Matsala: iyakar teburi
Tsarin Martingale da $10 a teburi mai iyakar $500. Sau nawa yake karyewa, nawa yake asara idan ya karye, kuma shin ɗaga iyakar yana taimakawa?
Asara shida a jere suna karya shi: 1.8% a kowane zagayen tsarin, kusan 60% na zaman wasa na juye-juye 100. Karyewar tana kashe $630. Ɗaga iyakar zuwa $5,000 yana sa karyewar ta zama ba kasafai ba (0.25% a kowane zagaye) amma mafi girma ($5,110). Asarar da ake tsammani a kan kowace dala da aka saka 2.70% ce a kowace iyaka.
Tambayar
Ɗan wasa yana saka $10 a kan jacacaJaCacar ɗaya zuwa ɗaya (even-money) a kan lambobi ja 18. A Faransanci: rouge.Karanta gaba ɗaya → kuma yana ninka kuɗin caca bayan kowace asara, sannan ya koma farko bayan nasara. Iyakar teburin $500 ce. (a) Asara nawa a jere ke karya jerin ƙarin cacadabaraJerin ƙarin cacaDuk wata ƙa'ida da ke canza kuɗin caca bisa sakamakon da ya gabata. Jerin ƙarin caca bayan asara yana ɗaga shi bayan an rasa, na bayan nasara kuma bayan an ci.Karanta gaba ɗaya →? (b) Mene ne yiwuwarlissafiYiwuwaDamar faruwar wani sakamako, daga 0 zuwa 1 ko daga 0% zuwa 100%. Lamba ɗaya: 1/37 = 2.70%.Karanta gaba ɗaya → a kowane zagaye? (c) Me ake asara a lokacin karyewa? (d) Ta yaya iyakar $5,000 ke canja amsoshin? (e) Shin akwai wata iyaka da za ta sa tsarin ya kawo riba?
Amsoshin (a)–(c)
Kuɗin caca: 10, 20, 40, 80, 160, 320. Na bakwai zai zama 640 > 500, don haka asara shida suna karya shi. (b) p(asara) = 19/37 a kan ja; (19/37)⁶ = 0.0183, kusan 1.8% a kowane zagaye. Zagaye yana ɗaukar kusan juye-juye biyu a matsakaici, don haka zaman wasa na juye-juye 100 yana da kusan zagaye 50 kuma p(aƙalla karyewa ɗaya) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, wato raka'akuɗiRaka'aKuɗin caca na tushe da tsari ko ɗan wasa ke aiki da shi. Ana bayyana jerin ƙarin caca a raka'a don su yi aiki a kowane tebur.Karanta gaba ɗaya → 63 na tushe, a kan $10 da zagayen ke ƙoƙarin ci.
Amsar (d)
A $5,000: an yarda da kuɗin caca har zuwa 2,560; kuɗin caca na goma, 5,120, ba a karɓa. Asara tara suna karya shi: (19/37)⁹ ≈ 0.25% a kowane zagaye, kusan 11% na zaman wasa na juye-juye 100. Asara a lokacin karyewa: 10 + 20 + … + 2,560 = $5,110. Ɗan wasan ya sayi bala'i da ba kasafai yake faruwa ba a farashin wanda ya fi girma. A cikin zaman wasa da yawa, kuɗin da ake tsammanin karyewa za ta kashe shi ne p × asara ≈ 0.0183 × 630 = $11.5 a kowane zagaye a iyakar $500 da 0.0025 × 5,110 = $12.8 a $5,000; kusan lamba ɗaya ce idan an zagaye ta, kuma ita ce asarar da ake tsammanilissafiAsarar da ake tsammaniƘimar da ake tsammani sau jimillar kuɗin da aka yi caca da su; matsakaicin adadin da ake rasawa a cikin zaman wasa ko wani lokaci.Karanta gaba ɗaya → ta zagayen tare da $10 da zagayen da suka ci nasara ke samu.
Amsar (e)
A'a. Asarar da ake tsammani a kowane zagaye = (jimillar kuɗin caca da ake tsammani a zagayen) / 37, a kowace iyaka, saboda kowace caca −1/37 ce ta kanta. Iyakar tana yanke ne kawai yadda asarar ke rarrabuwa tsakanin ƙananan nasarori da yawa da manyan asarori kaɗan. Ba tare da iyaka ba kuma da jarin cacakuɗiJarin cacaJimillar kuɗin da aka ware don wasa. Sarrafa jarin caca shi ne yanke shawarar nawa za a sa a haɗari a kowace caca da kowane zaman wasa.Karanta gaba ɗaya → mara iyaka, jerin ƙarin cacar zai “yi aiki”, ta ma'anar cewa ba zai taɓa karyewa ba, kuma zai saka adadi mara iyaka tare da asarar da ake tsammani ta 2.70% na wannan adadi mara iyaka. Ba iyakar teburiteburIyakar teburiMafi ƙanƙanta da mafi girman cacar da aka yarda da ita. Mafi girman ita ce ke dakatar da jerin ƙarin caca masu ninkawa.Karanta gaba ɗaya → ce ke hana MartingaledabaraMartingaleJerin ƙarin caca da ke ninka caca bayan kowace asara. Ku duba sashen “Dabarun caca na roulette” don ganin abin da yake yi da wanda ba ya yi.Karanta gaba ɗaya → aiki ba. Lissafi ne.
Teburai biyu na gaske
Misalin Thorp teburi ne na $1 mai iyakar $1,000, inda ɗan wasa ke fatan cin $30 don abincin dare dala ɗaya-ɗaya. Kuɗin caca 1, 2, 4 … 512 suna ba da damar cace-cace goma, don haka asara goma a jere suna karya jerin ƙarin cacar: sau ɗaya cikin zagaye 785 a dabaran Turai (sau ɗaya cikin 613 a na Amurka), a farashin $1,023. Zagaye talatin duk suna nasara 96.2% na lokaci. Sauran 3.8% suna asarar kusan $1,000, kuma matsakaicin sakamakon yammacin −$9.17 ne, wanda daidai yake da 1/37 na $339 da zagaye talatin ke sakawa a matsakaici. Teburi na $5 mai iyakar $2,000 ba shi da bambanci: kuɗin caca har zuwa $1,280 suna ba da damar cace-cace tara, kuma karyewa tana zuwa sau ɗaya cikin zagaye 403 a dabaran Turai (sau ɗaya cikin 323 a na Amurka) kuma tana kashe $2,555.
Ɗan wasa mai arziki
Ku ba jerin ƙarin cacar jarin caca mai zurfi maimakon ƙaramar iyaka: raka'a 1,000, tushe na raka'a ɗaya da juye-juye 1,000 a kan ja, ana sake farawa da raka'a ɗaya duk lokacin da ninki na gaba ya fi abin da ya rage. Mun yi kwaikwayon irin waɗannan yammaci miliyan ɗaya. A dabaran Turai 52% suna ƙarewa da riba, 21% suna asarar komai, kuma matsakaicin sakamako raka'a −124 ne. Ko a wasa marar sifilidabaranSifiliRamin kore guda ɗaya da babu cacar ɗaya zuwa ɗaya, ta dozin ko ta ginshiƙi da ke rufe shi; tushen fifikon gidan caca. Ana kuma kiransa: sifili ɗaya, 0.Karanta gaba ɗaya →, inda kowace caca ta adalci ce, kusan 60% kawai za su ƙare da riba. Yammacin da aka saba gani ƙaramar nasara ce; yammacin da ke ƙarewa a sifili ne ke biyan matsakaicin.
Darasin
Gidajen caca suna sa iyaka don su taƙaita bambancin sakamakolissafiBambancin sakamakoMa'aunin yadda sakamako ke yaɗuwa a kewayen ƙimar da ake tsammani. Yana da girma ga straight up, ƙanƙanta ga cace-cacen ɗaya zuwa ɗaya.Karanta gaba ɗaya → nasu, ba don su kayar da tsare-tsare ba. Kayar da Martingale wani sakamako ne na gefe da suke murna da shi, kuma lissafi zai kayar da shi ko ta yaya. Ɗan wasa da ke ganin iyakar ce matsalar ya gano bakin tudun da ake faɗuwa, amma ya ɗauka shi ne gangaren da ke kai mutum can.
