Inkinga: Umkhawulo wetafula
I-Martingale engu-$10 etafuleni elinomkhawulo ongu-$500. Iphuka kangaki, ilahlekelwa malini uma iphuka, futhi ingabe ukukhuphula umkhawulo kuyasiza?
Ukuhlulwa okuyisithupha ngokulandelana kuyayiphula: u-1.8% ngomjikelezo wohlelo ngamunye, cishe u-60% wamaseshini okujikeleza okungu-100. Ukuphuka kubiza u-$630. Ukukhuphula umkhawulo ube ngu-$5,000 kwenza ukuphuka kungavami (u-0.25% ngomjikelezo) kodwa kube kukhulu ($5,110). Ukulahlekelwa okulindelekile ngedola ngalinye elibhejiwe kungu-2.70% kunoma yimuphi umkhawulo.
Inkinga
Umdlali ubheja u-$10 kokubomvuukubhejaBomvuUkubheja okulingana nemali ezinombolweni eziyi-18 ezibomvu. NgesiFulentshi: rouge.Funda konke → bese ephinda kabili ngemuva kokuhlulwa ngakunye, abuyele enanini lokuqala ngemuva kokuwina. Umkhawulo wetafula ungu-$500. (a) Ukuhlulwa okungaki ngokulandelana okuphula ukwenyuswa kwenaniisuUkwenyuswa kwenaniNoma yimuphi umthetho oshintsha inani elibhejwayo ngokusekelwe emiphumeleni yangaphambili. Ukwenyuswa kwenani ngemuva kokuhlulwa kulikhuphula ngemuva kokulahlekelwa, ukwenyuswa kwenani ngemuva kokuwina kulikhuphula ngemuva kokuwina.Funda konke →? (b) Ayini amathubaizibaloAmathubaIthuba lomphumela, kusukela ku-0 kuya ku-1 noma ku-0% kuya ku-100%. Inombolo eyodwa: 1/37 = 2.70%.Funda konke → ngomjikelezo wohlelo? (c) Kulahleka malini lapho kuphuka? (d) Umkhawulo ongu-$5,000 uzishintsha kanjani izimpendulo? (e) Ingabe ukhona umkhawulo owenza uhlelo lube nenzuzo?
Izixazululo (a)–(c)
Amanani abhejwayo: 10, 20, 40, 80, 160, 320. Elesikhombisa belizoba ngu-640 > 500, ngakho ukuhlulwa okuyisithupha kuyayiphula. (b) p(ukuhlulwa) = 19/37 kokubomvu; (19/37)⁶ = 0.0183, cishe u-1.8% ngomjikelezo. Umjikelezo uhlala cishe ukujikeleza okubili ngokwesilinganiso, ngakho iseshini yokujikeleza okungu-100 inemijikelezo ecishe ibe ngu-50 futhi p(okungenani ukuphuka okukodwa) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, okungu-amayunithiimaliIyunithiInani elibhejwayo eliyisisekelo uhlelo noma umdlali asebenza ngalo. Ukwenyuswa kwenani kuchazwa ngamayunithi ukuze kusebenze kunoma yiliphi itafula.Funda konke → esisekelo angu-63, uma kuqhathaniswa no-$10 umjikelezo obuzama ukuwuwina.
Isixazululo (d)
Ku-$5,000: amanani abhejwayo kuze kufike ku-2,560 avunyelwe; inani leshumi, 5,120, liyenqatshwa. Ukuhlulwa okuyisishiyagalolunye kuyayiphula: (19/37)⁹ ≈ 0.25% ngomjikelezo, cishe u-11% wamaseshini okujikeleza okungu-100. Ukulahlekelwa lapho kuphuka: 10 + 20 + … + 2,560 = $5,110. Umdlali uthenge inhlekelele engavamile ngentengo yenhlekelele enkulu. Emaseshinini amaningi izindleko ezilindelekile zokuphuka zingu-p × ukulahlekelwa ≈ 0.0183 × 630 = $11.5 ngomjikelezo emkhawulweni ongu-$500 kanye no-0.0025 × 5,110 = $12.8 ku-$5,000; uma kusondezwa, kuyinombolo efanayo, futhi kungukulahlekelwa okulindelekileizibaloUkulahlekelwa okulindelekileInani elilindelekile liphindaphindwe ngemali yonke ebhejiwe; inani elilahlekayo ngokwesilinganiso phakathi neseshini noma esikhathini esithile.Funda konke → komjikelezo kuhlanganiswe no-$10 otholwa yimijikelezo ewinayo.
