Caɗeele: Keerol Taabal
Martingale $10 e taabal $500. No ɗum heewiri helde, ko foti ɗum majjinta so helii, e mbele ɓantude keerol ngol ina walla?
Majjeeje jeegom jokkondirɗe ina kela ɗum: 1,8% e daawal kala, hono 60% jonɗe taarle 100. Helde ngal ina coggoo $630. Ɓantude keerol ngol haa $5 000 ina ɓura ustude helde ngal (0,25% e daawal kala) ina ɓura mawninde ngal ($5 110). Majjere fodaande e dolaar kala pariyaaɗo ko 2,70% e kala keerol.
Caɗeele ɗee
Fijoowo ina waɗa pari $10 e boɗeerepariBoɗeerePari njoɓdi fotndi e limooje 18 boɗeeje. E Farayse: rouge.Tar fof →, ina ɓeyda ɗum laabi ɗiɗi caggal majjere kala, ina rutta e fuɗɗorde caggal dañal. Keerol dow taabal ngal ko $500. (a) Majjeeje jokkondirɗe hoto ngoni kelooje jokkondiral paridabareJokkondiral pariKala sariya waylowo pari e njeñtuɗi ɓennuɗi. Jokkondiral caggal majjere ina ɓeyda pari caggal majjere, caggal dañal ina ɓeyda ɗum caggal dañal.Tar fof → ngal? (b) Hol ko woni porobabilitematematikPorobabiliteSahaa njeñtudi, immorde 0 haa 1 walla 0% haa 100%. Limoore wootere: 1/37 = 2,70%.Tar fof → e daawal kala? (c) Hol ko majjata e helde ngal? (d) No keerol $5 000 waylitirta jaabawol ɗee? (e) Mbele keerol woodi waɗoowo sistem ngal ina rokka dañal?
Jaabawol (a)–(c)
Pariiji: 10, 20, 40, 80, 160, 320. Jeeɗiɗaɓo oo ina wonnoo 640 > 500, ɗum noon majjeeje jeegom ina kela ɗum. (b) p(majjere) = 19/37 e boɗeere; (19/37)⁶ = 0,0183, hono 1,8% e daawal kala. Daawal ina juuta hono taarle ɗiɗi e hakkundeere, ɗum noon jonnde taarle 100 ina jogii hono daawe 50 e p(helde gootol ko famɗi fof) ≈ 1 − 0,9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, ɗum ko initeejikaalisInitePari fuɗɗoode mo sistem walla fijoowo gollirta. Jokkondirɗe pari ina cifee e initeeji ngam ɗi kuutoroo e kala taabal.Tar fof → fuɗɗorde 63, e $10 ɗo daawal ngal ina etoo dañde.
Jaabawol (d)
E $5 000: pariiji haa 2 560 ina jaɓaa; pari sappoɓo oo, 5 120, ina salaa. Majjeeje jeenay ina kela ɗum: (19/37)⁹ ≈ 0,25% e daawal kala, hono 11% jonɗe taarle 100. Majjere e helde: 10 + 20 + … + 2 560 = $5 110. Fijoowo oo soodii masiibo ɓurɗo famɗude e coggu masiibo ɓurɗo mawnude. E nder jonɗe keewɗe, coggu fodaangu heleeji ɗii ko p × majjere ≈ 0,0183 × 630 = $11,5 e daawal kala e keerol $500 e 0,0025 × 5 110 = $12,8 e $5 000; ɗiɗi ɗii ina ɓadondiri kono fotaani, kala e mum ko majjere fodaandematematikMajjere fodaandeNjaru fodaandu fiyaandu e kaalis pariyaaɗo fof; ko hakkundeere kaalis majjataako nder jonnde walla nder sahaa.Tar fof → daawal ngal ɓeydi $10 ɗo daawe dañooje ɗee keɓata.
Jaabawol (e)
Alaa. Majjere fodaande e daawal kala = (pari fof fodaaɗo e nder daawal ngal) / 37, e kala keerol, sabu pari kala ko −1/37 mayri. Keerol ngol ina taƴa tan no majjere ndee senndirtee hakkunde dañe tokooseeje keewɗe e majjeeje mawɗe seeɗa. So alaa keerol 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 timmataa, jokkondiral ngal ina “gollira”, ɗum firti helataa abadaa, kono ina waɗa pari kaalis mo keerol alaa e majjere fodaande 2,70% kaalis mo keerol alaa oo. Wonaa keerol taabaltaabalKeerol taabalPari ɓurɗo famɗude e ɓurɗo mawnude jaɓaaɗo. Keerol dow ngol ko ngol dartinta jokkondirɗe pari ɓeydooje ɗiɗi laabi.Tar fof → haɗata MartingaledabareMartingaleJokkondiral pari ɓeydoowo pari ɗiɗi laabi caggal kala majjere. Ƴeew hello Dabareeji ngam anndude ko ngal waɗata e ko ngal waɗataa.Tar fof → gollude. Ko hiisa.
Taabalji goonga ɗiɗi
Misal Thorp ko taabal $1 jogiiɗo keerol dow $1 000, ɗo fijoowo ina tama dañde $30 ngam hiraande, dolaar e dolaar. Pariiji 1, 2, 4 … 512 ina njaɓa pariiji sappo, ɗum noon majjeeje sappo jokkondirɗe ina kela jokkondiral ngal: laabi gooto e nder daawe 785 e taarorde Orop (laabi gooto e nder 613 e taarorde Amerik), e coggu $1 023. Daawe capanɗe tati ɗee fof ina ndaña 96,2% sahaa. 3,8% goɗɗe ɗee ina majjina hono $1 000, njeñtudi hakkundeeri kiikiiɗe ɗee ko −$9,17, ɗum ko tigi 1/37 $339 ɗo daawe capanɗe tati ɗee ina mbaɗa pari e hakkundeere. Taabal $5 jogiiɗo keerol dow $2 000 ceertaani: pariiji haa $1 280 ina njaɓa pariiji jeenay, helde ngal ina waɗa laabi gooto e nder daawe 403 e taarorde Orop (laabi gooto e nder 323 e taarorde Amerik), ina coggoo $2 555.
Fijoowo galo
Rokk jokkondiral ngal kaalis pari keewɗo nden e keerol les: initeeji 1 000, fuɗɗorde inite wootere e taarle 1 000 e boɗeere, ruttoo e inite wootere kala nde ɓeydagol laabi ɗiɗi garoowo ngol ɓurii ko heddii. Men ndognii simulaasiyoŋ kiikiiɗe miliyoŋ hono ɗee. E taarorde Orop 52% ina timma e dañal, 21% ina majjina fof, njeñtudi hakkundeeri ndii ko −124 initeeji. Hay e fijo alaa 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 →, ɗo pari kala ko potndi, hono 60% tan timmata e dañal. Kiikiiɗe ɗe ɓuri heewde ko dañal tokooso; hakkundeere ndee ina yoɓee e kiikiiɗe timmotooɗe e seero.
Ekkitinol ngol
Kasinooji ina ngaɗa keeri ngam haɗde wariyaasmatematikWariyaasPeeta no njeñtuɗi saŋkitiri e njaru fodaandu. Ina toowi ngam limoore wootere, ina lesɗi ngam pariiji njoɓdi fotndi.Tar fof → mum en, wonaa ngam foolde sistemji. Foolde Martingale ko huunde ɓeydotoonde nde ɓe weltii heɓde, hiisa ngal kadi ina foolnoo ngal hay so alaa ɗum. Fijoowo sikkoowo keerol ngol ko caɗeele ɗee yiytii cakkol ngol, sikki ko ɗum woni tellorde nden.
Formil
Naamne
Ina waawna ma hiswude haa keerol ɓurngol toownude, ɗo helde ngal ɓuri famɗude e ɓuri mawnude. Majjere fodaande e dolaar kala waylitaaki.