Ƙa'idar lissafi
Tambayoyi
Yana ba ku damar tsira har zuwa iyaka mafi girma, inda karyewa ba kasafai take ba amma ta fi girma. Asarar da ake tsammani a kan kowace dala ba ta canjawa.
Kuɗin caca 10, 30, 70, 150, 310: ana kai iyakar $500 bayan asara biyar, kuma karyewar tana kashe $570. Har yanzu dai 2.70% ne.
Don kada ɗan wasa ɗaya ya saka fiye da abin da ajiyar gidan caca za ta iya biya cikin sauƙi. Sarrafa bambancin sakamako.
Nishaɗi da lissafi
Ku fara gwada waɗannan da kanku. Kowace mafita a naɗe take; ku buɗe ta idan kun shirya.
Tsarin MartingaledabaraMartingaleJerin ƙarin caca da ke ninka caca bayan kowace asara. Ku duba sashen “Dabarun caca na roulette” don ganin abin da yake yi da wanda ba ya yi.Karanta gaba ɗaya → yana farawa da $100 a kan jacacaJaCacar ɗaya zuwa ɗaya (even-money) a kan lambobi ja 18. A Faransanci: rouge.Karanta gaba ɗaya → kuma yana ba wa kansa cace-cace n kawai a kowane zagayen tsarin: bayan asara n a jere yana haƙura ya sake farawa. Ku rubuta matsakaicin sakamakon zagaye ɗaya a matsayin ƙa'idar lissafi ta n, kuma ku lissafa shi ga cace-cace 5, 6 da 7 a dabaran roulette na Turai.
Nuna mafita
Amsa Matsakaicin shi ne 100 × (1 − (38/37)ⁿ): −$14.26 ga cace-cace 5, −$17.35 ga 6 da −$20.52 ga 7. Ba da damar ƙarin cace-cace yana sa matsakaicin zagaye ya yi muni, ba ya inganta shi.
Zagaye yana cin $100 sai dai idan duka cace-cace n sun rasa, abin da ke faruwa da damar qⁿ, inda q = 19/37. Idan duka sun rasa, kuɗin caca 100, 200 … 100 × 2ⁿ⁻¹ suna haɗuwa zuwa 100 × (2ⁿ − 1). Don haka matsakaicin shi ne 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
A nan 2q = 38/37, ɗan fiye da 1, don haka (2q)ⁿ yana girma da n kuma matsakaicin yana faɗuwa: n = 5 yana ba da 1.1426 da −$14.26; n = 6 yana ba da 1.1735 da −$17.35; n = 7 yana ba da 1.2052 da −$20.52. Tsani mai tsawo yana rasawa sau ƙasa, a 3.6%, 1.8% da 0.9% na zagaye, amma kowace asara ta ninka girma biyu, kuma ƙarin kuɗi yana ketare teburin a kowane zagaye, a 2.70% na kowace caca.
A dabaran roulette na Amurka 2q = 40/38 kuma alƙaluman su ne −$29.24, −$36.04 da −$43.20 (Taylor, 2021). A wasan da babu sifilidabaranSifiliRamin kore guda ɗaya da babu cacar ɗaya zuwa ɗaya, ta dozin ko ta ginshiƙi da ke rufe shi; tushen fifikon gidan caca. Ana kuma kiransa: sifili ɗaya, 0.Karanta gaba ɗaya → 2q = 1, kuma kowane zagaye yana da matsakaicin sifili daidai ko yaya tsawon tsanin yake.
“Grand Martingale” yana ninka kuɗin caca biyu bayan asara kuma yana ƙara raka'akuɗiRaka'aKuɗin caca na tushe da tsari ko ɗan wasa ke aiki da shi. Ana bayyana jerin ƙarin caca a raka'a don su yi aiki a kowane tebur.Karanta gaba ɗaya → ɗaya, don haka kuɗin caca suna tafiya 1, 3, 7, 15, 31 … Nawa kuke a gaba idan nasara ta farko ta zo a caca ta 1, ta 2, ta 3 ko ta k? A raka'a ta $10 da iyakar $500, sau nawa zagaye ke karyewa, kuma nawa ne asarar zagaye a matsakaici?
