Matematika ʻo e Lulēti · Ngāueʻaki · miniti ʻe 13

Fakatuʻutāmaki ke ʻOsi ʻa e Paʻanga

Q: ʻOku fakasiʻisiʻi koā ʻe ha fakalakalaka peti ʻa e fakatuʻutāmaki ke ʻosi ʻa e paʻanga?

ʻOku fakasiʻisiʻi ia ʻe he Martingale ki he ngaahi taimi vaʻinga nounou pea fakalahi lahi ʻaupito ki he ngaahi taimi vaʻinga lōloa. Ko e peti tatau ʻaki ha iuniti lahi ange ʻa e founga faitotonu ke fakasiʻisiʻi ai.

Ko e ngalingali ke mole kotoa hoʻo paʻanga vaʻinga kimuʻa pea aʻu ki ha taumuʻa.

Tali nounou

Ki he ngaahi peti totongi tatau, ʻoku muimui ʻa e fakatuʻutāmaki ke ʻosi ʻa e paʻanga ki he fomula ʻo e gambler’s ruin. ʻAki ha paʻanga vaʻinga ʻo e $500, ngaahi peti ʻo e $10 mo ha taumuʻa ʻo e $600, ʻoku ke aʻu ki he taumuʻa ʻi he meimei 57% ʻo e taimi pea ʻosi kotoa hoʻo paʻanga ʻi he toenga.

fomula

gambler’s ruin

p (peti totongi tatau)

18/37

q/p

1.0556

ola motuʻa taha

Pascal–Fermat, 1656

Ko e fehuʻi

ʻOku ʻi ai hoʻo 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 a ʻo e 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 → pea ʻoku ke fie aʻu ki he iuniti ʻe a + b, ʻo peti ʻaki ha iuniti ʻe taha ʻi he taimi takitaha ki ha peti totongi tataupetiPeti totongi tatauKulokula/ʻuliʻuli, taʻehoa/hoa, māʻolunga/māʻulalo. Fika ʻe 18, totongi ʻo e 1:1.Lau kotoa → ʻoku ko 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 → ke ikuna ko e p = 18/37 pea ko e ngalingali ke mole ko e q = 19/37. Ko e hā ʻa e faingamālie ke ke aʻu ki he taumuʻa kimuʻa pea ke aʻu ki he noa? Naʻe fakafuofua ʻa e tali ʻe Pascal, Fermat mo Huygens ʻi he ngaahi taʻu 1650 pea ko e taha ia ʻo e ngaahi ola motuʻa taha ʻi he ngalingali.

Blaise Pascal: Ko e Talatupuʻa ʻo e Veʻeteka Lulēti

Ko e fomula ʻo e gambler’s ruin

P(aʻu ki he taumuʻa) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ). Ko e fakatuʻutāmaki ke ʻosi ʻa e paʻangamatematikaFakatuʻutāmaki ke ʻosi ʻa e paʻangaKo e ngalingali ke mole kotoa ʻa e paʻanga vaʻinga kimuʻa pea aʻu ki ha taumuʻa pe ki ha lahi ʻo e vilo.Lau kotoa → ko e taha toʻo ia. ʻAki ʻa e q/p = 19/18 ≈ 1.0556, ʻoku tupu fakaʻekisipōnenitiale ʻa e vahe ʻo fakatatau ki he lahi ʻo e ngaahi iuniti, ko ia ʻoku tautea lahi ʻe he fomula ʻa e ngaahi iuniti siʻisiʻi: ko e lahi ange ʻo e ngaahi peti ʻoku fiemaʻu ke aʻu ki he taumuʻa, ko e lahi ange ia ʻo e tuʻo ʻoku totongi ai ʻa e tukuhau 2.70% ʻi he hala.

paʻanga vaʻingaiunititaumuʻapeti ki he taumuʻap(aʻu ki he taumuʻa)fakatuʻutāmaki ke ʻosi ʻa e paʻanga

$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%

Ngaahi peti totongi tatau, peti tatau, ʻikai ha la partage. ʻOku fai ʻe he Meʻa Fika ki he Fakatuʻutāmaki ke ʻOsi ʻa e Paʻanga ʻa e fika ki ha ngaahi fika pē.