Isixazululo (e)
Cha. Ukulahlekelwa okulindelekile ngomjikelezo = (isamba esilindelekile samanani abhejwayo emjikelezweni) / 37, kunoma yimuphi umkhawulo, ngoba ukubheja ngakunye kungu-−1/37 wakho. Umkhawulo unquma kuphela ukuthi ukulahlekelwa kusatshalaliswa kanjani phakathi kokuwina okuncane okuningi nokulahlekelwa okukhulu okumbalwa. Ngaphandle komkhawulo nangemali yokubhejaimaliImali yokubhejaInani lonke lemali ebekelwe ukudlala. Ukuphatha imali yokubheja kusho ukunquma ukuthi ingakanani yayo ozoyibeka engcupheni ekubhejeni ngakunye nasesheshini ngayinye.Funda konke → engapheli ukwenyuswa kwenani bekungase “kusebenze”, ngomqondo wokuthi bekungeke kuphuke, kodwa bekungabheja imali engenamkhawulo nokulahlekelwa okulindelekile okungu-2.70% kwaleyo mali engenamkhawulo. Umkhawulo wetafulaitafulaUmkhawulo wetafulaInani elincane nelikhulu kakhulu lokubheja elivunyelwe. Umkhawulo ophezulu yiwona omisa ukwenyuswa kwenani okuphindaphinda kabili.Funda konke → akuwona ovimba uhlelo lwe-MartingaleisuMartingaleUkwenyuswa kwenani okuphinda kabili inani elibhejwayo ngemuva kokuhlulwa ngakunye. Bheka isigaba esithi “Amasu okubheja ku-roulette” ukuze ubone ukuthi kwenzani nokuthi yini engakwenzi.Funda konke → ukuthi lusebenze. Yizibalo.
Amatafula amabili angempela
Isibonelo sika-Thorp yitafula le-$1 elinomkhawulo ophezulu ongu-$1,000, lapho umdlali ethemba ukuwina u-$30 wesidlo sakusihlwa idola ngalinye ngesikhathi. Amanani abhejwayo angu-1, 2, 4 … 512 avumela ukubheja okuyishumi, ngakho ukuhlulwa okuyishumi ngokulandelana kuphula ukwenyuswa kwenani: kanye emijikelezweni engu-785 esondweni laseYurophu (kanye kwengu-613 kwelaseMelika), ngezindleko ezingu-$1,023. Imijikelezo engamashumi amathathu iphumelela yonke izikhathi ezingu-96.2%. Ezinye ezingu-3.8% zilahlekelwa cishe u-$1,000, futhi umphumela omaphakathi wobusuku ungu-−$9.17, okungu-1/37 ncamashi ka-$339 imijikelezo engamashumi amathathu eyibhejayo ngokwesilinganiso. Itafula le-$5 elinomkhawulo ophezulu ongu-$2,000 alihlukile: amanani abhejwayo afika ku-$1,280 avumela ukubheja okuyisishiyagalolunye, futhi ukuphuka kufika kanye emijikelezweni engu-403 esondweni laseYurophu (kanye kwengu-323 kwelaseMelika) futhi kubiza u-$2,555.
Umdlali ocebile
Nika ukwenyuswa kwenani imali yokubheja ejulile esikhundleni somkhawulo ophansi: amayunithi angu-1,000, isisekelo seyunithi elilodwa nokujikeleza okungu-1,000 kokubomvu, uqale kabusha ngeyunithi elilodwa noma nini lapho ukuphinda kabili okulandelayo kungaphezu kwalokho okusele. Silingise ubusuku obuyisigidi obunjalo. Esondweni laseYurophu u-52% wobusuku uphela umdlali esephambili, u-21% uphela elahlekelwe yikho konke, futhi umphumela omaphakathi ungamayunithi angu-−124. Ngisho nasemdlalweni ongenalo iqandaisondoIqandaIgodi elilodwa eliluhlaza elingambozwa ukubheja okulingana nemali, okwedazini noma okwekholamu; umthombo wenzuzo yekhasino. Kubuye kuthiwe: iqanda elilodwa, 0.Funda konke →, lapho konke ukubheja kufanele, cishe u-60% kuphela obebuzophela umdlali esephambili. Ubusuku obuvamile buwukuwina okuncane; isilinganiso sikhokhelwa ubusuku obuphela eqandeni.
Isifundo
Amakhasino abeka imikhawulo ukuze anciphise ukuguquguqukaizibaloUkuguquguqukaIsilinganiso sokuthi imiphumela ihlakazeke kangakanani ezungeze inani elilindelekile. Kuphezulu ku-straight up, kuphansi ekubhejeni okulingana nemali.Funda konke → kwawo, hhayi ukuze ahlule izinhlelo. Ukuhlula i-Martingale kungumphumela ongaqondiwe ajabula ngawo, futhi izibalo bezizoyihlula noma kunjalo. Umdlali ocabanga ukuthi umkhawulo uyinkinga uthole iwa wase ecabanga ukuthi ungumthambeka.
