Ikibazo: Urugero rw'Ameza
Martingale ya $10 ku meza afise urugero rwa $500. Ivunika incuro zingahe, ihomba angahe iyo ivunitse, kandi kwongereza urugero vyoba bifasha?
Ibihombo bitandatu bikurikirana biravuna: 1,8% kuri buri muzingi, hafi 60% vy'ibihe vy'ugukina vy'ukuzunguruka 100. Iryo vunika ritwara $630. Kwongereza urugero gushika $5.000 bituma ivunika riba gake (0,25% kuri buri muzingi) kandi rikaba rinini ($5.110). Igihombo citezwe kuri buri dorari riteze ni 2,70% ku rugero urwo ari rwo rwose.
Ikibazo
Umukinyi ategera $10 ku gitukuraitegoGitukuraItego rya rimwe kuri rimwe ku nomero 18 zitukura. Mu gifaransa: rouge.Soma vyose → agakuba kabiri inyuma ya buri gihombo, agasubira ku ntango inyuma y'itsinzi. Urugero rwo hejuru rw'ameza ni $500. (a) Ibihombo bingahe bikurikirana bivuna iyongerekanaingendo yo gutegaIyongerekanaItegeko ryose rihindura amahera y'itego hisunzwe ivyavuyemwo vyabanje. Iyongerekana rihakana riyongereza inyuma y'ibihombo, iryemeza inyuma y'itsinzi.Soma vyose →? (b) UgushobokaimibareUgushobokaUgushoboka kw'ivyavuyemwo, kuva kuri 0 gushika kuri 1 canke 0% gushika 100%. Inomero imwe: 1/37 = 2,70%.Soma vyose → kuri buri muzingi ni ukuhe? (c) Ni igiki gihomba ku ivunika? (d) Urugero rwa $5.000 ruhindura gute inyishu? (e) Hari urugero rutuma ubwo buryo buzana inyungu?
Inyishu (a)–(c)
Amahera y'itego: 10, 20, 40, 80, 160, 320. Irya indwi ryoba 640 > 500, ku buryo ibihombo bitandatu bivuna. (b) p(igihombo) = 19/37 ku gitukura; (19/37)⁶ = 0,0183, hafi 1,8% kuri buri muzingi. Umuzingi umara hafi ukuzunguruka kubiri ku mpuzandengo, ku buryo igihe c'ugukina c'ukuzunguruka 100 gifise hafi imizingi 50 kandi p(nibura ivunika rimwe) ≈ 1 − 0,9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, ari vyo ibiceamaheraIgiceAmahera y'itego y'ishingiro uburyo canke umukinyi akoresha. Iyongerekana ridondorwa mu bice kugira ngo rikore ku meza ayo ari yo yose.Soma vyose → 63 vy'intango, ugereranije na $10 umuzingi wariko urondera gutsinda.
Inyishu (d)
Kuri $5.000: amahera y'itego gushika 2.560 aremewe; itego rya cumi, 5.120, riranka. Ibihombo icenda biravuna: (19/37)⁹ ≈ 0,25% kuri buri muzingi, hafi 11% vy'ibihe vy'ugukina vy'ukuzunguruka 100. Igihombo ku ivunika: 10 + 20 + … + 2.560 = $5.110. Umukinyi yaguze ivyago biba gake ku giciro c'ivyago binini kurushiriza. Mu bihe vy'ugukina vyinshi, ico amavunika yitezwe gutwara ni p × igihombo ≈ 0,0183 × 630 = $11,5 kuri buri muzingi ku rugero rwa $500 na 0,0025 × 5.110 = $12,8 kuri $5.000; ni igitigiri kimwe, ukuyeko ugukomatanya igitigiri, kandi ni igihombo citezweimibareIgihombo citezweAgaciro kitezwe gakubwe n'amahera yose yateze; ni amahera y'impuzandengo atakara mu gihe c'ugukina canke mu gihe runaka.Soma vyose → c'umuzingi hamwe na $10 imizingi itsinda yinjiza.
