Chiopsezo Chotha Ndalama Zonse
Kuthekera kotaya ndalama zanu zonse zosewerera musanafike pa cholinga.
Pa mabetcha a even-money, chiopsezo chotha ndalama zonse chimatsatira fomula ya gambler’s ruin. Ndi ndalama zosewerera za $500, mabetcha a $10 ndi cholinga cha $600, mumafika pa cholinga pafupifupi 57% ya nthawi ndipo mumatha ndalama nthawi zotsala.
Funso
Muli ndi ndalama zosewererandalamaNdalama zosewereraNdalama zonse zimene mwazisiyira masewera. Kusamalira ndalama zosewerera ndiko kusankha kuchuluka kwake komwe mungaike pachiopsezo pa betcha lililonse ndi pa gawo lililonse la masewera.Werengani yonse → za ma unitndalamaUnitNdalama zobetcha zoyambira zimene dongosolo kapena wosewera amagwiritsa ntchito. Maprogression amafotokozedwa mu ma unit kuti agwire ntchito pa tebulo lililonse.Werengani yonse → a ndipo mukufuna kufika pa ma unit a + b, mukubetcha unit imodzi nthawi iliyonse pa betcha la even-moneybetchaBetcha la even-moneyZofiira/zakuda, osakhala awiriawiri/awiriawiri, okwera/otsika. Manambala 18, amalipira 1:1.Werengani yonse → lokhala ndi kuthekeramasamuKuthekeraMwayi wa zotsatira, kuchokera 0 mpaka 1 kapena 0% mpaka 100%. Nambala imodzi: 1/37 = 2.70%.Werengani yonse → kopambana p = 18/37 ndi kuthekera koluza q = 19/37. Kodi mwayi woti mufike pa cholinga musanafike pa ziro ndi wotani? Yankho linapezedwa ndi Pascal, Fermat ndi Huygens m’ma 1650 ndipo ndi limodzi mwa zotsatira zakale kwambiri mu kuthekera.
Fomula ya gambler’s ruin
P(kufika pa cholinga) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ). Chiopsezo chotha ndalama zonsemasamuChiopsezo chotha ndalama zonseKuthekera kotaya ndalama zonse zosewerera musanafike cholinga kapena chiwerengero cha kuzungulira.Werengani yonse → ndi chimodzi kuchotsa chimenecho. Ndi q/p = 19/18 ≈ 1.0556, chiyerekezo chimakula mofulumira kwambiri (exponentially) ndi chiwerengero cha ma unit, choncho fomulayo imalanga ma unit ang’onoang’ono koopsa: mabetcha ochuluka akafunika kuti mufike pa cholinga, nthawi zambiri msonkho wa 2.70% umalipidwa panjira.
Kuziwerenga
Ma unit ang’onoang’ono amapweteka. Kuyesa kuwirikiza $500 ndi mabetcha a $10 kumapambana 6% ya nthawi; ndi betcha limodzi la $500 kumapambana 48.6% ya nthawi. Kusewera molimba mtima (bold play), mabetcha ochepa ndi akuluakulu, kumakweza mwayi wofika pa cholinga m’masewera a mtengo woyembekezeka woipa, chifukwa kumachepetsa nthawi zimene phindu la kasino limagwira ntchito. Kumakwezanso kusinthasinthamasamuKusinthasinthaMuyeso wa kufalikira kwa zotsatira kuzungulira mtengo woyembekezeka. Kwakukulu pa straight up, kochepa pa mabetcha a even-money.Werengani yonse → kwa zimene zimakuchitikirani, ndipo kumathetsa madzulo msanga. Mfundo imeneyi imatchedwa “bold play is optimal” ndipo inatsimikiziridwa ndi Dubins ndi Savage mu 1965.
Kumene vutoli linayambira
Funsoli ndi lakale kuposa roulette. Mu 1656 Pascal anapatsa Fermat chithunzithunzi: osewera awiri amaponya madayisi atatu, aliyense akuyamba ndi zowerengera 12; kuponya 11 kumasuntha chowerengera mbali imodzi, kuponya 14 mbali ina. Onse awiri analithetsa, ndipo Huygens, amene anamva za ilo kudzera m’kalata ya katswiri wa masamu Pierre de Carcavi, analisindikiza mu 1657 ngati vuto lachisanu la buku loyamba la kuthekera, ndi yankho koma popanda umboni. Khumi ndi chimodzi chingaponyedwe m’njira 27 ndipo khumi ndi zinayi m’njira 15 zokha, choncho chowerengera chilichonse chimasunthira kumbali ya wosewera woyamba ka 9 pa 14, ndipo fomula ili pamwambayi, ndi zowerengera 12 aliyense, imamupatsa mwayi wopambana zonse wokwera pafupifupi kuwirikiza 1,157 kuposa kutaya zonse. Phindu la kasino la 2.70% ndi laling’ono kwambiri, koma fomula yomweyo imaonetsa zimene limachita pa mabetcha ambiri.