Pariiji 10, 30, 70, 150, 310: keerol $500 ina yettinee caggal majjeeje joy, helde ngal ina coggoo $570. 2,70% gooto oo.
Ngam fijoowo gooto waawaa waɗde pari ɓurɗo ko kaalis kasino reenaaɗo waawi yoɓde e newnude. Toppitagol wariyaas.
Weltaare e Matematik
Etu ɗum hoore maa adii. Kala jaabawol ina suuɗii; uddit ɗum so a heblii.
MartingaledabareMartingaleJokkondiral pari ɓeydoowo pari ɗiɗi laabi caggal kala majjere. Ƴeew hello Dabareeji ngam anndude ko ngal waɗata e ko ngal waɗataa.Tar fof → ina fuɗɗoo e $100 e boɗeerepariBoɗeerePari njoɓdi fotndi e limooje 18 boɗeeje. E Farayse: rouge.Tar fof → ina yamira hoore mum pariiji n tan e kala taarngal: caggal majjeeje n e jokkondiral ina accita ina fuɗɗitoo. Winndu njeñtudi hakkundeere taarngal gootal hono formil e n, limtu ɗum ngam pariiji 5, 6 e 7 e taarorde Orop.
Hollu jaabawol ngol
Jaabawol Hakkundeere ndee ko 100 × (1 − (38/37)ⁿ): −$14,26 ngam pariiji 5, −$17,35 ngam 6 e −$20,52 ngam 7. Yamirde pariiji ɓurɗi heewde ina bonna taarngal hakkundeejo, ina moƴƴintaa ɗum.
Taarngal ina daña $100 si wonaa so pariiji n ɗii fof majjii, ɗum ina waɗa e porobabilite qⁿ, ɗo q = 19/37. So ɗi fof majjii, pariiji 100, 200 … 100 × 2ⁿ⁻¹ ina ɓeydondira haa 100 × (2ⁿ − 1). Ɗum waɗi hakkundeere ndee ko 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Ɗoo 2q = 38/37, ko ɓuri 1 seeɗa, ɗum waɗi (2q)ⁿ ina ɓeydoo e n, hakkundeere ndee ina jippoo: n = 5 ina addi 1,1426 e −$14,26; n = 6 ina addi 1,1735 e −$17,35; n = 7 ina addi 1,2052 e −$20,52. Jokkondiral pari ɓurngal juutde ina ɓuri seeɗa majjude, e 3,6%, 1,8% e 0,9% taarɗe, kono kala majjere ina mawni laabi ɗiɗi, kaalis ɓurɗo heewde ina lolla e taabal ngal e kala taarngal, e 2,70% kala pari.
E taarorde Amerik 2q = 40/38, limooje ɗee ko −$29,24, −$36,04 e −$43,20 (Taylor, 2021). 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 2q = 1, kala taarngal ina jogii hakkundeere seero tigi, hay so jokkondiral pari ngal juutii no foti.
“Martingale mawngal” ina ɗiɗiɓina pari ngal caggal majjere ina ɓeyda initekaalisInitePari fuɗɗoode mo sistem walla fijoowo gollirta. Jokkondirɗe pari ina cifee e initeeji ngam ɗi kuutoroo e kala taabal.Tar fof → wootere, ɗum waɗi pariiji ɗii ina njaha 1, 3, 7, 15, 31 … Initeeji njelu ɓeydi-ɗaa so dañal adanngal arii e pari 1ɓiingal, 2ɓiingal, 3ɓiingal walla kɓiingal? E inite $10 e keerol dow $500, taarngal ina hella laabi njelu, taarngal ina coftina no foti e hakkundeere?
Hollu jaabawol ngol
Jaabawol Dañal e pari kɓiingal ina accoo ma ɓeydi initeeji k: +1, +2, +3 e ko nanndi e ɗum. E $10 e $500 taarngal ina hella caggal majjeeje joy, 3,57% taarɗe, ngam −$570. E hakkundeere taarngal ina majjina $2,32 e nder $85,73 pariyaaɗo: 2,70%.
