Nws tsis zoo li ntau tus neeg xav txog seb puas tuaj yeem suav cov xwm txheej uas ntau dua lossis tsawg dua. Hauv cov ntsiab lus yooj yim, nws puas muaj tseeb kom paub tias sab twg ntawm qhov tuag hauv cov tsuav yuav poob tawm tom ntej. Nws yog lo lus nug no uas ob tus kws tshawb fawb zoo nug, uas tau tsim lub hauv paus rau kev tshawb fawb xws li kev xav ntawm qhov tshwm sim, uas qhov tshwm sim ntawm qhov tshwm sim tau kawm ntau heev.
Keeb Kwm
Yog tias koj sim txhais lub tswv yim xws li qhov yuav tshwm sim txoj kev xav, koj tau txais cov hauv qab no: qhov no yog ib ceg ntawm kev ua lej uas kawm txog qhov xwm txheej tsis tu ncua. Tau kawg, lub tswv yim no tsis tshua nthuav qhia tag nrho cov ntsiab lus, yog li nws yuav tsum tau xav txog nws kom meej ntxiv.
Kuv xav pib nrog cov neeg tsim kev xav. Raws li tau hais los saum no, muaj ob ntawm lawv, cov no yog Pierre Fermat thiab Blaise Pascal. Nws yog lawv cov uas yog thawj tus uas sim xam qhov tshwm sim ntawm ib qho kev tshwm sim uas siv cov qauv thiab kev suav lej. Nyob rau hauv tag nrho, lub rudiments ntawm no science tau tshwm sim thaum ntxovNruab Nrab Hnub nyoog. Lub sijhawm ntawd, ntau tus neeg xav thiab cov kws tshawb fawb tau sim tshuaj xyuas kev twv txiaj, xws li roulette, craps, thiab lwm yam, yog li tsim kom muaj tus qauv thiab feem pua ntawm ib tus lej poob. Lub hauv paus tau muab tso rau hauv lub xyoo pua 17th los ntawm cov kws tshawb fawb tau hais tseg.
Thaum xub thawj, lawv txoj haujlwm tsis tuaj yeem suav tias yog qhov ua tau zoo hauv daim teb no, vim tias txhua yam lawv tau ua yog qhov tseeb tseeb, thiab cov kev sim tau teeb pom kev zoo, tsis tas siv cov qauv. Sij hawm dhau mus, nws tig tawm kom ua tiav cov txiaj ntsig zoo, uas tau tshwm sim los ntawm kev soj ntsuam qhov ntuav ntawm tsuav. Nws yog cov cuab yeej no uas tau pab kom tau txais thawj cov qauv paub tsis meej.
Associates
Nws tsis tuaj yeem hais tsis tau hais txog tus neeg zoo li Christian Huygens, nyob rau hauv tus txheej txheem ntawm kev kawm ib lub ntsiab lus hu ua "probability theory" (qhov tshwm sim ntawm qhov tshwm sim tau hais meej hauv qhov kev tshawb fawb no). Tus neeg no ntxim nyiam heev. Nws, zoo li cov kws tshawb fawb tau hais los saum toj no, sim ua kom tau qhov xwm txheej tsis tu ncua hauv cov qauv lej. Nws yog ib qho tseem ceeb uas nws tsis tau ua qhov no ua ke nrog Pascal thiab Fermat, uas yog, tag nrho nws tej hauj lwm tsis nyob rau hauv ib txoj kev cuam tshuam nrog cov hlwb. Huygens muab cov ntsiab lus tseem ceeb ntawm qhov kev xav tau.
Qhov tseeb nthuav yog tias nws txoj haujlwm tau tshwm sim ntev ua ntej cov txiaj ntsig ntawm cov neeg tho kev ua haujlwm, lossis zoo li, nees nkaum xyoo dhau los. Ntawm cov tswv yim tsim, nto moo tshaj plaws yog:
- lub tswv yim ntawm qhov tshwm sim raws li qhov loj ntawm lub caij nyoog;
- kev cia siab rau kev sib caiscas;
- theorems of multiplication and add of probabilities.
Nws tseem tsis tuaj yeem tsis nco qab Jacob Bernoulli, uas kuj tau pab txhawb rau kev kawm txog qhov teeb meem. Ua nws tus kheej cov kev xeem, tsis muaj leej twg, nws muaj peev xwm nthuav qhia cov ntaub ntawv pov thawj ntawm txoj cai ntawm cov neeg coob. Nyob rau hauv lem, cov kws tshawb fawb Poisson thiab Laplace, uas ua hauj lwm nyob rau hauv thaum pib ntawm lub XIX caug xyoo, muaj peev xwm ua pov thawj cov thawj theorems. Nws yog los ntawm lub sijhawm no uas qhov kev xav tau tshwm sim tau pib siv los txheeb xyuas qhov tsis raug hauv chav kawm ntawm kev soj ntsuam. Cov kws tshawb fawb Lavxias, lossis zoo li Markov, Chebyshev thiab Dyapunov, tsis tuaj yeem hla qhov kev tshawb fawb no. Raws li txoj haujlwm ua los ntawm cov neeg txawj ntse zoo, lawv kho qhov kev kawm no ua ib ceg ntawm kev ua lej. Cov duab no tau ua haujlwm nyob rau thaum xaus ntawm lub xyoo pua puv 19, thiab ua tsaug rau lawv txoj kev koom tes, phenomena xws li:
- lawv tus lej loj;
- Markov chain theory;
- central limit theorem.
Yog li, nrog rau keeb kwm ntawm kev yug los ntawm kev tshawb fawb thiab nrog cov neeg tseem ceeb uas cuam tshuam nws, txhua yam yog ntau dua lossis tsawg dua. Tam sim no nws yog lub sij hawm los concretize tag nrho cov tseeb.
Basic concepts
Ua ntej kov cov kev cai lij choj thiab theorems, nws tsim nyog kawm cov ntsiab lus tseem ceeb ntawm txoj kev xav yuav tshwm sim. Qhov kev tshwm sim coj lub luag haujlwm tseem ceeb hauv nws. Lub ntsiab lus no yog qhov muaj zog heev, tab sis yog tsis muaj nws nws yuav tsis tuaj yeem nkag siab txhua yam ntxiv.
Ib qho kev tshwm sim hauv qhov kev xav ntawm qhov tshwm sim yog ib qho ntawm cov txiaj ntsig ntawm kev sim. Tsis muaj ntau lub tswv yim ntawm qhov tshwm sim no. Yog li, tus kws tshawb fawb Lotman,ua hauj lwm hauv cheeb tsam no, hais tias nyob rau hauv cov ntaub ntawv no peb tab tom tham txog tej yam uas "tshwm sim, txawm hais tias nws yuav tsis tshwm sim."
Random txheej xwm (qhov kev xav tau tshwm sim tau them nyiaj tshwj xeeb rau lawv) yog lub tswv yim uas txhais tau tias txhua yam tshwm sim uas muaj peev xwm tshwm sim. Los yog, hloov pauv, qhov xwm txheej no yuav tsis tshwm sim thaum muaj ntau yam xwm txheej tau ntsib. Nws tseem tsim nyog paub tias nws yog qhov xwm txheej random uas ntes tag nrho cov ntim ntawm cov xwm txheej uas tau tshwm sim. Qhov kev xav tau tshwm sim qhia tau tias txhua yam xwm txheej tuaj yeem rov ua dua tas li. Nws yog lawv qhov kev coj ua uas hu ua "kev paub" lossis "kev xeem".
Ib qho xwm txheej yog ib qho uas yuav 100% tshwm sim hauv qhov kev sim muab. Raws li, qhov xwm txheej tsis tuaj yeem yog ib qho uas yuav tsis tshwm sim.
Kev sib xyaw ua ke ntawm ob qho kev ua (xws li rooj plaub A thiab rooj B) yog qhov tshwm sim uas tshwm sim ib txhij. Lawv raug xaiv ua AB.
Qhov sib npaug ntawm cov xwm txheej A thiab B yog C, hauv lwm lo lus, yog tias tsawg kawg yog ib qho tshwm sim (A lossis B), ces C yuav tau txais.: C=A + B.
Cov xwm txheej tsis sib xws hauv qhov kev xav tau tshwm sim txhais tau hais tias ob qhov xwm txheej yog sib koom ua ke. Lawv yeej tsis tuaj yeem tshwm sim tib lub sijhawm. Cov xwm txheej sib koom ua ke hauv txoj kev xav yuav tshwm sim yog lawv cov tshuaj tiv thaiv. Qhov no txhais tau tias yog A tshwm sim, ces nws tsis cuam tshuam nrog B.
Cov xwm txheej tawm tsam (qhov kev xav tau tshwm sim nrog lawv kom meej) yooj yim to taub. Nws yog qhov zoo tshaj rau kev sib piv nrog lawv. Lawv yuav luag tib yam lithiab cov xwm txheej tsis sib xws hauv qhov kev xav tau tshwm sim. Tab sis lawv qhov sib txawv nyob ntawm qhov tseeb tias ib qho ntawm ntau qhov xwm txheej yuav tsum tshwm sim lawm.
Cov xwm txheej sib npaug yog cov kev ua, qhov ua tau sib npaug. Ua kom pom tseeb dua, peb tuaj yeem xav txog qhov pov ntawm ib lub npib: lub caij nplooj zeeg ntawm ib sab yog qhov sib npaug yuav poob ntawm lwm qhov.
Kev tshwm sim muaj koob meej yog yooj yim pom nrog piv txwv. Cia peb hais tias muaj ntu B thiab ntu A. Qhov thib ib yog yob ntawm cov tsuav nrog cov tsos ntawm tus lej khib, thiab qhov thib ob yog qhov tshwm sim ntawm tus lej tsib ntawm qhov tuag. Tom qab ntawd nws hloov tawm tias A nyiam B.
Cov xwm txheej ywj pheej hauv txoj kev xav yuav tshwm sim tsuas yog tshwm sim ntawm ob lossis ntau qhov xwm txheej thiab hais txog kev ywj pheej ntawm kev ua los ntawm lwm qhov. Piv txwv li, A yog qhov poob ntawm tails thaum lub npib raug pov, thiab B yog qhov kos duab ntawm lub tsho los ntawm lub lawj. Lawv yog cov xwm txheej ywj pheej nyob rau hauv txoj kev xav yuav tshwm sim. Nrog lub sijhawm no nws tau pom meej dua.
Cov xwm txheej nyob hauv qhov kev xav tau tshwm sim kuj tuaj yeem lees paub tsuas yog rau lawv cov teeb tsa. Lawv hais txog kev vam khom ntawm ib leeg, uas yog, qhov tshwm sim B tuaj yeem tshwm sim tsuas yog A twb tau tshwm sim lossis, ntawm qhov tsis sib xws, tsis tau tshwm sim, thaum qhov no yog qhov tseem ceeb rau B.
Qhov tshwm sim ntawm qhov kev sim random uas muaj ib feem yog cov xwm txheej theem pib. Probability theory piav qhia tias qhov no yog qhov tshwm sim uas tshwm sim ib zaug xwb.
Basic formulas
Yog li, cov ntsiab lus ntawm "kev tshwm sim", "qhov kev xav tau tshwm sim",lub ntsiab txhais ntawm cov ntsiab lus tseem ceeb ntawm kev tshawb fawb no kuj tau muab. Tam sim no nws yog lub sijhawm kom paub txog cov qauv tseem ceeb. Cov kab lus no ua lej lees paub tag nrho cov ntsiab lus tseem ceeb hauv cov ntsiab lus nyuaj xws li qhov yuav tshwm sim txoj kev xav. Qhov tshwm sim ntawm qhov kev tshwm sim ua lub luag haujlwm loj ntawm no thiab.
Zoo pib nrog cov qauv tsim ntawm combinatorics. Thiab ua ntej yuav mus rau lawv, nws tsim nyog xav txog qhov nws yog dab tsi.
Combinatorics feem ntau yog ib ceg ntawm kev ua lej, nws cuam tshuam nrog kev kawm ntawm ntau tus lej, nrog rau ntau qhov kev hloov pauv ntawm tus lej lawv tus kheej thiab lawv cov ntsiab lus, ntau yam ntaub ntawv, thiab lwm yam, ua rau pom ntawm ib tug xov tooj ntawm kev sib txuas. Ntxiv nrog rau qhov yuav tshwm sim txoj kev xav, ceg no tseem ceeb rau kev txheeb cais, computer science thiab cryptography.
Tam sim no peb tuaj yeem txav mus los nthuav qhia cov qauv ntawm lawv tus kheej thiab txhais lawv.
thawj tus yuav yog qhov qhia rau tus lej ntawm kev hloov pauv, nws zoo li qhov no:
P_n=n ⋅ (n - 1) ⋅ (n - 2)…3 ⋅ 2 ⋅ 1=n!
Kev sib npaug tsuas yog siv yog cov ntsiab lus sib txawv nkaus xwb.
Tam sim no cov qauv kev tso kawm yuav raug txiav txim siab, nws zoo li qhov no:
A_n^m=n ⋅ (n - 1) ⋅ (n-2) ⋅ … ⋅ (n - m + 1)=n!: (n - m)!
Cov lus qhia no siv tsis tau rau qhov kev txiav txim ntawm lub caij, tab sis kuj rau nws cov muaj pes tsawg leeg.
Qhov thib peb kab zauv los ntawm combinatorics, thiab nws kuj yog qhov kawg, yog hu ua tus qauv rau tus lej ntawm kev sib txuas:
C_n^m=n !:((n -m))!:m !
Kev sib xyaw yog cov kev xaiv uas tsis raug xaj, raws li, thiab txoj cai no siv rau lawv.
Nws tau dhau los ua qhov yooj yim los txiav txim siab cov qauv ntawm kev sib xyaw ua ke, tam sim no peb tuaj yeem txav mus rau qhov txhais lus classical ntawm qhov yuav tshwm sim. Cov lus no zoo li no:
P(A)=m: n.
Nyob hauv cov qauv no, m yog tus naj npawb ntawm cov xwm txheej zoo rau kev tshwm sim A, thiab n yog tus naj npawb ntawm txhua qhov sib npaug thiab qhov tshwm sim hauv theem pib.
Muaj ntau cov lus hais, tsab xov xwm yuav tsis npog tag nrho cov ntawm lawv, tab sis qhov tseem ceeb tshaj plaws ntawm lawv yuav tsum tau kov, xws li, piv txwv li, qhov tshwm sim ntawm cov xwm txheej:
P(A + B)=P(A) + P(B) - qhov kev xav no yog rau ntxiv cov xwm txheej tsis sib xws;
P(A + B)=P(A) + P(B) - P(AB) - thiab qhov no tsuas yog ntxiv rau qhov sib xws xwb.
Qhov tshwm sim ntawm kev tsim khoom:
P(A ⋅ B)=P(A) ⋅ P(B) - qhov kev xav no yog rau cov xwm txheej ywj pheej;
(P(A ⋅ B)=P(A) ⋅ P(B∣A); P(A ⋅ B)=P(A) ⋅ P(A∣B)) - thiab qhov no yog rau neeg quav yeeb.
Cov qauv txheej xwm xaus rau daim ntawv teev npe. Qhov kev xav txog qhov tshwm sim qhia peb txog Bayes 'theorem, uas zoo li qhov no:
P(H_m∣A)=(P(H_m)P(A∣H_m)): (∑_(k=1)^n P(H_k)P(A∣H_k)), m=1, …, n
In no formula, H1, H2, …, H yog tus ua kom tiav pawg ntawm kev xav.
Cia li tso tseg ntawm no, tom qab ntawd cov piv txwv ntawm kev siv cov qauv los daws cov teeb meem tshwj xeeb los ntawm kev xyaum yuav raug txiav txim siab.
Examples
Yog koj ua tib zoo kawm ib ntulej, nws tsis ua yam tsis muaj kev ua si thiab cov qauv kev daws teeb meem. Yog li yog txoj kev xav ntawm qhov yuav tshwm sim: cov xwm txheej, piv txwv ntawm no yog ib qho tseem ceeb tivthaiv uas lees paub kev suav lej.
