Palopalema: Ko e Fakangatangata ʻo e Tēpile
Ko ha Martingale ʻo e $10 ʻi ha tēpile ʻo e $500. ʻOku faʻa fiha hono maumau, ʻoku mole ʻe fiha ʻi hono maumau, pea ʻoku tokoni nai ʻa e hiki hake ʻo e fakangatangata?
ʻOku maumau ia ʻi ha mole ʻe ono hokohoko: 1.8% ʻi he takai, ʻi he meimei 60% ʻo e ngaahi taimi vaʻinga ʻo e vilo ʻe 100. ʻOku fakamole ʻe he maumau ʻa e $630. ʻOku fakatātātaha ange ʻe he hiki hake ʻo e fakangatangata ki he $5,000 ʻa e maumau (0.25% ʻi he takai) pea lahi ange ia ($5,110). Ko e mole ʻamanekina ʻi he tola ʻe taha naʻe peti ko e 2.70% ʻi ha fakangatangata pē.
Ko e palopalema
ʻOku peti ʻe ha tokotaha vaʻinga ʻa e $10 ki he kulokulapetiKulokulaPeti totongi tatau ki he fika kulokula ʻe 18. Faka-Falanisē: rouge.Lau kotoa → pea liunga ua ʻi he hili ʻa e mole taki taha, ʻo toe kamata foʻou hili ha ikuna. Ko e peti lahi taha ʻo e tēpile ko e $500. (a) Ko e mole hokohoko ʻe fiha ʻoku maumau ai ʻa e fakalakalaka petifoungaFakalakalaka petiKo ha lao pē ʻoku ne liliu ʻa e lahi ʻo e peti ʻo fakatatau ki he ngaahi ola kimuʻa. Ko e ngaahi fakalakalaka fakafepaki ʻoku nau hiki hake ia hili ha mole, ko e ngaahi fakalakalaka fakapapau hili ha ikuna.Lau kotoa →? (b) Ko e hā ʻa e ngalingalimatematikaNgalingaliKo e faingamālie ʻo ha ola, mei he 0 ki he 1 pe 0% ki he 100%. Fika ʻe taha: 1/37 = 2.70%.Lau kotoa → ʻi he takai? (c) Ko e hā ʻoku mole ʻi he maumau? (d) ʻOku liliu fēfē ʻe ha fakangatangata ʻo e $5,000 ʻa e ngaahi tali? (e) ʻOku ʻi ai ha fakangatangata te ne ʻai ke maʻu tupu ʻa e sisitemi?
Ngaahi solovaʻanga (a)–(c)
Ngaahi peti: 10, 20, 40, 80, 160, 320. Ko hono fitu ʻe 640 > 500, ko ia ʻoku maumau ia ʻi ha mole ʻe ono. (b) p(mole) = 19/37 ʻi he kulokula; (19/37)⁶ = 0.0183, meimei 1.8% ʻi he takai. ʻOku lōloa ʻa e takai ʻi he meimei vilo ʻe ua ʻi he ʻavalisi, ko ia ʻoku ʻi ai ʻi ha taimi vaʻinga ʻo e vilo ʻe 100 ʻa e meimei takai ʻe 50 pea ko e p(maumau ʻo siʻi taha ʻi he tuʻo taha) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, ʻa ia ko e iunitipaʻangaIunitiKo e peti tefito ʻoku ngāueʻaki ʻe ha sisitemi pe ha tokotaha vaʻinga. ʻOku fakamatalaʻi ʻa e ngaahi fakalakalaka peti ʻaki ʻa e iuniti koeʻuhi ke ngāue ʻi ha tēpile pē.Lau kotoa → kamataʻanga ʻe 63, fakatatau ki he $10 naʻe feinga ʻa e takai ke ikunaʻi.
Solovaʻanga (d)
ʻI he $5,000: ʻoku ngofua ʻa e ngaahi peti ʻo aʻu ki he 2,560; ʻoku taʻofi ʻa hono hongofulu ʻo e peti, 5,120. ʻOku maumau ia ʻi ha mole ʻe hiva: (19/37)⁹ ≈ 0.25% ʻi he takai, meimei 11% ʻo e ngaahi taimi vaʻinga ʻo e vilo ʻe 100. Mole ʻi he maumau: 10 + 20 + … + 2,560 = $5,110. Kuo fakatau ʻe he tokotaha vaʻinga ha fakatuʻutāmaki tātātaha ange ʻi he totongi ʻo ha fakatuʻutāmaki lahi ange. ʻI ha ngaahi taimi vaʻinga lahi, ko e fakamole ʻamanekina ʻo e ngaahi maumau ko e p × mole ≈ 0.0183 × 630 = $11.5 ʻi he takai ʻi he fakangatangata ʻo e $500 pea 0.0025 × 5,110 = $12.8 ʻi he $5,000; ʻoku na ofi ka ʻoku ʻikai tatau, pea ko e taki taha ko e mole ʻamanekinamatematikaMole ʻamanekinaKo e mahuʻinga ʻamanekina liunga ʻaki ʻa e fakakātoa ʻo e paʻanga naʻe peti ʻaki; ko e ʻavalisi ʻo e paʻanga ʻoku mole ʻi ha taimi vaʻinga pe ha vahaʻa taimi.Lau kotoa → ia ʻo e takai fakataha mo e $10 ʻoku maʻu ʻe he ngaahi takai ikuna.
Solovaʻanga (e)
ʻIkai. Mole ʻamanekina ʻi he takai = (fakakātoa ʻamanekina ʻo e peti ʻi he takai) / 37, ʻi ha fakangatangata pē, koeʻuhí ko e peti taki taha ko e −1/37 ʻo ia tonu. ʻOku fakapapauʻi pē ʻe he fakangatangata ʻa e founga ʻoku tufaki ai ʻa e mole ʻi he vahaʻa ʻo ha ngaahi ikuna siʻi lahi mo ha ngaahi mole lahi siʻi. ʻI he ʻikai ha fakangatangata mo ha paʻanga vaʻingapaʻangaPaʻanga vaʻingaKo e fakakātoa ʻo e paʻanga kuo vahe ke vaʻinga ʻaki. Ko e puleʻi ʻo e paʻanga vaʻinga ko e fili pe ko e hā hono lahi ʻoku tuku ki he fakatuʻutāmaki ʻi he peti takitaha mo e taimi vaʻinga takitaha.Lau kotoa → taʻengata, ʻe “ngāue” ʻa e fakalakalaka peti, ʻi he ʻuhinga ʻe ʻikai ʻaupito maumau ia, pea te ne peti foki ha lahi taʻengata mo ha mole ʻamanekina ʻo e 2.70% ʻo e lahi taʻengata ko iá. ʻOku ʻikai ko e fakangatangata ʻo e tēpiletēpileFakangatangata ʻo e tēpileKo e peti siʻi taha mo e peti lahi taha ʻoku ngofua. Ko e peti lahi taha ʻoku ne taʻofi ʻa e ngaahi fakalakalaka peti fakaua.Lau kotoa → ʻoku ne taʻofi ʻa e MartingalefoungaMartingaleFakalakalaka peti ʻoku ne fakaua ʻa e peti hili ʻa e mole takitaha. Vakai ki he konga Ngaahi Founga ki he meʻa ʻoku ne fai mo e meʻa ʻoku ʻikai te ne fai.Lau kotoa → mei he ngāue. Ko e fika ia.
Tēpile moʻoni ʻe ua
Ko e sīpinga ʻa Thorp ko ha tēpile $1 mo ha peti lahi taha ʻo e $1,000, ʻa ia ʻoku ʻamanaki ai ha tokotaha vaʻinga ke ikunaʻi ʻa e $30 ki he kai efiafi ʻo tola ʻe taha ʻi he taimi. ʻOku ngofua ʻe he ngaahi peti 1, 2, 4 … 512 ha peti ʻe hongofulu, ko ia ʻoku maumau ʻa e fakalakalaka peti ʻi ha mole ʻe hongofulu hokohoko: tuʻo taha ʻi he takai ʻe 785 ʻi ha veʻeteka ʻIulope (tuʻo taha ʻi he 613 ʻi ha veʻeteka ʻAmelika), ʻi ha fakamole ʻo e $1,023. ʻOku lavameʻa kotoa ʻa e takai ʻe tolungofulu ʻi he 96.2% ʻo e taimi. ʻOku mole ʻa e meimei $1,000 ʻi he toenga 3.8%, pea ko e ola ʻavalisi ʻo e efiafi ko e −$9.17, ʻa ia ko e 1/37 tofu pē ia ʻo e $339 ʻoku peti ʻe he takai ʻe tolungofulu ʻi he ʻavalisi. ʻOku ʻikai kehe ha tēpile $5 mo ha peti lahi taha ʻo e $2,000: ʻoku ngofua ʻe he ngaahi peti ʻo aʻu ki he $1,280 ha peti ʻe hiva, pea ʻoku hoko ʻa e maumau ʻo tuʻo taha ʻi he takai ʻe 403 ʻi ha veʻeteka ʻIulope (tuʻo taha ʻi he 323 ʻi ha veʻeteka ʻAmelika) pea ʻoku fakamole ʻa e $2,555.
Ko ha tokotaha vaʻinga koloaʻia
ʻOange ki he fakalakalaka peti ha paʻanga vaʻinga lahi ʻo fetongi ʻaki ha fakangatangata māʻulalo: iuniti ʻe 1,000, kamataʻanga ʻo e iuniti ʻe taha mo ha vilo ʻe 1,000 ki he kulokula, ʻo toe kamata ʻi he iuniti ʻe taha ʻi he taimi kotoa pē ʻoku lahi ange ai ʻa e liunga ua hoko ʻi he meʻa ʻoku toe. Naʻa mau faʻifaʻitakiʻi ha efiafi pehē ʻe miliona ʻe taha. ʻI ha veʻeteka ʻIulope, ʻoku ngata ʻi muʻa ʻa e 52%, ʻoku mole kotoa ʻa e 21%, pea ko e ola ʻavalisi ko e iuniti ʻe −124. Neongo ʻi ha vaʻinga ʻoku ʻikai ha noaveʻetekaNoaKo e luo lanu mata pē ʻe taha ʻoku ʻikai kāpui ʻe ha peti totongi tatau, peti tasini pe peti kōlomu; ko e tupuʻanga ia ʻo e tuʻunga lelei ʻo e kasino. ʻOku toe ui: noa ʻe taha, 0.Lau kotoa →, ʻa ia ʻoku faitotonu ai ʻa e peti kotoa pē, ko e meimei 60% pē ʻe ngata ʻi muʻa. Ko e efiafi angamaheni ko ha kiʻi ikuna; ko e ʻavalisi ʻoku totongi ia ʻe he ngaahi efiafi ʻoku ngata ʻi he noa.
Ko e lēsoni
ʻOku fokotuʻu ʻe he ngaahi kasino ʻa e ngaahi fakangatangata ke fakangatangata ʻenau vēlianimatematikaVēlianiKo ha fua ʻo hono mafola ʻo e ngaahi ola takatakai ʻi he mahuʻinga ʻamanekina. Māʻolunga ki he peti fika ʻe taha, māʻulalo ki he ngaahi peti totongi tatau.Lau kotoa → pē ʻanautolu, ʻo ʻikai ke ikunaʻi ʻa e ngaahi sisitemi. Ko e ikunaʻi ʻo e Martingale ko ha ola fakatafaʻaki pē ʻoku nau fiefia ke maʻu, pea ʻe ikunaʻi pē ia ʻe he fika. Ko ha tokotaha vaʻinga ʻoku ne pehē ko e fakangatangata ʻa e palopalema, kuo ne ʻilo ʻa e mato ʻo e makatuʻu kae fakakaukau ko e hifo māmālie ia.
Fomula
Ngaahi fehuʻi
ʻOku ne ʻai ke ke moʻui ʻo aʻu ki ha fakangatangata māʻolunga ange, ʻa ia ʻoku tātātaha ange mo lahi ange ai ʻa e maumau. ʻOku ʻikai liliu ʻa e mole ʻamanekina ʻi he tola ʻe taha.
Ngaahi peti 10, 30, 70, 150, 310: ʻoku aʻu ki he fakangatangata ʻo e $500 hili ha mole ʻe nima, pea ʻoku fakamole ʻe he maumau ʻa e $570. Ko e 2.70% tatau pē.
Koeʻuhí ke ʻoua naʻa lava ha tokotaha vaʻinga ʻe taha ʻo peti lahi ange ʻi he meʻa ʻoku lava ʻe he paʻanga tauhi ʻa e kasino ʻo totongi faingofua. Puleʻi ʻo e vēliani.
Vaʻinga mo e Matematika
ʻUluaki feinga ke ke fakaleleiʻi ʻe koe. Kuo pelu ʻa e tali takitaha; fakaava ia ʻi hoʻo mateuteu.
ʻOku kamata ha MartingalefoungaMartingaleFakalakalaka peti ʻoku ne fakaua ʻa e peti hili ʻa e mole takitaha. Vakai ki he konga Ngaahi Founga ki he meʻa ʻoku ne fai mo e meʻa ʻoku ʻikai te ne fai.Lau kotoa → ʻi he $100 ki he kulokulapetiKulokulaPeti totongi tatau ki he fika kulokula ʻe 18. Faka-Falanisē: rouge.Lau kotoa → pea ʻoku ne ngofua pē ki ha peti ʻe n lahi taha ʻi he takai: hili ha mole ʻe n hokohoko ʻoku ne tukuange pea kamata foʻou. Tohi ʻa e ola ʻavalisi ʻo ha takai ʻe taha ko ha fomula ʻi he n, pea fakafuofua ia ki he peti ʻe 5, 6 mo e 7 ʻi ha veʻeteka ʻIulope.
Fakahā ʻa e tali
Tali Ko e ʻavalisi ko e 100 × (1 − (38/37)ⁿ): −$14.26 ki he peti ʻe 5, −$17.35 ki he 6 pea −$20.52 ki he 7. ʻOku kovi ange ʻa e takai ʻavalisi ʻi hono ngofua ha ngaahi peti lahi ange, ʻo ʻikai lelei ange.
ʻOku ikuna ʻe ha takai ʻa e $100 tukukehe kapau ʻe mole kotoa ʻa e peti ʻe n, ʻa ia ʻoku hoko ʻi he ngalingali ʻo e qⁿ, ʻa ia ko e q = 19/37. ʻI he mole kotoa, ʻoku tānaki ʻa e ngaahi peti 100, 200 … 100 × 2ⁿ⁻¹ ki he 100 × (2ⁿ − 1). Ko ia ko e ʻavalisi ko e 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
ʻI heni ko e 2q = 38/37, ʻoku siʻi lahi ange ʻi he 1, ko ia ʻoku tupu ʻa e (2q)ⁿ mo e n pea holo ʻa e ʻavalisi: ʻoku ʻomi ʻe he n = 5 ʻa e 1.1426 mo e −$14.26; ʻoku ʻomi ʻe he n = 6 ʻa e 1.1735 mo e −$17.35; ʻoku ʻomi ʻe he n = 7 ʻa e 1.2052 mo e −$20.52. ʻOku siʻi ange hono mole ʻo ha tuʻunga lōloa ange, ʻi he 3.6%, 1.8% mo e 0.9% ʻo e ngaahi takai, ka ʻoku liunga ua ʻa e lahi ʻo e mole takitaha, pea ʻoku lahi ange ʻa e paʻanga ʻoku kolosi ʻi he tēpile ʻi he takai takitaha, ʻi he 2.70% ʻo e peti kotoa pē.
ʻI ha veʻeteka ʻAmelika ko e 2q = 40/38 pea ko e ngaahi fika ko e −$29.24, −$36.04 mo e −$43.20 (Taylor, 2021). ʻI ha vaʻinga ʻoku ʻikai ha noaveʻetekaNoaKo e luo lanu mata pē ʻe taha ʻoku ʻikai kāpui ʻe ha peti totongi tatau, peti tasini pe peti kōlomu; ko e tupuʻanga ia ʻo e tuʻunga lelei ʻo e kasino. ʻOku toe ui: noa ʻe taha, 0.Lau kotoa → ko e 2q = 1, pea ko e ʻavalisi ʻo e takai kotoa pē ko e noa tofu pē neongo pe ʻoku lōloa fēfē ʻa e tuʻunga.
ʻOku fakaliunga ua ʻe he “Martingale lahi” (grand Martingale) ʻa e peti hili ha mole pea tānaki ha iunitipaʻangaIunitiKo e peti tefito ʻoku ngāueʻaki ʻe ha sisitemi pe ha tokotaha vaʻinga. ʻOku fakamatalaʻi ʻa e ngaahi fakalakalaka peti ʻaki ʻa e iuniti koeʻuhi ke ngāue ʻi ha tēpile pē.Lau kotoa → ʻe taha, ko ia ʻoku pehē ʻa e ngaahi peti 1, 3, 7, 15, 31 … ʻOku fiha hoʻo tupu ʻi he hoko mai ʻa e ʻuluaki ikuna ʻi he peti hono 1, 2, 3 pe k? ʻI ha iuniti ʻo e $10 mo ha lahi taha ʻo e $500, ʻoku maumau tuʻo fiha ha takai, pea ko e hā ʻoku fakamole ʻe ha takai ʻi he ʻavalisi?
Fakahā ʻa e tali
Tali Ko ha ikuna ʻi he peti hono k ʻoku tuku ai koe ke ke maʻu ʻa e iuniti ʻe k: +1, +2, +3 pea pehē atu. ʻI he $10 mo e $500 ʻoku maumau ʻa e takai hili ʻa e mole ʻe nima, 3.57% ʻo e ngaahi takai, ki he −$570. ʻI he ʻavalisi ʻoku mole ʻe ha takai ʻa e $2.32 mei he $85.73 naʻe peti: 2.70%.
Ko e peti hono k ko e iuniti ʻe 2ᵏ − 1: 1, 3, 7, 15, 31. ʻOku tānaki ʻa e ʻuluaki peti ʻe k − 1 ki he 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. ʻOku totongi ʻe ha ikuna ʻi he peti hono k ʻa e 2ᵏ − 1, ko ia ʻoku ngata ʻa e takai ʻi he (2ᵏ − 1) − (2ᵏ − k − 1) = iuniti ʻe k ʻoku tupu: ko ha iuniti ʻe taha lahi ange ki he mole kotoa pē ʻi muʻa ʻi he ikuna.
ʻI he $10 ʻi he iuniti, ko e ngaahi peti ko e $10, $30, $70, $150 mo e $310; ko e hoko, $630, ʻoku laka ia ʻi he lahi taha ʻo e $500. Ko e mole ʻe nima hokohoko, (19/37)⁵ = 3.57%, ʻoku fakamole ʻa e 10 + 30 + 70 + 150 + 310 = $570.
