Prɔblɛm: di tebul limit
Martingale wit $10 na tebul we gɛt $500 limit. Ɔmɔs tɛm i de brok, ɔmɔs i de lɔs wen i brok, ɛn if dɛn es di limit, dat de ɛp?
Siks lɔs wan afta di ɔda de brok am: 1.8% fɔ ɛni saykul, lɛk 60% pan sɛshɔn fɔ 100 spin. Di brok de kɔst $630. If dɛn es di limit to $5,000, di brok nɔ go kam bɔku (0.25% fɔ ɛni saykul) bɔt i go big pas ($5,110). Di lɔs we fɔ ɛkspɛkt fɔ ɛni dɔla we yu bɛt na 2.70% pan ɛni limit.
Di prɔblɛm
Pleya de bɛt $10 pan rɛdbɛtRɛdIvin-mɔni bɛt pan di 18 rɛd nɔmba. Na Frɛnch: rouge.Rid ɔl → ɛn i de dɔbul afta ɛni lɔs, ɛn i de bigin bak afta win. Di big limit fɔ di tebul na $500. (a) Ɔmɔs lɔs wan afta di ɔda de brok di prɔgrɛshɔnstratɛjiPrɔgrɛshɔnƐni rul we de chenj di bɛt mɔni bay di rizɔlt dɛn we dɔn pas. Nɛgativ prɔgrɛshɔn de es am afta lɔs, pɔzitiv wan afta win.Rid ɔl →? (b) Wetin na di prɔbabilitimatematiksPrɔbabilitiDi chans fɔ wan rizɔlt, frɔm 0 to 1 ɔ frɔm 0% to 100%. Wan nɔmba: 1/37 = 2.70%.Rid ɔl → fɔ ɛni saykul? (c) Wetin yu de lɔs wen i brok? (d) Aw $5,000 limit de chenj di ansa dɛn? (e) Ɛni limit de we go mek di sistɛm gi prɔfit?
Ansa (a)–(c)
Bɛt mɔni dɛn: 10, 20, 40, 80, 160, 320. Di sɛvin wan go bi 640 > 500, so siks lɔs de brok am. (b) p(lɔs) = 19/37 pan rɛd; (19/37)⁶ = 0.0183, lɛk 1.8% fɔ ɛni saykul. Wan saykul de las lɛk tu spin pan avrej, so sɛshɔn fɔ 100 spin gɛt lɛk 50 saykul ɛn p(wan brok ɔ mɔ) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, we na 63 bes yunitmɔniYunitDi bes bɛt mɔni we sistɛm ɔ pleya de wok wit. Dɛn de ɛksplen prɔgrɛshɔn wit yunit so dat i go wok na ɛni tebul.Rid ɔl →, we di saykul bin de tray fɔ win jɔs $10.
Ansa (d)
Wit $5,000: dɛn alaw bɛt mɔni te to 2,560; di tɛnt bɛt mɔni, 5,120, dɛn nɔ go tek am. Nayn lɔs de brok am: (19/37)⁹ ≈ 0.25% fɔ ɛni saykul, lɛk 11% pan sɛshɔn fɔ 100 spin. Lɔs wen i brok: 10 + 20 + … + 2,560 = $5,110. Di pleya dɔn bay big trɔbul we nɔ de kam bɔku, ɛn di prays na trɔbul we big pas. Ova bɔku sɛshɔn, di kɔst we fɔ ɛkspɛkt fɔ di brok dɛn na p × lɔs ≈ 0.0183 × 630 = $11.5 fɔ ɛni saykul wit di $500 limit ɛn 0.0025 × 5,110 = $12.8 wit $5,000; di tu nia bɔt dɛn nɔ ikwal, ɛn ɛni wan na di lɔs we fɔ ɛkspɛktmatematiksLɔs we fɔ ɛkspɛktValyu we fɔ ɛkspɛkt tɛm ɔl di mɔni we yu bɛt; di avrej mɔni we yu de lɔs insay wan sɛshɔn ɔ wan tɛm.Rid ɔl → fɔ di saykul plɔs di $10 we di saykul dɛn we win de gɛt.
