Imibare ya Ruleti · Ishirwa mu ngiro · iminota 13

Ingorane yo Gutakaza Vyose

Q: Mbega iyongerekana riragabanya ingorane yo gutakaza vyose?

Martingale irayigabanya mu bihe vy'ugukina bigufi kandi ikayongereza cane mu bihe birebire. Gutega amahera angana ku gice kinini ni co kintu c'ukuri co kuyigabanya.

Ugushoboka ko gutakaza amahera yawe yose yo gukinisha imbere yo gushika ku ntumbero.

Inyishu ngufi

Ku matego ya rimwe kuri rimwe, ingorane yo gutakaza vyose ikurikira ingingo y'imibare y'ugutakaza vyose kw'umutezi. N'amahera yo gukinisha ya $500, amatego ya $10 n'intumbero ya $600, ushika ku ntumbero nka 57% vy'incuro, izindi zose ugatakaza vyose.

ingingo y'imibare

ugutakaza vyose kw'umutezi

p (rimwe kuri rimwe)

18/37

q/p

1,0556

inyishu ya kera kuruta izindi

Pascal–Fermat, 1656

Ikibazo

Ufise ibiceamaheraIgiceAmahera y'itego y'ishingiro uburyo canke umukinyi akoresha. Iyongerekana ridondorwa mu bice kugira ngo rikore ku meza ayo ari yo yose.Soma vyose → a vy'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 → kandi ushaka gushika ku bice a + b, uteza igice kimwe kimwe ku itego rya rimwe kuri rimweitegoItego rya rimwe kuri rimweGitukura/cirabura, igiharwe/igiharuro, binini/bito. Inomero 18, riha 1:1.Soma vyose → rifise ugushobokaimibareUgushobokaUgushoboka kw'ivyavuyemwo, kuva kuri 0 gushika kuri 1 canke 0% gushika 100%. Inomero imwe: 1/37 = 2,70%.Soma vyose → ko gutsinda p = 18/37 n'ugushoboka ko guhomba q = 19/37. Ugushoboka ko gushika ku ntumbero imbere yo gushika kuri zeru ni ukuhe? Inyishu yatahuwe na Pascal, Fermat na Huygens mu myaka ya 1650, kandi ni imwe mu nyishu za kera kuruta izindi mu vy'ugushoboka.

Blaise Pascal: Umugani w'Uruziga rwa Ruleti

Ingingo y'imibare y'ugutakaza vyose kw'umutezi

P(gushika ku ntumbero) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ). Ingorane yo gutakaza vyoseimibareIngorane yo gutakaza vyoseUgushoboka kwo guhomba amahera yo gukinisha yose imbere yo gushika ku ntumbero canke ku gitigiri c'ukuzunguruka.Soma vyose → ni rimwe ukuyemwo ico gitigiri. Kubera q/p = 19/18 ≈ 1,0556, iryo gereranya rikura vuba cane (mu buryo bwa exponentiel) ukurikije igitigiri c'ibice, ku buryo iyo ngingo ihana cane ibice bito: uko amatego akenewe kugira ngo ushike ku ntumbero aba menshi, ni ko ikori rya 2,70% ririhwa incuro nyinshi mu nzira.

amahera yo gukinishaigiceintumberoamatego gushika ku ntumberop(gushika ku ntumbero)ingorane yo gutakaza vyose

$500$10$60010≈ 57%≈ 43%

$500$50$6002≈ 79%≈ 21%

$500$10$1.00050≈ 6%≈ 94%

$500$100$1.0005≈ 43%≈ 57%

$500$500$1.000148,6%51,4%

$100$10$20010≈ 37%≈ 63%

Amatego ya rimwe kuri rimwe, amahera y'itego angana, nta la partage. Imashini Iharura Ingorane yo Gutakaza Vyose ikora iyo harure ku bitigiri ivyo ari vyo vyose.

Ukuntu wobitahura

Ibice bito biragusesereza. Kugerageza gukuba kabiri $500 ukoresheje amatego ya $10 bikunda 6% vy'incuro; n'itego rimwe rya $500 bikunda 48,6% vy'incuro. Gukina ushize amanga, amatego make kandi manini, ni vyo bituma ugushoboka ko gushika ku ntumbero kuba kunini kuruta mu mukino ufise agaciro kitezwe kari munsi ya zeru, kuko bigabanya incuro inyungu ya kazino ishirwa mu ngiro. Kandi biratuma 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 → ry'ivyo bigushikira riba rinini cane, kandi umugoroba ukarangira vuba. Iyo nyigisho yitwa “gukina ushize amanga ni vyo vyiza kuruta” (bold play is optimal), kandi yemejwe na Dubins na Savage mu 1965.

