Nsogbu: Oke Tebụl
Martingale $10 na tebụl $500. Ugboro ole ka ọ na-agbaji, ego ole ka ọ na-atụfu mgbe ọ gbajiri, ịbuli oke ọ̀ na-enyere aka?
Mfu isii n’usoro na-agbaji ya: 1.8% kwa cycle, ihe dị ka 60% nke oge egwuregwu ntụgharị 100. Ngbaji ahụ na-efu $630. Ịbuli oke ruo $5,000 na-eme ka ngbaji ahụ ghara ime ọtụtụ mgbe (0.25% kwa cycle) ma buru ibu karịa ($5,110). Mfu a tụrụ anya kwa dọla a gbara bụ 2.70% n’oke ọ bụla.
Nsogbu ahụ
Onye egwuregwu na-agba $10 na uhienzọUhieNzọ ego hara nhata na nọmba uhie 18. French: rouge.Gụọ ya niile → ma na-amụba ya okpukpu abụọ mgbe mfu ọ bụla gasịrị, na-amalite ọzọ mgbe mmeri gasịrị. Oke kachasị elu nke tebụl bụ $500. (a) Mfu ole n’usoro na-agbaji usoro mmụba nzọatụmatụUsoro mmụba nzọIwu ọ bụla na-agbanwe ego nzọ dabere na nsonaazụ gara aga. Usoro na-adịghị mma na-ebuli ya mgbe mfu gasịrị, usoro dị mma mgbe mmeri gasịrị.Gụọ ya niile → ahụ? (b) Gịnị bụ oheremgbakọOhereOhere nke otu nsonaazụ, site na 0 ruo 1 ma ọ bụ 0% ruo 100%. Otu nọmba: 1/37 = 2.70%.Gụọ ya niile → kwa cycle? (c) Gịnị ka a na-atụfu mgbe ngbaji? (d) Kedu otu oke $5,000 si agbanwe azịza ndị ahụ? (e) Oke ọ bụla ọ̀ na-eme ka usoro ahụ wete uru?
Azịza (a)–(c)
Ego nzọ: 10, 20, 40, 80, 160, 320. Nke asaa ga-abụ 640 > 500, ya mere mfu isii na-agbaji ya. (b) p(mfu) = 19/37 na uhie; (19/37)⁶ = 0.0183, ihe dị ka 1.8% kwa cycle. Cycle na-adị ihe dị ka ntụgharị abụọ na nkezi, ya mere oge egwuregwu ntụgharị 100 nwere ihe dị ka cycle 50 na p(opekata mpe otu ngbaji) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, nke bụ unitegoUnitEgo nzọ ntọala nke otu usoro ma ọ bụ onye egwuregwu na-eji arụ ọrụ. A na-akọwa usoro mmụba nzọ na unit ka ha metụta tebụl ọ bụla.Gụọ ya niile → ntọala 63, megide $10 cycle ahụ na-agbalị imeri.
Azịza (d)
Na $5,000: a na-ekwe ego nzọ ruo 2,560; ego nzọ nke iri, 5,120, ka a jụrụ. Mfu itoolu na-agbaji ya: (19/37)⁹ ≈ 0.25% kwa cycle, ihe dị ka 11% nke oge egwuregwu ntụgharị 100. Mfu mgbe ngbaji: 10 + 20 + … + 2,560 = $5,110. Onye egwuregwu ahụ azụtala ọdachi na-adịkarịghị n’ọnụ ahịa nke ọdachi buru ibu karịa. N’ime ọtụtụ oge egwuregwu, mmefu a tụrụ anya nke ngbaji bụ p × mfu ≈ 0.0183 × 630 = $11.5 kwa cycle n’oke $500 na 0.0025 × 5,110 = $12.8 na $5,000; ọ bụ, n’ime nkwụkọ, otu ọnụọgụ ahụ, ọ bụkwa mfu a tụrụ anyamgbakọMfu a tụrụ anyaUru a tụrụ anya mụbara na ngụkọta ego e tinyere na nzọ; nkezi ego a na-atụfu n’otu oge egwuregwu ma ọ bụ n’otu oge.Gụọ ya niile → nke cycle ahụ gbakwunye $10 cycle ndị meriri na-enweta.
Azịza (e)
Mba. Mfu a tụrụ anya kwa cycle = (ngụkọta ego nzọ a tụrụ anya n’ime cycle) / 37, n’oke ọ bụla, n’ihi na nzọ ọ bụla bụ −1/37 nke onwe ya. Oke na-ekpebi naanị otu e si ekesa mfu ahụ n’etiti ọtụtụ obere mmeri na mfu ole na ole buru ibu. N’enweghị oke na ego egwuregwuegoEgo egwuregwuNgụkọta ego e debere maka igwu egwu. Njikwa ego egwuregwu bụ ikpebi ole n’ime ya a ga-etinye n’ihe egwu kwa nzọ na kwa oge egwuregwu.Gụọ ya niile → na-enweghị njedebe, usoro mmụba nzọ ahụ ga-“arụ ọrụ”, n’echiche na ọ gaghị agbaji mgbe ọ bụla, ọ ga-agbakwa ego na-enweghị oke ya na mfu a tụrụ anya nke 2.70% nke ego ahụ na-enweghị oke. Ọ bụghị oke tebụltebụlOke tebụlNzọ kacha nta na nke kachasị ukwuu a na-ekwe. Nke kachasị ukwuu bụ ihe na-akwụsị usoro mmụba nzọ okpukpu abụọ.Gụọ ya niile → na-egbochi MartingaleatụmatụMartingaleUsoro mmụba nzọ na-eme nzọ okpukpu abụọ mgbe mfu ọ bụla gasịrị. Lee ngalaba Atụmatụ maka ihe ọ na-eme na ihe ọ naghị eme.Gụọ ya niile → ịrụ ọrụ. Ọ bụ mgbakọ.
Tebụl abụọ n’ezie
Ihe atụ Thorp bụ tebụl $1 nwere oke kachasị elu nke $1,000, ebe onye egwuregwu na-atụ anya imeri $30 maka nri abalị otu dọla n’otu oge. Ego nzọ 1, 2, 4 … 512 na-ekwe nzọ iri, ya mere mfu iri n’usoro na-agbaji usoro mmụba nzọ ahụ: otu ugboro n’ime cycle 785 na wiil Europe (otu ugboro n’ime 613 na nke America), na ọnụ ahịa $1,023. Cycle iri atọ niile na-aga nke ọma 96.2% nke oge. 3.8% nke ọzọ na-atụfu ihe dị ka $1,000, nsonaazụ nkezi nke mgbede ahụ bụkwa −$9.17, nke bụ kpọmkwem 1/37 nke $339 cycle iri atọ ahụ na-etinye na nkezi. Tebụl $5 nwere oke kachasị elu nke $2,000 adịghị iche: ego nzọ ruo $1,280 na-ekwe nzọ itoolu, ngbaji ahụ na-abịakwa otu ugboro n’ime cycle 403 na wiil Europe (otu ugboro n’ime 323 na nke America) ma na-efu $2,555.
Onye egwuregwu bara ọgaranya
Nye usoro mmụba nzọ ahụ ego egwuregwu miri emi kama oke dị ala: unit 1,000, ntọala otu unit na ntụgharị 1,000 na uhie, na-amalite ọzọ n’otu unit mgbe ọ bụla mmụba okpukpu abụọ na-esote karịrị ihe fọdụrụ. Anyị mere ka simulator gbaa otu nde mgbede dị otu a. Na wiil Europe, 52% na-emecha n’uru, 21% na-atụfu ihe niile, nsonaazụ nkezi bụkwa unit −124. Ọbụna n’egwuregwu na-enweghị efuwiilEfuOtu oghere akwụkwọ ndụ nke nzọ ego hara nhata, dozen ma ọ bụ column ọ bụla na-ekpuchighị; isi mmalite uru ụlọ. A na-akpọkwa ya: single zero, 0.Gụọ ya niile →, ebe nzọ ọ bụla ziri ezi, naanị ihe dị ka 60% ga-emecha n’uru. Mgbede a na-ahụkarị bụ obere mmeri; mgbede ndị na-ejedebe na efu na-akwụ ụgwọ nkezi ahụ.
Ihe mmụta ahụ
Kasino na-etinye oke iji kpachie mgbasamgbakọMgbasaIhe ọtụtụ na-egosi otu nsonaazụ si gbasaa gburugburu uru a tụrụ anya. Ọ dị elu maka straight up, dị ala maka nzọ ego hara nhata.Gụọ ya niile → nke ha, ọ bụghị iji merie usoro. Imeri Martingale bụ ihe na-esote ya ha nwere obi ụtọ inwe, mgbakọ ga-emerikwa ya n’agbanyeghị. Onye egwuregwu chere na oke bụ nsogbu ahụ achọtala ọnụ ugwu dị elu ma were ya maka mkpọda.
Fọmụla
Ajụjụ
Ọ na-enye gị ohere ịlanarị ruo oke dị elu karịa, ebe ngbaji na-adịkarịghị ma buru ibu karịa. Mfu a tụrụ anya kwa dọla agbanweghị.
Ego nzọ 10, 30, 70, 150, 310: a na-eru oke $500 mgbe mfu ise gasịrị, ngbaji ahụ na-efukwa $570. Otu 2.70% ahụ.
Ka otu onye egwuregwu ghara ịgba karịa ihe ego nchekwa kasino nwere ike ịkwụ n’enweghị nsogbu. Njikwa mgbasa.
Egwuregwu Mgbakọ
Buru ụzọ nwaa ha n’onwe gị. A pịachiri azịza ọ bụla; meghee ya mgbe ị dị njikere.
MartingaleatụmatụMartingaleUsoro mmụba nzọ na-eme nzọ okpukpu abụọ mgbe mfu ọ bụla gasịrị. Lee ngalaba Atụmatụ maka ihe ọ na-eme na ihe ọ naghị eme.Gụọ ya niile → na-amalite na $100 n’uhienzọUhieNzọ ego hara nhata na nọmba uhie 18. French: rouge.Gụọ ya niile → ma na-ekwe onwe ya ọ kacha nzọ n kwa cycle: mgbe mfu n n’usoro gasịrị, ọ na-akwụsị ma malite ọzọ. Dee nkezi nsonaazụ otu cycle dịka usoro mgbakọ na n, ma gbakọọ ya maka nzọ 5, 6 na 7 na wiil Europe.
Gosi azịza ya
Azịza Nkezi ya bụ 100 × (1 − (38/37)ⁿ): −$14.26 maka nzọ 5, −$17.35 maka 6, −$20.52 maka 7. Ikwe nzọ ndị ọzọ na-eme ka nkezi cycle ka njọ, ọ bụghị ka mma.
Cycle na-emeri $100 ma ọ bụrụ na nzọ n niile atụfughị, nke na-eme na ohere qⁿ, ebe q = 19/37. Mgbe ha niile tụfuru, ego nzọ 100, 200 … 100 × 2ⁿ⁻¹ na-agbakọta ruo 100 × (2ⁿ − 1). Ya mere nkezi bụ 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Ebe a 2q = 38/37, ntakịrị karịa 1, ya mere (2q)ⁿ na-eto na n, nkezi na-adakwa: n = 5 na-enye 1.1426 na −$14.26; n = 6 na-enye 1.1735 na −$17.35; n = 7 na-enye 1.2052 na −$20.52. Ubube dị ogologo na-efu obere ugboro, na 3.6%, 1.8% na 0.9% nke cycle, mana mfu ọ bụla dị okpukpu abụọ, ego ka ukwuu na-agafekwa n’ofe tebụl na cycle ọ bụla, na 2.70% nke nzọ ọ bụla.
Na wiil America 2q = 40/38, ọnụọgụ ndị ahụ bụkwa −$29.24, −$36.04 na −$43.20 (Taylor, 2021). N’egwuregwu na-enweghị efuwiilEfuOtu oghere akwụkwọ ndụ nke nzọ ego hara nhata, dozen ma ọ bụ column ọ bụla na-ekpuchighị; isi mmalite uru ụlọ. A na-akpọkwa ya: single zero, 0.Gụọ ya niile → 2q = 1, cycle ọ bụla na-enwekwa nkezi kpọmkwem efu n’agbanyeghị ogologo ubube ahụ.
“Grand Martingale” na-amụba ego nzọ okpukpu abụọ mgbe mfu gasịrị ma gbakwunye otu unitegoUnitEgo nzọ ntọala nke otu usoro ma ọ bụ onye egwuregwu na-eji arụ ọrụ. A na-akọwa usoro mmụba nzọ na unit ka ha metụta tebụl ọ bụla.Gụọ ya niile →, ya mere ego nzọ na-aga 1, 3, 7, 15, 31 … Ole ka ị nọ n’elu mgbe mmeri mbụ bịara na nzọ nke 1, nke 2, nke 3 ma ọ bụ nke k? Na unit $10 na oke $500, ugboro ole ka cycle na-agbaji, ego ole ka cycle na-efu na nkezi?
Gosi azịza ya
Azịza Mmeri na nzọ nke k na-ahapụ gị unit k n’elu: +1, +2, +3, wdg. Na $10 na $500, cycle na-agbaji mgbe mfu ise gasịrị, 3.57% nke cycle, maka −$570. Na nkezi cycle na-efu $2.32 n’ime $85.73 e tinyere: 2.70%.
Ego nzọ nke k bụ unit 2ᵏ − 1: 1, 3, 7, 15, 31. Ego nzọ k − 1 mbụ na-agbakọta ruo 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Mmeri na nzọ nke k na-akwụ 2ᵏ − 1, ya mere cycle ahụ na-akwụsị (2ᵏ − 1) − (2ᵏ − k − 1) = unit k n’elu: otu unit ọzọ maka mfu ọ bụla tupu mmeri ahụ.
Na $10 kwa unit, ego nzọ bụ $10, $30, $70, $150 na $310; nke na-esote, $630, karịrị oke $500. Mfu ise n’usoro, (19/37)⁵ = 3.57%, na-efu 10 + 30 + 70 + 150 + 310 = $570.
Nkezi: mmeri ndị ahụ na-eweta $10 × k na ohere (19/37)ᵏ⁻¹ × 18/37 maka k = 1 ruo 5, ngbaji ahụ na-ewe $570 na ohere 3.57%; ọnụ −$2.32. Ego e tinyere bụ nkezi $85.73 kwa cycle, 2.32/85.73 = 2.70%. Martingale nkịtị na otu tebụl ahụ na-agbaji mgbe mfu isii gasịrị, 1.83% nke cycle, maka −$630, ma na-efu $1.74 n’ime $64.20 e tinyere: 2.70% ọzọ.
Onye egwuregwu na-eme atụmatụ Martingale n’uhie na nzọ mbụ $1 na tebụl nwere oke $1,000, ya mere nzọ iri ga-ekwe omume: $1, $2, $4, wdg ruo $512. Ugboro ole ka cycle na-atụfu iri niile, ogologo oge ole ka nke ahụ na-ewe na nkezi, ugboro ole ka mgbede nke nzọ 200 na-ejedebe otu ahụ?
Gosi azịza ya
Azịza Cycle na-atụfu iri niile ihe dị ka otu ugboro n’ime 784, nke na-efu $1,023. Nke ahụ bụ ihe dị ka nzọ 1,600 na nkezi, mgbede nke nzọ 200 na-akụkwa mgbidi ahụ ihe dị ka 11% nke oge. Cycle ka na-efu $0.31 na nkezi n’ime $11.31 e tinyere: 2.70%.
Nzọ n’uhie na-efu na ohere 19/37, gụnyere efu. Iri n’usoro: (19/37)¹⁰ ≈ 0.00128, ihe dị ka 1 n’ime 784. Ego nzọ 1 + 2 + 4 + … + 512 na-agbakọta ruo $1,023.
Cycle na-ewe nzọ 2.05 na nkezi, ya mere cycle 784 na-ewe ihe dị ka nzọ 1,600. Ma a gụọ ya ntụgharị na ntụgharị, usoro mfu iri na-apụta n’ebe ọ bụla n’ime nzọ 200 11.3% nke oge.
N’etiti ọdachi, onye egwuregwu ahụ na-emeri $1 kwa cycle: ihe dị ka $783 tupu nke ọjọọ ahụ, mgbe ahụ −$1,023, ihe dị ka −$240 n’ime oge ahụ niile. Kwa cycle: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Ego e tinyere kwa cycle: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Ma −0.31/11.31 = −2.70%.
Tụnyere nzọ mbụ $10 na oke $500, nzọ isii: mgbidi ahụ na-abịa otu ugboro n’ime cycle 55, mgbede nke nzọ 200 na-ezute ya 84% nke oge. Oke dị elu na-eme ka ọdịda ahụ dị ụkọ ma buo ibu, ọ bụghị ka ọ dị ọnụ ala.
Ihe ọchị
Ọ na-adịkarịghị ma na-ebu ibu karịa, n’oke ọ bụla. Ngwakọta ha na-anọgide otu.
Isi Mmalite
- 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.