Imibare ya Roulette · Ikoreshwa · iminota 13

Ibyago byo Gushira Igishoro

Q: Ese uruhererekane rw'amatega rugabanya ibyago byo gushira igishoro?

Martingale ibigabanya ku mikino migufi kandi ikabyongera cyane ku mikino miremire. Gutega itega rihoraho ku nite nini ni bwo buryo bw'ukuri bwo kubigabanya.

Amahirwe yo gutakaza igishoro cyawe cyose cyo gukina mbere yo kugera ku ntego.

Igisubizo kigufi

Ku matega yishyura kimwe ku kindi, ibyago byo gushira igishoro bikurikiza formula ya gambler’s ruin. Ufite igishoro cya $500, amatega ya $10 n'intego ya $600, ugera ku ntego hafi 57% by'igihe naho ibindi bihe igishoro kigashira.

formula

gambler’s ruin

p (kimwe ku kindi)

18/37

q/p

1,0556

igisubizo cya kera kurusha ibindi

Pascal–Fermat, 1656

Ikibazo

Ufite initeamafarangaIniteItega ry'ibanze uburyo cyangwa umukinnyi akoresha. Impererekane z'amatega zisobanurwa mu nite kugira ngo zikoreshwe ku meza ayo ari yo yose.Soma byose → a z'igishoro cyo gukinaamafarangaIgishoro cyo gukinaAmafaranga yose yashyizwe ku ruhande yo gukina. Gucunga igishoro ni ukwemeza ingano yacyo uzashyira mu kaga kuri buri tega no kuri buri mukino.Soma byose → kandi ushaka kugera ku nite a + b, utega inite imwe icyarimwe ku itega ryishyura kimwe ku kindiitegaItega ryishyura kimwe ku kindiUmutuku/umukara, itabangikanye/ibangikanye, minini/mito. Imibare 18, ryishyura 1:1.Soma byose → rifite amahirweimibareAmahirweAmahirwe y'igisubizo, kuva kuri 0 kugeza kuri 1 cyangwa kuva kuri 0% kugeza kuri 100%. Umubare umwe: 1/37 = 2,70%.Soma byose → yo gutsinda p = 18/37 n'amahirwe yo gutsindwa q = 19/37. Amahirwe yo kugera ku ntego mbere yo kugera kuri zeru ni angahe? Igisubizo cyabonywe na Pascal, Fermat na Huygens mu myaka ya 1650 kandi ni kimwe mu bisubizo bya kera kurusha ibindi mu mibare y'amahirwe.

Blaise Pascal: Umugani w'Uruziga rwa Roulette

Formula ya gambler’s ruin

P(kugera ku ntego) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ). Ibyago byo gushira igishoroimibareIbyago byo gushira igishoroAmahirwe yo gutakaza igishoro cyose cyo gukina mbere yo kugera ku ntego cyangwa ku mubare w'amazunguruka.Soma byose → ni rimwe ukuyemo ibyo. Hamwe na q/p = 19/18 ≈ 1,0556, igereranya rikura mu buryo bwikuba hakurikijwe umubare w'inite, bityo formula ihana cyane inite nto: uko amatega akenewe ngo ugere ku ntego aba menshi, ni ko umusoro wa 2,70% wishyurwa inshuro nyinshi mu nzira.

igishoroiniteintegoamatega kugera ku ntegop(kugera ku ntego)ibyago byo gushira igishoro

$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%

Amatega yishyura kimwe ku kindi, itega rihoraho, nta la partage. Mubazi y'Ibyago byo Gushira Igishoro ikora ibarura ku mibare iyo ari yo yose.

Kubisoma

Inite nto zirababaza. Kugerageza gukuba $500 kabiri ukoresheje amatega ya $10 bigerwaho 6% by'igihe; ukoresheje itega rimwe rya $500 bigerwaho 48,6% by'igihe. Gukina ubutwari (bold play), amatega make kandi manini, byongera cyane amahirwe yo kugera ku ntego mu mukino ufite ibiteganyijwe biri munsi ya zeru, kuko bigabanya inshuro inyungu ya kazino ikoreshwa. Binongera cyane ihindagurikaimibareIhindagurikaIgipimo cy'uko ibisubizo bikwirakwira kure y'agaciro gateganyijwe. Riri hejuru kuri straight up, hasi ku matega yishyura kimwe ku kindi.Soma byose → ry'ibikubaho, kandi birangiza umugoroba vuba. Iyo theorem yitwa “bold play is optimal” kandi yagaragajwe na Dubins na Savage mu 1965.

