhov kwv yees hais tias tus xov tooj yog txawm. Yog tsis nyuaj rau xam hais tias txawm xov tooj ntawm plaub, thiab nws yog peb m, tag nrho cov cuaj tau xaiv, uas yog, m = 9. Ces qhov yuav tshwm yog sib npaug zos rau 0,44 los yog 4/9.

Peb xav txog lub rooj thib ob, tus xov tooj ntawm variants ntawm cuaj, thiab ib tug muaj kev vam meej lub sij hawm tsis tau rau tag nrho cov, hais tias yog, m yog pes tsawg. Qhov ntau hais tias cov elongated card yuav muaj ib tug ob tus lej, raws li pes tsawg.