Qhov sib txawv txheem hauv roulette
Qhov sib txawv txheem hauv roulette, lub hauv paus plaub fab (square root) ntawm kev hloov pauvlejKev hloov pauv (variance)Ib qho kev ntsuas tias cov qhov tshwm sim tawg dav npaum li cas ncig tus nqi xav tias yuav tau. Siab rau twv ib tus lej, qis rau kev twv them sib npaug.Nyeem tag nrho →, yog qhov deb ib txwm ntawm ib qho tshwm sim thiab qhov xav tias yuav tau.
Rau txhua $1 uas twv, σ yog 5.84 rau twv ib tus lej thiab 1.00 rau liab. Muab sib khoo nrog cov nyiaj twv thiab nrog lub hauv paus plaub fab ntawm tus naj npawb zaus kiv kom tau rau ib zaus ua si.
Txoj cai yooj yim
Kwv yees li ob feem peb ntawm cov zaus ua si poob rau hauv ib σ ntawm qhov xav tias yuav tau thiab 95% poob rau hauv ob npaug ntawd. Rau 100 zaus kiv twv liabkev twvLiabKev twv them sib npaug rau 18 tus lej liab. Lus Fabkis: rouge.Nyeem tag nrho → ntawm $10 (xav tias yuav tau −$27, σ = $100) feem ntau ntawm cov zaus ua si poob rau nruab nrab −$127 thiab +$73, thiab yuav luag tag nrho poob rau nruab nrab −$227 thiab +$173. Ib qho tshwm sim sab nraud ntawm qhov ntawd tsis yog pov thawj ntawm dab tsi li tsuas yog tias ib zaus ua si hauv nees nkaum zaus ua li ntawd.
Vim li cas thiaj yog √N
Nyiaj yuav poob raws nruab nrablejNyiaj yuav poob raws nruab nrabNqi xav tias yuav tau muab sib npaug rau tag nrho cov nyiaj twv; cov nyiaj nruab nrab uas poob hauv ib zaug ua si lossis ib lub sij hawm.Nyeem tag nrho → loj zuj zus raws N; qhov nthuav tawm loj zuj zus raws √N. Thaum N me, qhov nthuav tawm loj dua thiab cov zaus ua si zoo li pov npib uas qaij me ntsis. Thaum N loj, qhov poob khiav dhau qhov nthuav tawm. Qhov hla, uas nyiaj yuav poob raws nruab nrab sib npaug ib qhov sib txawv txheemlejQhov sib txawv txheemCag plaub fab ntawm kev hloov pauv (variance); qhov deb ib txwm ntawm ib qhov tshwm sim ntawm tus nqi xav tias yuav tau.Nyeem tag nrho →, nyob ntawm N = (σ/feem tsev twv txiaj)² = (1/0.027)² ≈ 1,370 zaus kiv rau them sib npaug thiab (5.84/0.027)² ≈ 46,800 rau twv ib tus lejkev twvTwv ib tus lejIb tus lej xwb, them 35:1.Nyeem tag nrho →. Qis dua cov lej ntawd, hmoov txiav txim; siab dua lawv, feem tsev twv txiaj txiav txim.
Siv nws
Yuav txiav txim seb ib zaus ua si puas txawv txawv, suav (qhov tshwm sim − EV)/σ. Ib tus neeg twv liab $10 uas xaus 100 zaus kiv ntawm +$150 nyob ntawm (150 + 27)/100 = 1.77σ, ib qho tshwm sim ib zaug hauv 25: muaj hmoov, tsis yog qhov txawv. Ib tus neeg twv ib tus lej ntawm tib +$150 nyob ntawm 0.3σ: ib hmo li qub. Tib qho tshwm sim ua duas las txhais txawv rau cov kev twv txawv.
Qhov yuam kev txheem ntawm ib feem
Tib qho kev suav no qhia koj tias yuav tsum muaj pes tsawg zaus kiv thiaj pom feem tsev twv txiaj hauv cov ntaub ntawv. Qhov yuam kev txheem ntawm ib feem yeej uas pom yog √(p(1 − p)/N). Rau liab nrog N = 1,000: √(0.4865 × 0.5135 / 1000) ≈ 1.6%. Qhov sib txawv ntawm ib lub npib pov ncaj ncees thiab liab yog 1.35%, yog li tom qab 1,000 zaus kiv koj tseem qhia tsis tau tseeb tias lawv txawv. Tom qab 10,000 qhov yuam kev txheem yog 0.5% thiab koj qhia tau. Qhov no kuj yog vim li cas kev nrhiav lub log tsis ncaj thiaj yuav tsum muaj ntau zaus kiv heev.