Ugushoboka ko guhindura $500 ikaba $1.000 kuri gitukura%

itego rimwe rya $500

48,6%

amatego ya $100

≈ 43%

amatego ya $50

≈ 37%

amatego ya $20

≈ 21%

amatego ya $10

≈ 6%

amatego ya $5

≈ 0,5%

024,3048,6%

Uko amatego akenewe aba menshi, ni ko inyungu ya kazino iriwa kenshi. Gukina utinya birahenda.

Aho ikibazo catanguriye

Iki kibazo ni ica kera kuruta ruleti. Mu 1656 Pascal yahaye Fermat igisokozo: abakinyi babiri bataba amadado atatu, umwe wese atangura afise ibimenyetso 12; ugutaba kuvamwo 11 kwimura ikimenyetso kimwe mu ruhande rumwe, ukuvamwo 14 mu rundi. Bompi barakiboneye inyishu, kandi Huygens, uwakimenye biciye mu ikete ry'umuhinga mu mibare Pierre de Carcavi, yagicapuye mu 1657 nk'ikibazo ca gatanu c'igitabu ca mbere kivuga ivy'ugushoboka, gifise inyishu ariko ata ndemezo. Cumi na kumwe gishobora kuvamwo mu buryo 27 kandi cumi na kane mu buryo 15 gusa, ku buryo ikimenyetso cose cimuka mu ruhande rw'umukinyi wa mbere incuro 9 muri 14, kandi ingingo y'imibare iri hejuru, umwe wese afise ibimenyetso 12, ituma ashobora gutsinda vyose incuro nka 1.157 kuruta gutakaza vyose. Inyungu ya kazino ya 2,70% ni ntoya cane, ariko ingingo y'imibare nyene irerekana ico ikora mu matego menshi.

Mu gisokozo ca Pascal, ugutaba kwose kw'amadado atatu kwimura ikimenyetso kimwe hagati y'abakinyi.

Itego nyakuri ryo gushira amanga

Muri Ndamukiza 2004 Ashley Revell, umusore w'Umwongereza, yaragurishije hafi ivyo yari atunze vyose, harimwo n'impuzu ziwe, ajana $135.300 kuri Plaza i Las Vegas. Inyuma y'amatego mato makeyi, yashize amahera yose kuri gitukuraitegoGitukuraItego rya rimwe kuri rimwe ku nomero 18 zitukura. Mu gifaransa: rouge.Soma vyose →, rimwe gusa. UmupiraibikoreshoUmupiraUmupira mutoyi uterwa mu nzira uhushanye na rotor. Imipira y'ubu ikozwe muri ivorine canke Teflon, 18–21 mm.Soma vyose → waguye kuri 7 itukura, asohoka afise $270.600. Ku ruziga rufise 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 → zibiri, ukwo kuzunguruka kumwe kwari gufise ugushoboka kwa 47,37% kandi kwamutwaye $7.121 ku mpuzandengo, ni ukuvuga inyungu ya kazino ya 5,26% ku itego rimwe. Iyo yokwiragije ayo mahera mu matego ya $10, iyo ntumbero nyene yari kuba hafi kudashoboka. Gukina ushize amanga ntibituma itego riba ryiza; bituma rimara igihe gito gishoboka.

Amahera yose yo gukinisha ya Ashley Revell ari ku kuzunguruka kumwe gusa kw'uruziga rufise zeru zibiri.

Ingorane yo gutakaza vyose ata ntumbero

Nimba ukina gusa gushika utakaje vyose canke ugahagarara inyuma y'ukuzunguruka N, ikibazo kiba “ni ugushoboka kungana iki ko amahera yanje yo gukinisha ashika kuri zeru mu kuzunguruka N?”. Nta nyishu yuzuye y'ingingo y'imibare ku N iyo ari yo yose, ariko ibikoresho vyo kwigana biri kuri uru rubuga birabiharura: amahera yo gukinisha ya $500 ku $10 kuri gitukura ntakunda gushira mu kuzunguruka 100 (munsi ya 1%), ashira nka 9% vy'incuro mu kuzunguruka 500, na nka 95% vy'incuro mu kuzunguruka 5.000. Iyo nta tegeko ryo guhagarara, ugushoboka kwegera 100%, ari yo ndaba y'igihe kirekire y'inyungu ya kazinoimibareInyungu ya kazinoIgice c'amahera y'itego kazino igumana ku mpuzandengo: 2,70% muri ruleti y'i Buraya.Soma vyose →.

