Faayre Majjinde Kaalis Fof
Porobabilite majjinde kaalis pari maa fof ado hewtaade faandaare.
E pariiji njoɓdi fotndi, faayre majjinde kaalis fof ina rewi formil juumre fijoowo kaalis. E kaalis pari $500, pariiji $10 e faandaare $600, a hewtan faandaare ndee e hedde 57% sahaaji, a majjina fof e ko heddii.
Naamnal ngal
A jogii a initekaalisInitePari fuɗɗoode mo sistem walla fijoowo gollirta. Jokkondirɗe pari ina cifee e initeeji ngam ɗi kuutoroo e kala taabal.Tar fof → kaalis parikaalisKaalis pariKaalis fof ko seŋaa ngam fijde. Toppitagol kaalis pari ko ñaawde ko foti waɗeede e kala pari e kala jonnde.Tar fof → a yiɗi hewtude a + b inite, a waɗa pari inite wootere e kala laawol e pari njoɓdi fotndipariPari njoɓdi fotndiBoɗeere/ɓaleere, peccotaake/peccotoonde, mawɗe/tokooseeje. Limooje 18, ina yoɓa 1:1.Tar fof → jogiingal porobabilitematematikPorobabiliteSahaa njeñtudi, immorde 0 haa 1 walla 0% haa 100%. Limoore wootere: 1/37 = 2,70%.Tar fof → dañal p = 18/37 e porobabilite majjere q = 19/37. Ko honɗum woni sahaa hewtude faandaare ndee ado hewtude meere? Jaabawol ngol Pascal, Fermat e Huygens njiytii ɗum e duuɓi 1650, ɗum ko gooto e njeñtuɗi ɓurɗi naywude e porobabilite.
Formil juumre fijoowo kaalis
P(hewtude faandaare) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ). Faayre majjinde kaalis fofmatematikFaayre majjinde kaalis fofPorobabilite majjinde kaalis pari fof hade yettaade faandaare walla limoore taarle.Tar fof → ko gooto ustaa ɗum. E q/p = 19/18 ≈ 1,0556, fotondiral ngal ina ɓeydoo no feewi e limoore initeeji, ɗuum waɗi formil ngal ina lepta initeeji tokooɗi no feewi: nde pariiji ɓeydotoo ngam hewtude faandaare, ko ɓuri heewde laabi ɗi njoɓdi 2,70% ndii yoɓetee e laawol.
Janngude ɗum
Initeeji tokooɗi ina lorna. Etaade ɗiɗiɓnude $500 e pariiji $10 ina hewta e 6% sahaaji; e pari gootal $500 ina hewta e 48,6% sahaaji. Fijo cuusal, pariiji seeɗa mawɗi, ina ɓurna sahaa hewtude faandaare e fijo ngo fodaare mum ko majjere, sabu ina famɗina laabi ɗi ɓural ngal huutortee. Ina ɓurna kadi wariyaasmatematikWariyaasPeeta no njeñtuɗi saŋkitiri e njaru fodaandu. Ina toowi ngam limoore wootere, ina lesɗi ngam pariiji njoɓdi fotndi.Tar fof → ko heɓata ma, ina timmina jemma oo law. Teoreem ngal ina noddee “fijo cuusal ko ɓuri moƴƴude”, Dubins e Savage tabitinii ɗum e 1965.
Ɗo caɗeele ɗee fuɗɗori
Naamnal ngal ina ɓuri ruleet naywude. E 1656 Pascal rokki Fermat caɗeele: fijooɓe ɗiɗo ina njana deeji tati, kala e maɓɓe ina fuɗɗoo e jetoŋji 12; jaltinngol 11 ina dirna jetoŋ gooto e sera gooto, jaltinngol 14 e sera goɗɗo. Ɓe ɗiɗo fof ñiiɓtii ɗum, Huygens, nannguɗo ɗum e bataake ganndo limto Pierre de Carcavi, muulnii ɗum e 1657 hono caɗeele joyaɓe e deftere aranndeere e porobabilite, e jaabawol tawa alaa seedee. Sappo e gooto ina waawi jaltude e laabi 27 tawa sappo e nay ko 15 tan, ɗuum waɗi kala jetoŋ ina dirna e sera fijoowo aranɗo laabi 9 e nder 14, formil dow ngol kadi, e jetoŋji 12 kala gooto, ina waɗa mo ina ɓura laabi 1 157 hedde sahaa dañde fof e majjinde fof. Ɓural 2,70% ina ɓura famɗude no feewi, kono formil gootol ngol ina holla ko ngal waɗata e pariiji keewɗi.