Formula rau tus naj npawb ntawm permutations
Cia hais tias muaj peb caug daim npav hauv ib lub lawj ntawm daim npav, pib nrog lub ntsej muag tus nqi ib. Cov lus nug tom ntej. Muaj pes tsawg txoj hauv kev los teeb lub lawj kom daim npav nrog lub ntsej muag tus nqi ntawm ib thiab ob tsis nyob ib sab?
Txoj hauj lwm tau teem tseg, tam sim no cia peb mus daws nws. Ua ntej koj yuav tsum tau txiav txim seb tus naj npawb ntawm permutations ntawm peb caug lub ntsiab, rau qhov no peb coj cov qauv saum toj no, nws hloov tawm P_30=30!.
Raws li txoj cai no, peb yuav pom tias muaj pes tsawg txoj kev xaiv los tuav lub lawj nyob rau hauv ntau txoj kev, tab sis peb yuav tsum tau rho tawm ntawm lawv cov uas thawj thiab thib ob daim npav tom ntej. Txhawm rau ua qhov no, cia peb pib nrog qhov kev xaiv thaum thawj zaug saum toj thib ob. Nws hloov tawm tias thawj daim npav tuaj yeem nqa nees nkaum cuaj qhov chaw - los ntawm thawj zaug mus rau nees nkaum cuaj, thiab daim npav thib ob los ntawm qhov thib ob mus rau peb caug, nws hloov tawm nees nkaum cuaj qhov chaw rau ob daim npav. Nyob rau hauv lem, tus so yuav coj nees nkaum yim qhov chaw, thiab nyob rau hauv tej kev txiav txim. Ntawd yog, rau kev hloov pauv ntawm nees nkaum yim daim npav, muaj nees nkaum yim xaiv P_28=28!
Vim li ntawd, nws hloov tawm tias yog peb xav txog kev daws teeb meem thaum thawj daim npav dhau qhov thib ob, muaj 29 ⋅ 28 qhov muaj peev xwm ntxiv!=29!
Siv tib txoj kev, koj yuav tsum xam cov kev xaiv rov qab rau rooj plaub thaum thawj daim npav nyob hauv qab thib ob. Nws kuj hloov tawm 29 ⋅ 28!=29!
Nws ua raws tias muaj 2 ⋅ 29 kev xaiv ntxiv!, thaum muaj 30 txoj hauv kev yuav tsum tau tsim lub lawj! - 2 ⋅ 29!. Tsuas yog suav xwb.
30! xwb=29! ⋅ 30; 30!-2 ⋅29! xwb=29! ⋅ (30 - 2)=29! ⋅ 28
Tam sim no koj yuav tsum muab txhua tus lej los ntawm ib mus rau nees nkaum cuaj ua ke, thiab tom qab ntawd thaum kawg muab txhua yam los ntawm 28. Cov lus teb yog 2, 4757335 ⋅〖10〗^32
Kev daws qhov piv txwv. Formula for Placement Number
Nyob hauv qhov teeb meem no, koj yuav tsum nrhiav kom paub tias muaj pes tsawg txoj hauv kev muab kaum tsib ntim rau ntawm ib lub txee, tab sis nyob rau hauv cov xwm txheej uas muaj peb caug ntim tag nrho.
Qhov teeb meem no muaj kev daws yooj yim dua li yav dhau los. Siv cov qauv uas twb paub lawm, nws yog ib qho tsim nyog los xam tag nrho cov chaw ntawm peb caug ntim ntawm kaum tsib.
A_30^15=30 ⋅ 29 ⋅ 28⋅… ⋅ (30 - 15 + 1)=30 ⋅ 29 ⋅ 28 ⋅ … ⋅ 16=202 843 204 70 30
teb, raws, yuav yog 202 843 204 931 727 360 000.
Tam sim no cia peb ua txoj haujlwm nyuaj dua me ntsis. Koj yuav tsum paub seb muaj pes tsawg txoj hauv kev los npaj peb caug phau ntawv ntawm ob lub txee, yog tias tsuas yog kaum tsib phau ntawv tuaj yeem nyob ntawm ib lub txee.
Ua ntej pib txoj kev daws teeb meem, Kuv xav qhia meej tias qee qhov teeb meem daws tau ntau txoj hauv kev, yog li muaj ob txoj hauv kev no, tab sis tib lub mis siv ob qho tib si.
Hauv qhov teeb meem no, koj tuaj yeem nqa cov lus teb los ntawm qhov dhau los, vim tias muaj peb suav pes tsawg zaus koj tuaj yeem sau lub txee nrog kaum tsib phau ntawv rau-txawv. A_30^15=30 ⋅ 29 ⋅ 28 ⋅ … ⋅ (30 - 15 + 1)=30 ⋅ 29 ⋅ 28 ⋅ …⋅ 16.
Peb yuav suav lub txee thib ob uas siv cov qauv kev hloov pauv, vim tias kaum tsib phau ntawv tau muab tso rau hauv nws, thaum tsuas yog kaum tsib nyob. Siv tus qauv P_15=15!.
Nws hloov tawm tias tag nrho yuav yog A_30^15 ⋅ P_15 txoj hauv kev, tab sis, ntxiv rau, cov khoom ntawm txhua tus lej ntawm peb caug mus rau kaum rau yuav tsum tau muab faib los ntawm cov khoom ntawm cov lej ntawm ib mus rau kaum tsib, xws li qhov tshwm sim, cov khoom ntawm txhua tus lej ntawm ib mus rau peb caug, yog li cov lus teb yog 30!
Tab sis qhov teeb meem no tuaj yeem daws tau los ntawm txoj kev sib txawv - yooj yim dua. Ua li no, koj tuaj yeem xav txog tias muaj ib lub txee rau peb caug phau ntawv. Tag nrho cov ntawm lawv tau muab tso rau hauv lub dav hlau no, tab sis txij li qhov xwm txheej yuav tsum muaj ob lub rhawv, peb txiav ib qho ntev ib nrab, nws hloov tawm ob kaum tsib txhua. Los ntawm qhov no nws hloov tawm tias qhov kev xaiv tso kawm tuaj yeem yog P_30=30!.
Kev daws qhov piv txwv. Formula rau tus lej ua ke
Tam sim no peb yuav xav txog qhov sib txawv ntawm qhov teeb meem thib peb los ntawm combinatorics. Koj yuav tsum paub seb muaj pes tsawg txoj hauv kev los npaj kaum tsib phau ntawv, yog tias koj yuav tsum xaiv peb caug zoo ib yam nkaus.
Rau qhov kev daws, tau kawg, cov qauv rau cov lej ntawm kev sib txuas yuav raug siv. Los ntawm cov xwm txheej nws pom tseeb tias qhov kev txiav txim ntawm kaum tsib phau ntawv zoo tib yam tsis tseem ceeb. Yog li ntawd, thawj zaug koj yuav tsum nrhiav kom paub tag nrho cov kev sib txuas ntawm peb caug phau ntawv ntawm kaum tsib.
C_30^15=30 !: ((30-15)) ib.:15 ib!=155 117 520
Tseem. Siv cov qauv no, hauv lub sijhawm luv tshaj plaws nws ua taudaws qhov teeb meem no, cov lus teb, feem, yog 155 117 520.
Kev daws qhov piv txwv. Classic txhais ntawm qhov yuav tshwm sim
Nrog cov qauv saum toj no, koj tuaj yeem pom cov lus teb rau qhov teeb meem yooj yim. Tab sis nws yuav pab kom pom kev pom thiab ua raws li qhov kev ua.
Nws muab rau hauv qhov teeb meem tias muaj kaum lub pob zoo ib yam hauv lub urn. Ntawm no, plaub yog daj thiab rau yog xiav. Ib lub pob yog muab los ntawm lub qhov cub. Koj yuav tsum paub qhov tshwm sim ntawm qhov tau txais xiav.
