Ingxaki: umda wetafile
I-Martingale ye-$10 kwitafile ye-$500. Yaphuka kangaphi, ilahlekelwa malini xa isenzeka, kwaye ingaba ukunyusa umda kuyanceda?
Iilahleko ezintandathu zilandelelana ziyayaphula: i-1.8% kumjikelo wenkqubo ngamnye, malunga nama-60% eeseshoni zemijikelo eli-100. Ukwaphuka kubiza i-$630. Ukunyusa umda ukuya kwi-$5,000 kwenza ukwaphuka kunqabe (0.25% kumjikelo wenkqubo) kwaye kube kukhulu ($5,110). Ilahleko elindelekileyo kwidola nganye ebhejiweyo yi-2.70% kuwo nawuphi na umda.
Ingxaki
Umdlali ubheja i-$10 kobomvuibhejiObomvuIbheji yemali elinganayo kumanani ali-18 abomvu. NgesiFrentshi: rouge.Funda yonke → aze aphinde kabini emva kwelahleko nganye, abuyele ekuqaleni emva kokuphumelela. Ubuninzi betafile yi-$500. (a) Zingaphi iilahleko ezilandelelanayo ezaphula ukunyuswa kweebhejiisicwangcisoUkunyuswa kweebhejiNawuphi na umthetho otshintsha ibheji ngokweziphumo zangaphambili. Ukunyuswa okubi kuyinyusa emva kwelahleko, okuhle emva kokuphumelela.Funda yonke →? (b) Ngawaphi amathubaizibaloAmathubaIthuba lesiphumo, ukusuka ku-0 ukuya ku-1 okanye ku-0% ukuya ku-100%. Inani elinye: 1/37 = 2.70%.Funda yonke → kumjikelo wenkqubo ngamnye? (c) Kulahleka malini ekwaphukeni? (d) Umda we-$5,000 uzitshintsha njani iimpendulo? (e) Ingaba ukhona umda owenza inkqubo ibe nenzuzo?
Izisombululo (a)–(c)
Iibheji: 10, 20, 40, 80, 160, 320. Eyesixhenxe ibiya kuba ngu-640 > 500, ngoko iilahleko ezintandathu ziyayaphula. (b) p(ilahleko) = 19/37 kobomvu; (19/37)⁶ = 0.0183, malunga ne-1.8% kumjikelo wenkqubo. Umjikelo wenkqubo uhlala malunga nemijikelo emibini ngokomndilili, ngoko iseshoni yemijikelo eli-100 inemijikelo yenkqubo emalunga nama-50 kunye ne-p(ubuncinane ukwaphuka okukodwa) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, eziiyunithi ezisisisekoimaliIyunithiIbheji esisiseko esetyenziswa yinkqubo okanye ngumdlali. Ukunyuswa kweebheji kuchazwa ngeeyunithi ukuze kusebenze kuyo nayiphi na itafile.Funda yonke → ezingama-63, ngokuchasene ne-$10 umjikelo wenkqubo ebewuzama ukuyiphumelela.
Isisombululo (d)
Kwi-$5,000: iibheji ukuya ku-2,560 zivunyelwe; ibheji yeshumi, 5,120, iyaliwa. Iilahleko ezilithoba ziyayaphula: (19/37)⁹ ≈ 0.25% kumjikelo wenkqubo, malunga ne-11% yeeseshoni zemijikelo eli-100. Ilahleko ekwaphukeni: 10 + 20 + … + 2,560 = $5,110. Umdlali uthenge intlekele enqabileyo ngexabiso lentlekele enkulu. Kwiiseshoni ezininzi iindleko ezilindelekileyo zokwaphuka zi-p × ilahleko ≈ 0.0183 × 630 = $11.5 kumjikelo wenkqubo kumda we-$500 kunye no-0.0025 × 5,110 = $12.8 kwi-$5,000; ngaphakathi kokujikeleza, linani elifanayo, kwaye yilahleko elindelekileyoizibaloIlahleko elindelekileyoIxabiso elilindelekileyo liphindwe ngesixa esipheleleyo esibhejiweyo; isixa esiphakathi esilahlekayo kwiseshoni okanye kwixesha elithile.Funda yonke → yomjikelo wenkqubo kunye ne-$10 efunyanwa yimijikelo yenkqubo ephumeleleyo.