Ceev faj txog kev qaij
Cov qhov tshwm sim ntawm twv ib tus lej nyob deb ntawm kev faib tawm normal hauv ib zaus ua si luv: feem ntau ntawm cov zaus ua si yog poob −$10 txhua zaus kiv nrog qee zaum dhia +$350. Txoj cai ±1σ piav qhov nthuav tawm yog tab sis piav tus duab tsis zoo kom txog thaum N muaj ntau pua. Rau kev twv them sib npaugkev twvKev twv them sib npaugLiab/dub, khib/txawm, siab/qis. 18 tus lej, them 1:1.Nyeem tag nrho → kev kwv yees normal zoo txij li kwv yees 30 zaus kiv.
Qauv lej
Cov lus nug
Nws yog pov thawj tias koj tau ua si kwv yees li 40 zaus, lossis koj yog tus muaj hmoov ib zaug hauv plaub caug. Ua dua.
Nws nce nqis tsawg zuj zus: qhov yuam kev txheem ntawm qhov nruab nrab yog σ/√N. Thaum N = 10,000 nws yog 1% ntawm cov nyiaj twv.
Lom zem nrog lej
Sim ua koj tus kheej ua ntej. Txhua lo lus teb raug quav cia; qhib nws thaum koj npaj txhij.
Dhau 100 zaus kiv ntawm lub log roulette Europe, ib chav twvnyiajChav twvNyiaj twv pib uas ib txheej txheem los yog ib tus neeg ua si siv. Kev nce nyiaj twv raug piav ua chav twv kom siv tau ntawm txhua lub rooj.Nyeem tag nrho → rau liabkev twvLiabKev twv them sib npaug rau 18 tus lej liab. Lus Fabkis: rouge.Nyeem tag nrho → tau nyiaj nrog tsawg kawg 51 zaug yeej, thiab ib chav twv rau twv ib tus lejkev twvTwv ib tus lejIb tus lej xwb, them 35:1.Nyeem tag nrho → tau nyiaj yog nws tus lej tawm tsawg kawg 3 zaug. Siv tus kab tswb (bell curve) kwv yees ob qho feem, ces piv lawv nrog cov lus teb tseeb.
Qhia lus teb
Lus teb Liab: tus kab tswb muab 35.6% thiab cov lus teb tseeb yog 35.5%. Twv ib tus lej: tus kab tswb muab 55.0% thiab cov lus teb tseeb yog 50.9%. Txoj kev luv siv tau rau liab thiab plam rau ib tus lej.
Liab: cov zaug yeej nruab nrab yog 100 × 18/37 = 48.6, nrog kev sib kis √(100 × 18/37 × 19/37) = 5.0. Tau nyiaj txhais tau 51 lossis ntau dua; qhov cim ib nrab, 50.5, yog 0.37 kev sib kis siab dua qhov nruab nrab, thiab tus kab tswb muab 35.6% siab dua ntawd. Muab cov feem tseeb ntawm 51, 52 … 100 zaug yeej sib ntxiv muab 35.5%.
Twv ib tus lej: cov zaug raug nruab nrab yog 100/37 = 2.70, nrog kev sib kis √(100 × 1/37 × 36/37) = 1.62. Peb zaug raug lossis ntau dua txhais tau siab dua 2.5, 0.12 kev sib kis qis dua qhov nruab nrab: 55.0% los ntawm tus kab tswb. Raws tseeb, nws yog 1 rho cov feem ntawm 0, 1 thiab 2 zaug raug, 6.5% + 17.9% + 24.7%, uas tshuav 50.9%.
Tus kab tswb xav tau ntau qhov tshwm sim ob sab ntawm qhov nruab nrab. Liab muaj lawv tom qab 100 zaus kiv; ib tus lej uas raug tsawg dua peb zaug raws qhov nruab nrab tsis muaj. Tus kab txawm muab 8.7% rau “tsis raug li”, qhov uas feem tseeb yog 6.5%. Taylor (2021) muab binomial thiab tus kab tswb tso ib sab ib sab ib yam li ntawd.
Lus lom zem
Ib zaus ua si hauv nees nkaum zaus nyob sab nraud ob sigma. Ib txwm muaj ib tus nyob ntawm tus tw.
Qhov chaw los
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed.