Hevi: Mak bilong Tebol
Martingale long $10 long tebol i gat mak $500. Hamas taim em i bruk, hamas mani em i lusim taim em i bruk, na sapos mak i go antap, em i helpim?
Sikispela lus wanpela bihain long narapela i brukim: 1.8% long wanpela saikel, klostu 60% bilong ol sesen bilong 100 tanim. Bruk i kostim $630. Sapos mak i go antap long $5,000, bruk i no save kamap planti (0.25% long wanpela saikel), tasol em i bikpela moa ($5,110). Avarej lus long wanwan dola yu betim em 2.70% long wanem mak.
Hevi
Wanpela pleia i betim $10 long retbetRetBet wan-long-wan long 18 namba ret. Tok Frans: rouge.Ritim olgeta → na i mekim dabel bihain long wanwan lus, na i go bek long stat bihain long win. Mak antap bilong tebol em $500. (a) Hamas lus wanpela bihain long narapela i brukim progresenstratejiProgresenWanem lo i senisim mani bilong bet long ol resal i go pinis. Negative progresen i mekim bet i go antap bihain long lus, positive progresen bihain long win.Ritim olgeta →? (b) Wanem sansmatematikSansSans bilong wanpela resal, long 0 i go 1 o 0% i go 100%. Wanpela namba: 1/37 = 2.70%.Ritim olgeta → long wanpela saikel? (c) Hamas i lus taim em i bruk? (d) Olsem wanem mak bilong $5,000 i senisim ol bekim? (e) I gat wanpela mak i mekim sistem i givim win?
Bekim (a)–(c)
Ol mani bilong bet: 10, 20, 40, 80, 160, 320. Namba seven bai 640 > 500, olsem na sikispela lus i brukim. (b) p(lus) = 19/37 long ret; (19/37)⁶ = 0.0183, klostu 1.8% long wanpela saikel. Wanpela saikel i stap klostu tupela tanim long avarej. Olsem na sesen bilong 100 tanim i gat klostu 50 saikel, na p(wanpela bruk o moa) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630. Em 63 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 stat bet, na saikel i bin traim long winim $10 tasol.
Bekim (d)
Long $5,000: ol mani bilong bet i go inap 2,560 i orait; namba ten, 5,120, ol i no orait. Nainpela lus i brukim: (19/37)⁹ ≈ 0.25% long wanpela saikel, klostu 11% bilong ol sesen bilong 100 tanim. Lus taim i bruk: 10 + 20 + … + 2,560 = $5,110. Pleia i baim pinis bagarap i no save kamap planti, tasol em i bikpela moa. Long planti sesen, avarej pe bilong ol bruk em p × lus ≈ 0.0183 × 630 = $11.5 long wanpela saikel long mak $500, na 0.0025 × 5,110 = $12.8 long $5,000. Sapos yu raunim, em wankain namba. Na em i avarej lusmatematikAvarej lusEkspektet velyu taims olgeta mani yu betim; avarej mani yu lusim long wanpela sesen o long wanpela taim.Ritim olgeta → bilong saikel wantaim $10 ol saikel i win i kisim.
Bekim (e)
Nogat. Avarej lus long wanpela saikel = (avarej olgeta mani bilong bet long saikel) / 37, long wanem mak, long wanem wanwan bet em −1/37 bilong em yet. Mak i makim tasol olsem wanem lus i stap namel long planti liklik win na sampela bikpela lus. Sapos nogat mak na 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 gat arere, progresen bai “wok” long wanpela we: em i no inap bruk. Tasol em bai betim mani i no gat arere tu, wantaim avarej lus 2.70% bilong dispela mani i no gat arere. Mak bilong teboltebolMak bilong tebolBet liklik tru na bet bikpela tru ol i larim. Mak antap em i samting i stopim ol progresen i save dabel.Ritim olgeta → i no samting i pasim 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 →. Matematik i pasim.