Ifomula
Imibuzo
Ikuvumela ukuthi usinde kuze kufike emkhawulweni ophezulu, lapho ukuphuka kungavamile kodwa kukukhulu. Ukulahlekelwa okulindelekile ngedola ngalinye akushintshi.
Amanani abhejwayo angu-10, 30, 70, 150, 310: umkhawulo ongu-$500 ufinyelelwa ngemuva kokuhlulwa okuhlanu, futhi ukuphuka kubiza u-$570. U-2.70% ofanayo.
Ukuze umdlali oyedwa angakwazi ukubheja ngaphezu kwalokho imali egciniwe yekhasino engakukhokha kalula. Ukuphatha ukuguquguquka.
Ubumnandi bezibalo
Zama lezi ngokwakho kuqala. Isixazululo ngasinye sifihliwe; sivule uma usukulungile.
I-MartingaleisuMartingaleUkwenyuswa kwenani okuphinda kabili inani elibhejwayo ngemuva kokuhlulwa ngakunye. Bheka isigaba esithi “Amasu okubheja ku-roulette” ukuze ubone ukuthi kwenzani nokuthi yini engakwenzi.Funda konke → iqala ngo-$100 ku-bomvuukubhejaBomvuUkubheja okulingana nemali ezinombolweni eziyi-18 ezibomvu. NgesiFulentshi: rouge.Funda konke → futhi izivumela ukubheja okungadluli ku-n emjikelezweni wohlelo: ngemuva kokuhlulwa okungu-n ngokulandelana iyayeka bese iqala kabusha. Bhala umphumela omaphakathi womjikelezo owodwa njengefomula ku-n, bese uwubala ngokubheja okungu-5, 6 no-7 esondweni laseYurophu.
Bonisa isixazululo
Impendulo Isilinganiso singu-100 × (1 − (38/37)ⁿ): −$14.26 ngokubheja okungu-5, −$17.35 ngokungu-6 no-−$20.52 ngokungu-7. Ukuvumela ukubheja okuningi kwenza umjikelezo omaphakathi ube mubi nakakhulu, hhayi ube ngcono.
Umjikelezo uwina u-$100 ngaphandle uma konke ukubheja okungu-n kuhlulwa, okwenzeka ngamathuba angu-qⁿ, lapho q = 19/37. Uma konke kuhlulwa, amanani abhejiwe angu-100, 200 … 100 × 2ⁿ⁻¹ ahlangana abe ngu-100 × (2ⁿ − 1). Ngakho isilinganiso singu-100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Lapha 2q = 38/37, ngaphezudlwana kuka-1, ngakho u-(2q)ⁿ ukhula no-n bese isilinganiso sehla: n = 5 unikeza u-1.1426 no-−$14.26; n = 6 unikeza u-1.1735 no-−$17.35; n = 7 unikeza u-1.2052 no-−$20.52. Isitebhisi eside asilahlekelwa kaningi, ku-3.6%, 1.8% no-0.9% wemijikelezo, kodwa ukulahlekelwa ngakunye kuphindwe kabili ngobukhulu, futhi imali eningi iwela itafula emjikelezweni ngamunye, ngo-2.70% wakho konke ukubheja.
Esondweni laseMelika 2q = 40/38 futhi izibalo zingu-−$29.24, −$36.04 no-−$43.20 (Taylor, 2021). Emdlalweni ongenalo iqandaisondoIqandaIgodi elilodwa eliluhlaza elingambozwa ukubheja okulingana nemali, okwedazini noma okwekholamu; umthombo wenzuzo yekhasino. Kubuye kuthiwe: iqanda elilodwa, 0.Funda konke → 2q = 1, futhi yonke imijikelezo inesilinganiso esingelutho ncamashi noma ngabe isitebhisi side kangakanani.
I-“grand Martingale” iphinda kabili inani elibhejwayo ngemuva kokuhlulwa bese yengeza iyunithiimaliIyunithiInani elibhejwayo eliyisisekelo uhlelo noma umdlali asebenza ngalo. Ukwenyuswa kwenani kuchazwa ngamayunithi ukuze kusebenze kunoma yiliphi itafula.Funda konke → elilodwa, ngakho amanani ahamba ngo-1, 3, 7, 15, 31 … Usuphambili ngamalini uma ukuwina kokuqala kufika ekubhejeni kwe-1, kwe-2, kwe-3 noma kwe-k? Ngeyunithi ye-$10 nomkhawulo ophezulu we-$500, umjikelezo uphuka kangaki, futhi umjikelezo ubiza malini ngokwesilinganiso?