Betcha lenileni lolimba mtima
Mu April 2004 Ashley Revell, mnyamata wa ku England, anagulitsa pafupifupi chilichonse chimene anali nacho, kuphatikizapo zovala zake, ndipo anatenga $135,300 kupita nazo ku Plaza ku Las Vegas. Pambuyo pa mabetcha ang’onoang’ono anaika ndalama zonse pa zofiirabetchaZofiiraBetcha la even-money pa manambala 18 ofiira. Mu Chifalansa: rouge.Werengani yonse →, kamodzi. MpirazipangizoMpiraMpira waung’ono umene umaponyedwa mu njira motsutsana ndi rotor. Mipira yamakono ndi ya ivorine kapena Teflon, 18–21 mm.Werengani yonse → unagwera pa 7 yofiira ndipo anatuluka ndi $270,600. Pa gudumu la zirogudumuZiroKabowo kamodzi kobiriwira kamene betcha la even-money, dazeni kapena mzati siliphimba; gwero la phindu la kasino. Imatchedwanso: ziro imodzi, 0.Werengani yonse → ziwiri kuzungulira kumodzi kumeneko kunali mwayi wa 47.37% ndipo kunamudula $7,121 pa avareji, phindu la kasino la 5.26% pa betcha limodzi. Atagawa m’mabetcha a $10, cholinga chomwecho chikanakhala pafupifupi chosatheka. Kusewera molimba mtima sikupangitsa betcha kukhala labwino; kumapangitsa kukhala lalifupi momwe kungathekere.
Chiopsezo chotha ndalama zonse popanda cholinga
Ngati mungosewera mpaka ndalama zitatha kapena kuleka pambuyo pa kuzungulira N, funso limakhala “kodi mwayi woti ndalama zanga zosewerera zifike pa ziro mkati mwa kuzungulira N ndi wotani?”. Palibe fomula yotsekedwa ya N iliyonse koma zoyerekezera za patsamba lino zimawerengera: ndalama zosewerera za $500 pa $10 pa zofiira pafupifupi sizitha konse pa kuzungulira 100 (pansi pa 1%), zimatha pafupifupi 9% ya nthawi pa kuzungulira 500, ndipo pafupifupi 95% ya nthawi pa 5,000. Popanda lamulo loleka kuthekera kumafika ku 100%, umene ndi mtundu wa nthawi yayitali wa phindu la kasinomasamuPhindu la kasinoGawo la ndalama zobetcha zilizonse limene kasino imasunga pa avareji: 2.70% mu roulette ya ku Ulaya.Werengani yonse →.
- p10 −$500
- p25 −$500
- pakati −$500
- p75 +$0
- p90 +$350
Kuyandikira pa Nthawi Yayitali: Lamulo la Manambala Akuluakulu mu Roulette
Nthawi yotani, ndi patsogolo motani
Zotsatira zina ziwiri zakale zimayankha mafunso amene osewera amafunsadi. Popanda cholinga, mabetcha a even-money osasintha amakhalitsa pa avareji kuzungulira 37 pa unit iliyonse ya ndalama zosewerera pa gudumu la ku Ulaya ndi 19 pa la ku America, chifukwa kuzungulira kulikonse kumatenga 1/37 ya unit pa avareji (1/19 pa gudumu la ku America): ma unit 100 amakhalitsa kuzungulira 3,700 pa avareji, ngakhale theka la magawo otere limatha mkati mwa 3,134 (onani Kutalika kwa Gawo la Masewera ndi Mtengo Wake). Ndipo ngakhale ndalama zosewerera zikhale zochuluka bwanji, mwayi pa gudumu la ku Ulaya woti mukhale patsogolo ndi ma unit k nthawi ina iliyonse ndi (18/19)ᵏ. Zimenezo ndi 95% pa unit imodzi ndi 58% pa khumi, koma palibe ndalama zosewerera, ngakhale zochuluka bwanji, zimene zimapereka mwayi wofanana woti mukhale patsogolo ndi ma unit 13; pa gudumu la ku America mwayi ndi (9/10)ᵏ ndipo umatsika pansi pa theka pa ma unit 7.