Pari cuusal goonga
E Abriil 2004 Ashley Revell, gorko Angalteer sukaajo, yeeyii hedde ko o jogii fof, haa e comci makko, ŋabbi $135 300 haa Plaza to Las Vegas. Caggal pariiji tokooɗi seeɗa o waɗii kaalis ngal fof e boɗeerepariBoɗeerePari njoɓdi fotndi e limooje 18 boɗeeje. E Farayse: rouge.Tar fof →, laawol gootol. BalkuutorɗeBalBal tokoosal werlaangal e laawol ngol e laawol rotoor ngol. Balli jamaanu hannde ko ivorine walla Teflon, 18–21 mm.Tar fof → ngal yani e boɗeere 7 o yaltii e $270 600. E taarorde seerotaarordeSeeroSuudu haako wootere nde hay pari njoɓdi fotndi, sappo e ɗiɗi walla kolon kuuɓataa; ko ɗum woni iwdi ɓural kasino. Ina noddee kadi: seero gooto, 0.Tar fof → ɗiɗi, ngol taarol gootol ko sahaa 47,37%, coggi mo $7 121 e hakkundeere, ɓural 5,26% e pari gootal. So feccaama e pariiji $10, faandaare wootere ndee ina sikkitinnoo hay sahaa. Fijo cuusal waɗataa pari ngal moƴƴude; ina waɗa ngal ina ɓura gaɗaade.
Faayre majjinde kaalis fof tawa alaa faandaare
So a fijii tan haa a majjina fof walla a dartoo caggal taarle N, naamnal ngal ina wayloo “ko honɗum woni sahaa kaalis pari am hewtoo meere nder taarle N?”. Alaa formil timmungol ngam kala N kono simulateeruuji e ngal lowre ina kiisa ɗum: kaalis pari $500 e $10 e boɗeere hedde majjinaa fof abada e taarle 100 (les 1%), ina majjina fof e hedde 9% sahaaji e taarle 500, e hedde 95% e 5 000. So alaa sariya darnde, porobabilite ndee ina yaha haa 100%, ɗum ko mbaadi juuta ɓural kasinomatematikƁural kasinoGeɗal kala pari ngal kasino jogiiyata e hakkundeere: 2,70% e ruleet Orop.Tar fof →.
- p10 −$500
- p25 −$500
- medyan −$500
- p75 +$0
- p90 +$350
No ɗum ɓooyata, e no a ardata
Njeñtuɗi aadaaji ɗiɗi goɗɗi ina njaaboo naamne ɗe fijooɓe naamnotoo goonga. Tawa alaa faandaare, pariiji njoɓdi fotndi fotɗi ina ɓooya e hakkundeere taarle 37 e kala inite kaalis pari e taarorde Orop e 19 e taarorde Amerik, sabu kala taarol ina ƴetta 1/37 inite e hakkundeere (1/19 e taarorde Amerik): inite 100 ina ɓooya taarle 3 700 e hakkundeere, hay so feccere jonɗe ɗeen ina timmoo nder 3 134 (ƴeew Juutal e Coggu Jonnde). Kadi hay so kaalis pari ina mawni no feewi, sahaa e taarorde Orop wonde a ardii k inite hay laawol gootol ko (18/19)ᵏ. Ɗum ko 95% e inite wootere e 58% e sappo, kono alaa kaalis pari, hay so ina luggi, rokkata sahaa feccere-feccere wonde a ardii inite 13 hay laawol gootol; e taarorde Amerik sahaa oo ko (9/10)ᵏ ina jippoo les feccere e inite 7.