Tupela tebol tru
Eksampel bilong Thorp em tebol bilong $1 wantaim mak antap $1,000. Wanpela pleia i laik winim $30 bilong kaikai long nait, wanpela dola long wanpela taim. Ol mani bilong bet 1, 2, 4 … 512 i givim ten bet, olsem na ten lus wanpela bihain long narapela i brukim progresen. Em i kamap wanpela taim long 785 saikel long wil bilong Yurop (wanpela taim long 613 long wil bilong Amerika), na i kostim $1,023. Tetipela saikel olgeta i win 96.2% bilong taim. Narapela 3.8% i lusim klostu $1,000, na avarej risalt bilong nait em −$9.17. Em stret 1/37 bilong $339 tetipela saikel i betim long avarej. Tebol bilong $5 wantaim mak antap $2,000 i no narakain: ol mani bilong bet i go inap $1,280 i givim nainpela bet. Bruk i kamap wanpela taim long 403 saikel long wil bilong Yurop (wanpela taim long 323 long wil bilong Amerika), na i kostim $2,555.
Wanpela pleia i gat planti mani
Givim progresen bikpela mani bilong pilai, i no mak i daun: 1,000 yunit, stat bet bilong wanpela yunit, na 1,000 tanim long ret. Stat gen long wanpela yunit taim dabel i kam i winim mani i stap yet. Mipela i ranim simulesen bilong wan milion kain nait olsem. Long wil bilong Yurop, 52% i pinis wantaim win, 21% i lusim olgeta, na avarej risalt em −124 yunit. Maski long gem i no gat 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 →, we olgeta bet i stret, klostu 60% tasol bai pinis wantaim win. Nait i save kamap planti em liklik win. Ol nait i pinis long siro i baim avarej.
Skul bilong en
Ol kasino i putim mak bilong banisim 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 ol yet, i no bilong winim ol sistem. Winim Martingale em samting i kamap wantaim na ol i amamas long en, tasol matematik bai winim em maski. Pleia i ting mak em hevi, em i painim pinis klif na i ting em i rot i go daun isi.
Fomula
Ol askim
Em i helpim yu long stap inap long mak i antap moa, we bruk i no save kamap planti tasol em i bikpela moa. Avarej lus long wanwan dola i no senis.
Ol mani bilong bet 10, 30, 70, 150, 310: yu kisim mak $500 bihain long faivpela lus, na bruk i kostim $570. Wankain 2.70%.
Bai wanpela pleia i no ken betim moa long mani kasino i holim inap long baim isi. Em lukautim varians.
Amamas wantaim Matematik
Traim ol dispela yu yet pastaim. Bekim bilong wanwan i hait; opim taim yu redi.
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 stat long $100 long retbetRetBet wan-long-wan long 18 namba ret. Tok Frans: rouge.Ritim olgeta → na i larim em yet n bet tasol long wanwan saikel: bihain long n lus i bihainim lus em i lusim na stat gen. Raitim avarej risal bilong wanpela saikel olsem fomula long n, na kaunim long 5, 6 na 7 bet long wil bilong Yurop.
Soim bekim
Bekim Avarej em 100 × (1 − (38/37)ⁿ): −$14.26 long 5 bet, −$17.35 long 6 na −$20.52 long 7. Sapos yu larim moa bet, avarej saikel i kamap nogut moa, i no gutpela moa.
Saikel i winim $100 sapos olgeta n bet i no lus, na dispela i kamap wantaim sans qⁿ, we q = 19/37. Taim olgeta i lus, ol bet 100, 200 … 100 × 2ⁿ⁻¹ i bung i kamap 100 × (2ⁿ − 1). Olsem na avarej em 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Hia 2q = 38/37, liklik moa long 1, olsem na (2q)ⁿ i go antap wantaim n na avarej i go daun: n = 5 i givim 1.1426 na −$14.26; n = 6 i givim 1.1735 na −$17.35; n = 7 i givim 1.2052 na −$20.52. Lada i longpela moa i lus liklik taim moa, long 3.6%, 1.8% na 0.9% bilong ol saikel, tasol wanwan lus i bikpela tupela taim, na planti mani moa i go long tebol long wanwan saikel, long 2.70% bilong olgeta bet.
Long wil bilong Amerika 2q = 40/38 na ol namba em −$29.24, −$36.04 na −$43.20 (Taylor, 2021). 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 → 2q = 1, na olgeta saikel i gat avarej siro stret maski lada i longpela olsem wanem.
“Grand Martingale” i dabolim bet bihain long lus na i putim wanpela 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 → moa, olsem na ol bet i go 1, 3, 7, 15, 31 … Yu stap antap hamas taim win namba wan i kamap long bet namba 1, 2, 3 o k? Long yunit $10 na bikpela bet $500, hamas taim saikel i bruk, na wanpela saikel i kostim hamas long avarej?
Soim bekim
Bekim Win long bet namba k i lusim yu k yunit antap: +1, +2, +3 na i go olsem. Long $10 na $500 saikel i bruk bihain long faivpela lus, 3.57% bilong ol saikel, long −$570. Long avarej wanpela saikel i lusim $2.32 long $85.73 yu betim: 2.70%.
Bet namba k em 2ᵏ − 1 yunit: 1, 3, 7, 15, 31. Ol k − 1 bet pastaim i bung i kamap 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Win long bet namba k i pe 2ᵏ − 1, olsem na saikel i pinis (2ᵏ − 1) − (2ᵏ − k − 1) = k yunit antap: wanpela yunit moa long wanwan lus bipo long win.
Long $10 long wanpela yunit ol bet em $10, $30, $70, $150 na $310; bet bihain, $630, i winim bikpela bet $500. Faivpela lus i bihainim lus, (19/37)⁵ = 3.57%, i kostim 10 + 30 + 70 + 150 + 310 = $570.
Avarej: ol win i givim $10 × k wantaim sans (19/37)ᵏ⁻¹ × 18/37 long k = 1 i go 5, na bruk i kisim $570 wantaim sans 3.57%; wantaim −$2.32. Mani yu betim i gat avarej $85.73 long wanwan saikel, na 2.32/85.73 = 2.70%. Nomol Martingale long wankain tebol i bruk bihain long sikispela lus, 1.83% bilong ol saikel, long −$630, na i lusim $1.74 long $64.20 yu betim: 2.70% gen.
Wanpela pleia i plen long pilai Martingale long ret wantaim bet namba wan $1 long tebol i gat mak $1,000, olsem na ten bet inap: $1, $2, $4 na i go antap inap $512. Hamas taim wanpela saikel i lusim olgeta ten, hamas taim dispela i kisim long avarej, na hamas taim nait bilong 200 bet i pinis olsem?
Soim bekim
Bekim Wanpela saikel i lusim olgeta ten klostu wanpela taim long 784, na dispela i kostim $1,023. Em klostu 1,600 bet long avarej, na nait bilong 200 bet i paitim banis klostu 11% bilong taim. Wanpela saikel i lusim yet $0.31 long avarej long $11.31 yu betim: 2.70%.
Bet long ret i lus wantaim sans 19/37, wantaim siro. Ten i bihainim narapela: (19/37)¹⁰ ≈ 0.00128, klostu 1 long 784. Ol bet 1 + 2 + 4 + … + 512 i bung i kamap $1,023.
Wanpela saikel i kisim 2.05 bet long avarej, olsem na 784 saikel i kisim klostu 1,600 bet. Sapos yu kaunim tanim bihain long tanim, ran bilong ten lus i kamap long wanpela hap long 200 bet 11.3% bilong taim.
Namel long ol bikpela lus, pleia i winim $1 long wanwan saikel: klostu $783 pastaim long saikel nogut, na bihain −$1,023, klostu −$240 long olgeta taim. Long wanwan saikel: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Mani yu betim long wanwan saikel: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Na −0.31/11.31 = −2.70%.
Skelim wantaim bet namba wan $10 long mak $500, sikispela bet: banis i kamap wanpela taim long 55 saikel, na nait bilong 200 bet i paitim em 84% bilong taim. Mak i antap moa i mekim bikpela lus i no save kamap planti na i bikpela moa, i no kostim liklik moa.
Liklik lap
I no save kamap planti na i bikpela moa, long olgeta mak. Tupela i bung i no senis.
Ol As bilong Save
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Thorp, E. O. “Optimal gambling systems for favorable games.”
- Thorp, E. O. The Mathematics of Gambling.
- Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed.