Lub hom phiaj ntawm qhov muaj tseeb. Yuav muaj kev tshwm sim, nyob kev tshwm sim (probability kev tshawb xav). Independent thiab incompatible uas nyob rau hauv cov kev tshawb xav ntawm ntau yam

Nws yog tsis zoo li hais tias muaj ntau cov neeg xav hais tias nws yog ua tau rau suav txheej xwm, uas rau ib co raws li lo. Yuav kom muab tso rau hauv cov lus yooj yim, yog nws muaj tiag paub uas sab ntawm lub thawv muaj rau sab nyob rau hauv lub dice yuav poob tom ntej no lub sij hawm. Nws yog lo lus nug no nug ob tug poj zaum, nteg lub lub hauv paus rau qhov kev kawm, cov kev tshawb xav ntawm ntau yam, qhov tshwm sim rau cov kev tshwm sim nyob rau hauv uas lub kawm feem txaus.

tiam

Yog hais tias koj sim los txhais tau xws li ib tug lub tswv yim raws li lub hom phiaj ntawm qhov muaj tseeb, peb tau txais hauv qab no: qhov no yog ib tug ntawm cov ceg ntawm kev kawm txog zauv uas kawm cov constancy ntawm random txheej xwm. Kom meej meej, qhov no tswvyim yeej tsis qhia tus essence, yog li ntawd koj yuav tsum xav txog nws nyob rau hauv ntau yam.

Kuv xav pib nrog lub founders ntawm cov kev tshawb xav. Raws li twb hais saum toj no, muaj ob, uas ib Ferma thiab Blez Paskal. Lawv yog cov thawj sim siv cov qauv thiab zauv los mus laij lub sij hawm ntawm ib tug tshwm sim. Nyob rau hauv dav dav, lub rudiments ntawm no science yog txawm nyob rau hauv cov Nrab Hnub nyoog. Thaum ntau yam xav thiab zaum tau sim rau kev tsom xam cov twv txiaj yuam pov ua si xws li roulette, craps, thiab thiaj li nyob, li no mus nrhiav kom tau ib tug qauv, thiab qhov feem pua tsis ib tug xov tooj. Lub hauv paus twb tseem pw nyob rau hauv lub thib kaum xya xyoo pua nws yog tus aforementioned scholars.

Chiv, lawv ua hauj lwm yuav tsis tau ntaus nqi mus rau qhov zoo kawg thiab ua tau nyob rau hauv daim teb no, tom qab tag nrho, lawv tau ua, lawv tsuas empirical tseeb thiab thwmsim tau kom meej meej tsis siv cov qauv. Thaum lub sij hawm, nws tig mus rau tau poj ntsuam, uas nyob li ib tug tshwm sim ntawm kev soj xyuas tus cam khwb cia ntawm lub pob txha. Nws yog qhov ntsuas tau pab coj cov thawj cov mis.

los ntawm UFW

Tsis hais xws li ib tug txiv neej li Christiaan Huygens, nyob rau hauv tus txheej txheem ntawm kev kawm cov kev kawm bears lub npe ntawm "probability kev tshawb xav" (probability ntawm cov kev tshwm sim tseem ceeb nyob rau hauv no science). Tus neeg no yog heev nthuav. Nws, raws li zoo raws li cov kws tshawb fawb hais saum toj no yog sim nyob rau hauv daim ntawv ntawm cov formulas lej rau deduce ib tug qauv ntawm random txheej xwm. Nws yog noteworthy hais tias nws tsis muab nws qhia rau Pascal thiab Fermat, uas yog tag nrho nws ua hauj lwm tsis sib tshooj nrog cov minds. Huygens muab lub tswv yim ntawm ntau yam kev tshawb xav.

Ib tug interesting tseeb yog hais tias nws ua hauj lwm tuaj ntev ua ntej qhov kev tshwm sim ntawm tej hauj lwm ntawm cov tho kev, yuav tsum tau pes tsawg, nees nkaum xyoo ua ntej lawm. Muaj tsuas yog ntawm cov tswv yim uas pom tias yog:

• raws li lub tswvyim ntawm qhov muaj tseeb qhov tseem ceeb lub sij hawm;
• expectation rau lub discrete ntaub ntawv;
• theorems ntxiv thiab npaug ntawm probabilities.