Kaalis pari ngam sahaa feccere-feccere
Huccitin naamnal ngal: no foti ko a foti addude ngam jogaade hay seeɗa sahaa feccere-feccere dañde kaalis ganndaaɗo ado majjinde fof? Ngam dañal tokoosal, wonaa keewɗum. Ngam ɓurngal mawnude kaalis pari soklaaɗo ina ɓeydoo no feewi, caggal nokku alaa kaalis pari heewata, sabu hay mo timmataa ina rokka ko les sahaa feccere-feccere wonde a ardan ndeen ɓadaade. Taylor (2021) ina wiya ɗum e laawol goɗngol: e taarorde Amerik, ngam jogaade sahaa feccere-feccere hewtude inite 100 a foti fuɗɗaade e 94, faayre 94 ngam dañde 6. E taarorde Orop a sokliino 88.
Jaŋde kuutoraade
Taƴ ko etotoɗaa waɗde. So faandaare ndee ko dañal ganndaangal, waɗ pariiji mawɗi seeɗa. So faandaare ndee ko jemma juutɗo, waɗ pariiji tokooɗi jaɓaa wonde ɓural ngal ina ƴetta 2,70% mum e kaalis pariyaaɗo fofkaalisKaalis pariyaaɗo fofKawrital kaalis pariyaaɗo nder jonnde walla sahaa, limtaa kadi kaalis dañaaɗo pariyaaɗo kadi. Ina noddee kadi: handle, action.Tar fof → mawɗo. So faandaare ndee ko ɗiɗi ɗii fof, huunde ina foti accaade, ko heewi ko faandaare ndee.
Formil
Naamne
MartingaledabareMartingaleJokkondiral pari ɓeydoowo pari ɗiɗi laabi caggal kala majjere. Ƴeew hello Dabareeji ngam anndude ko ngal waɗata e ko ngal waɗataa.Tar fof → ina ustoo ɗum e jonɗe gaɗaaɗe ina ɓeyda ɗum no feewi e juutɗe. Pari fotɗidabarePari fotɗiWaɗde pari gooto kala taarol. Ko ɗum kala jokkondiral pari kawrondirtee.Tar fof → e inite ɓurnde mawnude ko laawol goongaangol ustaade ɗum.
Formil dow ngol ko ngam njoɓdi fotndi. E pariiji wariyaas toowɗo, majjinde fof ina ɓura sahaa e pariiji tokooɗi, faandaare kadi ina hewtee e laawol jaaɓotoongol; huutoro limorde ndee.
Kala daawal ngam faandaare ina famɗi sahaa seeɗa e kala daawal ngam majjinde fof; fotondiral ngal ina ɓeydondira e limoore daawe.
Weltaare e Matematik
Etu ɗum hoore maa adii. Kala jaabawol ina suuɗii; uddit ɗum so a heblii.
A jogii $100 a yiɗi $1 000, a waɗata pari e boɗeerepariBoɗeerePari njoɓdi fotndi e limooje 18 boɗeeje. E Farayse: rouge.Tar fof → tan. Feere adannde: waɗ pari $1 e kala laabi. Feere ɗimmere: waɗ pari e ko jogaa fof haa a ɓurta $500, ɓaawo ɗuum ko foti tan ngam hewtude $1 000. Feere honde ɓuri, ɓurirde no foti?
Hollu jaabawol ngol
Jaabawol Feere ɗimmere, ɓural mawngal. Ina hewta $1 000 hono 9,0% sahaa, takko 10% potɗo. Feere adannde ina hewta ɗum hono laabi 7 e nder 10²².
E fijo potngomatematikFijo potngoFijo ngo hay sera wootere alaa ɓural: kala pari ina jogii njaru fodaandu potndu e seero.Tar fof → a hewtayno $1 000 iwde e $100 laabi gooto e nder sappo, ko waɗuɗaa fof, sabu njaru hakkundeere kaalis maa ina heddoo e $100.
Ɓural kasino ina ɓetta 2,70% e kala pari, ɗum waɗi ko ɓuri famɗude ko pariyaa fof, ko ɓuri ɓadaade gooto e nder sappo. Feere ɗimmere ina taƴa haala kaa e pariiji seeɗa: $100 ina wonta $200, $400, $800, ɓaawo ɗuum $200 ina yonii. Ina hewta hono 9,0% sahaa. Feere adannde ina naamnoo pariiji ujunnaaje hade yahdu ndu gasude, ɓural ngal, famɗungal e kala e majji, ina ɓeydoo haa ina wona majjinde kaalis fof hono goonga.
Ɗum ko njeñtudi Dubins–Savage e mbaadi mum ɓurndi weeɓde (Epstein, 2013): e fijo ɗo kasino ɓuri, waɗ pari ko foti ma fof, e laabi ɓurɗi famɗude ɗi mbaawɗaa.
A jogii $500 a yiɗi $1 000, a darata ko a hewti ɗum tan. A suɓa pari gootal a fillitoo ngal haa ngal daña laabi gooto walla kaalis ngal gasa. Pari honngal ɓuri rokkude porobabilite?
Hollu jaabawol ngol
Jaabawol Pari gootal $500 e boɗeere: 48,6%. Kala feere senndoore kaalis ngal e pariiji ɓurɗi famɗude ina ɓuri bonde.
Suɓo kala pari no dañal gootal yonirta: $30 e limoore wooterepariLimoore wootereLimoore wootere tan, ina yoɓa 35:1.Tar fof → ina daña $1 050, $60 e limooje ɗiɗipariLimooje ɗiɗiLimooje ɗiɗi takkondirɗe, ina yoɓa 17:1.Tar fof → ina daña $1 020, e ko nanndi e ɗum. Ndeen limtu porobabilite nde pari gootal tal dañata hade kaalis ngal gasude. Pariiji limoore wootere sappo e jeegom $30 fof majjude ina jogii porobabilite (36/37)¹⁶ = 64,5%, ɗum waɗi ɗi kewtat $1 000 35,5% sahaa.
Tabalo oo ina holla kala pari. Boɗeere ɓuri sabu ina hewta e pari gootal, ɗum waɗi ɓural ngal ina yoɓee laabi gooto, e $500. Limooje jeegompariLimooje jeegomLimooje jeegom e lelɗe ɗiɗi takkondirɗe, ina yoɓa 5:1. E Farayse: sixain.Tar fof → ina ɗimmoo sabu ina naamnoo tati tan. Pari gootal e sappo e ɗiɗipariSappo e ɗiɗi1–12, 13–24 walla 25–36, ina yoɓa 2:1.Tar fof → ɓuri bonde, porobabilite 12/37; hay so ina fijee e initeejikaalisInitePari fuɗɗoode mo sistem walla fijoowo gollirta. Jokkondirɗe pari ina cifee e initeeji ngam ɗi kuutoroo e kala taabal.Tar fof → $250 haa a jogii $1 000 walla ko ɓuri, sappo e ɗiɗi ina hewta 46,4% sahaa tan.
Bollman (2024) ina mahi tabalo gootal ngal ngam taarorde Amerik. Lelnde mum sappo e ɗiɗi ina jogii juumre: ina foti wonde 12/38 = 31,6%.
paripari × limoore pariijiporobabilite hewtude $1 000Limoore wootere$30 × 1635,5%Limooje ɗiɗi$60 × 835,9%Limooje tati$90 × 534,5%Limooje nay$125 × 436,7%Limooje jeegom$166, $166, $16841,2%Sappo e ɗiɗi$500 × 132,4%Boɗeere$500 × 148,6%A waɗa pari inite wootere e boɗeere, taarol e taarol, e kaalis parikaalisKaalis pariKaalis fof ko seŋaa ngam fijde. Toppitagol kaalis pari ko ñaawde ko foti waɗeede e kala pari e kala jonnde.Tar fof → mo gasataa fey. Caggal taarol adanngol a ɓurii inite wootere walla a ustii inite wootere. Hol porobabilite nde ngartataa e fotde tigi kadi?