ʻAvalisi: ʻoku ʻomi ʻe he ngaahi ikuna ʻa e $10 × k ʻi he ngalingali ʻo e (19/37)ᵏ⁻¹ × 18/37 ki he k = 1 ki he 5, pea ʻoku toʻo ʻe he maumau ʻa e $570 ʻi he ngalingali ʻo e 3.57%; fakataha ko e −$2.32. Ko e ʻavalisi ʻo e paʻanga ʻoku peti ko e $85.73 ʻi he takai, pea ko e 2.32/85.73 = 2.70%. ʻOku maumau ʻa e Martingale angamaheni ʻi he tēpile tatau hili ʻa e mole ʻe ono, 1.83% ʻo e ngaahi takai, ki he −$630, pea mole ʻa e $1.74 mei he $64.20 naʻe peti: 2.70% foki.
ʻOku palani ha tokotaha vaʻinga ki ha Martingale ki he kulokula mo ha ʻuluaki peti ʻo e $1 ʻi ha tēpile ʻoku ko e $1,000 hono fakangatangata, ko ia ʻoku lava ʻa e peti ʻe hongofulu: $1, $2, $4 pea pehē atu ʻo aʻu ki he $512. ʻOku mole tuʻo fiha ʻe ha takai ʻa e hongofulu kotoa, ʻoku fuoloa fēfē ia ʻi he ʻavalisi, pea ʻoku ngata pehē tuʻo fiha ha efiafi ʻo e peti ʻe 200?
Fakahā ʻa e tali
Tali ʻOku mole ʻe ha takai ʻa e hongofulu kotoa tuʻo taha nai ʻi he 784, ʻa ia ʻoku fakamole ai ʻa e $1,023. Ko e peti ʻe 1,600 nai ia ʻi he ʻavalisi, pea ʻoku tau ha efiafi ʻo e peti ʻe 200 ki he ʻā ʻi he 11% nai ʻo e taimi. ʻOku kei mole pē ʻe ha takai ʻa e $0.31 ʻi he ʻavalisi mei he $11.31 naʻe peti: 2.70%.
ʻOku mole ha peti ki he kulokula ʻi he ngalingali ʻo e 19/37, kau ai ʻa e noa. Hongofulu hokohoko: (19/37)¹⁰ ≈ 0.00128, ʻo ofi ki he 1 ʻi he 784. ʻOku tānaki ʻa e ngaahi peti 1 + 2 + 4 + … + 512 ki he $1,023.
ʻOku fiemaʻu ʻe ha takai ʻa e peti ʻe 2.05 ʻi he ʻavalisi, ko ia ʻoku fiemaʻu ʻe he takai ʻe 784 ʻa e peti ʻe 1,600 nai. ʻO lau ʻi he vilo ki he vilo, ʻoku hā ha hokohoko ʻo e mole ʻe hongofulu ʻi ha feituʻu ʻi he peti ʻe 200 ʻi he 11.3% ʻo e taimi.
ʻI he vahaʻa ʻo e ngaahi fakatamaki ʻoku ikuna ʻe he tokotaha vaʻinga ʻa e $1 ʻi he takai: ʻo ofi ki he $783 ʻi muʻa ʻi he takai kovi, pea −$1,023, ʻo ofi ki he −$240 ʻi he vahaʻa taimi kotoa. ʻI he takai: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Paʻanga ʻoku peti ʻi he takai: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Pea ko e −0.31/11.31 = −2.70%.
Fakatatau mo ha ʻuluaki peti ʻo e $10 ʻi ha fakangatangata ʻo e $500, peti ʻe ono: ʻoku hoko mai ʻa e ʻā tuʻo taha ʻi he takai ʻe 55, pea ʻoku fetaulaki mo ia ha efiafi ʻo e peti ʻe 200 ʻi he 84% ʻo e taimi. ʻOku ngaohi ʻe ha fakangatangata māʻolunga ange ʻa e tō ke tātātaha ange mo lahi ange, ʻo ʻikai ke maʻamaʻa ange.
Meʻa fakaoli
Tātātaha ange mo lahi ange, ʻi he fakangatangata kotoa pē. ʻOku ʻikai liliu ʻa e ola ʻo hono fakaliunga.
Ngaahi Maʻuʻanga
- 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.