Ko hono lau

ʻOku fakamamahi ʻa e ngaahi iuniti siʻisiʻi. Ko e feinga ke fakaua ʻa e $500 ʻaki ha ngaahi peti ʻo e $10 ʻoku lavameʻa ʻi he 6% ʻo e taimi; ʻaki ha peti ʻe taha ʻo e $500 ʻoku lavameʻa ʻi he 48.6% ʻo e taimi. Ko e vaʻinga taʻemanavahē, ko e peti siʻi ange mo lahi ange, ʻoku ne fakalahi taha ʻa e faingamālie ke aʻu ki ha taumuʻa ʻi ha vaʻinga ʻoku fakafiemoʻua hono ʻamanekina, he ʻoku ne fakasiʻisiʻi taha ʻa e tuʻo ʻoku ngāue ai ʻa e tuʻunga lelei. ʻOku ne toe fakalahi taha foki ʻa e 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 → ʻo e meʻa ʻe hoko kiate koe, pea ʻoku ne fakaʻosi vave ʻa e efiafi. ʻOku ui ʻa e teolēmi ko e “ko e vaʻinga taʻemanavahē ʻa e lelei taha” pea naʻe fakamoʻoniʻi ia ʻe Dubins mo Savage ʻi he 1965.

Faingamālie ke liliu ʻa e $500 ki he $1,000 ʻi he kulokula%

peti ʻe taha ʻo e $500

48.6%

ngaahi peti ʻo e $100

≈ 43%

ngaahi peti ʻo e $50

≈ 37%

ngaahi peti ʻo e $20

≈ 21%

ngaahi peti ʻo e $10

≈ 6%

ngaahi peti ʻo e $5

≈ 0.5%

024.3048.6%

Ko e lahi ange ʻo e ngaahi peti ʻoku fiemaʻu, ko e lahi ange ia ʻo e tuʻo ʻoku totongi ai ʻa e tuʻunga lelei. ʻOku mamafa ʻa e vaʻinga manavahē.

Ko e feituʻu naʻe kamata ai ʻa e palopalema

ʻOku motuʻa ange ʻa e fehuʻi ʻi he lulēti. ʻI he 1656 naʻe ʻoatu ʻe Pascal kia Fermat ha palopalema: ʻoku lī ʻe ha tokotaha vaʻinga ʻe toko ua ha taisi ʻe tolu, ʻo kamata takitaha ʻaki ha meʻa lau ʻe 12; ko ha lī ʻo e 11 ʻoku ne ʻave ha meʻa lau ki he tafaʻaki ʻe taha, ko ha lī ʻo e 14 ki he tafaʻaki ʻe taha. Naʻa na solova fakatouʻosi, pea naʻe paaki ia ʻe Huygens, ʻa ia naʻa ne fanongo ki ai ʻi ha tohi mei he tokotaha fika ko Pierre de Carcavi, ʻi he 1657 ko e palopalema hono nima ʻo e ʻuluaki tohi ki he ngalingali, mo e tali kae ʻikai ha fakamoʻoni. ʻE lava ke lī ʻa e hongofulu mā taha ʻi ha founga ʻe 27 pea ko e hongofulu mā fā ʻi he founga ʻe 15 pē, ko ia ʻoku ʻalu ʻa e meʻa lau takitaha ki he tafaʻaki ʻo e ʻuluaki tokotaha vaʻinga tuʻo 9 ʻi he 14, pea ʻoku ʻai ʻe he fomula ʻi ʻolunga, mo e meʻa lau ʻe 12 takitaha, ke meimei liunga 1,157 ʻa e ngalingali ke ne ikuna ʻa e meʻa kotoa ʻi he mole ʻa e meʻa kotoa. Ko ha tuʻunga lelei ʻo e 2.70% ʻoku siʻi ange ʻaupito, ka ʻoku fakahā ʻe he fomula tatau ʻa e meʻa ʻoku ne fai ʻi ha ngaahi peti lahi.

ʻI he palopalema ʻa Pascal, ʻoku ʻave ʻe he lī takitaha ʻo e taisi ʻe tolu ha meʻa lau ʻe taha ʻi he vahaʻa ʻo e ongo tokotaha vaʻinga.

Ko ha peti taʻemanavahē moʻoni

ʻI ʻEpeleli 2004 naʻe fakatau atu ʻe Ashley Revell, ko ha talavou ʻIngilani, ʻa e meimei meʻa kotoa naʻa ne maʻu, kau ai hono ngaahi vala, pea ne ʻave ʻa e $135,300 ki he Plaza ʻi Las Vegas. Hili ha ngaahi kiʻi peti siʻisiʻi naʻa ne tuku ʻa e lahi kotoa ki he kulokulapetiKulokulaPeti totongi tatau ki he fika kulokula ʻe 18. Faka-Falanisē: rouge.Lau kotoa →, tuʻo taha. Naʻe tō ʻa e pulumeʻangāuePuluKo e kiʻi pulu siʻisiʻi ʻoku tolongi ʻi he hala ʻo fehangahangai mo e konga vilo. Ko e ngaahi pulu ʻo onopooni ʻoku ngaohi ʻaki ʻa e ivorine pe Teflon, 18–21 mm.Lau kotoa → ki he kulokula 7 pea naʻa ne ʻalu ki tuʻa mo e $270,600. ʻI ha veʻeteka 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 → ua, ko e vilo ʻe taha ko ia ko ha faingamālie ʻo e 47.37% pea naʻe fakamole ai ʻa e $7,121 ʻi he ʻavalisi, ko e tuʻunga lelei ʻo e 5.26% ʻi ha peti ʻe taha. Kapau naʻe vahevahe ki he ngaahi peti ʻo e $10, ko e taumuʻa tatau ʻe meimei taʻeʻamanaki. ʻOku ʻikai ʻai ʻe he vaʻinga taʻemanavahē ʻa e peti ke lelei; ʻoku ne ʻai pē ke nounou taha.

Ko e paʻanga vaʻinga kotoa ʻa Ashley Revell ʻoku tuku ki ha vilo pē ʻe taha ʻo ha veʻeteka noa ua.

Fakatuʻutāmaki ke ʻosi ʻa e paʻanga ʻo ʻikai ha taumuʻa

Kapau ʻoku ke vaʻinga pē kae ʻoua kuo ʻosi kotoa hoʻo paʻanga pe ke taʻofi hili ha vilo ʻe N, ʻoku hoko ʻa e fehuʻi ko e “ko e hā ʻa e faingamālie ke aʻu ʻeku paʻanga vaʻinga ki he noa ʻi loto ʻi ha vilo ʻe N?”. ʻOku ʻikai ha fomula tāpuni ki ha N fakalūkufua ka ʻoku fakafuofua ia ʻe he ngaahi meʻa faʻifaʻitaki ʻi he uepisaiti ni: ko ha paʻanga vaʻinga ʻo e $500 ʻi he $10 ki he kulokula ʻoku meimei ʻikai ʻosi kotoa ʻi ha vilo ʻe 100 (siʻi hifo ʻi he 1%), ʻoku ʻosi kotoa ʻi he meimei 9% ʻo e taimi ʻi ha vilo ʻe 500, pea meimei 95% ʻo e taimi ʻi ha 5,000. ʻO ʻikai ha lao ke taʻofi, ʻoku hehema ʻa e ngalingali ki he 100%, ʻa ia ko e faʻahinga ʻi he taimi lōloa ʻo e tuʻunga lelei ʻo e kasinomatematikaTuʻunga lelei ʻo e kasinoKo e konga ʻo e peti takitaha ʻoku tauhi ʻe he kasino ʻi he ʻavalisi: 2.70% ʻi he lulēti ʻIulope.Lau kotoa →.

Taimi vaʻinga ʻe hongofulu, paʻanga vaʻinga ʻo e $500, $25 ki he kulokula, vilo ʻe 200ola ʻo fakahoa ki he kamataʻanga

taimi vaʻinga ʻe 1,000 × vilo ʻe 500 · vaioleti p10/p25/p75/p90, pulū maama fika lotoloto (median), sīpinga vaivai ʻe 81 · ʻave ki he kalafi ke sio ki he ngaahi kuanitaila ʻi ha vilo

p10 −$500
p25 −$500
fika lotoloto (median) −$500
p75 +$0
p90 +$350

taimi vaʻinga ʻe 1,000 × vilo ʻe 500 · kuo kosi ʻa e ʻakisi-y ki he pēsenitaila 1–99 (ngaahi ngataʻanga −$500 ki he +$1,500) · vilo ʻe 500 · tefito $25 · totongi tatau

Ko e ngaahi laine ʻoku ala ki lalo kuo ʻosi kotoa ʻenau paʻanga. ʻAki ha iuniti siʻi ange ʻe lōloa ange ʻenau tuʻu pea ngata ʻi he feituʻu tatau.