Amahirwe yo guhindura $500 ikaba $1.000 ku mutuku%

itega rimwe rya $500

48,6%

amatega ya $100

≈ 43%

amatega ya $50

≈ 37%

amatega ya $20

≈ 21%

amatega ya $10

≈ 6%

amatega ya $5

≈ 0,5%

024,3048,6%

Uko amatega aba menshi, ni ko inyungu ya kazino yishyurwa kenshi. Gukina utinya birahenda.

Aho ikibazo cyatangiriye

Iki kibazo kiruta roulette ubukuru. Mu 1656 Pascal yahaye Fermat igisakuzo: abakinnyi babiri bajugunya udusoro dutatu, buri wese atangira afite udukoresho 12 two kubara; ijugunywa rya 11 ryimura agakoresho mu cyerekezo kimwe, irya 14 mu kindi. Bombi barakibonye, kandi Huygens, wabimenye abibwiwe mu ibaruwa y'umuhanga mu mibare Pierre de Carcavi, yagicapye mu 1657 nk'ikibazo cya gatanu cy'igitabo cya mbere ku mahirwe, gifite igisubizo nta gihamya. Cumi n'umwe ishobora kujugunywa mu nzira 27 naho cumi n'ine mu nzira 15 gusa, bityo buri gakoresho kimukira ku ruhande rw'umukinnyi wa mbere inshuro 9 muri 14, kandi formula iri hejuru, hamwe n'udukoresho 12 kuri buri wese, imuha amahirwe agera ku nshuro 1.157 yo gutsindira byose kurusha gutakaza byose. Inyungu ya kazino ya 2,70% ni nto cyane, ariko formula imwe yerekana icyo ikora mu matega menshi.

Mu gisakuzo cya Pascal, buri jugunywa ry'udusoro dutatu ryimura agakoresho kamwe hagati y'abakinnyi.

Itega ry'ubutwari nyaryo

Muri Mata 2004 Ashley Revell, umusore w'Umwongereza, yagurishije hafi ibyo yari atunze byose, harimo n'imyenda ye, maze ajyana $135.300 muri Plaza i Las Vegas. Nyuma y'amatega mato, yashyize amafaranga yose ku mutukuitegaUmutukuItega ryishyura kimwe ku kindi ku mibare 18 y'umutuku. Mu Gifaransa: rouge.Soma byose →, rimwe. AgapiraibikoreshoAgapiraAgapira gato koherezwa mu nzira kerekeza mu cyerekezo kinyuranye n'icya rotor. Udupira tw'iki gihe dukozwe muri ivorine cyangwa Teflon, mm 18–21.Soma byose → kaguye kuri 7 itukura maze asohoka afite $270.600. Ku ruziga rwa zeruuruzigaZeruAkazu kamwe k'icyatsi katafatwa n'itega ryishyura kimwe ku kindi, irya dazeni cyangwa iry'inkingi; ni ho inyungu ya kazino ikomoka. Nanone: zeru imwe, 0.Soma byose → ebyiri iryo zunguruka rimwe ryari amahirwe ya 47,37% kandi ryamutwaye $7.121 ku mpuzandengo, inyungu ya kazino ya 5,26% ku itega rimwe. Iyo bigabanywa mu matega ya $10, iyo ntego imwe yari kuba hafi idashoboka. Gukina ubutwari ntibituma itega riba ryiza; bituma riba rigufi uko bishoboka kose.

Igishoro cyose cya Ashley Revell kiri ku izunguruka rimwe ry'uruziga rwa zeru ebyiri.