Ansa (e)
Nɔ. Lɔs we fɔ ɛkspɛkt fɔ ɛni saykul = (tɔtal bɛt mɔni we fɔ ɛkspɛkt insay di saykul) / 37, pan ɛni limit, bikɔs ɛni bɛt na −1/37 pan insɛf. Di limit jɔs de disayd aw di lɔs de sheb bitwin bɔku smɔl win ɛn sɔm smɔl big lɔs. If limit nɔ de ɛn gambul mɔnimɔniGambul mɔniƆl di mɔni we yu dɔn put asay fɔ ple. Fɔ manej yu gambul mɔni min fɔ disayd ɔmɔs pan am yu go risk fɔ ɛni bɛt ɛn fɔ ɛni sɛshɔn.Rid ɔl → nɔ gɛt ɛnd, di prɔgrɛshɔn go “wok”, in di sɛns se i nɔ go ɛva brok, ɛn i go bak bɛt mɔni we nɔ gɛt ɛnd, wit lɔs we fɔ ɛkspɛkt fɔ 2.70% pan dat mɔni we nɔ gɛt ɛnd. Nɔto di tebul limittebulTebul limitDi smɔl ɛn di big bɛt we dɛn alaw. Na di big limit de stɔp prɔgrɛshɔn dɛn we de dɔbul.Rid ɔl → de stɔp MartingalestratɛjiMartingalePrɔgrɛshɔn we de dɔbul di bɛt afta ɛni lɔs. Luk di Stratɛji pat fɔ si wetin i de du ɛn wetin i nɔ de du.Rid ɔl → fɔ wok. Na di kalkuleshɔn.
Tu rial tebul
Thorp in ɛgzampul na $1 tebul wit $1,000 big limit, usay pleya de op fɔ win $30 fɔ it dina, wan dɔla wan dɔla. Di bɛt mɔni dɛn 1, 2, 4 … 512 de alaw tɛn bɛt, so tɛn lɔs wan afta di ɔda de brok di prɔgrɛshɔn: wan tɛm insay 785 saykul pan Yuropian wil (wan tɛm insay 613 pan Amɛrikan wan), ɛn i de kɔst $1,023. Tati saykul ɔl de wok 96.2% pan di tɛm. Di ɔda 3.8% de lɔs lɛk $1,000, ɛn di avrej rizɔlt fɔ di ivintɛm na −$9.17, we na ɛgzaktli 1/37 pan di $339 we di tati saykul de bɛt pan avrej. $5 tebul wit $2,000 big limit nɔ difrɛn: bɛt mɔni te to $1,280 de alaw nayn bɛt, ɛn di brok de kam wan tɛm insay 403 saykul pan Yuropian wil (wan tɛm insay 323 pan Amɛrikan wan) ɛn i de kɔst $2,555.
Pleya we jɛntri
Gi di prɔgrɛshɔn big gambul mɔni instɛd fɔ smɔl limit: 1,000 yunit, bes fɔ wan yunit ɛn 1,000 spin pan rɛd, ɛn bigin bak wit wan yunit ɛnitɛm we di nɛks dɔbul pas wetin lɛf. Wi simyulet wan milyɔn ivintɛm lɛk dat. Pan Yuropian wil, 52% de dɔn wit prɔfit, 21% de lɔs ɔltin, ɛn di avrej rizɔlt na −124 yunit. Ivin na gem we nɔ gɛt zirowilZiroDi wan grin pɔkit we nɔ ivin-mɔni, dɔzin ɔ kɔlɔm bɛt nɔ kɔba; na usay di kasino advantej kɔmɔt. Dɛn kin kɔl am bak: wan ziro, 0.Rid ɔl →, usay ɛni bɛt fɛr, na jɔs lɛk 60% go dɔn wit prɔfit. Di nɔmal ivintɛm na smɔl win; na di ivintɛm dɛn we dɔn na ziro de pe fɔ di avrej.
Wetin wi lan
Kasino dɛn de sɛt limit fɔ kɔntrol dɛn yon variansmatematiksVariansWan mɛzhɔ fɔ aw di rizɔlt dɛn sprɛd rawnd di valyu we fɔ ɛkspɛkt. I ay fɔ bɛt pan wan nɔmba, i lɔ fɔ ivin-mɔni bɛt.Rid ɔl →, nɔto fɔ bit sistɛm dɛn. Fɔ bit Martingale na tin we jɔs apin, we dɛn gladi fɔ, ɛn di kalkuleshɔn go bit am ɛni we. Pleya we tink se na di limit na di prɔblɛm dɔn si di klif ɛn i tink se na di slop.
Fɔmula
Kwɛstyɔn dɛn
I de mek yu sav te yu rich limit we ay pas, usay di brok nɔ de kam bɔku bɔt i big pas. Di lɔs we fɔ ɛkspɛkt fɔ ɛni dɔla nɔ chenj.
Bɛt mɔni dɛn 10, 30, 70, 150, 310: i de nak di $500 limit afta fayv lɔs, ɛn di brok de kɔst $570. Di sem 2.70%.
So dat wan pleya nɔ go bɛt mɔ pas wetin di kasino in rizav mɔni kin pe izi. Fɔ manej varians.
Ple wit Mat
Tray dɛn yusɛf fɔs. Ɛni ansa de ayd; opin am we yu rɛdi.
MartingalestratɛjiMartingalePrɔgrɛshɔn we de dɔbul di bɛt afta ɛni lɔs. Luk di Stratɛji pat fɔ si wetin i de du ɛn wetin i nɔ de du.Rid ɔl → de bigin wit $100 pan rɛdbɛtRɛdIvin-mɔni bɛt pan di 18 rɛd nɔmba. Na Frɛnch: rouge.Rid ɔl → ɛn i de alaw insɛf n bɛt fɔ di mɔs insay wan saykul: afta n lɔs wan afta di ɔda, i de lɛf ɛn bigin bak. Rayt di avrej rizɔlt fɔ wan saykul as fɔmyula insay n, ɛn kɔnt am fɔ 5, 6 ɛn 7 bɛt pan Yuropian wil.
Sho di ansa
Ansa Di avrej na 100 × (1 − (38/37)ⁿ): −$14.26 fɔ 5 bɛt, −$17.35 fɔ 6 ɛn −$20.52 fɔ 7. If yu alaw mɔ bɛt, di avrej saykul de wɔs, i nɔ de bɛtɛ.
Saykul de win $100 pas if ɔl n bɛt lɔs, ɛn dat de apin wit chans qⁿ, usay q = 19/37. We ɔl lɔs, di bɛt dɛn 100, 200 … 100 × 2ⁿ⁻¹ de ad ɔp to 100 × (2ⁿ − 1). So di avrej na 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Ya 2q = 38/37, smɔl mɔ pas 1, so (2q)ⁿ de gro wit n ɛn di avrej de fɔdɔm: n = 5 de gi 1.1426 ɛn −$14.26; n = 6 de gi 1.1735 ɛn −$17.35; n = 7 de gi 1.2052 ɛn −$20.52. Lada we lɔng pas de lɔs lɛs tɛm, insay 3.6%, 1.8% ɛn 0.9% pan di saykul dɛn, bɔt ɛni lɔs big tu tɛm mɔ, ɛn mɔ mɔni de krɔs di tebul ɛni saykul, bay 2.70% pan ɛni bɛt.
Pan Amɛrikan wil 2q = 40/38 ɛn di nɔmba dɛn na −$29.24, −$36.04 ɛn −$43.20 (Taylor, 2021). Insay gem we nɔ gɛt zirowilZiroDi wan grin pɔkit we nɔ ivin-mɔni, dɔzin ɔ kɔlɔm bɛt nɔ kɔba; na usay di kasino advantej kɔmɔt. Dɛn kin kɔl am bak: wan ziro, 0.Rid ɔl → 2q = 1, ɛn ɛni saykul in avrej na ziro rayt rayt ilɛk aw lɔng di lada.
Di “grand Martingale” de dɔbul di bɛt afta lɔs ɛn ad wan yunitmɔniYunitDi bes bɛt mɔni we sistɛm ɔ pleya de wok wit. Dɛn de ɛksplen prɔgrɛshɔn wit yunit so dat i go wok na ɛni tebul.Rid ɔl →, so di bɛt dɛn de go 1, 3, 7, 15, 31 … Ɔmɔs yu de ɔp we di fɔs win kam pan bɛt nɔmba 1, 2, 3 ɔ k? Wit $10 yunit ɛn $500 fɔ di mɔs, ɔmɔs tɛm saykul de brok, ɛn ɔmɔs saykul de kɔst pan avrej?