Tsis tas li ntawd, ib tug yuav tsis nco qab Yakoba Bernulli, uas kuj pab mus rau txoj kev tshawb no ntawm qhov teeb meem. Los ntawm lawv tus kheej, tej yam uas leej twg yog cov ywj siab kev ntsuam xyuas, nws muaj peev xwm mus muab pov thawj ntawm cov kev cai lij choj ntawm cov coob. Nyob rau hauv lem, zaum Poisson thiab Laplace, uas ua hauj lwm nyob rau hauv thaum ntxov thib kaum cuaj xyoo pua, twb tau los ua pov thawj tus thawj theorem. Los ntawm lub sij hawm ntawd mus tsom xam uas tsis nyob rau hauv lub tswvyim peb pib siv ntau yam kev tshawb xav. Tog nyob ib ncig ntawm no science yuav tsis thiab Lavxias teb sab kws tshawb fawb, es Markov, Chebyshev thiab Dyapunov. Lawv yog raws li nyob rau hauv lub chaw ua hauj lwm ua zoo geniuses, ruaj qhov kev kawm raws li ib tug ceg ntawm kev kawm txog zauv. Peb ua hauj lwm cov nuj nqis rau ntawm lub kawg ntawm lub puas tawm xyoo, thiab ua tsaug rau lawv pab, muaj lawm proven phenomena xws li:

• kev cai lij choj ntawm loj tus xov tooj;
• Kev tshawb xav ntawm Markov chains;
• Lub hauv paus txwv theorem.

Yog li ntawd, lub keeb kwm ntawm cov me nyuam yug ntawm science thiab nrog rau cov loj personalities uas pab rau nws, txhua yam yog xav paub ntau los yog tsawg dua ntshiab. Tam sim no nws yog lub sij hawm rau cov nqaij tawm tag nrho qhov tseeb.

tswv yim

Ua ntej koj kov cov kev cai thiab theorems yuav tsum kawm lub tswv yim ntawm ntau yam kev tshawb xav. Cov kev tshwm sim nws occupies ib tug tseem ceeb luag hauj lwm. Cov npe no yog theej nws kim heev, tab sis yuav tsis tau to taub tag nrho tus so tsis tau nws.

Cov kev tshwm sim nyob rau hauv ntau yam kev tshawb xav - nws Tej txheej tshwm sim ntawm qhov xyaum ua tej. Ntsiab ntawm no phenomenon muaj tsis txaus. Yog li, Lotman paub txog kev ua hauj lwm nyob rau hauv cov cheeb tsam no, tau qhia tias nyob rau hauv cov ntaub ntawv no peb yuav sib tham txog dab tsi "tshwm sim, txawm hais tias nws yuav tsis tshwm sim."

Random txheej xwm (probability kev tshawb xav them tshwj xeeb mloog mus rau lawv) - yog ib lub tswvyim uas yuav kiag li tej tshwm sim muaj tau tshwm sim. Los yog, on qhov tsis tooj, qhov no scenario yuav tsis tshwm sim nyob rau hauv qhov kev kawm ntawm ib tug ntau yam ntawm tej yam kev mob. Nws tseem yog nqi paub tias nyob rau hauv tag nrho lub ntim ntawm lub phenomena tshwm sim cia li random txheej xwm. Muaj tseeb kev tshawb xav qhia tias tag nrho cov tej yam kev mob yuav tau rov qab tu li. Nws yog lawv coj tau raug hu ua "kev" los yog "kev kuaj."

Tseem ceeb tshwm sim - qhov no yog ib tug tshwm sim uas yog ib puas feem pua nyob rau hauv qhov kev xeem no tshwm sim. Raws li, qhov tsis yooj yim sua tshwm sim - qhov no yog ib yam dab tsi uas tsis tshwm sim.

Combining officers Action (conventionally rooj plaub A thiab cov ntaub ntawv B) yog ib tug tshwm sim uas tshwm sim ib txhij. Lawv raug xa mus rau raws li AB.