Ibyago byo gushira igishoro nta ntego

Niba ukina gusa kugeza igishoro gishize cyangwa ugahagarara nyuma y'amazunguruka N, ikibazo kiba “amahirwe y'uko igishoro cyanjye gikora kuri zeru mu mazunguruka N ni angahe?”. Nta formula ifunze ibaho kuri N iyo ari yo yose ariko amagerageza ari kuri uru rubuga arabibara: igishoro cya $500 ku $10 ku mutuku hafi ntikijya gishira mu mazunguruka 100 (munsi ya 1%), gishira hafi 9% by'igihe mu mazunguruka 500, na hafi 95% by'igihe muri 5.000. Nta tegeko ryo guhagarara, amahirwe yegera 100%, ari yo miterere y'igihe kirekire y'inyungu ya kazinoimibareInyungu ya kazinoIgice cya buri tega kazino igumana ku mpuzandengo: 2,70% muri roulette y'i Burayi.Soma byose →.

Imikino icumi, igishoro cya $500, $25 ku mutuku, amazunguruka 200igisubizo ugereranyije n'intangiriro

imikino 1.000 × amazunguruka 500 · ibara rya move p10/p25/p75/p90, ubururu bworoheje mediane, ingero 81 zoroheje · nyura ku gishushanyo urebe quantile ku izunguruka

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

imikino 1.000 × amazunguruka 500 · umurongo uhagaze waciwe kuri percentile ya 1–99 (impera −$500 kugeza +$1.500) · amazunguruka 500 · ibanze $25 · kimwe ku kindi

Imirongo ikora hasi ni iy'imikino yashizemo igishoro. Hamwe n'inite nto yamara igihe kirekire ikarangirira aho hamwe.

Kwegerana mu Gihe Kirekire: Itegeko ry'Imibare Minini muri Roulette

Igihe bimara, n'aho inyungu ishobora kugera

Ibindi bisubizo bibiri bya kera bisubiza ibibazo abakinnyi babaza koko. Nta ntego, amatega ahoraho yishyura kimwe ku kindi amara ku mpuzandengo amazunguruka 37 kuri buri nite y'igishoro ku ruziga rw'i Burayi na 19 ku rw'Abanyamerika, kuko buri zunguruka ritwara 1/37 y'inite ku mpuzandengo (1/19 ku ruziga rw'Abanyamerika): inite 100 zimara amazunguruka 3.700 ku mpuzandengo, nubwo kimwe cya kabiri cy'imikino nk'iyo irangira mbere ya 3.134 (reba Uburebure bw'Umukino n'Ikiguzi Cyawo). Kandi uko igishoro cyaba kingana kose, amahirwe ku ruziga rw'i Burayi yo kugira inyungu y'inite k igihe runaka ni (18/19)ᵏ. Ni 95% ku nite imwe na 58% ku icumi, ariko nta gishoro, uko cyaba kinini kose, gitanga amahirwe angana yo kugira inyungu y'inite 13; ku ruziga rw'Abanyamerika amahirwe ni (9/10)ᵏ kandi ajya munsi ya kimwe cya kabiri ku nite 7.

inite z'inyungu, igihe icyo ari cyo cyoseIy'i BurayiIy'Abanyamerika

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%

Amatega yishyura kimwe ku kindi, itega rihoraho, igishoro kinini ku buryo kitazigera gishira. Amahirwe ni (18/19)ᵏ ku ruziga rwa zeru imwe na (9/10)ᵏ ku rwa zeru ebyiri.

Uburebure bw'Umukino n'Ikiguzi Cyawo Uruziga n'Amategeko bya Roulette y'Abanyamerika

Igishoro cy'amahirwe angana

Hindura ikibazo: ugomba kuzana angahe kugira ngo ugire nibura amahirwe angana yo gutsindira amafaranga runaka mbere yo gutakaza byose? Ku ntsinzi nto, si menshi. Ku nini kurushaho igishoro gikenewe kizamuka cyane, kandi kuva ku rugero runaka nta gishoro gihagije, kuko n'igishoro kitagira iherezo gitanga amahirwe ari munsi y'angana yo kugera kure gutyo mu nyungu. Taylor (2021) abivuga mu bundi buryo: ku ruziga rw'Abanyamerika, kugira ngo ugire amahirwe angana yo kugera ku nite 100 ugomba gutangira ufite 94, ushyira 94 mu kaga ngo utsindire 6. Ku ruziga rw'i Burayi wakenera 88.

gutsindiraIy'i BurayiIy'Abanyamerika

inite 122

inite 233

inite 345

inite 468

inite 5712

inite 6921

inite 712nta gishoro gihagije

inite 815nta gishoro gihagije

inite 1024nta gishoro gihagije

inite 1246nta gishoro gihagije

inite 13 cyangwa zirenganta gishoro gihagijenta gishoro gihagije

Igishoro gito kurusha ibindi, mu nite, gitanga nibura amahirwe angana yo gutsindira ayo mafaranga mbere yo gutakaza igishoro, hakoreshejwe amatega ahoraho yishyura kimwe ku kindi. Ibarura ryacu rikoresheje formula iri hejuru.