Pari kɓiingal ko initeeji 2ᵏ − 1: 1, 3, 7, 15, 31. Pariiji k − 1 adiiɗi ɗii ina ɓeydondira haa 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Dañal e pari kɓiingal ina yoɓa 2ᵏ − 1, ɗum waɗi taarngal ina gasa (2ᵏ − 1) − (2ᵏ − k − 1) = initeeji k ɓeydiiɗi: inite ɓeydaande wootere e kala majjere hade dañal ngal.
E $10 e kala inite, pariiji ɗii ko $10, $30, $70, $150 e $310; garoowal ngal, $630, ina ɓuri keerol dow $500. Majjeeje joy e jokkondiral, (19/37)⁵ = 3,57%, ina coftina 10 + 30 + 70 + 150 + 310 = $570.
Hakkundeere: dañe ɗee ina ngaddi $10 × k e porobabilite (19/37)ᵏ⁻¹ × 18/37 ngam k = 1 haa 5, hellugol ngol ina ɓetta $570 e porobabilite 3,57%; kawral −$2,32. Kaalis pariyaaɗo ina hakkundeere $85,73 e kala taarngal, 2,32/85,73 = 2,70%. Martingale aadanteengal e taabal gootal ngal ina hella caggal majjeeje jeegom, 1,83% taarɗe, ngam −$630, ina majjina $1,74 e nder $64,20 pariyaaɗo: 2,70% kadi.
Fijoowo ina faandii Martingale e boɗeere e pari adanngal $1 e taabal mo keerol mum woni $1 000, ɗum waɗi pariiji sappo ina mbaawi waɗude: $1, $2, $4 e ko nanndi e ɗum haa $512. Taarngal ina majjina sappo ɗii fof laabi njelu, ɗum ina naamnoo no foti e hakkundeere, kiikiiɗe pariiji 200 ina gasira nii laabi njelu?
Hollu jaabawol ngol
Jaabawol Taarngal ina majjina sappo ɗii fof hono laabi gooto e nder 784, ɗum ina coftina $1 023. Ɗum ko hono pariiji 1 600 e hakkundeere, kiikiiɗe pariiji 200 ina fiyoo e keerol ngol hono 11% sahaa. Taarngal ina majjina tan $0,31 e hakkundeere e nder $11,31 pariyaaɗo: 2,70%.
Pari e boɗeere ina majja e porobabilite 19/37, seero ina e nder. Sappo e jokkondiral: (19/37)¹⁰ ≈ 0,00128, hono 1 e nder 784. Pariiji 1 + 2 + 4 + … + 512 ina ɓeydondira haa $1 023.
Taarngal ina naamnoo pariiji 2,05 e hakkundeere, ɗum waɗi taarɗe 784 ina naamnoo hono pariiji 1 600. So limtaama taarol e taarol, jokkondiral majjeeje sappo ina feeñoo nokku e nder pariiji 200 11,3% sahaa.
Hakkunde musibaaji ɗii fijoowo oo ina daña $1 e kala taarngal: hono $783 hade bonngal ngal, ndeen −$1 023, hono −$240 e juuteendi ndii fof. E kala taarngal: 1 × (1 − 0,00128) − 1 023 × 0,00128 ≈ −$0,31. Kaalis pariyaaɗo e kala taarngal: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11,31. E −0,31/11,31 = −2,70%.
Yerondir pari adanngal $10 e keerol $500, pariiji jeegom: keerol ngol ina fiyoo laabi gooto e nder taarɗe 55, kiikiiɗe pariiji 200 ina hawra e mum 84% sahaa. Keerol ɓurngol toowde ina waɗa hellugol ngol ɓuri seeɗa e ɓuri mawnude, ɓurataa hoyde.
Ƴeewndo weltinoowo
Ɓuri famɗude e ɓuri mawnude, e kala keerol. Fiilsugol mum ina tabiti.
Iwdi
- 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.