Kutalika kwa Gawo la Masewera ndi Mtengo WakeGudumu ndi Malamulo a Roulette ya ku America
Ndalama zosewerera za mwayi wa theka ndi theka
Tembenuzani funsolo: kodi muyenera kubweretsa zingati kuti mukhale ndi mwayi wa theka ndi theka pang’ono wopambana ndalama zinazake musanataye zonse? Pa kupambana kwazing’ono, osati zambiri. Pa kwakukulu ndalama zosewerera zofunikira zimakwera kwambiri, ndipo kupitirira malo ena palibe ndalama zosewerera zokwanira, chifukwa ngakhale zopanda malire zimapereka mwayi wochepera theka ndi theka woti mufike patsogolo motero. Taylor (2021) amanena mwanjira ina: pa gudumu la ku America, kuti mukhale ndi mwayi wa theka ndi theka wofika pa ma unit 100 muyenera kuyamba ndi 94, mukuika pangozi 94 kuti mupambane 6. Pa gudumu la ku Ulaya mukanafunika 88.
Phunziro lothandiza
Sankhani zimene mukuyesa kuchita. Ngati cholinga ndi kupambana kwinakwake, betchani zazikulu ndi kochepa. Ngati cholinga ndi madzulo aatali, betchani zazing’ono ndipo vomerezani kuti phindu la kasino lidzatenga 2.70% yake ya ndalama zonse zobetchedwandalamaNdalama zonse zobetchedwaNdalama zonse zobetchedwa mu gawo la masewera kapena nthawi inayake, kuphatikizapo ndalama zopambana zobetchedwanso. Zimatchedwanso: handle, action.Werengani yonse → zochuluka. Ngati cholinga ndi zonse ziwiri, china chake chiyenera kusiyidwa, ndipo kawirikawiri ndi cholinga.
Fomula
Mafunso
MartingalenjiraMartingaleProgression yowirikiza betcha pambuyo poluza kulikonse. Onani gawo la Njira kuti mudziwe zimene imachita ndi zimene siimachita.Werengani yonse → imachichepetsa pa magawo afupiafupi ndipo imachikweza kwambiri pa aatali. Kubetcha mofanananjiraKubetcha mofananaKubetcha ndalama zofanana pa kuzungulira kulikonse. Muyeso umene progression iliyonse imayerekezeredwa nawo.Werengani yonse → ndi unit yaikulu ndiyo njira yowona yochichepetsera.
Fomula ili pamwambayi ndi ya even money. Pa mabetcha a kusinthasintha kokwera kutha ndalama n’kotheka kwambiri pa ndalama zobetcha zazing’ono ndipo cholinga chimafikiridwa modumphadumpha; gwiritsani ntchito chowerengera.
Sitepe iliyonse yolowera ku cholinga ili ndi mwayi wochepa pang’ono kuposa sitepe iliyonse yolowera ku kutha ndalama; chiyerekezo chimachulukana pa chiwerengero cha masitepe.
Masamu Osangalatsa
Yesani nokha choyamba. Yankho lililonse labisika; litsegulireni mukakonzeka.
Muli ndi $100 ndipo mukufuna $1,000, mukubetcha pa zofiirabetchaZofiiraBetcha la even-money pa manambala 18 ofiira. Mu Chifalansa: rouge.Werengani yonse → zokha. Dongosolo loyamba: betchani $1 pa nthawi. Dongosolo lachiwiri: betchani zonse zimene muli nazo mpaka mutapitirira $500, kenako zokwanira kungofika $1,000. Ndi dongosolo liti labwino, ndipo ndi zingati?
Onetsani yankho
Yankho Dongosolo lachiwiri, kutali kwambiri. Limafika $1,000 pafupifupi 9.0% ya nthawi, pafupi ndi 10% yachilungamo. Dongosolo loyamba limakwaniritsidwa pafupifupi ka 7 pa 10²².
M’masewera achilungamomasamuMasewera achilungamoMasewera amene palibe mbali yokhala ndi mwayi: betcha lililonse lili ndi mtengo woyembekezeka wa ziro.Werengani yonse → mukanafika $1,000 kuchokera pa $100 kamodzi pa kakhumi chilichonse chimene mungachite, chifukwa mtengo wa avareji wa ndalama zanu umakhalabe $100.
Phindu la kasino limatenga 2.70% ya betcha lililonse, choncho mukabetcha zochepa zonse pamodzi, mumakhala pafupi kwambiri ndi kamodzi pa kakhumi. Dongosolo lachiwiri limathetsa nkhaniyi m’mabetcha ochepa: $100 imakhala $200, $400, $800, kenako $200 ikukwanira. Limakwaniritsidwa pafupifupi 9.0% ya nthawi. Dongosolo loyamba limatenga mabetcha masauzande ambiri ulendowo usanathe, ndipo phindu la kasino, laling’ono pa lililonse la iwo, limasonkhana kufika pa kutha ndalama kotsala pang’ono kutsimikizika.
Izi ndi zotsatira za Dubins–Savage m’njira yake yosavuta kwambiri (Epstein, 2013): m’masewera amene amakomera kasino, betchani zochuluka monga mukufunira, kangapo kochepa monga momwe mungathere.
Muli ndi $500 ndipo mukufuna $1,000, kusiya mukangozipeza. Mukusankha mtundu umodzi wa betcha ndipo mukuubwereza mpaka utapambana kamodzi kapena ndalama zitatha. Ndi mtundu uti umene umapereka mwayi wabwino kwambiri?
Onetsani yankho
Yankho Betcha limodzi la $500 pa zofiira: 48.6%. Dongosolo lililonse logawa ndalama m’mabetcha ang’onoang’ono limachita moipirapo.
Sankhani ndalama zobetcha zilizonse kuti kupambana kamodzi kukwanire: $30 pa straight upbetchaStraight upNambala imodzi, amalipira 35:1.Werengani yonse → imapambana $1,050, $60 pa splitbetchaSplitManambala awiri oyandikana, amalipira 17:1.Werengani yonse → imapambana $1,020, ndi zina zotero. Kenako werengani kuthekera kwakuti betcha limodzi kapena kuposerapo lipambane ndalama zisanathe. Ma straight up khumi ndi asanu ndi limodzi a $30 onse amaphonya ndi kuthekera kwa (36/37)¹⁶ = 64.5%, choncho amafika $1,000 35.5% ya nthawi.
Tebulo likusonyeza betcha lililonse. Zofiira zimachita bwino kwambiri chifukwa zimafika kumeneko ndi betcha limodzi, choncho phindu la kasino limalipidwa kamodzi, pa $500. Six linebetchaSix lineManambala asanu ndi limodzi m’mizere iwiri yoyandikana, amalipira 5:1. Mu Chifalansa: sixain.Werengani yonse → imabwera yachiwiri chifukwa imafunika atatu okha. Betcha limodzi la dazenibetchaDazeni1–12, 13–24 kapena 25–36, amalipira 2:1.Werengani yonse → ndiloipa kwambiri, mwayi wa 12/37; ngakhale litaseweredwa m’ma unitndalamaUnitNdalama zobetcha zoyambira zimene dongosolo kapena wosewera amagwiritsa ntchito. Maprogression amafotokozedwa mu ma unit kuti agwire ntchito pa tebulo lililonse.Werengani yonse → a $250 mpaka mutakhala ndi $1,000 kapena kuposerapo, dazeni limafika kumeneko 46.4% yokha ya nthawi.
Bollman (2024) amapanga tebulo lomwelo la gudumu la ku America. Mzere wake wa dazeni uli ndi cholakwika: uyenera kunena 12/38 = 31.6%.
betchandalama zobetcha × chiwerengero cha mabetchakuthekera kofika $1,000Straight up$30 × 1635.5%Split$60 × 835.9%Street$90 × 534.5%Kona$125 × 436.7%Six line$166, $166, $16841.2%Dazeni$500 × 132.4%Zofiira$500 × 148.6%Mukubetcha unit imodzi pa zofiira, kuzungulira pambuyo pa kuzungulira, ndi ndalama zosewererandalamaNdalama zosewereraNdalama zonse zimene mwazisiyira masewera. Kusamalira ndalama zosewerera ndiko kusankha kuchuluka kwake komwe mungaike pachiopsezo pa betcha lililonse ndi pa gawo lililonse la masewera.Werengani yonse → zimene sizitha konse. Pambuyo pa kuzungulira koyamba muli patsogolo ndi unit imodzi kapena m’mbuyo ndi unit imodzi. Kodi kuthekera kwakuti mudzakhalanso pa ziro ndendende ndi kotani?
