Hevi long Lusim Olgeta Mani
Sans bilong lusim olgeta mani bilong pilai bilong yu bipo long yu kamap long mak yu laik kisim.
Long ol bet wan-long-wan, hevi long lusim olgeta mani i bihainim fomula bilong gambler’s ruin. Wantaim $500 mani bilong pilai, ol bet bilong $10 na mak bilong $600, yu kamap long mak klostu 57% bilong ol taim, na long ol narapela taim yu lusim olgeta.
Askim
Yu gat a yunitmaniYunitMani bilong bet bilong stat we wanpela sistem o wanpela pleia i wok long en. Ol i stori long ol progresen long yunit, olsem na ol i ken wok long wanem tebol.Ritim olgeta → bilong mani bilong pilaimaniMani bilong pilaiOlgeta mani yu putim long sait bilong pilai. Lukautim mani bilong pilai em i makim hamas long en yu redi long lusim long wanpela bet na long wanpela sesen.Ritim olgeta → na yu laik kamap long a + b yunit. Yu putim wanpela yunit long wanpela taim long bet wan-long-wanbetBet wan-long-wanRet/blak, od/iven, antap/daunbilo. 18 namba, pe em i 1:1.Ritim olgeta → wantaim sansmatematikSansSans bilong wanpela resal, long 0 i go 1 o 0% i go 100%. Wanpela namba: 1/37 = 2.70%.Ritim olgeta → bilong win p = 18/37 na sans bilong lus q = 19/37. Sans olsem yu kamap long mak bipo long yu kamap long siro em i wanem? Pascal, Fermat na Huygens i painim bekim long ol yia 1650, na em i wanpela bilong ol bekim i lapun moa long matematik bilong sans.
Fomula bilong gambler’s ruin
P(kamap long mak) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ). Hevi long lusim olgeta manimatematikHevi long lusim olgeta maniSans long lusim olgeta mani bilong pilai paslain long yu kamap long mak yu laik kisim o long namba bilong ol tanim.Ritim olgeta → em i wan maines dispela. Wantaim q/p = 19/18 ≈ 1.0556, skel i go antap kwik tru wantaim hamas yunit, olsem na fomula i pairapim strong ol liklik yunit: taim yu nidim planti bet moa long kamap long mak, yu baim takis bilong 2.70% planti taim moa long rot.
Olsem wanem long ritim
Ol liklik yunit i bagarapim yu. Sapos yu traim dabelim $500 wantaim ol bet bilong $10, yu win 6% bilong ol taim; wantaim wanpela bet bilong $500, yu win 48.6% bilong ol taim. Pilai strong, liklik bet tasol na bikpela, i mekim sans bilong kamap long mak i go antap moa long pilai we avarej i nogut, long wanem em i daunim hamas taim edj i wok. Tasol em i mekim variansmatematikVariansMak bilong hamas ol resal i kalap nabaut long ekspektet velyu. I bikpela long bet long wanpela namba, i liklik long bet wan-long-wan.Ritim olgeta → bilong samting i kamap long yu i go antap moa, na nait bilong yu i pinis kwik. Ol i kolim dispela tok “pilai strong em i gutpela moa”, na Dubins na Savage i soim i tru long 1965.
Hevi i stat we
Dispela askim i lapun moa long rulet. Long 1656 Pascal i givim wanpela pasol long Fermat: tupela pleia i tromoi tripela dais, na wan wan i stat wantaim 12 kaunta. Sapos tromoi i kamap 11, wanpela kaunta i muv long wanpela sait; sapos 14, long narapela sait. Tupela i painim bekim. Huygens i harim long wanpela pas bilong saveman bilong matematik Pierre de Carcavi, na em i prinim long 1657 olsem namba faiv hevi long namba wan buk long sans, wantaim bekim tasol i no gat tok i soim. Yu inap tromoi ilevan long 27 rot na fotin long 15 rot tasol, olsem na olgeta kaunta i muv long sait bilong namba wan pleia 9 taim long 14. Na fomula antap, wantaim 12 kaunta long wan wan, i mekim em i gat sans klostu 1,157 taim moa long winim olgeta samting long lusim olgeta samting. Edj bilong 2.70% i liklik moa tru, tasol wankain fomula i soim wanem samting em i mekim long planti bet.