Isisombululo (e)
Hayi. Ilahleko elindelekileyo kumjikelo wenkqubo = (isixa esipheleleyo sebheji esilindelekileyo kumjikelo wenkqubo) / 37, kuwo nawuphi na umda, kuba ibheji nganye ingu-−1/37 wayo. Umda ugqiba kuphela indlela ilahleko esasazwa ngayo phakathi kweempumelelo ezininzi ezincinci neelahleko ezimbalwa ezinkulu. Ngaphandle komda kunye nemali yokudlalaimaliImali yokudlalaIsixa esipheleleyo esibekelwe ukudlala. Ulawulo lwemali yokudlala kukugqiba ukuba ingakanani kuyo oyibeka emngciphekweni kwibheji nganye nakwiseshoni nganye.Funda yonke → engapheliyo ukunyuswa kweebheji bekuya “kusebenza”, ngengqiqo yokuba bekungasoze kwaphuke, kwaye bekuya kubheja nesixa esingenamda kunye nelahleko elindelekileyo ye-2.70% yeso sixa singenamda. Umda wetafileitafileUmda wetafileIbheji encinci nenkulu evumelekileyo. Ubuninzi yinto emisa ukunyuswa kweebheji okuphindaphindayo.Funda yonke → asiyonto emisa i-MartingaleisicwangcisoMartingaleUkunyuswa kweebheji okuphindaphinda ibheji emva kokulahlekelwa ngakunye. Bona icandelo lezicwangciso ukuze wazi oko ikwenzayo nengakwenziyo.Funda yonke → ekusebenzeni. Zizibalo.
Iitafile ezimbini zokwenyani
Umzekelo kaThorp yitafile ye-$1 enobuninzi be-$1,000, apho umdlali ethemba ukuphumelela i-$30 yesidlo sangokuhlwa idola ngedola. Iibheji 1, 2, 4 … 512 zivumela iibheji ezilishumi, ngoko iilahleko ezilishumi zilandelelana zaphula ukunyuswa kweebheji: kanye kwimijikelo yenkqubo engama-785 kwivili laseYurophu (kanye kwengama-613 kwelaseMelika), ngeendleko ze-$1,023. Imijikelo yenkqubo engamashumi amathathu iphumelela yonke ama-96.2% exesha. Ezinye i-3.8% zilahlekelwa malunga ne-$1,000, kwaye isiphumo somndilili sobusuku yi-−$9.17, eyiyo kanye i-1/37 ye-$339 ebhejwa yimijikelo yenkqubo engamashumi amathathu ngokomndilili. Itafile ye-$5 enobuninzi be-$2,000 ayahlukanga: iibheji ukuya kwi-$1,280 zivumela iibheji ezilithoba, kwaye ukwaphuka kufika kanye kwimijikelo yenkqubo engama-403 kwivili laseYurophu (kanye kwengama-323 kwelaseMelika) kwaye kubiza i-$2,555.
Umdlali osisityebi
Nika ukunyuswa kweebheji imali yokudlala enzulu endaweni yomda ophantsi: iiyunithi eziyi-1,000, isiseko seyunithi enye kunye nemijikelo eyi-1,000 kobomvu, uqala kwakhona ngeyunithi enye nanini na xa ukuphinda kabini okulandelayo kungaphezu kwento eseleyo. Silinganise ubusuku obunjalo obusisigidi. Kwivili laseYurophu ama-52% agqiba ephambili, ama-21% alahlekelwa yonke into, kwaye isiphumo somndilili ziiyunithi ezili-−124. Nakumdlalo ongenalo iqandaiviliIqandaIsikhewu esinye esiluhlaza esingagubungelwa yibheji yemali elinganayo, yedazini okanye yekholamu; umthombo wenzuzo yekhasino. Kwakhona: iqanda elinye, 0.Funda yonke →, apho yonke ibheji ifanelekile, malunga nama-60% kuphela abeya kugqiba ephambili. Ubusuku obuqhelekileyo yimpumelelo encinci; umndilili uhlawulwa bubusuku obuphela kwiqanda.
Isifundo
Iikhasino zibeka imida ukuze zinqande ukwahlukaizibaloUkwahlukaUmlinganiselo wokuba iziphumo zisasazeke kangakanani zijikeleze ixabiso elilindelekileyo. Kuphezulu kwibheji yenani elinye, kuphantsi kwiibheji zemali elinganayo.Funda yonke → kwazo, hayi ukoyisa iinkqubo. Ukoyisa i-Martingale sisiphumo esisecaleni ezivuyela ukuba naso, kwaye izibalo bezinokuyoyisa nangona kunjalo. Umdlali ocinga ukuba umda yingxaki ulifumene iliwa waza walithatha njengethambeka.