Fakaofi ʻi he Taimi Lōloa: Ko e Lao ʻo e Ngaahi Fika Lalahi ʻi he Lulēti

ʻE fēfē hono lōloa, pea ʻe fēfē hono mamaʻo ki muʻa

ʻOku tali ʻe ha toe ola motuʻa ʻe ua ʻa e ngaahi fehuʻi ʻoku fehuʻi moʻoni ʻe he kau vaʻinga. ʻO ʻikai ha taumuʻa, ʻoku tuʻu ʻa e ngaahi peti totongi tatau tatau ʻi he ʻavalisi ki he vilo ʻe 37 ki he iuniti takitaha ʻo e paʻanga vaʻinga ʻi ha veʻeteka ʻIulope pea 19 ʻi ha veʻeteka ʻAmelika, he ʻoku toʻo ʻe he vilo takitaha ʻa e 1/37 ʻo ha iuniti ʻi he ʻavalisi (1/19 ʻi ha veʻeteka ʻAmelika): ko e iuniti ʻe 100 ʻoku tuʻu ki he vilo ʻe 3,700 ʻi he ʻavalisi, neongo ko e vaeua ʻo e ngaahi taimi vaʻinga pehē ʻoku ʻosi ʻi loto ʻi he 3,134 (sio ki he Lōloa mo e Fakamole ʻo ha Taimi Vaʻinga). Pea neongo pe ʻoku lahi fēfē ʻa e paʻanga vaʻinga, ko e faingamālie ʻi ha veʻeteka ʻIulope ke ke ʻi muʻa ʻaki ha iuniti ʻe k ʻi ha taimi ko e (18/19)ᵏ. Ko e 95% ia ki he iuniti ʻe taha pea 58% ki he hongofulu, ka ʻoku ʻikai ha paʻanga vaʻinga, neongo pe ʻoku loloto fēfē, te ne ʻoatu ha faingamālie vaeua ke ke ʻi muʻa ʻaki ha iuniti ʻe 13; ʻi ha veʻeteka ʻAmelika ko e faingamālie ko e (9/10)ᵏ pea ʻoku tō ki lalo ʻi he vaeua ʻi he iuniti ʻe 7.

iuniti ʻi muʻa, ʻi ha taimi pēʻIulopeʻAmelika

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%

Ngaahi peti totongi tatau, peti tatau, ha paʻanga vaʻinga lahi feʻunga ke ʻoua ʻe ʻosi. Ko e faingamālie ko e (18/19)ᵏ ʻi ha veʻeteka noa ʻe taha pea (9/10)ᵏ ʻi ha veʻeteka noa ua.

Lōloa mo e Fakamole ʻo ha Taimi Vaʻinga Veʻeteka mo e Ngaahi Lao ʻo e Lulēti ʻAmelika

Ko ha paʻanga vaʻinga ki ha faingamālie vaeua

Fulihi ʻa e fehuʻi: ko e hā hono lahi ʻoku pau ke ke ʻomi ke maʻu ha faingamālie vaeua pe lahi ange ke ikuna ha lahi pau kimuʻa pea mole ʻa e meʻa kotoa? Ki ha kiʻi ikuna siʻi, ʻoku ʻikai lahi. Ki ha ikuna lahi ange ʻoku kaka vave ʻa e paʻanga vaʻinga ʻoku fiemaʻu, pea hili ha tuʻunga ʻoku ʻikai feʻunga ha paʻanga vaʻinga, he neongo ha paʻanga taʻengata ʻoku siʻi hifo hono faingamālie ʻi he vaeua ke ke aʻu ki muʻa pehē. ʻOku fakamatalaʻi ʻe Taylor (2021) ʻi ha founga kehe: ʻi ha veʻeteka ʻAmelika, ke maʻu ha faingamālie vaeua ke aʻu ki he iuniti ʻe 100 kuo pau ke ke kamata ʻaki ʻa e 94, ʻo fakatuʻutāmakiʻi ʻa e 94 ke ikuna ʻa e 6. ʻI ha veʻeteka ʻIulope ʻe fiemaʻu ʻa e 88.