Ibihe vy'ugukina cumi, amahera yo gukinisha $500, $25 kuri gitukura, ukuzunguruka 200ivyavuyemwo ugereranije n'intango

ibihe vyo gukina 1.000 × ukuzunguruka 500 · ibara ry'urwungu p10/p25/p75/p90, ubururu bwerurutse hagati, uturorero 81 tudasa · nyuza akamenyetso hejuru y'igishushanyo kugira ubone ibipimo ku kuzunguruka

p10 −$500
p25 −$500
hagati −$500
p75 +$0
p90 +$350

ibihe vyo gukina 1.000 × ukuzunguruka 500 · umurongo uhagaze ugarukira ku ijanisha rya 1–99 (impera −$500 gushika +$1.500) · ukuzunguruka 500 · intango $25 · rimwe kuri rimwe

Imirongo ishika hasi yatakaje vyose. N'igice gitoyi, yomara igihe kirekire ariko igaheraheza ahantu hamwe.

Ugushikana mu Gihe Kirekire: Itegeko ry'Ibitigiri Vyinshi muri Ruleti

Igihe bimara, n'ukuntu ushobora kuja imbere

Inyishu zibiri za kera zinyishura ibibazo abakinyi babaza vy'ukuri. Iyo nta ntumbero, amatego ya rimwe kuri rimwe angana amara ku mpuzandengo ukuzunguruka 37 kuri buri gice c'amahera yo gukinisha ku ruziga rw'i Buraya na 19 ku rw'Amerika, kuko ukuzunguruka kwose gutwara 1/37 c'igice ku mpuzandengo (1/19 ku ruziga rw'Amerika): ibice 100 bimara ukuzunguruka 3.700 ku mpuzandengo, naho igice c'ibihe nk'ivyo vy'ugukina birangira mu kuzunguruka 3.134 (raba Uburebure bw'Igihe c'Ugukina n'Ico Kigutwara). Kandi naho amahera yo gukinisha yoba angana gute, ugushoboka ku ruziga rw'i Buraya ko ugeze ukaba imbere n'ibice k ni (18/19)ᵏ. Ni 95% ku gice kimwe na 58% ku bice cumi, ariko nta mahera yo gukinisha, naho yoba ari menshi gute, aha ugushoboka kungana ku kuba ugeze imbere n'ibice 13; ku ruziga rw'Amerika ugushoboka ni (9/10)ᵏ kandi kiramanuka munsi y'igice ku bice 7.

ibice uri imbere, igihe ico ari co coseI BurayaAmerika

194,7%90,0%

576,3%59,0%

1058,2%34,9%

1349,5%25,4%

2033,9%12,2%

506,7%0,5%

1000,45%0,003%

Amatego ya rimwe kuri rimwe, amahera y'itego angana, amahera yo gukinisha menshi ku buryo atazoshira. Ugushoboka ni (18/19)ᵏ ku ruziga rufise zeru imwe na (9/10)ᵏ ku rufise zeru zibiri.

Uburebure bw'Igihe c'Ugukina n'Ico Kigutwara Uruziga n'Amategeko vya Ruleti y'Amerika

Amahera yo gukinisha ku gushoboka kwa mirongo itanu kuri mirongo itanu

Hindura ikibazo: ukwiye kuzana amahera angana iki kugira ngo ugire nibura ugushoboka kwa mirongo itanu kuri mirongo itanu ko gutsinda amahera runaka imbere yo gutakaza vyose? Ku itsinzi ntoya, si menshi. Ku nini, amahera yo gukinisha akenewe aduga cane, kandi iyo urenze aho, nta mahera yo gukinisha ahagije, kuko n'atagira iherezo atanga ugushoboka kuri munsi ya mirongo itanu kuri mirongo itanu ko uzoshika imbere gutyo. Taylor (2021) abivuga mu bundi buryo: ku ruziga rw'Amerika, kugira ngo ugire ugushoboka kwa mirongo itanu kuri mirongo itanu ko gushika ku bice 100, utegerezwa gutangura ufise 94, ugashira mu kaga 94 kugira ngo utsinde 6. Ku ruziga rw'i Buraya wokenera 88.