Tus nqi ntawm cov officers ntawm cov txheej xwm A thiab B - C yog, nyob rau hauv lwm yam lus, yog muaj ib tug ntawm lawv yuav (A los yog B), koj yuav tau ib C. Cov mis piav phenomenon yog sau raws li C = A + B.

Incompatible uas nyob rau hauv cov kev tshawb xav ntawm ntau yam implies hais tias ob tus neeg mob yog ob leeg kom. Nyob rau tib lub sij hawm lawv nyob rau hauv txhua rooj plaub yuav tsis muaj tshwm sim. Joint txheej xwm nyob rau hauv ntau yam kev tshawb xav - nws yog lawv antipode. Lub implication yog hais tias yog ib tug tshwm sim, nws tsis txwv C.

Tawm tsam muaj cov kev tshwm sim (probability kev tshawb xav considers lawv nyob rau hauv zoo meej), yog ib qho yooj yim to taub. Nws yog qhov zoo tshaj plaws nrog lawv nyob rau hauv kev sib piv. Lawv yog cov yuav luag tib yam li incompatible uas nyob rau hauv cov kev tshawb xav ntawm ntau yam. Txawm li cas los, lawv sib txawv yog hais tias ib tug ntawm ib tug plurality ntawm phenomena nyob rau hauv txhua rooj plaub yuav tsum tshwm sim.

Attendance yuav cov txheej xwm - cov kev ua, tau repetition yog sib npaug zos. Yuav kom ua kom nws ntshiab, koj yuav tau xav txog tej yam pov ib npib: tsis ntawm ib qho ntawm nws cov sab yog Attendance tej zaum tsis muaj lwm yam.

nws yog yooj yim los mus xav txog cov piv txwv ntawm favouring cov kev tshwm sim. Piv txwv muaj ib tug rov nyob rau hauv lub rov A. Tus thawj - ib tug yob ntawm ib tug tuag nrog lub advent ntawm ib tug khib tooj, thiab lub thib ob - cov tsos ntawm tus xov tooj tsib nyob rau hauv tsuav. Ces nws hloov tawm hais tias ib tug yog favored V.

Independent txheej xwm nyob rau hauv ntau yam kev tshawb xav yog npaj tsuas yog nyob rau ob los yog ntau tshaj lub sij hawm ntawd thiab cia ywj siab ntawm kev txiav txim los ntawm lwm yam. Piv txwv li, A - thaum tsis tails npib tossing, thiab B - dostavanie jack los ntawm lub lawj. Lawv muaj ywj siab cov txheej xwm nyob rau hauv ntau yam kev tshawb xav. Los ntawm lub sij hawm no nws los ua tseeb.

Yus cov txheej xwm nyob rau hauv ntau yam kev tshawb xav no kuj yog permissible tsuas rau lawv cov teeb. Lawv hais dependence ntawm ib tug nyob rau lwm yam, uas yog, lub phenomenon yuav tshwm sim nyob rau hauv xwb nyob rau hauv lub rooj plaub thaum A twb tshwm sim los yog, nyob rau hauv tsis tooj, tsis tshwm sim thaum twg nws yog - lub ntsiab mob rau B.

Lub sij hawm ntawm qhov random xyaum ua tej yam muaj ib zaug xwb tivthaiv - nws yog elementary cov txheej xwm. Muaj tseeb kev tshawb xav hais tias nws yog ib tug tshwm sim hais tias yog ua li cas ib zaug xwb.

yooj yim mis

Yog li, lub saum toj no raug xam tias yog lub tswvyim ntawm "kev tshwm sim", "muaj tseeb kev tshawb xav", txhais cov ntsiab lus tseem ceeb hauv cov nqe lus ntawm no science twb tseem muab. Tam sim no nws yog lub sij hawm mus paub nws tus kheej nrog qhov tseem ceeb cov qauv. Cov kab zauv yog lej paub tseeb hais tias tag nrho cov ntsiab tswv yim nyob rau hauv xws li ib tug yooj yim kev kawm raws li lub hom phiaj ntawm qhov muaj tseeb. Yuav muaj kev tshwm sim thiab ua ib tug lossis loj lub luag hauj lwm.

Zoo pib nrog qhov yooj yim qauv ntawm combinatorics. Thiab ua ntej koj pib rau lawv, nws yog tsim nyog xav tias nws yog dab tsi.