Ifomula
Imibuzo
Ikuvumela ukuba usinde ukuya kumda ophezulu, apho ukwaphuka kunqabile kwaye kukhulu. Ilahleko elindelekileyo kwidola nganye ayitshintshi.
Iibheji 10, 30, 70, 150, 310: umda we-$500 ufikelelwa emva kweelahleko ezintlanu, kwaye ukwaphuka kubiza i-$570. I-2.70% efanayo.
Ukuze umdlali omnye angabheji ngaphezu kwento enokuhlawulwa ngokulula yimali egciniweyo yekhasino. Ulawulo lokwahluka.
Ukonwaba ngezibalo
Qala uzizame ngokwakho. Isisombululo ngasinye sisongiwe; sivule xa ukulungele.
I-MartingaleisicwangcisoMartingaleUkunyuswa kweebheji okuphindaphinda ibheji emva kokulahlekelwa ngakunye. Bona icandelo lezicwangciso ukuze wazi oko ikwenzayo nengakwenziyo.Funda yonke → iqala nge-$100 kobomvuibhejiObomvuIbheji yemali elinganayo kumanani ali-18 abomvu. NgesiFrentshi: rouge.Funda yonke → kwaye izivumela ubuninzi beebheji ezili-n ngomjikelo wenkqubo: emva kwelahleko ezili-n ngokulandelelana iyayeka iqale kwakhona. Bhala isiphumo esiphakathi somjikelo wenkqubo omnye njengefomula ku-n, uze usibale kwiibheji ezi-5, 6 nezi-7 kwivili laseYurophu.
Bonisa isisombululo
Impendulo Umndilili ngu-100 × (1 − (38/37)ⁿ): −$14.26 kwiibheji ezi-5, −$17.35 kwezi-6 kunye no-−$20.52 kwezi-7. Ukuvumela iibheji ezingakumbi kwenza umjikelo wenkqubo ophakathi ube mbi ngakumbi, hayi ngcono.
Umjikelo wenkqubo uzuza i-$100 ngaphandle kokuba zonke iibheji ezili-n ziyalahleka, nto leyo eyenzeka ngethuba eli-qⁿ, apho q = 19/37. Xa zonke zilahleka, iibheji eziyi-100, 200 … 100 × 2ⁿ⁻¹ zidibana zibe ngu-100 × (2ⁿ − 1). Ngoko umndilili ngu-100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Apha 2q = 38/37, ngaphezulu kancinci ko-1, ngoko u-(2q)ⁿ ukhula no-n kwaye umndilili uyehla: n = 5 unika u-1.1426 no-−$14.26; n = 6 unika u-1.1735 no-−$17.35; n = 7 unika u-1.2052 no-−$20.52. Ileli ende ilahlekelwa kancinci, kwi-3.6%, 1.8% nakwi-0.9% yemijikelo yenkqubo, kodwa ilahleko nganye iphindwe kabini ngobukhulu, kwaye imali eninzi idlula etafileni kumjikelo wenkqubo ngamnye, nge-2.70% yayo yonke ibheji.
Kwivili laseMelika 2q = 40/38 kwaye amanani ngu-−$29.24, −$36.04 no-−$43.20 (Taylor, 2021). Kumdlalo ongenalo iqandaiviliIqandaIsikhewu esinye esiluhlaza esingagubungelwa yibheji yemali elinganayo, yedazini okanye yekholamu; umthombo wenzuzo yekhasino. Kwakhona: iqanda elinye, 0.Funda yonke → 2q = 1, kwaye wonke umjikelo wenkqubo unomndilili olingana nonothi ngqo nokuba ileli inde kangakanani.
I-“Martingale enkulu” iphindaphinda ibheji kabini emva kwelahleko kwaye yongeza iyunithiimaliIyunithiIbheji esisiseko esetyenziswa yinkqubo okanye ngumdlali. Ukunyuswa kweebheji kuchazwa ngeeyunithi ukuze kusebenze kuyo nayiphi na itafile.Funda yonke → enye, ngoko iibheji ziba ngu-1, 3, 7, 15, 31 … Uphambili ngamalini xa impumelelo yokuqala ifika kwibheji ye-1, ye-2, ye-3 okanye ye-k? Ngeyunithi ye-$10 nobuninzi be-$500, umjikelo wenkqubo waphuka kangaphi, kwaye umjikelo wenkqubo ubiza malini ngomndilili?