gutsindaI BurayaAmerika

igice 122

ibice 233

ibice 345

ibice 468

ibice 5712

ibice 6921

ibice 712nta mahera ahagije

ibice 815nta mahera ahagije

ibice 1024nta mahera ahagije

ibice 1246nta mahera ahagije

ibice 13 canke birenganta mahera ahagijenta mahera ahagije

Amahera yo gukinisha make cane, mu bice, aha nibura ugushoboka kwa mirongo itanu kuri mirongo itanu ko gutsinda ayo mahera imbere yo gutakaza amahera yo gukinisha, ku matego ya rimwe kuri rimwe angana. Harure yacu ivuye ku ngingo y'imibare iri hejuru.

Icigwa c'ingirakamaro

Fata ingingo y'ico ugerageza gukora. Nimba intumbero ari itsinzi runaka, teza amahera menshi incuro nkeyi. Nimba intumbero ari umugoroba muremure, teza amahera make kandi wemere ko inyungu ya kazino izotwara 2,70% vyayo ku giteranyo c'amategoamaheraIgiteranyo c'amategoIgiteranyo c'amahera yose yateze mu gihe c'ugukina canke mu kiringo, harimwo n'ivyatsinzwe vyongeye gutegwa. Kandi bita: handle, action.Soma vyose → kinini. Nimba ushaka vyompi, hari ikizotegerezwa kurekurwa, kandi kenshi ni intumbero.

Ingingo y'imibare

P(intumbero) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ)

ingorane yo gutakaza vyose = 1 − P(intumbero)

q/p = 19/18 ku rimwe kuri rimwe

impuzandengo y'ukuzunguruka gushika utakaje vyose, ata ntumbero = 37 × ibice

P(kuba igihe kimwe imbere n'ibice k) = (18/19)ᵏ

Ibibazo

Mbega iyongerekana riragabanya ingorane yo gutakaza vyose?

Martingaleingendo yo gutegaMartingaleIyongerekana rikuba kabiri itego inyuma y'igihombo cose. Raba igice c'Ingendo zo gutega umenye ivyo bukora n'ivyo budakora.Soma vyose → irayigabanya mu bihe vy'ugukina bigufi kandi ikayongereza cane mu bihe birebire. Gutega amahera anganaingendo yo gutegaGutega amahera anganaGutega amahera angana kuri buri kuzunguruka. Ni urugero iyongerekana ryose rigereranywa na rwo.Soma vyose → ku gice kinini ni co kintu c'ukuri co kuyigabanya.

Na ho amatego ya straight up?

Ingingo y'imibare iri hejuru ni iy'amatego ya rimwe kuri rimwe. Ku matego afise ihindagurika rinini, gutakaza vyose birashoboka cane ku mahera y'itego mato kandi intumbero ishikwako mu buryo budasanzwe; koresha imashini iharura.

Ni kubera iki q/p ari co gitigiri nyamukuru?

Intambwe yose yerekeza ku ntumbero ntishoboka gatoyi kuruta intambwe yose yerekeza ku gutakaza vyose; iryo gereranya ririyongerekanya ukurikije igitigiri c'intambwe.

Twinezwe n'Imibare

Banza ubigerageze wewe nyene. Inyishu yose irapfutse; yugurure igihe witeguye.