ke ikunaʻIulopeʻAmelika

iuniti ʻe 122

iuniti ʻe 233

iuniti ʻe 345

iuniti ʻe 468

iuniti ʻe 5712

iuniti ʻe 6921

iuniti ʻe 712ʻoku ʻikai feʻunga ha paʻanga vaʻinga

iuniti ʻe 815ʻoku ʻikai feʻunga ha paʻanga vaʻinga

iuniti ʻe 1024ʻoku ʻikai feʻunga ha paʻanga vaʻinga

iuniti ʻe 1246ʻoku ʻikai feʻunga ha paʻanga vaʻinga

iuniti ʻe 13 pe lahi angeʻoku ʻikai feʻunga ha paʻanga vaʻingaʻoku ʻikai feʻunga ha paʻanga vaʻinga

Ko e paʻanga vaʻinga siʻi taha, ʻi he iuniti, ʻoku ne ʻoatu ha faingamālie vaeua pe lahi ange ke ikuna ʻa e lahi kimuʻa pea mole ʻa e paʻanga vaʻinga, ʻaki ha ngaahi peti totongi tatau tatau. Ko ʻemau fakafuofua mei he fomula ʻi ʻolunga.

Ko e lēsoni ʻaonga

Fili pe ko e hā ʻoku ke feinga ke fai. Kapau ko e taumuʻa ko ha ikuna pau, peti lahi pea siʻi. Kapau ko e taumuʻa ko ha efiafi lōloa, peti siʻisiʻi pea tali ʻe toʻo ʻe he tuʻunga lelei ʻa hono 2.70% ʻo ha fakakātoa ʻo e petipaʻangaFakakātoa ʻo e petiKo e fakakātoa ʻo e paʻanga naʻe peti ʻaki ʻi ha taimi vaʻinga pe vahaʻa taimi, kau ai ʻa e paʻanga naʻe ikunaʻi kuo toe peti ʻaki. ʻOku toe ui: handle, action.Lau kotoa → lahi. Kapau ko e taumuʻa ko e ongo meʻa fakatouʻosi, kuo pau ke tukuange ha meʻa, pea ko e faʻa taumuʻa ia.

Fomula

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

fakatuʻutāmaki ke ʻosi ʻa e paʻanga = 1 − P(taumuʻa)

q/p = 19/18 ki he peti totongi tatau

ʻavalisi ʻo e ngaahi vilo ke ʻosi ʻa e paʻanga, ʻikai ha taumuʻa = 37 × iuniti

P(ʻi muʻa ʻaki ha iuniti ʻe k ʻi ha taimi) = (18/19)ᵏ

Ngaahi fehuʻi

ʻOku fakasiʻisiʻi koā ʻe ha fakalakalaka peti ʻa e fakatuʻutāmaki ke ʻosi ʻa e paʻanga?

ʻOku fakasiʻisiʻi ia ʻe he 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 → ki he ngaahi taimi vaʻinga nounou pea fakalahi lahi ʻaupito ki he ngaahi taimi vaʻinga lōloa. Ko e peti tataufoungaPeti tatauPeti ʻaki ʻa e paʻanga tatau ʻi he vilo kotoa pē. Ko e tefito ia ʻoku fakahoa ki ai ʻa e fakalakalaka peti kotoa pē.Lau kotoa → ʻaki ha iuniti lahi ange ʻa e founga faitotonu ke fakasiʻisiʻi ai.

Fēfē ʻa e ngaahi peti fika ʻe taha?

Ko e fomula ʻi ʻolunga ki he peti totongi tatau. Ki he ngaahi peti vēliani māʻolunga ʻoku lahi ange ʻa e ngalingali ke ʻosi kotoa ʻa e paʻanga ʻi he ngaahi peti siʻisiʻi pea ʻoku aʻu ki he taumuʻa ʻi ha founga taʻemolemole ange; ngāueʻaki ʻa e meʻa fika.

Ko e hā ʻoku mahuʻinga taha ai ʻa e q/p?

Ko e sitepu takitaha ki he taumuʻa ʻoku kiʻi siʻi ange hono ngalingali ʻi he sitepu takitaha ki he ʻosi ʻa e paʻanga; ʻoku fakaliunga ʻa e vahe ʻi he lahi ʻo e ngaahi sitepu.

Vaʻinga mo e Matematika

ʻUluaki feinga ke ke fakaleleiʻi ʻe koe. Kuo pelu ʻa e tali takitaha; fakaava ia ʻi hoʻo mateuteu.

ʻOku ʻi ai hoʻo $100 pea ʻoku ke fie maʻu ʻa e $1,000, ʻo peti pē ki he kulokulapetiKulokulaPeti totongi tatau ki he fika kulokula ʻe 18. Faka-Falanisē: rouge.Lau kotoa →. ʻUluaki palani: peti ʻa e $1 ʻi he taimi takitaha. Palani hono ua: peti ʻa e meʻa kotoa ʻoku ʻi ai kiate koe kae ʻoleva ke ke laka hake ʻi he $500, pea toki peti leva pē ʻa e meʻa ʻoku feʻunga ke aʻu ki he $1,000. Ko fē ʻa e palani ʻoku lelei ange, pea ʻoku lelei ange ʻaki ʻa e hā?
Fakahā ʻa e tali
Tali Ko e palani hono ua, ʻo mamaʻo ʻaupito. ʻOku aʻu ia ki he $1,000 ʻi he 9.0% nai ʻo e taimi, ʻo ofi ki he 10% faitotonu. ʻOku lavameʻa ʻa e ʻuluaki palani ʻi he tuʻo 7 nai ʻi he 10²².
ʻI ha vaʻinga faitotonumatematikaVaʻinga faitotonuKo ha vaʻinga ʻoku ʻikai maʻu ʻe ha tafaʻaki ha tuʻunga lelei: ko e peti kotoa pē ʻoku noa hono mahuʻinga ʻamanekina.Lau kotoa → te ke aʻu ki he $1,000 mei he $100 tuʻo taha ʻi he tuʻo hongofulu neongo pē ko e hā hoʻo fai, koeʻuhi ʻoku nofo maʻu ʻa e mahuʻinga ʻavalisi ʻo hoʻo paʻanga ʻi he $100.
ʻOku toʻo ʻe he tuʻunga lelei ʻo e kasino ʻa e 2.70% ʻo e peti kotoa pē, ko ia ko e siʻi ange hoʻo peti fakakātoa, ko e ofi ange ia hoʻo nofo ki he taha ʻi he hongofulu. ʻOku fakaʻosi ʻe he palani hono ua ʻa e meʻa ʻi ha kiʻi peti siʻi pē: ʻoku hoko ʻa e $100 ko e $200, $400, $800, pea feʻunga leva ʻa e $200. ʻOku lavameʻa ia ʻi he 9.0% nai ʻo e taimi. ʻOku fiemaʻu ʻe he ʻuluaki palani ha peti ʻe afe ʻi muʻa pea ngata ʻa e fononga, pea ko e tuʻunga lelei ʻo e kasino, ʻa ia ʻoku siʻi ʻi he peti takitaha, ʻoku tānaki ia ʻo ofi ke pau ai ke ʻosi ʻa e paʻanga.
Ko e ola ʻeni ʻa Dubins–Savage ʻi hono anga faingofua taha (Epstein, 2013): ʻi ha vaʻinga ʻoku lelei ange ki he kasino, peti ʻa e lahi ʻoku ke fiemaʻu, ʻi he siʻi taha ʻo e ngaahi taimi ʻe lava.
ʻOku ʻi ai hoʻo $500 pea ʻoku ke fie maʻu ʻa e $1,000, ʻo taʻofi leva ʻi he taimi pē ʻoku ke maʻu ai ia. ʻOku ke fili ha faʻahinga peti ʻe taha pea toe fai ia kae ʻoleva ke ikuna tuʻo taha pe ʻosi ʻa e paʻanga. Ko e faʻahinga fē ʻoku ne ʻomi ʻa e faingamālie lelei taha?
Fakahā ʻa e tali
Tali Peti $500 ʻe taha ki he kulokula: 48.6%. ʻOku kovi ange ʻa e palani kotoa pē ʻoku ne vahevahe ʻa e paʻanga ki ha ngaahi peti siʻi ange.
Fili ʻa e lahi ʻo e peti takitaha koeʻuhi ke feʻunga ha ikuna ʻe taha: ko e $30 ki ha peti fika ʻe tahapetiPeti fika ʻe tahaFika pē ʻe taha, totongi ʻo e 35:1.Lau kotoa → ʻoku ikuna ʻa e $1,050, $60 ki ha peti fika ʻe uapetiPeti fika ʻe uaFika ʻe ua ʻoku fehokotaki, totongi ʻo e 17:1.Lau kotoa → ʻoku ikuna ʻa e $1,020, pea pehē atu. Pea lau leva ʻa e ngalingali ke ikuna ha peti ʻe taha pe lahi ange ʻi muʻa pea ʻosi ʻa e paʻanga. ʻOku hala kotoa ʻa e peti fika ʻe taha ʻe hongofulu-mā-ono ʻo e $30 ʻi he ngalingali ʻo e (36/37)¹⁶ = 64.5%, ko ia ʻoku nau aʻu ki he $1,000 ʻi he 35.5% ʻo e taimi.
ʻOku fakahā ʻe he tēpile ʻa e peti takitaha. ʻOku lelei taha ʻa e kulokula koeʻuhi ʻoku aʻu ia ʻi ha peti ʻe taha, ko ia ʻoku totongi ʻa e tuʻunga lelei ʻo e kasino tuʻo taha pē, ʻi he $500. ʻOku hoko hono ua ʻa e peti fika ʻe onopetiPeti fika ʻe onoFika ʻe ono ʻi ha laine fakafaʻalalo ʻe ua ʻoku fehokotaki, totongi ʻo e 5:1. Faka-Falanisē: sixain.Lau kotoa → koeʻuhi ʻoku fiemaʻu pē ʻe ia ʻa e peti ʻe tolu. Ko ha peti tasinipetiTasini1–12, 13–24 pe 25–36, totongi ʻo e 2:1.Lau kotoa → ʻe taha ʻoku kovi taha, ko ha faingamālie ʻo e 12/37; neongo ʻoku vaʻinga ʻi he 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 → ʻo e $250 kae ʻoleva ke ʻi ai hoʻo $1,000 pe lahi ange, ʻoku aʻu pē ʻa e tasini ʻi he 46.4% ʻo e taimi.
ʻOku langa ʻe Bollman (2024) ʻa e tēpile tatau maʻa ha veʻeteka ʻAmelika. ʻOku ʻi ai ha kiʻi fehalaaki ʻi heʻene laine ʻo e tasini: ʻoku totonu ke pehē ko e 12/38 = 31.6%.
petilahi ʻo e peti × lahi ʻo e ngaahi petingalingali ke aʻu ki he $1,000
Peti fika ʻe taha$30 × 1635.5%
Peti fika ʻe ua$60 × 835.9%
Peti fika ʻe tolu$90 × 534.5%
Peti tuliki$125 × 436.7%
Peti fika ʻe ono$166, $166, $16841.2%
Tasini$500 × 132.4%
Kulokula$500 × 148.6%
ʻOku ke peti ha iuniti ʻe taha ki he kulokula, ʻi he vilo ki he vilo, 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 → ʻoku ʻikai ʻaupito ʻosi. Hili ʻa e ʻuluaki vilo ʻoku ke maʻu iuniti ʻe taha pe mole iuniti ʻe taha. Ko e hā ʻa e ngalingali te ke toe tatau pē tonu ʻi ha taimi?
Fakahā ʻa e tali
Tali 36 ʻi he 37, 97.3%. ʻI ha veʻeteka ʻAmelika ko e 36 ʻi he 38, 94.7%.
Kapau ʻe ikuna ʻa e ʻuluaki vilo (tuʻo 18 ʻi he 37), ʻoku ke maʻu iuniti ʻe taha, pea ʻoku toho hifo ʻe he tuʻunga lelei ʻo e kasino ʻa e taimi vaʻinga taʻengata kotoa pē, ko ia ʻi ha taimi te ke toe foki ʻo tatau pē: ʻoku pau ia. Kapau ʻe mole ia (tuʻo 19 ʻi he 37), kuo pau ke ke kaka hake ʻaki ha iuniti ʻe taha, pea ko e ngalingali ke ke ʻi muʻa ʻaki ha iuniti ʻe taha ʻi ha taimi ko e 18/19 (vakai ki he tēpile ʻi ʻolunga).
Fakataha: 18/37 + 19/37 × 18/19 = 18/37 + 18/37 = 36/37.
Fakalūkufua, ko e ngalingali ko e 1 toʻo ʻa e vahaʻa ʻi he ngalingali ke ikuna mo e ngalingali ke mole ha vilo: 1 − 1/37 ʻi ha veʻeteka ʻIulope, 1 − 2/38 ʻi ha veʻeteka ʻAmelika. 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 → ʻoku ʻikai ha vahaʻa pea ko e ngalingali ko e 1: ʻoku foki maʻu pē ʻa e lau ki he tatau pē. ʻOku ʻomi ʻe Epstein (2013) ʻa e lao fakalūkufua.
