Inkinga: Umkhawulo welitafula
I-Martingale ya-$10 etafuleni le-$500. Iphuka kangakanani, ilahlekelwa malini nayiphuka, futsi kukhuphula umkhawulo kuyasita yini?
Kulahlekelwa kwemahlandla lasitfupha kulandzelana kuyayiphula: 1.8% emjikeletweni weluhlelo ngamunye, cishe 60% wetiseshini tekujikeleta lokungu-100. Kuphuka kubita $630. Kukhuphula umkhawulo uye ku-$5,000 kwenta kuphuka kungavami (0.25% emjikeletweni ngamunye) futsi kube kukhulu ($5,110). Kulahlekelwa lokulindzelekile ngeludola ngalinye lolubhejiwe kungu-2.70% kunobe ngumuphi umkhawulo.
Inkinga
Umdlali ubheja $10 kulokubovukubhejaBovuKubheja lokulingana nemali etinombolweni letingu-18 letibovu. NgesiFulentji: rouge.Fundza konkhe → bese uphindza kabili ngemuva kwekulahlekelwa ngakunye, abuyele ekucaleni ngemuva kwekuwina. Linani lelikhulu letafula ngu-$500. (a) Kulahlekelwa lokungakanani lokulandzelanako lokuphula kwenyuswa kwelinanilisuKwenyuswa kwelinaniNoma ngumuphi umtsetfo loshintja linani lelibhejwako ngekuya kwemiphumela yangaphambilini. Kwenyuswa kwelinani ngemuva kwekulahlekelwa kulikhuphula ngemuva kwekulahlekelwa, kwenyuswa kwelinani ngemuva kwekuwina kulikhuphula ngemuva kwekuwina.Fundza konkhe →? (b) Lingakanani litfubatibaloLitfubaLitfuba lekutsi umphumela wenteke, kusukela ku-0 kuya ku-1 noma ku-0% kuya ku-100%. Inombolo yinye: 1/37 = 2.70%.Fundza konkhe → emjikeletweni ngamunye? (c) Kulahlekelwa malini nakuphuka? (d) Umkhawulo wa-$5,000 utigucula njani timphendvulo? (e) Ukhona yini umkhawulo lowenta luhlelo lube nenzuzo?
Tisombululo (a)–(c)
Emanani labhejwako: 10, 20, 40, 80, 160, 320. Lelesikhombisa belitawuba ngu-640 > 500, ngako kulahlekelwa lokusitfupha kuyakuphula. (b) p(kulahlekelwa) = 19/37 kulokubovu; (19/37)⁶ = 0.0183, cishe 1.8% emjikeletweni ngamunye. Umjikeleto uhlala cishe kujikeleta lokumbili ngesilinganiso, ngako iseshini yekujikeleta lokungu-100 inemijikeleto lecishe ibe ngu-50 futsi p(kuphuka kunye nobe ngetulu) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, lokungu-63 emayunithiimaliLiyunithiLinani lelibhejwako lelisisekelo luhlelo noma umdlali lasebenta ngalo. Kwenyuswa kwelinani kuchazwa ngemayunithi kute kusebente kunoma ngumuphi litafula.Fundza konkhe → lasisekelo, nangabe kucatsaniswa ne-$10 lebewuzama kuyiwina umjikeleto.
Sisombululo (d)
Nga-$5,000: emanani labhejwako kuye ku-2,560 avumelekile; lelishumi, 5,120, liyaliwa. Kulahlekelwa lokuyimfica kuyakuphula: (19/37)⁹ ≈ 0.25% emjikeletweni ngamunye, cishe 11% wetiseshini tekujikeleta lokungu-100. Kulahlekelwa nakuphuka: 10 + 20 + … + 2,560 = $5,110. Umdlali utsengile inhlekelele lengavami ngentsengo yenhlekelele lenkhulu. Etiseshinini letinyenti indleko lelindzelekile yekuphuka ingu-p × kulahlekelwa ≈ 0.0183 × 630 = $11.5 emjikeletweni ngamunye emkhawulweni we-$500 kanye na-0.0025 × 5,110 = $12.8 ku-$5,000; kokubili kuyasondzelana kodvwa akulingani, futsi ngakunye kulahlekelwa lokulindzelekiletibaloKulahlekelwa lokulindzelekileLinani lelilindzelekile liphindzaphindzwe ngemali yonkhe lebhejiwe; linani lesilinganiso lelilahlekako eseshinini noma esikhatsini lesitsite.Fundza konkhe → kwemjikeleto kanye ne-$10 leyitfolwa yimijikeleto lewinako.