Inyishu (e)
Oya. Igihombo citezwe kuri buri muzingi = (amahera yose y'itego yitezwe mu muzingi) / 37, ku rugero urwo ari rwo rwose, kubera ko itego ryose ari −1/37 y'agaciro karyo. Urugero rugena gusa ukuntu igihombo kigabanywa hagati y'itsinzi ntoyi nyinshi n'ibihombo binini bike. Ata rugero kandi ufise amahera yo gukinishaamaheraAmahera yo gukinishaAmahera yose yashizwe ku ruhande yo gukina. Gucunga amahera yo gukinisha ni uguhitamwo ayo ushobora gushira mu kaga kuri buri tego no kuri buri gihe c'ugukina.Soma vyose → atagira iherezo, iyongerekana “ryokora”, mu buryo ritovunika na rimwe, kandi ryoteza n'amahera atagira iherezo afise igihombo citezwe ca 2,70% vy'ayo mahera atagira iherezo. Urugero rw'amezaamezaUrugero rw'amezaItego ritoya n'irinini kuruta ayandi vyemewe. Urugero rwo hejuru ni rwo ruhagarika iyongerekana rikuba kabiri.Soma vyose → si rwo rubuza Martingaleingendo yo gutegaMartingaleIyongerekana rikuba kabiri itego inyuma y'igihombo cose. Raba igice c'Ingendo zo gutega umenye ivyo bukora n'ivyo budakora.Soma vyose → gukora. Ni imibare.
Ameza abiri nyayo
Akarorero ka Thorp ni ameza ya $1 afise urugero rwo hejuru rwa $1.000, aho umukinyi yizera gutsinda $30 y'imfungurwa z'umugoroba idorari rimwe rimwe. Amahera y'itego 1, 2, 4 … 512 yemera amatego cumi, ku buryo ibihombo cumi bikurikirana bivuna iyongerekana: incuro imwe mu mizingi 785 ku ruziga rwa ruleti y'i Buraya (incuro imwe muri 613 ku ruziga rwa ruleti y'Amerika), bigatwara $1.023. Imizingi mirongo itatu yose iraroranirwa incuro 96,2%. Izindi 3,8% zihomba hafi $1.000, kandi ivyavuyemwo ku mpuzandengo vy'umugoroba ni −$9,17, ari vyo neza neza 1/37 ya $339 imizingi mirongo itatu iteza ku mpuzandengo. Ameza ya $5 afise urugero rwo hejuru rwa $2.000 nta co atandukaniyeko: amahera y'itego gushika $1.280 yemera amatego icenda, kandi ivunika riza incuro imwe mu mizingi 403 ku ruziga rwa ruleti y'i Buraya (incuro imwe muri 323 ku ruziga rwa ruleti y'Amerika) rigatwara $2.555.
Umukinyi w'umutunzi
Ha iyongerekana amahera yo gukinisha menshi aho kuba urugero ruri hasi: ibice 1.000, intango y'igice kimwe n'ukuzunguruka 1.000 ku gitukura, usubira ku gice kimwe igihe cose ugukuba kabiri gukurikira kurenze ivyo usigaranye. Twiganye imigoroba nk'iyo miliyoni. Ku ruziga rwa ruleti y'i Buraya, 52% irangira iri imbere, 21% ihomba vyose, kandi ivyavuyemwo ku mpuzandengo ni ibice −124. No mu mukino utagira zeruuruzigaZeruAkazu kamwe k'icatsi kibisi kadapfukwa n'itego rya rimwe kuri rimwe, ry'umurwi canke ry'inkingi; ni yo soko y'inyungu ya kazino. Kandi bita: zeru imwe, 0.Soma vyose →, aho itego ryose ritabogamye, hafi 60% gusa ni yo yorangira iri imbere. Umugoroba usanzwe ni itsinzi ntoyi; impuzandengo irihwa n'imigoroba irangira kuri zeru.
Icigwa
Amakazino ashiraho urugero kugira ngo agabanye ihindagurika ry'ivyavuyemwoimibareIhindagurika ry'IvyavuyemwoIgipimo c'ukuntu ivyavuyemwo bikwiragira kure y'agaciro kitezwe. Kinini kuri straight up, gito ku matego ya rimwe kuri rimwe.Soma vyose → ryayo, si ukugira ngo atsinde uburyo bwo gutega. Gutsinda Martingale ni ingaruka y'iyongera banezerwa kugira, kandi imibare yoyitsinze uko biri kwose. Umukinyi yiyumvira ko urugero ari co kibazo aba yabonye amanga akayitiranya n'umusozi umanuka.