Isomo rifatika

Fata icyemezo ku cyo ugerageza gukora. Niba intego ari intsinzi runaka, tega menshi kandi make. Niba intego ari umugoroba muremure, tega gake kandi wemere ko inyungu ya kazino izatwara 2,70% yayo ku mafaranga yose yategeweamafarangaAmafaranga yose yategeweAmafaranga yose yategewe mu mukino umwe cyangwa mu gihe runaka, harimo n'intsinzi zongeye gutegwa. Nanone: handle, action.Soma byose → menshi. Niba intego ari byombi, hari ikigomba kurekurwa, kandi akenshi ni intego.

Formula

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

ibyago byo gushira igishoro = 1 − P(intego)

q/p = 19/18 ku matega yishyura kimwe ku kindi

impuzandengo y'amazunguruka kugeza igishoro gishize, nta ntego = 37 × inite

P(kugira inyungu y'inite k igihe runaka) = (18/19)ᵏ

Ibibazo

Ese uruhererekane rw'amatega rugabanya ibyago byo gushira igishoro?

MartingaleingambaMartingaleUruhererekane rw'amatega rukuba itega kabiri nyuma ya buri gutsindwa. Reba igice cy'Ingamba umenye icyo rukora n'icyo rudakora.Soma byose → ibigabanya ku mikino migufi kandi ikabyongera cyane ku mikino miremire. Gutega itega rihorahoingambaGutega itega rihorahoGutega amafaranga angana kuri buri zunguruka. Ni yo ngero buri ruhererekane rw'amatega rugereranywa na yo.Soma byose → ku nite nini ni bwo buryo bw'ukuri bwo kubigabanya.

Bite ku matega ya straight up?

Formula iri hejuru ni iy'amatega yishyura kimwe ku kindi. Ku matega afite ihindagurika rinini, gushira igishoro bishoboka cyane ku matega mato kandi intego igerwaho mu buryo budahwanye; koresha mubazi.

Kuki q/p ari wo mubare w'ingenzi?

Buri ntambwe igana ku ntego ifite amahirwe make gato kurusha buri ntambwe igana ku gushira igishoro; igereranya ryiyongera ku bwinshi bw'intambwe.

Kwishimisha n'Imibare

Banza ubigerageze ubwawe. Buri gisubizo kirapfunyitse; ugifungure igihe witeguye.