Ko e palopalema ʻa Pascal ʻi he 1656: ʻoku lī ʻe ha tokotaha vaʻinga ʻe toko ua ha taisi ʻe tolu, ʻo kamata takitaha ʻaki ha meʻa lau ʻe 12. ʻOku ʻave ʻe he 11 kotoa pē ha meʻa lau ʻe taha ki he ʻuluaki tokotaha vaʻinga, ʻoku ʻave ʻe he 14 kotoa pē ha taha ki hono ua; ʻoku ʻikai lau ʻa e ngaahi lī kehe. ʻOku ngalingali fēfē ke ikuna ʻe he ʻuluaki tokotaha vaʻinga ʻa e 24 kotoa? Pea kapau te ke peti ʻa e iuniti ʻe 1 ki he kulokula ʻi he vilo takitaha ʻi he fāitaha mo ha kaungāmeʻa ʻoku ngāue ko e pangikē, ko e iuniti ʻe 12 takitaha, ʻoku ngalingali fēfē te ke ikuna ʻa e 24 kotoa?
Fakahā ʻa e tali
Tali ʻOku ikuna ʻa e ʻuluaki tokotaha vaʻinga ʻo ofi ki he tuʻo 1,157 ʻi he 1,158. Ko koe, ʻi he kulokula, ʻoku ke ikuna ʻa e iuniti kotoa ʻe 24 ʻi he 34% nai pē ʻo e taimi.
ʻOku faʻu ʻe he taisi ʻe tolu ʻa e 11 ʻi he founga ʻe 27 mo e 14 ʻi he founga ʻe 15, ko ia ʻoku ʻalu ʻa e meʻa lau takitaha ʻoku hiki ki he ʻuluaki tokotaha vaʻinga tuʻo 27 ʻi he 42, tuʻo 9 ʻi he 14. ʻI he kamata ʻa e tafaʻaki takitaha ʻi he 12, ko e ngalingali ke toʻo ʻa e meʻa kotoa ko e 1/(1 + r¹²), ʻa ia ko e r ko e vahevahe ʻo e ngalingali ʻo e tafaʻaki ʻe taha ʻaki hoʻo ngalingali.
Ki he ʻuluaki tokotaha vaʻinga ko e r = 5/9, pea ko e (5/9)¹² ≈ 0.00086, ko ia ko e ngalingali ko e 0.99914 nai: ko e faingamālie ʻo e 9¹² ki he 5¹², pe 282,429,536,481 ki he 244,140,625, ʻo ofi ki he 1,157 ki he 1. Naʻe paaki tonu ʻe Huygens ʻa e ngaahi fika ko ia ʻi he 1657.
ʻI he kulokula, ko e r = 19/18 koeʻuhi ko e noa, pea ko e (19/18)¹² ≈ 1.913: ko hoʻo ngalingali ko e 1/2.913 ≈ 34.3%. ʻE ʻomi ʻe ha paʻanga lī ʻa e 50%. ʻOku tānaki ʻa e tuʻunga lelei ʻo e 2.70% ʻi he peti takitaha, ʻi he ngaahi peti lahi ʻoku fiemaʻu, ki ha vahaʻa ʻo e poini ʻe 16.

Meʻa fakaoli

$Ko e vaʻinga taʻemanavahē ʻa e lelei taha. ʻOku ʻosi vave foki ia.$

Ko e vaʻinga taʻemanavahē ʻa e lelei taha. ʻOku ʻosi vave foki ia.

Ngaahi Maʻuʻanga

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1. Wiley, paaki hono 3, 1968. Gambler’s ruin, vahe 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. Tohi kia Huygens, 28 Sepitema 1656, fekauʻaki mo e palopalema naʻe ʻoatu ʻe Pascal kia Fermat. Œuvres complètes de Christiaan Huygens, vol. 1, 1888, pp. 492–494.
Huygens, C. De Ratiociniis in Ludo Aleae. 1657. Ko e fakamatala paaki muʻomuʻa taha ki he ʻamanekina.
Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010. Vahe 13, lulēti.
Reuters. “Briton bets it all on roulette spin, and wins.” ʻEpeleli 2004.
Las Vegas Sun. “Man wagers life’s worth.” 12 ʻEpeleli 2004.

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