Wanpela bet strong tru
Long Epril 2004, Ashley Revell, wanpela yangpela man bilong Inglan, i salim klostu olgeta samting bilong em, wantaim ol klos bilong em, na em i kisim $135,300 i go long Plaza long Las Vegas. Bihain long sampela liklik bet, em i putim olgeta mani long retbetRetBet wan-long-wan long 18 namba ret. Tok Frans: rouge.Ritim olgeta →, wanpela taim. BaltulBalLiklik bal dila i sutim long rot i go long narapela sait long rota. Ol nupela bal i ivorine o Teflon, 18–21 mm.Ritim olgeta → i pundaun long ret 7 na em i go ausait wantaim $270,600. Long wil i gat dabel sirowilSiroWanpela grinpela hul we i no gat bet wan-long-wan, dosen o kolam i karamapim; em i as bilong edj bilong kasino. Narapela nem: wanpela siro, 0.Ritim olgeta →, dispela wanpela tanim i gat sans bilong 47.37% na em i baim $7,121 long avarej, edj bilong 5.26% long wanpela bet. Sapos em i brukim long ol bet bilong $10, wankain mak bai i klostu i no gat sans olgeta. Pilai strong i no mekim bet i kamap gutpela; em i mekim bet i sot tru tasol.
Hevi long lusim olgeta mani taim i no gat mak
Sapos yu pilai tasol inap yu lusim olgeta o yu stop bihain long N tanim, askim i kamap “sans olsem mani bilong pilai bilong mi i kamap siro insait long N tanim em i wanem?”. I no gat wanpela isi fomula long olgeta N, tasol ol simulata long dispela sait i kaunim: $500 mani bilong pilai wantaim $10 long ret klostu i no save lus olgeta long 100 tanim (aninit long 1%), em i lus olgeta klostu 9% bilong ol taim long 500 tanim, na klostu 95% bilong ol taim long 5,000. Sapos i no gat lo bilong stop, sans i go klostu long 100%, na dispela em i pasin bilong edj bilong kasinomatematikEdj bilong kasinoHap bilong olgeta mani bilong bet kasino i holim long avarej: 2.70% long rulet bilong Yurop.Ritim olgeta → long longpela taim.
- p10 −$500
- p25 −$500
- namba long namel −$500
- p75 +$0
- p90 +$350
I Kam Klostu long Longpela Taim: Lo bilong Planti Namba long Rulet
Hamas taim, na hamas antap
Tupela narapela bekim i lapun i bekim ol askim ol pleia i save askim tru. Sapos i no gat mak, ol bet wan-long-wan wankain oltaim i stap long avarej 37 tanim long olgeta yunit bilong mani bilong pilai long wil bilong Yurop, na 19 long wil bilong Amerika, long wanem olgeta tanim i kisim 1/37 bilong wanpela yunit long avarej (1/19 long wil bilong Amerika): 100 yunit i stap 3,700 tanim long avarej, tasol hap bilong kain sesen olsem i pinis insait long 3,134 (lukim “Longpela bilong Sesen na Pe Yu Lusim”). Na maski mani bilong pilai i bikpela olsem wanem, sans long wil bilong Yurop long kamap k yunit antap long wanpela taim em i (18/19)ᵏ. Em i 95% long wanpela yunit na 58% long tenpela, tasol i no gat mani bilong pilai, maski em i bikpela tru, i givim sans hap-hap long kamap 13 yunit antap; long wil bilong Amerika sans em i (9/10)ᵏ na em i go daun aninit long hap long 7 yunit.
Longpela bilong Sesen na Pe Yu LusimWil na Lo bilong Rulet bilong Amerika
Mani bilong pilai bilong sans hap-hap
Tanim askim i go baksait: hamas mani yu mas kisim i kam long gat sans hap-hap o moa long winim wanpela mak bipo long yu lusim olgeta? Long liklik win, i no planti. Long bikpela win, mani bilong pilai yu nidim i go antap kwik tru, na bihain long wanpela mak i no gat mani bilong pilai inap, long wanem maski mani i no save pinis, em i givim sans i daun long hap-hap long kamap antap olsem. Taylor (2021) i tok long narapela rot: long wil bilong Amerika, long gat sans hap-hap long kamap long 100 yunit, yu mas stat wantaim 94, na yu putim 94 long hevi bilong winim 6. Long wil bilong Yurop yu bai nidim 88.
Skul bilong wok tru
Makim wanem samting yu traim long mekim. Sapos yu laik winim wanpela mak, putim liklik bet tasol na bikpela. Sapos yu laik wanpela longpela nait, putim liklik bet na save olsem edj bai kisim 2.70% bilong bikpela olgeta mani yu betimmaniOlgeta mani yu betimOlgeta mani yu betim long wanpela sesen o long wanpela taim, wantaim mani yu win na yu betim gen. Narapela nem: handle, action.Ritim olgeta →. Sapos yu laik tupela wantaim, wanpela samting mas givap, na planti taim em i mak.