Combinatorics - yog feem ntau ib tug ceg ntawm kev kawm txog zauv, nws tau raug kawm ib tug lossis loj tus naj npawb ntawm cov zauv, thiab ntau yam permutations ntawm ob lub xov tooj thiab lawv cov ntsiab, ntau yam ntaub ntawv, thiab lwm yam, ua rau ib tug xov tooj ntawm ob peb ua ke ... Nyob rau hauv tas li ntawd mus rau lub hom phiaj ntawm qhov muaj tseeb, qhov kev lag luam yog ib qho tseem ceeb rau cov statistics, computer science thiab cryptography.

Yog li ntawd tam sim no koj yuav tsiv mus nyob rau qhov kev nthuav qhia ntawm lawv tus kheej thiab lawv txhais cov qauv.

Cov ntawm no yog cov kev qhia rau tus xov tooj ntawm permutations, nws yog raws li nram no:

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

Kab zauv siv xwb nyob rau hauv rooj plaub yog hais tias tus hais txawv xwb nyob rau hauv qhov kev txiav txim ntawm kev kho.

Tam sim no qhov kev tso kawm mis, nws zoo li no yuav pom tau tias:

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

Qhov no qhia yog muaj feem xyuam tsis tsuas mus rau lub tsuas caij ntawm kev txiav txim qhov kev tso kawm, tab sis kuj rau nws muaj pes tsawg leeg.

Qhov thib peb kab zauv ntawm combinatorics, thiab nws yog tus tom kawg, hu ua cov mis rau tus naj npawb ntawm ob peb ua ke:

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

Thaum lawv tseem hu ua piv txwv, uas yog tsis tau kom, ntsig txog, rau thiab thov txoj cai no.

Nrog cov qauv ntawm combinatorics tuaj kom to taub tau yooj yim, koj yuav tam sim no mus rau lub classical txhais ntawm qhov muaj tseeb. Nws zoo li no qhia raws li nram no:

P (A) = m: n.

Nyob rau hauv cov mis no, m - yog tus naj npawb ntawm tej yam kev mob conducive mus rau lub kev tshwm sim A, thiab n - tus naj npawb ntawm Attendance thiab tag tag nrho cov elementary cov txheej xwm.

Muaj ntau ntau cov kab zauv nyob rau hauv tsab xov xwm yuav tsis muab suav hais tias dab tsi tab sis cuam tshuam yuav tsum yog tus tseem ceeb tshaj plaws uas sawv daws yuav xws li, piv txwv li, qhov yuav tshwm ntawm cov txheej xwm nqi:

P (A + B) = P (A) + P (B) - qhov no theorem rau ntxiv xwb nrog kom cov txheej xwm;

P (A + B) = P (A) + P (B) - P (AB) - tab sis qhov no yog tsuas yog rau cov ntxiv tau tshaj.

Qhov ntau yam ntawm cov kev tshwm sim tej hauj lwm:

P (A ⋅ B) = P (A) ⋅ P (B) - qhov no theorem rau ywj siab cov txheej xwm;

(P (A ⋅ B) = P (A) ⋅ P (B | A); P (A ⋅ B) = P (A) ⋅ P (A | B)) - thiab qhov no rau lub nyob.

Twb daim ntawv teev cov txheej xwm mis. Lub hom phiaj ntawm qhov muaj tseeb qhia rau peb theorem Bayes, uas zoo li no:

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

Nyob rau hauv cov mis no, H _1, H _2, ..., H _n - yog ib tug ua tiav set ntawm hypotheses.

Thaum no nres, kuaj qauv daim ntawv thov yuav tam sim no yuav tsum tau kev pab raws qib los ntawm kev xyaum.

piv txwv

Yog hais tias koj ua tib zoo kawm tej ceg ntawm kev kawm txog zauv, nws tsis yog tsis muaj ce thiab cov qauv kev daws teeb meem. Thiab lub hom phiaj ntawm qhov muaj tseeb: cov txheej xwm, piv txwv no yog ib feem ntawm kev pom zoo scientific suav.