Ufite $100 kandi ushaka $1.000, utega ku mutukuitegaUmutukuItega ryishyura kimwe ku kindi ku mibare 18 y'umutuku. Mu Gifaransa: rouge.Soma byose → gusa. Umugambi wa mbere: tega $1 buri gihe. Umugambi wa kabiri: tega ibyo ufite byose kugeza urenze $500, hanyuma utege gusa ibihagije ngo ugere kuri $1.000. Ni uwuhe mugambi mwiza, kandi ku rugero rungana iki?
Erekana igisubizo
Igisubizo Umugambi wa kabiri, kure cyane. Ugera kuri $1.000 hafi 9,0% by'igihe, hafi ya 10% iboneye. Umugambi wa mbere ugenda neza hafi incuro 7 kuri 10²².
Mu mukino uboneyeimibareUmukino utabogamyeUmukino aho nta ruhande rufite inyungu: buri tega rifite agaciro gateganyijwe kangana na zeru.Soma byose → wagera kuri $1.000 uvuye kuri $100 rimwe mu icumi icyo wakora cyose, kuko agaciro k'impuzandengo k'amafaranga yawe kaguma kuri $100.
Inyungu ya kazino ifata 2,70% bya buri tega, bityo uko utega bike muri rusange, ni ko uguma hafi ya rimwe mu icumi. Umugambi wa kabiri ukemura ikibazo mu matega make: $100 iba $200, $400, $800, hanyuma $200 ikaba ihagije. Ugenda neza hafi 9,0% by'igihe. Umugambi wa mbere ufata ibihumbi by'amatega mbere y'uko urugendo rurangira, kandi inyungu ya kazino, nto kuri buri rimwe muri yo, iteranya ikagera hafi ku gushira igishoro kudashidikanywaho.
Iki ni igisubizo cya Dubins–Savage mu buryo bwacyo bworoshye (Epstein, 2013): mu mukino uhengamiye ku nzu, tega ibyo ukeneye byose, incuro nke uko ushoboye.
Ufite $500 kandi ushaka $1.000, uhagarara ukimara kuyabona. Uhitamo ubwoko bumwe bw'itega maze ukabusubiramo kugeza butsinze rimwe cyangwa amafaranga agashira. Ni ubuhe bwoko butanga amahirwe meza kurusha ubundi?
Erekana igisubizo
Igisubizo Itega rimwe rya $500 ku mutuku: 48,6%. Buri mugambi ukwirakwiza amafaranga mu matega mato ukora nabi kurushaho.
Hitamo buri tega ku buryo intsinzi imwe ihagije: $30 kuri straight upitegaStraight upUmubare umwe, yishyura 35:1.Soma byose → itsinda $1.050, $60 kuri splititegaSplitImibare ibiri yegeranye, yishyura 17:1.Soma byose → itsinda $1.020, n'ibindi. Hanyuma ubare amahirwe y'uko nibura itega rimwe ritsinda mbere y'uko amafaranga ashira. Straight up cumi n'esheshatu za $30 zose zihusha ku mahirwe ya (36/37)¹⁶ = 64,5%, bityo zigera kuri $1.000 35,5% by'igihe.
Imbonerahamwe yerekana buri tega. Umutuku ukora neza kurusha kuko uhagera mu itega rimwe, bityo inyungu ya kazino yishyurwa rimwe, kuri $500. Six lineitegaSix lineImibare itandatu iri mu mirongo ibiri yegeranye, yishyura 5:1. Mu Gifaransa: sixain.Soma byose → iza ku mwanya wa kabiri kuko ikenera atatu gusa. Itega rimwe kuri dazeniitegaDazeni1–12, 13–24 cyangwa 25–36, yishyura 2:1.Soma byose → ni ryo ribi kurusha, amahirwe ya 12/37; ndetse rikinwe mu niteamafarangaIniteItega ry'ibanze uburyo cyangwa umukinnyi akoresha. Impererekane z'amatega zisobanurwa mu nite kugira ngo zikoreshwe ku meza ayo ari yo yose.Soma byose → za $250 kugeza ugize $1.000 cyangwa arenga, dazeni ihagera 46,4% by'igihe gusa.
Bollman (2024) yubaka imbonerahamwe nk'iyo ku ruziga rw'Abanyamerika. Umurongo we wa dazeni ufite ikosa: wagombye kuba 12/38 = 31,6%.
itegaitega × umubare w'amategaamahirwe yo kugera kuri $1.000
Straight up$30 × 1635,5%
Split$60 × 835,9%
Street$90 × 534,5%
Inguni$125 × 436,7%
Six line$166, $166, $16841,2%
Dazeni$500 × 132,4%
Umutuku$500 × 148,6%