Nuna mafita
Amsa Nasara a caca ta k tana barin ku a gaba da raka'a k: +1, +2, +3 da sauransu. A $10 da $500 zagayen yana karyewa bayan asara biyar, 3.57% na zagaye, a −$570. A matsakaici zagaye yana rasa $2.32 daga cikin $85.73 da aka sa: 2.70%.
Kuɗin caca na k shi ne raka'a 2ᵏ − 1: 1, 3, 7, 15, 31. Kuɗin caca k − 1 na farko suna haɗuwa zuwa 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Nasara a caca ta k tana biyan 2ᵏ − 1, don haka zagayen yana ƙarewa (2ᵏ − 1) − (2ᵏ − k − 1) = raka'a k a gaba: ƙarin raka'a ɗaya ga kowace asara kafin nasarar.
A $10 a kowace raka'a kuɗin caca su ne $10, $30, $70, $150 da $310; na gaba, $630, ya wuce iyakar $500. Asara biyar a jere, (19/37)⁵ = 3.57%, suna jawo asarar 10 + 30 + 70 + 150 + 310 = $570.
Matsakaici: nasarorin suna kawo $10 × k da damar (19/37)ᵏ⁻¹ × 18/37 ga k = 1 zuwa 5, karyewar kuma tana ɗaukar $570 da damar 3.57%; tare −$2.32. Kuɗin da aka sa yana da matsakaicin $85.73 a kowane zagaye, kuma 2.32/85.73 = 2.70%. Martingale na yau da kullum a teburi iri ɗaya yana karyewa bayan asara shida, 1.83% na zagaye, a −$630, kuma yana rasa $1.74 daga cikin $64.20 da aka sa: 2.70% kuma.
Wani ɗan wasa ya shirya tsarin Martingale a kan ja da cacar farko ta $1 a teburin da iyakarsa $1,000 ce, don haka cace-cace goma suna yiwuwa: $1, $2, $4 da sauransu har zuwa $512. Sau nawa zagaye ke rasa duka goma, tsawon nawa hakan ke ɗauka a matsakaici, kuma sau nawa maraice na cace-cace 200 ke ƙarewa haka?
Nuna mafita
Amsa Zagaye yana rasa duka goma kusan sau ɗaya cikin 784, abin da ke jawo asarar $1,023. Wannan kusan cace-cace 1,600 ne a matsakaici, kuma maraice na cace-cace 200 yana bugun bango kusan kashi 11% na lokaci. Zagaye har yanzu yana rasa $0.31 a matsakaici daga cikin $11.31 da aka sa: 2.70%.
Caca a kan ja tana rasawa da damar 19/37, tare da sifili. Goma a jere: (19/37)¹⁰ ≈ 0.00128, kusan 1 cikin 784. Kuɗin caca 1 + 2 + 4 + … + 512 suna haɗuwa zuwa $1,023.
Zagaye yana ɗaukar cace-cace 2.05 a matsakaici, don haka zagaye 784 suna ɗaukar kusan cace-cace 1,600. Idan aka ƙirga juyi bayan juyi, jerin asara goma yana bayyana a wani wuri a cikin cace-cace 200 kashi 11.3% na lokaci.
Tsakanin bala'o'i ɗan wasan yana cin $1 a kowane zagaye: kusan $783 kafin mummunan, sannan −$1,023, kusan −$240 a duk tsawon lokacin. A kowane zagaye: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Kuɗin da aka sa a kowane zagaye: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Kuma −0.31/11.31 = −2.70%.
Ku kwatanta da cacar farko ta $10 a iyakar $500, cace-cace shida: bangon yana zuwa sau ɗaya cikin zagaye 55, kuma maraice na cace-cace 200 yana haɗuwa da shi kashi 84% na lokaci. Iyaka mafi girma tana sa faɗuwar ta yi ƙaranci kuma ta fi girma, ba ta sa ta yi arha ba.
Ɗan barkwanci
Ba kasafai ba amma mafi girma, a kowace iyaka. Yiwuwa sau tsada ba ya canjawa.
Majiyoyi
- 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.