Ingingo y'imibare
Ibibazo
Agufasha kurokoka gushika ku rugero ruri hejuru, aho ivunika riba gake kandi rikaba rinini. Igihombo citezwe kuri buri dorari ntigihinduka.
Amahera y'itego 10, 30, 70, 150, 310: urugero rwa $500 rushikwako inyuma y'ibihombo bitanu, kandi ivunika ritwara $570. Ni 2,70% nyene.
Kugira ngo umukinyi umwe ntashobore gutega ibirenze ivyo ububiko bwa kazino bushobora kuriha ata ngorane. Ni ugucunga ihindagurika.
Twinezwe n'Imibare
Banza ubigerageze wewe nyene. Inyishu yose irapfutse; yugurure igihe witeguye.
Martingaleingendo yo gutegaMartingaleIyongerekana rikuba kabiri itego inyuma y'igihombo cose. Raba igice c'Ingendo zo gutega umenye ivyo bukora n'ivyo budakora.Soma vyose → itangura kuri $100 kuri gitukuraitegoGitukuraItego rya rimwe kuri rimwe ku nomero 18 zitukura. Mu gifaransa: rouge.Soma vyose → kandi yiha amatego n ntarengwa kuri buri muzingi: inyuma y'ibihombo n bikurikirana irareka igasubira gutangura. Andika ivyavuyemwo vy'impuzandengo vy'umuzingi umwe nk'ingingo y'imibare muri n, uharure ku matego 5, 6 na 7 ku ruziga rw'i Buraya.
Erekana inyishu
Inyishu Impuzandengo ni 100 × (1 − (38/37)ⁿ): −$14,26 ku matego 5, −$17,35 kuri 6 na −$20,52 kuri 7. Kwemera amatego menshi bituma umuzingi w'impuzandengo urushiriza kuba mubi, si mwiza.
Umuzingi utsinda $100 kiretse amatego yose n ahomvye, ivyo biba ku gushoboka kwa qⁿ, aho q = 19/37. Iyo yose ahomvye, amahera y'itego 100, 200 … 100 × 2ⁿ⁻¹ ateranirizwa akaba 100 × (2ⁿ − 1). Rero impuzandengo ni 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Ng'aha 2q = 38/37, birenze gato 1, ku buryo (2q)ⁿ ikura uko n ikura n'impuzandengo ikamanuka: n = 5 itanga 1,1426 na −$14,26; n = 6 itanga 1,1735 na −$17,35; n = 7 itanga 1,2052 na −$20,52. Urwego rurerure ruhomba gake, muri 3,6%, 1,8% na 0,9% vy'imizingi, ariko igihombo cose kiba kinini incuro zibiri, kandi amahera menshi aca ku meza kuri buri muzingi, kuri 2,70% vya buri tego.
Ku ruziga rw'Amerika 2q = 40/38 n'ibitigiri ni −$29,24, −$36,04 na −$43,20 (Taylor, 2021). Mu mukino utagira zeruuruzigaZeruAkazu kamwe k'icatsi kibisi kadapfukwa n'itego rya rimwe kuri rimwe, ry'umurwi canke ry'inkingi; ni yo soko y'inyungu ya kazino. Kandi bita: zeru imwe, 0.Soma vyose → 2q = 1, kandi umuzingi wose ugira impuzandengo ya zeru neza neza, urwego rwoba rurerure gute.
“Martingale nini” ikuba kabiri amahera y'itego inyuma y'igihombo ikongerako igiceamaheraIgiceAmahera y'itego y'ishingiro uburyo canke umukinyi akoresha. Iyongerekana ridondorwa mu bice kugira ngo rikore ku meza ayo ari yo yose.Soma vyose → kimwe, ku buryo amahera y'itego aba 1, 3, 7, 15, 31 … Uri imbere angahe iyo itsinzi rya mbere rije ku tego rya 1, rya 2, rya 3 canke rya k? Ku gice ca $10 n'urugero runini rwa $500, umuzingi umeneka kangahe, kandi umuzingi utwara iki ku mpuzandengo?