Fomula
Ol askim
MartingalestratejiMartingaleProgresen i mekim bet i dabel bihain long olgeta lus. Lukim hap Strateji long wanem samting em i mekim na i no mekim.Ritim olgeta → i daunim long ol sot sesen na i apim bikpela tru long ol longpela sesen. Bet wankain oltaimstratejiBet wankain oltaimBetim wankain mani long olgeta tanim. Olgeta progresen ol i skelim wantaim dispela.Ritim olgeta → wantaim bikpela yunit em i stretpela rot long daunim.
Fomula antap em i bilong bet wan-long-wan. Long ol bet i gat bikpela varians, lusim olgeta i gat sans moa long ol liklik bet, na yu kamap long mak long rot i kalap nabaut; yusim kalkuleta.
Olgeta step i go long mak i gat sans i daun liklik long olgeta step i go long lusim olgeta; skel i bung i go antap long hamas step.
Amamas wantaim Matematik
Traim ol dispela yu yet pastaim. Bekim bilong wanwan i hait; opim taim yu redi.
Yu gat $100 na yu laik kisim $1,000, na yu betim retbetRetBet wan-long-wan long 18 namba ret. Tok Frans: rouge.Ritim olgeta → tasol. Plen namba wan: betim $1 long wanwan taim. Plen namba tu: betim olgeta mani yu gat inap yu winim $500, na bihain betim inap tasol long kisim $1,000. Wanem plen i gutpela moa, na i winim hamas?
Soim bekim
Bekim Plen namba tu, i winim tru. Em i kisim $1,000 klostu 9.0% bilong taim, klostu long stretpela 10%. Plen namba wan i win klostu 7 taim long 10²².
Long pilai i stretmatematikPilai i stretPilai we i no gat wanpela sait i gat edj: olgeta bet i gat ekspektet velyu siro.Ritim olgeta → yu bai kisim $1,000 long $100 wanpela taim long ten maski yu mekim wanem, long wanem avarej velyu bilong mani bilong yu i stap $100.
Edj i kisim 2.70% bilong olgeta bet, olsem na sapos yu betim liklik mani long olgeta, yu stap klostu moa long wanpela long ten. Plen namba tu i pinisim samting long sampela bet tasol: $100 i kamap $200, $400, $800, na bihain $200 i inap. Em i win klostu 9.0% bilong taim. Plen namba wan i kisim planti tausen bet pastaim long wokabaut i pinis, na edj, liklik long wanwan, i bung na kamap klostu olsem yu mas lusim olgeta mani.
Em i risal bilong Dubins–Savage long isi pasin tru (Epstein, 2013): long pilai em kasino i gat edj, betim planti olsem yu nidim, long liklik taim tru olsem yu inap.
Yu gat $500 na yu laik kisim $1,000, na yu stop taim yu kisim. Yu makim wanpela kain bet na mekim gen inap em i win wanpela taim o mani i pinis. Wanem kain i givim sans i gutpela tru?
Soim bekim
Bekim Wanpela bet $500 long ret: 48.6%. Olgeta plen i tilim mani long ol liklik bet i nogut moa.
Makim wanwan bet olsem wanpela win i inap: $30 long bet long wanpela nambabetWanpela nambaWanpela namba tasol, pe em i 35:1.Ritim olgeta → i winim $1,050, $60 long bet long tupela nambabetTupela nambaTupela namba i stap klostu, pe em i 17:1.Ritim olgeta → i winim $1,020, na i go olsem. Bihain kaunim sans olsem wanpela bet o moa i win pastaim long mani i pinis. Sikstinpela bet $30 long wanpela namba i no win olgeta wantaim sans (36/37)¹⁶ = 64.5%, olsem na ol i kisim $1,000 35.5% bilong taim.
Namba tebol i soim wanwan bet. Ret i gutpela tru long wanem em i kamap long wanpela bet, olsem na yu peim edj wanpela taim, long $500. Bet long sikispela nambabetBet long sikispela nambaSikispela namba long tupela lain i stap klostu, pe em i 5:1. Tok Frans: sixain.Ritim olgeta → i namba tu long wanem em i nidim tripela tasol. Wanpela bet dosenbetDosen1–12, 13–24 o 25–36, pe em i 2:1.Ritim olgeta → i nogut tru, sans 12/37; maski yu pilai long yunitmaniYunitMani bilong bet bilong stat we wanpela sistem o wanpela pleia i wok long en. Ol i stori long ol progresen long yunit, olsem na ol i ken wok long wanem tebol.Ritim olgeta → bilong $250 inap yu gat $1,000 o moa, dosen i kamap 46.4% bilong taim tasol.