Sho di ansa
Ansa Win pan bɛt nɔmba k de lɛf yu k yunit ɔp: +1, +2, +3 ɛn so so. Wit $10 ɛn $500 di saykul de brok afta fayv lɔs, 3.57% pan di saykul dɛn, fɔ −$570. Pan avrej saykul de lɔs $2.32 pan $85.73 we yu bɛt: 2.70%.
Bɛt nɔmba k na 2ᵏ − 1 yunit: 1, 3, 7, 15, 31. Di fɔs k − 1 bɛt dɛn de ad ɔp to 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Win pan bɛt nɔmba k de pe 2ᵏ − 1, so di saykul de dɔn (2ᵏ − 1) − (2ᵏ − k − 1) = k yunit ɔp: wan yunit mɔ fɔ ɛni lɔs bifo di win.
Bay $10 ɛni yunit di bɛt dɛn na $10, $30, $70, $150 ɛn $310; di nɛks wan, $630, pas di $500 limit. Fayv lɔs wan afta di ɔda, (19/37)⁵ = 3.57%, de kɔst 10 + 30 + 70 + 150 + 310 = $570.
Avrej: di win dɛn de briŋ $10 × k wit chans (19/37)ᵏ⁻¹ × 18/37 fɔ k = 1 te to 5, ɛn di brok de tek $570 wit chans 3.57%; togɛda −$2.32. Di mɔni we yu bɛt na $85.73 pan avrej ɛni saykul, ɛn 2.32/85.73 = 2.70%. Di nɔmal Martingale na di sem tebul de brok afta siks lɔs, 1.83% pan di saykul dɛn, fɔ −$630, ɛn i de lɔs $1.74 pan $64.20 we yu bɛt: 2.70% bak.
Pleya plan fɔ ple Martingale pan rɛd wit $1 fɔs bɛt na tebul we in limit na $1,000, so tɛn bɛt pɔsibul: $1, $2, $4 ɛn so so te to $512. Ɔmɔs tɛm saykul de lɔs ɔl tɛn, aw lɔng dat de tek pan avrej, ɛn ɔmɔs tɛm ivintɛm fɔ 200 bɛt de ɛnd so?
Sho di ansa
Ansa Saykul de lɔs ɔl tɛn lɛk wan tɛm insay 784, ɛn dat de kɔst $1,023. Dat na lɛk 1,600 bɛt pan avrej, ɛn ivintɛm fɔ 200 bɛt de jam di wɔl lɛk 11% pan di tɛm. Saykul stil de lɔs $0.31 pan avrej pan $11.31 we yu bɛt: 2.70%.
Bɛt pan rɛd de lɔs wit chans 19/37, wit di ziro insay. Tɛn wan afta di ɔda: (19/37)¹⁰ ≈ 0.00128, lɛk 1 insay 784. Di bɛt dɛn 1 + 2 + 4 + … + 512 de ad ɔp to $1,023.
Saykul de tek 2.05 bɛt pan avrej, so 784 saykul de tek lɛk 1,600 bɛt. If yu kɔnt spin bay spin, strik fɔ tɛn lɔs de kam sɔmsay insay 200 bɛt 11.3% pan di tɛm.
Bitwin di bad tin dɛn, di pleya de win $1 ɛni saykul: lɛk $783 bifo di bad wan, dɔn −$1,023, lɛk −$240 oba ɔl di tɛm. Pan ɛni saykul: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Mɔni we yu bɛt ɛni saykul: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Ɛn −0.31/11.31 = −2.70%.
Kɔmpia $10 fɔs bɛt wit $500 limit, siks bɛt: di wɔl de kam wan tɛm insay 55 saykul, ɛn ivintɛm fɔ 200 bɛt de mit am 84% pan di tɛm. Limit we ay pas de mek di krash nɔ kin apin bɔku bɔt i big pas, i nɔ de chip pas.
Smɔl jok fɔ ɛz di maynd
I nɔ de kam bɔku bɔt i big pas, pan ɛni limit. Di tu mɔltiplay togɛda nɔ de chenj.
Usay wi gɛt am frɔm
- 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.