Yuav daws tau qhov teeb meem, nws yog ib qho tsim nyog los xaiv kom tau lub pob xiav raws li kev tshwm sim A. Qhov kev paub no tuaj yeem muaj kaum qhov tshwm sim, uas, dhau los, yog theem pib thiab qhov tshwm sim sib npaug. Nyob rau tib lub sijhawm, tawm ntawm kaum, rau yog qhov zoo rau kev tshwm sim A. Peb daws raws li cov qauv:
P(A)=6: 10=0, 6
Kev siv cov qauv no, peb pom tias qhov tshwm sim ntawm qhov tau txais pob xiav yog 0.6.
Kev daws qhov piv txwv. Qhov tshwm sim ntawm qhov suav ntawm cov xwm txheej
Tam sim no qhov sib txawv yuav raug nthuav tawm, uas tau daws los ntawm kev siv cov qauv rau qhov tshwm sim ntawm cov lej ntawm cov xwm txheej. Yog li, nyob rau hauv cov xwm txheej uas muaj ob lub thawv, thawj muaj ib lub grey thiab tsib lub pob dawb, thiab lub thib ob muaj yim grey thiab plaub lub pob dawb. Yog li ntawd, ib qho ntawm lawv tau raug coj los ntawm thawj thiab thib ob lub thawv. Koj yuav tsum paub seb qhov twg yog lub caij nyoog uas cov khoom koj tau txais yuav grey thiab dawb.
Yuav daws qhov teeb meem no, koj yuav tsum sau cov xwm txheej.
- Yog li, A - nqa lub pob grey ntawm thawj lub thawv: P(A)=1/6.
- A' – nqa ib pob dawb los ntawm thawj lub thawv: P(A')=5/6.
- B – lub pob grey twb tau muab tshem tawm ntawm lub thawv thib ob: P(B)=2/3.
- B' – nqa lub pob grey ntawm lub thawv thib ob: P(B')=1/3.
Raws li qhov xwm txheej ntawm qhov teeb meem, ib qho tshwm sim yuav tsum tshwm sim: AB' lossis A'B. Siv cov formula, peb tau txais: P(AB')=1/18, P(A'B)=10/18.
Tam sim no qhov tshwm sim qhov sib npaug tau raug siv. Tom ntej no, txhawm rau paub cov lus teb, koj yuav tsum siv qhov sib npaug rau lawv qhov sib ntxiv:
P=P(AB' + A'B)=P(AB') + P(A'B)=11/18.
Qhov no yog li cas, siv cov qauv, koj tuaj yeem daws cov teeb meem zoo sib xws.
Result
Cov kab lus tau muab cov ntaub ntawv hais txog "Probability Theory", uas qhov tshwm sim ntawm qhov tshwm sim ua lub luag haujlwm tseem ceeb. Tau kawg, tsis yog txhua yam raug coj mus rau hauv tus account, tab sis, raws li cov lus qhia, ib tus tuaj yeem paub txog qhov kev ua lej no. Cov kev tshawb fawb hauv nqe lus nug tuaj yeem muaj txiaj ntsig tsis yog hauv kev ua haujlwm nkaus xwb, tab sis kuj tseem nyob hauv lub neej txhua hnub. Nrog nws cov kev pab, koj tuaj yeem xam qhov ua tau ntawm txhua qhov xwm txheej.
Cov ntawv nyeem kuj tau chwv cov hnub tseem ceeb hauv keeb kwm ntawm kev tsim cov kev xav tau zoo li kev tshawb fawb, thiab cov npe ntawm cov neeg uas nws ua haujlwm tau nqis peev hauv nws. Qhov no yog li cas tib neeg xav paub coj mus rau qhov tseeb tias tib neeg tau kawm los suav cov xwm txheej random. Ib zaug lawv tsuas xav txog nws, tab sis hnub no sawv daws twb paub txog nws. Thiab tsis muaj leej twg yuav hais dab tsi uas tos peb nyob rau yav tom ntej, dab tsi lwm qhov kev tshawb pom ci ntsa iab cuam tshuam nrog txoj kev xav hauv kev txiav txim siab yuav ua. Tab sis ib yam yog qhov tseeb - kev tshawb fawb tsis nyob twj ywm!