Cov mis rau tus xov tooj ntawm permutations

Piv txwv li, nyob rau hauv ib daim ntawv lawj muaj peb caug phaib, pib nrog lub nominal ib. Tom ntej no lo lus nug. Yuav ua li cas muaj ntau txoj kev quav lub lawj yog li ntawd cov ntawv teev lus nrog ib lub ntsej muag tus nqi ntawm ib tug thiab ob tug twb tsis nyob tom ntej no?

Cov neeg ua hauj lwm yog teeb, tam sim no cia tsiv mus rau nrog nws. Ua ntej koj yuav los mus txiav txim tus naj npawb ntawm permutations ntawm peb caug hais, rau lub hom phiaj no peb siv cov saum toj no mis, nws puv P_30 = 30!.

Raws li txoj cai no, peb paub yuav ua li cas muaj ntau yam kev xaiv muaj rau nteg cia lub lawj nyob rau hauv ntau txoj kev, tab sis peb yuav tsum tau muab rho tawm los ntawm lawv yog cov neeg nyob rau hauv uas tus thawj thiab thib ob daim ntawv yuav yuav tom ntej no. Ua li no, pib nrog ib tug variant, thaum thawj yog nyob rau hauv lub thib ob. Nws hloov tawm hais tias tus thawj daim ntawv qhia tej zaum yuav muab nees nkaum cuaj qhov chaw - los ntawm tus thawj rau lub nees nkaum-cuaj, thiab lub thib ob daim ntawv los ntawm lub thib ob mus rau lub peb caug, puv nees nkaum cuaj rooj rau officers ntawm phaib. Nyob rau hauv lem, lwm tus yuav coj nees nkaum-yim rooj, thiab nyob rau hauv tej kev txiav txim. Hais tias yog, rau lub rearrangement ntawm nees nkaum-yim phaib tau nees nkaum yim xaiv P_28 = 28!

Cov no hais tias yog hais tias peb xav txog qhov kev txiav txim, thaum thawj card yog rau lub thib ob ntxiv caij nyoog tau 29 ⋅ 28! = 29!

Siv tib txoj kev, koj yuav tsum xam tus naj npawb ntawm redundant xaiv rau rooj plaub thaum thawj card yog nyob rau hauv lub thib ob. Tsis tas li ntawd tau 29 ⋅ 28! = 29!

Los ntawm no nws raws li nram no uas tus ntxiv kev xaiv 2 ⋅ 29!, Thaum lub sij hawm tsim nyog txhais tau tias ntawm sau lub lawj 30! - 2 ⋅ 29!. Nws tseem raug nyob tsuas xam.

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

Tam sim no peb yuav tsum tau muab ua ke tag nrho cov ntawm cov zauv los ntawm ib tug mus rau nees nkaum cuaj, thiab ces nyob rau hauv lub kawg ntawm tag nrho cov khoo los ntawm 28. Cov lus teb tau 2,4757335 ⋅ 〖〗 10 ^ 32

Piv txwv ntawm cov kev daws. Cov mis rau tus xov tooj ntawm cov kev pab

Nyob rau hauv qhov teeb meem no, koj yuav tsum paub ntau npaum li cas muaj ntau txoj kev rau muab tso rau lub kaum tsib tagnrho rau ib tug txee, tab sis nyob rau hauv cov mob uas tsuas peb caug tagnrho.

Nyob rau hauv no ua hauj lwm, cov kev txiav txim siab ib tug me ntsis yooj yim dua yav dhau los. Siv cov twb paub mis, nws yog tsim nyog los xam tag nrho cov ntawm peb caug qhov chaw kaum tsib tagnrho.

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

Lo lus teb, ntsig txog, yuav muaj sib npaug zos rau 202 843 204 931 727 360 000.

Tam sim no siv rau qhov hauj lwm ntau me ntsis yooj yim. Koj yuav tsum paub ntau npaum li cas muaj ntau txoj kev mus teem lub peb caug-ob phau ntawv rau cov txee, nrog rau cov proviso tias tsuas yog kaum tsib tagnrho yuav tsum nyob rau ntawm tib lub txee.