Ufise $100 ushaka $1.000, uteza kuri gitukuraitegoGitukuraItego rya rimwe kuri rimwe ku nomero 18 zitukura. Mu gifaransa: rouge.Soma vyose → gusa. Umugambi wa mbere: teza $1 incuro imwe imwe. Umugambi wa kabiri: teza ivyo ufise vyose gushika urenze $500, hanyuma uteze gusa ivyo bikenewe kugira ngo ushike kuri $1.000. Ni uwuhe mugambi mwiza kuruta uwundi, kandi n'ingero zingana iki?
Erekana inyishu
Inyishu Umugambi wa kabiri, kure cane. Ushika kuri $1.000 nka 9,0% vy'incuro, hafi ya 10% itabogamye. Umugambi wa mbere ukunda nka incuro 7 muri 10²².
Mu mukino utabogamyeimibareUmukino utabogamyeUmukino aho nta ruhande rufise inyungu: itego ryose rifise agaciro kitezwe ka zeru.Soma vyose → woshika kuri $1.000 uvuye kuri $100 incuro imwe mu cumi ico wokora cose, kuko agaciro k'impuzandengo k'amahera yawe kaguma kuri $100.
Inyungu ya kazino itwara 2,70% vya buri tego, ku buryo uko uteza make mu giteranyo, ni ko uguma hafi ya rimwe mu cumi. Umugambi wa kabiri uca urubanza mu matego makeyi: $100 aba $200, $400, $800, hanyuma $200 arahagije. Ukunda nka 9,0% vy'incuro. Umugambi wa mbere usaba amatego ibihumbi imbere y'uko urugendo rurangira, kandi inyungu ya kazino, ntoya kuri buri tego, iteranirizwa hamwe igashika hafi ku gutakaza vyose bidakekeka.
Iyi ni inyishu ya Dubins–Savage mu buryo bwayo bworoshe kuruta (Epstein, 2013): mu mukino ushigikira kazino, teza ivyo ukeneye vyose, incuro nkeyi zishoboka.
Ufise $500 ushaka $1.000, ugahagarara ukimara kuyaronka. Uhitamwo ubwoko bumwe bw'itego ukabusubiramwo gushika butsinze rimwe canke amahera ashize. Ni ubuhe bwoko butanga ugushoboka kwiza kuruta?
Erekana inyishu
Inyishu Itego rimwe rya $500 kuri gitukura: 48,6%. Umugambi wose ukwiragiza amahera mu matego mato urushirizaho kuba mubi.
Hitamwo amahera ya buri tego ku buryo itsinzi imwe ihagije: $30 kuri straight upitegoStraight upInomero imwe gusa, iriha 35:1.Soma vyose → atsinda $1.050, $60 kuri splititegoSplitInomero zibiri zifatanye, iriha 17:1.Soma vyose → atsinda $1.020, n'ibindi. Hanyuma harura ugushoboka ko nibura itego rimwe ritsinda imbere y'uko amahera ashira. Amatego cumi n'atandatu ya straight up ya $30 yose ahusha ku gushoboka kwa (36/37)¹⁶ = 64,5%, ku buryo ashika kuri $1.000 35,5% vy'incuro.
Igishushanyo cerekana itego ryose. Gitukura ni co kiza imbere kuko gishikayo mu tego rimwe, ku buryo inyungu ya kazino iriwa rimwe, kuri $500. Six lineitegoSix lineInomero zitandatu ziri mu mirongo ibiri ifatanye, iriha 5:1. Mu gifaransa: sixain.Soma vyose → iza ku wa kabiri kuko ikenera amatego atatu gusa. Itego rimwe ku murwiitegoUmurwi1–12, 13–24 canke 25–36, riha 2:1.Soma vyose → ni ryo ribi kuruta, ugushoboka kwa 12/37; naho ryokinwa mu biceamaheraIgiceAmahera y'itego y'ishingiro uburyo canke umukinyi akoresha. Iyongerekana ridondorwa mu bice kugira ngo rikore ku meza ayo ari yo yose.Soma vyose → vya $250 gushika ufise $1.000 canke birenga, umurwi ushikayo 46,4% vy'incuro gusa.
Bollman (2024) yubaka igishushanyo nk'ico ku ruziga rw'Amerika. Umurongo wiwe w'umurwi ufise ikosa: wari ukwiye kwandika 12/38 = 31,6%.
itegoamahera y'itego × igitigiri c'amategougushoboka ko gushika kuri $1.000
Straight up$30 × 1635,5%
Split$60 × 835,9%
Street$90 × 534,5%
Corner$125 × 436,7%
Six line$166, $166, $16841,2%
Umurwi$500 × 132,4%
Gitukura$500 × 148,6%