Bollman (2024) i wokim wankain namba tebol long wil bilong Amerika. Lain bilong dosen bilong em i gat rong: em i mas tok 12/38 = 31.6%.
betbet × hamas betsans long kisim $1,000Wanpela namba$30 × 1635.5%Tupela namba$60 × 835.9%Bet long tripela namba$90 × 534.5%Fopela namba$125 × 436.7%Bet long sikispela namba$166, $166, $16841.2%Dosen$500 × 132.4%Ret$500 × 148.6%Yu betim wanpela yunit long ret, tanim bihain long tanim, wantaim mani bilong pilaimaniMani bilong pilaiOlgeta mani yu putim long sait bilong pilai. Lukautim mani bilong pilai em i makim hamas long en yu redi long lusim long wanpela bet na long wanpela sesen.Ritim olgeta → i no save pinis. Bihain long tanim namba wan yu gat wanpela yunit antap o wanpela yunit daun. Wanem sans olsem yu bai kamap no win no lus stret gen?
Soim bekim
Bekim 36 long 37, 97.3%. Long wil bilong Amerika em 36 long 38, 94.7%.
Sapos tanim namba wan i win (18 taim long 37), yu wanpela yunit antap, na edj i pulim olgeta sesen i no save pinis i go daun, olsem na bihain yu bai go bek long no win no lus: dispela em i tru olgeta. Sapos em i lus (19 taim long 37), yu mas go antap wanpela yunit, na sans long kamap wanpela yunit antap em 18/19 (lukim namba tebol antap).
Wantaim: 18/37 + 19/37 × 18/19 = 18/37 + 18/37 = 36/37.
Long olgeta, sans em 1 rausim senis namel long sans long win na sans long lus long wanpela tanim: 1 − 1/37 long wil bilong Yurop, 1 − 2/38 long wil bilong Amerika. Long pilai i nogat sirowilSiroWanpela grinpela hul we i no gat bet wan-long-wan, dosen o kolam i karamapim; em i as bilong edj bilong kasino. Narapela nem: wanpela siro, 0.Ritim olgeta →, senis em nating na sans em 1: mak oltaim i kam bek long no win no lus. Epstein (2013) i givim lo bilong olgeta.
Hevi bilong Pascal long 1656: tupela pleia i tromoi tripela daiz, wanwan i stat wantaim 12 mak. Olgeta 11 i muvim wanpela mak i go long pleia namba wan, olgeta 14 i muvim wanpela i go long namba tu; ol narapela tromoi i no kaunim. Sans bilong pleia namba wan long winim olgeta 24 em hamas? Na sapos yu betim 1 yunit long ret long wanwan tanim long pren i wok olsem benk, 12 yunit wanwan, sans bilong yu long winim olgeta 24 em hamas?
Soim bekim
Bekim Pleia namba wan i win klostu 1,157 taim long 1,158. Yu, long ret, i winim olgeta 24 yunit klostu 34% bilong taim tasol.
Tripela daiz i kamapim 11 long 27 rot na 14 long 15 rot, olsem na wanwan mak i muv i go long pleia namba wan 27 taim long 42, 9 taim long 14. Taim wanwan sait i stat long 12, sans long kisim olgeta em 1/(1 + r¹²), we r em sans bilong narapela sait tilim long sans bilong yu.
Long pleia namba wan r = 5/9, na (5/9)¹² ≈ 0.00086, olsem na sans em klostu 0.99914: ods 9¹² long 5¹², o 282,429,536,481 long 244,140,625, klostu 1,157 long 1. Huygens i prinim stret ol dispela namba long 1657.
Long ret, r = 19/18 long wanem siro, na (19/18)¹² ≈ 1.913: sans bilong yu em 1/2.913 ≈ 34.3%. Koin bai givim 50%. Edj 2.70% long wanwan bet i bung, long planti bet em i nidim, na i kamap senis bilong 16 poin.
Liklik lap
Pilai strong em i gutpela moa. Na em i pinis kwik tu.
Ol As bilong Save
- 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. Pas i go long Huygens, 28 Septemba 1656, long hevi Pascal i givim long 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.”