Bonisa isisombululo
Impendulo Impumelelo kwibheji ye-k ikushiya uphambili ngeeyunithi ezili-k: +1, +2, +3 njalo njalo. Kwi-$10 nakwi-$500 umjikelo wenkqubo uyaphuka emva kwelahleko ezintlanu, i-3.57% yemijikelo yenkqubo, nge-−$570. Ngomndilili umjikelo wenkqubo ulahlekelwa yi-$2.32 kwi-$85.73 ebhejiweyo: 2.70%.
Ibheji ye-k ziiyunithi ezingu-2ᵏ − 1: 1, 3, 7, 15, 31. Iibheji zokuqala ezingu-k − 1 zidibana zibe ngu-2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Impumelelo kwibheji ye-k ihlawula u-2ᵏ − 1, ngoko umjikelo wenkqubo uphela uphambili ngeeyunithi ezingu-(2ᵏ − 1) − (2ᵏ − k − 1) = k: iyunithi enye eyongezelelweyo kuyo yonke ilahleko ngaphambi kwempumelelo.
Nge-$10 ngeyunithi iibheji zi-$10, $30, $70, $150 kunye ne-$310; elandelayo, i-$630, idlula ubuninzi be-$500. Ilahleko ezintlanu ngokulandelelana, (19/37)⁵ = 3.57%, zibiza u-10 + 30 + 70 + 150 + 310 = $570.
Umndilili: iimpumelelo zingenisa i-$10 × k ngethuba (19/37)ᵏ⁻¹ × 18/37 ku-k = 1 ukuya ku-5, kwaye ukuphuka kuthatha i-$570 ngethuba le-3.57%; kunye −$2.32. Imali ebhejiweyo inomndilili we-$85.73 ngomjikelo wenkqubo, kwaye 2.32/85.73 = 2.70%. I-Martingale eqhelekileyo kwitafile efanayo iyaphuka emva kwelahleko ezintandathu, i-1.83% yemijikelo yenkqubo, nge-−$630, kwaye ilahlekelwa yi-$1.74 kwi-$64.20 ebhejiweyo: 2.70% kwakhona.
Umdlali uceba i-Martingale kobomvu ngebheji yokuqala ye-$1 kwitafile enomda we-$1,000, ngoko kunokwenzeka iibheji ezilishumi: $1, $2, $4 njalo njalo ukuya kwi-$512. Umjikelo wenkqubo ulahlekelwa zizo zonke ezilishumi kangaphi, oko kuthatha ixesha elingakanani ngomndilili, kwaye ingokuhlwa kweebheji ezingama-200 kuphela ngolo hlobo kangaphi?
Bonisa isisombululo
Impendulo Umjikelo wenkqubo ulahlekelwa zizo zonke ezilishumi malunga kanye kwezingama-784, nto leyo ebiza i-$1,023. Oko malunga neebheji eziyi-1,600 ngomndilili, kwaye ingokuhlwa kweebheji ezingama-200 ibetha udonga malunga ne-11% yexesha. Umjikelo wenkqubo usalahlekelwa yi-$0.31 ngomndilili kwi-$11.31 ebhejiweyo: 2.70%.
Ibheji kobomvu ilahlekelwa ngethuba eli-19/37, kuquka iqanda. Ezilishumi ngokulandelelana: (19/37)¹⁰ ≈ 0.00128, malunga no-1 kwezingama-784. Iibheji eziyi-1 + 2 + 4 + … + 512 zidibana zibe yi-$1,023.
Umjikelo wenkqubo uthatha iibheji ezi-2.05 ngomndilili, ngoko imijikelo yenkqubo engama-784 ithatha malunga neebheji eziyi-1,600. Xa kubalwa umjikelo ngomjikelo, uthotho lwelahleko ezilishumi luvela kwindawo ethile kwiibheji ezingama-200 kwi-11.3% yexesha.
Phakathi kweentlekele umdlali uzuza i-$1 ngomjikelo wenkqubo: malunga ne-$783 ngaphambi kombi, emva koko −$1,023, malunga ne-−$240 kulo lonke ixesha. Ngomjikelo wenkqubo: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Imali ebhejiweyo ngomjikelo wenkqubo: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Kwaye −0.31/11.31 = −2.70%.
Thelekisa nebheji yokuqala ye-$10 kumda we-$500, iibheji ezintandathu: udonga lufika kanye kwimijikelo yenkqubo engama-55, kwaye ingokuhlwa kweebheji ezingama-200 iludibana kwi-84% yexesha. Umda ophezulu wenza ukuwa kunqabe kwaye kube kukhulu, hayi kube nexabiso eliphantsi.
Uncumo
Kunqabile kwaye kukhulu, kuwo wonke umda. Imveliso ayitshintshi.
Imithombo
- 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.