Utega inite imwe ku mutuku, izunguruka ku rindi, ufite igishoro cyo gukinaamafarangaIgishoro cyo gukinaAmafaranga yose yashyizwe ku ruhande yo gukina. Gucunga igishoro ni ukwemeza ingano yacyo uzashyira mu kaga kuri buri tega no kuri buri mukino.Soma byose → kitigera gishira. Nyuma y'izunguruka rya mbere uba uri imbere ho inite imwe cyangwa inyuma ho inite imwe. Ni ayahe mahirwe y'uko uzigera wongera kuringanira neza?
Erekana igisubizo
Igisubizo 36 muri 37, 97,3%. Ku ruziga rw'Abanyamerika ni 36 muri 38, 94,7%.
Niba izunguruka rya mbere ritsinze (incuro 18 muri 37), uba uri imbere ho inite imwe, kandi inyungu ya kazino imanura buri mukino utagira iherezo, bityo amaherezo usubira ku kuringanira: ibyo ntibishidikanywaho. Niba ritsinzwe (incuro 19 muri 37), ugomba kuzamuka inite imwe, kandi amahirwe yo kwigera uba imbere ho inite imwe ni 18/19 (reba imbonerahamwe iri hejuru).
Hamwe: 18/37 + 19/37 × 18/19 = 18/37 + 18/37 = 36/37.
Muri rusange amahirwe ni 1 ukuyemo ikinyuranyo kiri hagati y'amahirwe yo gutsinda n'ayo gutsindwa izunguruka: 1 − 1/37 ku ruziga rw'i Burayi, 1 − 2/38 ku rw'Abanyamerika. Mu mukino utagira zeruuruzigaZeruAkazu kamwe k'icyatsi katafatwa n'itega ryishyura kimwe ku kindi, irya dazeni cyangwa iry'inkingi; ni ho inyungu ya kazino ikomoka. Nanone: zeru imwe, 0.Soma byose →, ikinyuranyo ni ubusa kandi amahirwe ni 1: amanota ahora agaruka ku kuringanira. Epstein (2013) atanga itegeko rusange.
Ikibazo cya Pascal cyo mu 1656: abakinnyi babiri batera udusoro dutatu, buri wese atangirana udukoresho two kubara 12. Buri 11 yimurira agakoresho ku mukinnyi wa mbere, buri 14 ikimurira kamwe ku wa kabiri; andi matera ntabarwa. Ni ayahe mahirwe y'uko umukinnyi wa mbere atsinda twose uko ari 24? Kandi niba aho kubikora utega inite 1 ku mutuku kuri buri zunguruka uhanganye n'inshuti ikora nka banki, inite 12 kuri buri wese, ni ayahe mahirwe yawe yo gutsinda zose uko ari 24?
Erekana igisubizo
Igisubizo Umukinnyi wa mbere atsinda hafi incuro 1.157 muri 1.158. Wowe, ku mutuku, utsinda inite zose 24 hafi 34% by'igihe gusa.
Udusoro dutatu dukora 11 mu buryo 27 na 14 mu buryo 15, bityo buri gakoresho kimuwe kajya ku mukinnyi wa mbere incuro 27 muri 42, incuro 9 muri 14. Buri ruhande rutangiye kuri 12, amahirwe yo gufata byose ni 1/(1 + r¹²), aho r ari igipimo cy'amahirwe y'urundi ruhande ugereranyije n'ayawe.
Ku mukinnyi wa mbere r = 5/9, kandi (5/9)¹² ≈ 0,00086, bityo amahirwe ni hafi 0,99914: odds za 9¹² kuri 5¹², cyangwa 282.429.536.481 kuri 244.140.625, hafi 1.157 kuri 1. Huygens yacapye iyo mibare neza mu 1657.
Ku mutuku, r = 19/18 kubera zeru, kandi (19/18)¹² ≈ 1,913: amahirwe yawe ni 1/2,913 ≈ 34,3%. Igiceri cyatanga 50%. Inyungu ya kazino ya 2,70% kuri buri tega iteranya, mu matega menshi bisaba, ikinyuranyo cy'amanota 16.

Akanya k'urwenya

$Gukina ubutwari ni byo byiza kurusha. Ariko birangira vuba.$

Gukina ubutwari ni byo byiza kurusha. Ariko birangira vuba.

Inkomoko

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1. Wiley, 3rd ed., 1968. Gushira igishoro k'umukinnyi w'urusimbi, igice cya 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. Letter to Huygens, 28 September 1656, on the problem Pascal posed to Fermat. Œuvres complètes de Christiaan Huygens, vol. 1, 1888, pp. 492–494.
Huygens, C. De Ratiociniis in Ludo Aleae. 1657. Inyandiko ya mbere icapwe ku gaciro gateganyijwe.
Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010. Igice cya 13, roulette.
Reuters. “Briton bets it all on roulette spin, and wins.” Mata 2004.
Las Vegas Sun. “Man wagers life’s worth.” 12 Mata 2004.

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