Hollu jaabawol ngol
Jaabawol 36 e nder 37, 97,3%. E taarorde Amerik ko 36 e nder 38, 94,7%.
So taarol adanngol dañii (laabi 18 e nder 37), a ɓurii inite wootere, ɓural ngal ina foodda kala jonnde nde gasataa les, ɗum waɗi ko ɓooyi ko ɓooyaani a rewtoto e fotde: ɗum ko goonga tigi. So majjii (laabi 19 e nder 37), ina foti ma ƴentude inite wootere, porobabilite ɓurtude inite wootere ko 18/19 (ƴeew tabalo dow oo).
Kawral: 18/37 + 19/37 × 18/19 = 18/37 + 18/37 = 36/37.
E kuuɓal porobabilite ndee ko 1 ustaa ceerondiral hakkunde porobabilite dañde e majjude taarol: 1 − 1/37 e taarorde Orop, 1 − 2/38 e taarorde Amerik. E fijo ɗo seerotaarordeSeeroSuudu haako wootere nde hay pari njoɓdi fotndi, sappo e ɗiɗi walla kolon kuuɓataa; ko ɗum woni iwdi ɓural kasino. Ina noddee kadi: seero gooto, 0.Tar fof → alaa, ceerondiral ngal alaa, porobabilite ndee ko 1: limoore ndee ina ruttoo e fotde sahaa fof. Epstein (2013) ina rokka sariya kuuɓal ngal.
Caɗeele Pascal 1656: fijooɓe ɗiɗo ina werla dee tati, kala e maɓɓe ina fuɗɗoo e limorɗe 12. Kala 11 ina rokka limorgal gootal fijoowo adanɗo, kala 14 ina rokka gootal ɗimmo; werle goɗɗe limtaaka. Hol porobabilite fijoowo adanɗo dañde 24 ɗee fof? So a waɗii pari inite 1 e boɗeere e kala taarol e yeeso sehil maa mo wonata banke, initeeji 12 kala e mon, hol porobabilite maa dañde 24 ɗee fof?
Hollu jaabawol ngol
Jaabawol Fijoowo adanɗo ina daña hono laabi 1 157 e nder 1 158. Aan, e boɗeere, a dañata initeeji 24 ɗee fof hono 34% sahaa tan.
Dee tati ina ngaddi 11 e laabi 27, 14 e laabi 15, ɗum waɗi kala limorgal dirtungal ina yaha e fijoowo adanɗo laabi 27 e nder 42, laabi 9 e nder 14. So kala gedal fuɗɗorii e 12, porobabilite ɓettude fof ko 1/(1 + r¹²), ɗo r woni feccirde porobabilite gedal goɗngal e maa.
Ngam fijoowo adanɗo r = 5/9, (5/9)¹² ≈ 0,00086, ɗum waɗi porobabilite ndee ko hono 0,99914: kot 9¹² e 5¹², walla 282 429 536 481 e 244 140 625, hono 1 157 e 1. Huygens fiyii limooje ɗee tigi e hitaande 1657.
E boɗeere, r = 19/18 sabu seero, (19/18)¹² ≈ 1,913: porobabilite maa ko 1/2,913 ≈ 34,3%. Kaalis werlaaɗo ina rokkayno 50%. Ɓural 2,70% e kala pari ina ɓeydoo, e pariiji keewɗi ɗi ɗum naamnotoo, haa ina wona ceerondiral poyntuuji 16.
Ƴeewndo weltinoowo
Fijo cuusal ko ɓuri moƴƴude. Ina timma law kadi.
Iwdi
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed.
- Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed.
- Carcavi, P. de. Bataake to Huygens, 28 Septaambar 1656, e caɗeele ɗe Pascal rokki Fermat.
- Huygens, C. De Ratiociniis in Ludo Aleae.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Reuters. “Briton bets it all on roulette spin, and wins.”
- Las Vegas Sun. “Man wagers life’s worth.”