Uteza igice kimwe kuri gitukura, ukuzunguruka ku kundi, 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 → atigera ashira. Inyuma y'ukuzunguruka kwa mbere, uri imbere n'igice kimwe canke inyuma n'igice kimwe. Ugushoboka ko uzosubira kunganya neza neza ni ukuhe?
Erekana inyishu
Inyishu 36 muri 37, 97,3%. Ku ruziga rw'Amerika ni 36 muri 38, 94,7%.
Iyo ukuzunguruka kwa mbere gutsinze (incuro 18 muri 37), uri imbere n'igice kimwe, kandi inyungu ya kazino ikwega hasi igihe c'ugukina cose kitagira iherezo, ku buryo bwanka bwakunda usubira ukaca ku kunganya: ivyo ntibikekeka. Iyo guhomvye (incuro 19 muri 37), utegerezwa kuduga igice kimwe, kandi ugushoboka ko ugeze ukaba imbere n'igice kimwe ni 18/19 (raba igishushanyo kiri hejuru).
Hamwe: 18/37 + 19/37 × 18/19 = 18/37 + 18/37 = 36/37.
Muri rusangi, ugushoboka ni 1 ukuyemwo itandukaniro hagati y'ugushoboka ko gutsinda n'uko guhomba ukuzunguruka kumwe: 1 − 1/37 ku ruziga rw'i Buraya, 1 − 2/38 ku rw'Amerika. 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 → itandukaniro ni ubusa kandi ugushoboka ni 1: amanota yama asubira ku kunganya. Epstein (2013) atanga itegeko rusangi.
Igisokozo ca Pascal co mu 1656: abakinyi babiri bataba amadado atatu, umwe wese atangura afise ibimenyetso 12. Igihe cose havuyemwo 11 ikimenyetso kimwe kija ku mukinyi wa mbere, igihe cose havuyemwo 14 kimwe kija ku wa kabiri; ugutaba kundi nta co guharura. Umukinyi wa mbere afise ugushoboka kungana iki ko gutsinda vyose uko ari 24? Kandi iyo woteza igice 1 kuri gitukura kuri buri kuzunguruka uhanganye n'umugenzi wawe akora nka banki, ibice 12 umwe wese, ufise ugushoboka kungana iki ko gutsinda vyose uko ari 24?
Erekana inyishu
Inyishu Umukinyi wa mbere atsinda nka 1.157 muri 1.158. Wewe, kuri gitukura, utsinda ibice vyose 24 nka 34% vy'incuro gusa.
Amadado atatu atanga 11 mu buryo 27 na 14 mu buryo 15, ku buryo ikimenyetso cose cimuka kija ku mukinyi wa mbere incuro 27 muri 42, incuro 9 muri 14. Uruhande rwose rutangura kuri 12, ugushoboka ko gutwara vyose ni 1/(1 + r¹²), aho r ari igereranya ry'ugushoboka kw'urundi ruhande n'ukwawe.
Ku mukinyi wa mbere r = 5/9, kandi (5/9)¹² ≈ 0,00086, ku buryo ugushoboka ari nka 0,99914: igipimo c'ugushoboka ca 9¹² kuri 5¹², canke 282.429.536.481 kuri 244.140.625, nka 1.157 kuri 1. Huygens yacapuye ivyo bitigiri nyene mu 1657.
Kuri gitukura, r = 19/18 kubera zeru, kandi (19/18)¹² ≈ 1,913: ugushoboka kwawe ni 1/2,913 ≈ 34,3%. Igiceri coguha 50%. Inyungu ya kazino ya 2,70% kuri buri tego iteranirizwa hamwe, mu matego menshi bisaba, igashika ku itandukaniro ry'amanota 16.

Agakuru ko gutwenga

$Gukina ushize amanga ni vyo vyiza kuruta. Kandi birangira vuba.$

Gukina ushize amanga ni vyo vyiza kuruta. Kandi birangira vuba.

Aho Twakuye

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1. Wiley, 3rd ed., 1968. Ugutakaza vyose kw'umutezi, igice ca XIV.
Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed. Academic Press, 2013.
Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed. CRC Press, 2021.
Carcavi, P. de. Ikete ryandikiwe Huygens, 28 Nyakanga 1656, ku kibazo Pascal yahaye Fermat. Œuvres complètes de Christiaan Huygens, vol. 1, 1888, urup. 492–494.
Huygens, C. De Ratiociniis in Ludo Aleae. 1657. Inyandiko ya mbere yacapuwe ivuga ivy'agaciro kitezwe.
Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010. Igice ca 13, ruleti.
Reuters. “Briton bets it all on roulette spin, and wins.” Ndamukiza 2004.
Las Vegas Sun. “Man wagers life’s worth.” 12 Ndamukiza 2004.

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