Ua ntej qhov pib ntawm qhov kev txiav txim yuav nyiam hais tias ib txhia ntawm cov teeb meem yuav daws tau nyob rau hauv ob peb txoj kev, thiab nyob rau hauv no muaj ob txoj kev, tab sis nyob rau hauv ob qho tib si ib tug thiab tib mis yog thov.

Nyob rau hauv no ua hauj lwm, koj muaj peev xwm coj cov lus teb los ntawm lub yav dhau los ib tug, vim hais tias muaj peb tau xam tus naj npawb ntawm cov sij hawm koj yuav tau sau tawm hauv lub txee rau kaum tsib phau ntawv nyob rau hauv ntau txoj kev. Nws muab A_30 ^ 15 = 30 ⋅ 29 ⋅ 28 ⋅ ... ⋅ (30 - 15 + 1) = 30 ⋅ 29 ⋅ 28 ⋅ ... ⋅ 16.

Qhov thib ob tso kom muaj kab muaj kev xam los ntawm cov mis reshuffle, vim hais tias nws yog muab tso rau kaum tsib phau ntawv, thaum lub sij hawm uas tseem tshuav ntawm kaum tsib. Peb siv mis P_15 = 15!.

Nws hloov tawm hais tias lub sum yuav A_30 ^ 15 ⋅ P_15 txoj kev, tab sis, nyob rau hauv tas li ntawd, cov khoom ntawm tag nrho cov zauv los ntawm peb caug rau kaum yuav tsum khoo los ntawm cov khoom ntawm cov zauv los ntawm ib tug mus rau kaum tsib, nyob rau hauv lub kawg tig tawm cov khoom ntawm tag nrho cov zauv ntawm ib tug mus peb caug, uas yog cov lus teb 30!

Tab sis qhov teeb meem no yuav tsum daws nyob rau hauv ib tug txawv txoj kev - yooj yim. Ua li no, koj yuav tau xav txog tej yam uas muaj yog ib tug txee rau peb caug phau ntawv. Tag nrho cov ntawm lawv yog muab tso rau hauv no dav hlau, tab sis vim hais tias cov kev mob yuav tsum tau hais tias muaj ob txee, ib tug ntev peb sawing nyob rau hauv ib nrab, ob lem kaum tsib. Los ntawm no nws hloov tawm hais tias rau qhov no yuav ua tau P_30 = 30!.

Piv txwv ntawm cov kev daws. Cov mis rau tus xov tooj ntawm ob peb ua ke ntawm

Leej twg yog ib tug variant ntawm peb cov teeb meem ntawm combinatorics. Koj yuav tsum paub ntau npaum li cas txoj kev muaj los npaj kaum tsib phau ntawv nyob rau cov mob uas koj yuav tsum tau xaiv los ntawm peb caug raws nraim tib yam.

Rau qhov kev txiav txim yuav, ntawm chav kawm, thov lub mis rau tus xov tooj ntawm ob peb ua ke. Los ntawm cov mob hais tias nws yuav paub tseeb hais tias qhov kev txiav txim ntawm lub tib lub kaum tsib phau ntawv yog tsis tseem ceeb. Yog li ntawd chiv koj yuav tsum paub tag nrho cov xov tooj ntawm cov ob peb ua ke ntawm peb caug kaum tsib phau ntawv.

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

Hais tias yog tag nrho. Siv cov mis, nyob rau hauv lub sij hawm shortest tau mus daws tau tej teeb meem, cov lus teb, ntsig txog, sib npaug zos rau 155.117.520.

Piv txwv ntawm cov kev daws. Lub classic txhais ntawm qhov muaj tseeb

Siv cov mis muab saum toj no, ib tug yuav nrhiav ib lo lus teb nyob rau hauv ib tug yooj yim ua hauj lwm. Tab sis nws yuav kom meej meej pom thiab ua raws li cov chav kawm ntawm kev txiav txim.

Cov neeg ua hauj lwm muab hais tias nyob rau hauv ib tug urn muaj kaum kiag tib yam khoom. Ntawm no, plaub daj thiab rau xiav. Los ntawm lub urn ib lub pob. Nws yog tsim nyog los paub qhov yuav tshwm dostavaniya xiav.