Erekana inyishu
Inyishu Itsinzi ku tego rya k rigusiga uri imbere n'ibice k: +1, +2, +3 n'ibindi. Kuri $10 na $500 umuzingi umeneka inyuma y'ibihombo bitanu, 3,57% vy'imizingi, kuri −$570. Ku mpuzandengo umuzingi uhomba $2,32 muri $85,73 yateze: 2,70%.
Itego rya k ni 2ᵏ − 1 vy'ibice: 1, 3, 7, 15, 31. Amatego k − 1 ya mbere ateranirizwa akaba 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Itsinzi ku tego rya k riha 2ᵏ − 1, ku buryo umuzingi urangira uri imbere (2ᵏ − 1) − (2ᵏ − k − 1) = k vy'ibice: igice kimwe c'iyongera kuri buri gihombo imbere y'itsinzi.
Kuri $10 igice, amahera y'itego ni $10, $30, $70, $150 na $310; irikurikira, $630, rirenza urugero runini rwa $500. Ibihombo bitanu bikurikirana, (19/37)⁵ = 3,57%, bitwara 10 + 30 + 70 + 150 + 310 = $570.
Impuzandengo: amatsinzi azana $10 × k ku gushoboka kwa (19/37)ᵏ⁻¹ × 18/37 kuri k = 1 gushika 5, n'ukumeneka gutwara $570 ku gushoboka kwa 3,57%; hamwe −$2,32. Amahera yateze agira impuzandengo ya $85,73 ku muzingi, na 2,32/85,73 = 2,70%. Martingale isanzwe ku meza nyene imeneka inyuma y'ibihombo bitandatu, 1,83% vy'imizingi, kuri −$630, igahomba $1,74 muri $64,20 yateze: 2,70% kandi.
Umukinyi ateganya Martingale kuri gitukura n'itego rya mbere rya $1 ku meza afise urugero rwa $1.000, ku buryo amatego cumi ashoboka: $1, $2, $4 n'ibindi gushika $512. Umuzingi uhomba ayo cumi yose kangahe, bifata igihe kingana iki ku mpuzandengo, kandi umugoroba w'amatego 200 urangira gutyo kangahe?
Erekana inyishu
Inyishu Umuzingi uhomba yose cumi nka rimwe muri 784, ivyo bigatwara $1.023. Ivyo ni nka matego 1.600 ku mpuzandengo, kandi umugoroba w'amatego 200 ushikira urwo rukuta nka 11% vy'incuro. Umuzingi uracahomba $0,31 ku mpuzandengo muri $11,31 yateze: 2,70%.
Itego kuri gitukura rihomba ku gushoboka kwa 19/37, zeru irimwo. Cumi bikurikirana: (19/37)¹⁰ ≈ 0,00128, nka 1 muri 784. Amahera y'itego 1 + 2 + 4 + … + 512 ateranirizwa akaba $1.023.
Umuzingi ufata amatego 2,05 ku mpuzandengo, ku buryo imizingi 784 ifata nka matego 1.600. Uharuye ukuzunguruka ku kundi, umurongo w'ibihombo cumi uboneka ahantu mu matego 200 11,3% vy'incuro.
Hagati y'ivyago, umukinyi atsinda $1 kuri buri muzingi: nka $783 imbere y'umuzingi mubi, hanyuma −$1.023, nka −$240 mu gihe cose. Kuri buri muzingi: 1 × (1 − 0,00128) − 1.023 × 0,00128 ≈ −$0,31. Amahera yateze kuri buri muzingi: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11,31. Na −0,31/11,31 = −2,70%.
Gereranya n'itego rya mbere rya $10 ku rugero rwa $500, amatego atandatu: urukuta ruza rimwe mu mizingi 55, kandi umugoroba w'amatego 200 uruhura na rwo 84% vy'incuro. Urugero rwo hejuru rutuma ugusenyuka kuba gake kandi kunini, si uguhendutse.
Agakuru ko gutwenga
Gake kandi binini, ku rugero urwo ari rwo rwose. Igikubo ntigihinduka.
Aho Twakuye
- 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.