Onetsani yankho
Yankho 36 pa 37, 97.3%. Pa gudumu la ku America ndi 36 pa 38, 94.7%.
Ngati kuzungulira koyamba kwapambana (ka 18 pa 37), muli patsogolo ndi unit imodzi, ndipo phindu la kasino limakokera pansi gawo la masewera lililonse losatha, choncho posachedwa kapena patapita nthawi mumadutsanso pa ziro: zimenezo n’zotsimikizika. Ngati kwaluza (ka 19 pa 37), muyenera kukwera unit imodzi, ndipo kuthekera kokhala patsogolo ndi unit imodzi nthawi ina iliyonse ndi 18/19 (onani tebulo pamwambapa).
Pamodzi: 18/37 + 19/37 × 18/19 = 18/37 + 18/37 = 36/37.
Mwachizolowezi kuthekera kumeneku ndi 1 kuchotsa kusiyana pakati pa kuthekera kopambana ndi kuluza kuzungulira: 1 − 1/37 pa gudumu la ku Ulaya, 1 − 2/38 pa la ku America. M’masewera opanda zirogudumuZiroKabowo kamodzi kobiriwira kamene betcha la even-money, dazeni kapena mzati siliphimba; gwero la phindu la kasino. Imatchedwanso: ziro imodzi, 0.Werengani yonse → kusiyana kulibe ndipo kuthekera ndi 1: chiwerengero nthawi zonse chimabwerera pa ziro. Epstein (2013) akupereka lamulo lonse.
Chithunzithunzi cha Pascal cha 1656: osewera awiri akuponya madayisi atatu, aliyense akuyamba ndi zowerengera 12. Nthawi iliyonse 11 ikatuluka chowerengera chimodzi chimapita kwa wosewera woyamba, 14 ikatuluka chimodzi chimapita kwa wachiwiri; kuponya kwina sikuwerengedwa. Kodi wosewera woyamba angapambane zonse 24 ndi kuthekera kotani? Ndipo ngati m’malo mwake mubetcha unit 1 pa zofiira pa kuzungulira kulikonse polimbana ndi mnzanu yemwe ndi banki, ma unit 12 aliyense, mungapambane zonse 24 ndi kuthekera kotani?
Onetsani yankho
Yankho Wosewera woyamba amapambana pafupifupi ka 1,157 pa 1,158. Inu, pa zofiira, mumapambana ma unit onse 24 pafupifupi 34% yokha ya nthawi.
Madayisi atatu amapanga 11 m’njira 27 ndi 14 m’njira 15, choncho chowerengera chilichonse chimene chikusuntha chimapita kwa wosewera woyamba ka 27 pa 42, ka 9 pa 14. Mbali iliyonse ikuyamba pa 12, kuthekera kotenga zonse ndi 1/(1 + r¹²), pamene r ndi chiyerekezo cha kuthekera kwa mbali ina ndi kwanu.
Kwa wosewera woyamba r = 5/9, ndipo (5/9)¹² ≈ 0.00086, choncho kuthekera ndi pafupifupi 0.99914: mwayi wa 9¹² ku 5¹², kapena 282,429,536,481 ku 244,140,625, pafupifupi 1,157 ku 1. Huygens anasindikiza manambala amenewo ndendende mu 1657.
Pa zofiira, r = 19/18 chifukwa cha ziro, ndipo (19/18)¹² ≈ 1.913: kuthekera kwanu ndi 1/2.913 ≈ 34.3%. Ndalama yoponya ikanapereka 50%. Phindu la kasino la 2.70% pa betcha lililonse limasonkhana, pa mabetcha ambiri amene amafunika, kufika pa kusiyana kwa mapointi 16.
Nthabwala yopumitsa
Kusewera molimba mtima ndiko kwabwino kwambiri. Ndiponso kumatha msanga.
Magwero
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed.
- Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed.
- Carcavi, P. de. Kalata kwa Huygens, 28 September 1656, yokhudza vuto limene Pascal anapatsa Fermat.
- Huygens, C. De Ratiociniis in Ludo Aleae.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Reuters. “Briton bets it all on roulette spin, and wins.”
- Las Vegas Sun. “Man wagers life’s worth.”