Bonisa isixazululo
Impendulo Ukuwina ekubhejeni kwe-k kukushiya uphambili ngamayunithi angu-k: +1, +2, +3 njalo njalo. Ngo-$10 no-$500 umjikelezo uphuka ngemuva kokuhlulwa okuhlanu, ku-3.57% wemijikelezo, ngo-−$570. Ngokwesilinganiso umjikelezo ulahlekelwa u-$2.32 ku-$85.73 obhejiwe: 2.70%.
Inani elibhejwayo le-k lingamayunithi angu-2ᵏ − 1: 1, 3, 7, 15, 31. Amanani okuqala angu-k − 1 ahlangana abe ngu-2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Ukuwina ekubhejeni kwe-k kukhokha u-2ᵏ − 1, ngakho umjikelezo uphela usuphambili ngamayunithi angu-(2ᵏ − 1) − (2ᵏ − k − 1) = k: iyunithi elilodwa eyengeziwe ngakho konke ukuhlulwa ngaphambi kokuwina.
Ngo-$10 ngeyunithi amanani abhejwayo angu-$10, $30, $70, $150 no-$310; elilandelayo, u-$630, lidlula umkhawulo ophezulu we-$500. Ukuhlulwa okuhlanu ngokulandelana, (19/37)⁵ = 3.57%, kubiza u-10 + 30 + 70 + 150 + 310 = $570.
Isilinganiso: ukuwina kuletha u-$10 × k ngamathuba angu-(19/37)ᵏ⁻¹ × 18/37 ku-k = 1 kuya ku-5, kanti ukuphuka kuthatha u-$570 ngamathuba angu-3.57%; konke ndawonye −$2.32. Imali ebhejwayo inesilinganiso esingu-$85.73 emjikelezweni ngamunye, futhi 2.32/85.73 = 2.70%. I-Martingale ejwayelekile etafuleni elifanayo iphuka ngemuva kokuhlulwa okuyisithupha, ku-1.83% wemijikelezo, ngo-−$630, futhi ilahlekelwa u-$1.74 ku-$64.20 obhejiwe: 2.70% futhi.
Umdlali uhlela i-Martingale kokubomvu ngokubheja kokuqala kwe-$1 etafuleni elinomkhawulo we-$1,000, ngakho kungenzeka ukubheja okuyishumi: $1, $2, $4 njalo njalo kuze kube ngu-$512. Umjikelezo ulahlekelwa konke okuyishumi kangaki, lokho kuthatha isikhathi esingakanani ngokwesilinganiso, futhi kusihlwa kokubheja okungu-200 kuphela ngaleyo ndlela kangaki?
Bonisa isixazululo
Impendulo Umjikelezo ulahlekelwa konke okuyishumi cishe kanye kokungu-784, okubiza u-$1,023. Lokho kucishe kube wukubheja okungu-1,600 ngokwesilinganiso, futhi kusihlwa kokubheja okungu-200 kushayisa odongeni cishe ku-11% wezikhathi. Umjikelezo usalahlekelwa u-$0.31 ngokwesilinganiso ku-$11.31 obhejiwe: 2.70%.
Ukubheja kokubomvu kuhlulwa ngamathuba angu-19/37, iqanda lifakiwe. Okuyishumi ngokulandelana: (19/37)¹⁰ ≈ 0.00128, cishe 1 ku-784. Amanani angu-1 + 2 + 4 + … + 512 ahlangana abe ngu-$1,023.
Umjikelezo uthatha ukubheja okungu-2.05 ngokwesilinganiso, ngakho imijikelezo engu-784 ithatha ukubheja okucishe kube ngu-1,600. Uma kubalwa ukujikeleza ngakunye, uchungechunge lokuhlulwa okuyishumi luvela endaweni ethile ekubhejeni okungu-200 ku-11.3% wezikhathi.
Phakathi kwezinhlekelele umdlali uwina u-$1 emjikelezweni ngamunye: cishe u-$783 ngaphambi kwalowo omubi, bese −$1,023, cishe −$240 esikhathini sonke. Emjikelezweni ngamunye: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Imali ebhejwayo emjikelezweni ngamunye: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Futhi −0.31/11.31 = −2.70%.
Qhathanisa nokubheja kokuqala kwe-$10 nomkhawulo we-$500, ukubheja okuyisithupha: udonga lufika kanye emijikelezweni engu-55, futhi kusihlwa kokubheja okungu-200 kuluhlangabezana ku-84% wezikhathi. Umkhawulo ophezulu wenza ukuwa kungavami futhi kube kukhulu, kodwa hhayi kushibhe.
Okokuhlekisa
Kuyaqabuka futhi kukhule, kunoma yimuphi umkhawulo. Umphumela wokuphindaphinda awushintshi.
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, uhlelo lwesi-2.