Yuav kom daws tau qhov teeb meem nws yog tsim nyog los xaiv dostavanie xiav pob tshwm sim A. Qhov no tej zaum yuav muaj kaum tshwm sim, uas, nyob rau hauv lem, elementary thiab Attendance yuav. Nyob rau tib lub sij hawm, rau ntawm lub kaum yog dej siab rau cov kev tshwm sim A. Daws cov nram qab no mis:

P (A) = 6: 10 = 0.6

Thov cov mis no, peb tau kawm hais tias tus tau dostavaniya xiav pob yog 0.6.

Piv txwv ntawm cov kev daws. Qhov ntau yam ntawm cov txheej xwm npaum li cas

Leej twg yuav tau ib tug variant uas yog solved los ntawm kev siv cov mis ntawm ntau yam ntawm cov txheej xwm npaum li cas. Yog li ntawd, muab cov mob uas muaj ob tus neeg, ib tug thawj yog grey thiab tsib dawb khoom, thaum lub sij hawm thib ob - yim grey thiab plaub dawb npas. Raws li ib tug tshwm sim, tus thawj thiab thib ob thawv muaj npaum li cas rau ib tug ntawm lawv. Nws yog tsim nyog los nrhiav tau tawm yog dab tsi yog qhov uas yuav hais tias Corporate cov khoom yog grey thiab dawb.

Yuav kom daws tau qhov teeb meem no, nws yog tsim nyog los qhia cov kev tshwm sim.

• Yog li, A - peb muaj ib tug grey pob ntawm cov thawj box: P (A) = 1/6.
• A '- dawb noob kuj noj los ntawm tus thawj box: P (A') = 5/6.
• Lub - twb rho gray pob ntawm lub thib ob conduit: P (B) = 2/3.
• B '- coj ib tug grey pob ntawm lub thib ob lub tub rau khoom: P (B') = 1/3.

Raws li qhov teeb meem nws yog tsim nyog hais tias ib tug ntawm cov phenomena tshwm sim: AB 'los yog' B. Siv cov mis, peb tau: P (AB ') = 1/18, P (A'B) = 10/18.

Tam sim no cov mis ntawm txawj qhov yuav tshwm tau siv. Tom ntej no, mus nrhiav tau tawm lub teb, koj yuav tsum mus thov lawv cov kab zauv ntxiv:

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

Qhov ntawd yog yuav ua li cas, siv cov mis, koj yuav kom daws tau tej teeb meem.

tshwm sim

Cov ntawv tau qhia rau cov ntaub ntawv rau "probability kev tshawb xav", qhov tshwm sim rau cov txheej xwm uas ua si ib qho tseem ceeb luag hauj lwm. Ntawm cov hoob kawm, tsis txhua yam tau raug suav hais tias, tab sis nyob rau hauv lub hauv paus ntawm cov ntawv nyeem hais, koj muaj peev xwm raws tau ntsib nrog rau qhov no ceg ntawm kev kawm txog zauv. Xam tias yog science yuav pab tau tsis tsuas nyob rau hauv cov kev ua lag ua luam, tab sis kuj nyob rau hauv lub neej txhua hnub. Koj muaj peev xwm siv nws los xam tej tau ib tug tshwm sim.

Cov ntawv nyeem twb tseem cuam tshuam los ntawm tseem ceeb hnub nyob rau hauv lub keeb kwm ntawm txoj kev loj hlob ntawm ntau yam kev tshawb xav raws li ib tug science, thiab lub npe ntawm tus neeg uas nws ua hauj lwm tau raug muab tso rau hauv nws. Qhov ntawd yog yuav ua li cas tib neeg xav tau coj mus rau lub fact tias cov neeg tau kawm suav, txawm random txheej xwm. Thaum lawv yog cov cia li xav nyob rau hauv no, tab sis hnub no nws yog twb paub tag nrho. Thiab tsis muaj ib tug yuav hais li cas yuav tshwm sim rau peb nyob rau hauv lub neej yav tom ntej, yog dab tsi lwm yam ci ntsa iab discoveries muaj feem xyuam rau cov kev tshawb xav nyob rau hauv kev saib xyuas, yuav tsum tau cog lus. Tab sis ib tug tshaj plaws yog tseeb - txoj kev kawm no tseem yog tsis nqi nws!