Sisombululo (e)
Cha. Kulahlekelwa lokulindzelekile emjikeletweni ngamunye = (linani lonkhe lelibhejwako lelilindzelekile emjikeletweni) / 37, kunobe ngumuphi umkhawulo, ngobe kubheja ngakunye kungu-−1/37 wako. Umkhawulo uncuma kuphela kutsi kulahlekelwa kwabiwa njani emkhatsini wekuwina lokuncane lokunyenti nekulahlekelwa lokukhulu lokumbalwa. Ngaphandle kwemkhawulo kanye nemali yekubhejaimaliImali yekubhejaImali yonkhe lebekelwe kudlala. Kuphatsa imali yekubheja kusho kunquma kutsi ungafaka engcupheni lengakanani kuko kubheja ngakunye nakuyo iseshini ngayinye.Fundza konkhe → lengapheli, kwenyuswa kwelinani “bekutawusebenta”, ngemcondvo wekutsi bekungeke kuphuke, futsi bekutawubheja nemali lengenamkhawulo nekulahlekelwa lokulindzelekile lokungu-2.70% walelo linani lelingenamkhawulo. Umkhawulo welitafulalitafulaUmkhawulo welitafulaLinani lelincane nalelikhulu kakhulu lekubheja lelivunyelwe. Umkhawulo lophakeme ngiwo lomisa kwenyuswa kwelinani lokuphindza kabili.Fundza konkhe → akusiwo lowenta i-MartingalelisuMartingaleKwenyuswa kwelinani lokuphindza kubheja kabili ngemuva kwekulahlekelwa ngakunye. Buka sigaba semasu kute ubone kutsi kwentani nekutsi akwenti ini.Fundza konkhe → ingasebenti. Tibalo ngito.
Ematafula lamabili langempela
Sibonelo sa-Thorp litafula le-$1 lelinemkhawulo we-$1,000, lapho umdlali atsemba kuwina $30 wesidlo sakusihlwa ludola ngeludola. Emanani 1, 2, 4 … 512 avumela kubheja kalishumi, ngako kulahlekelwa kalishumi kulandzelana kuphula kwenyuswa kwelinani: kanye emijikeletweni lengu-785 esondvweni laseYurophu (kanye ku-613 kulelaseMelika), ngendleko ye-$1,023. Imijikeleto lengemashumi lamatsatfu iphumelela yonkhe ku-96.2% wetikhatsi. Le-3.8% lesele ilahlekelwa cishe $1,000, futsi umphumela ngesilinganiso webusuku ngu-−$9.17, lokungu-1/37 ncamashi we-$339 lebhejwa yimijikeleto lengemashumi lamatsatfu ngesilinganiso. Litafula le-$5 lelinemkhawulo we-$2,000 alehlukile: emanani kuya ku-$1,280 avumela kubheja kayimfica, futsi kuphuka kufika kanye emijikeletweni lengu-403 esondvweni laseYurophu (kanye ku-323 kulelaseMelika) futsi kubita $2,555.
Umdlali lonjingile
Nika kwenyuswa kwelinani imali yekubheja lenkhulu esikhundleni semkhawulo lophansi: emayunithi langu-1,000, lisisekelo leliyunithi linye kanye nekujikeleta lokungu-1,000 kulokubovu, ucale kabusha ngeliyunithi linye njalo nangabe kuphindza kabili lokulandzelako kungetulu kwaloko lokusele. Silingise busuku lobunjalo lobusigidzi. Esondvweni laseYurophu 52% buphetsa buphambili, 21% bulahlekelwa konkhe, futsi umphumela ngesilinganiso ngu-−124 emayunithi. Nanoma kungaba semdlalweni longenalo licandzalisondvoLicandzaUmgodzi munye loluhlata longambonjwa kubheja lokulingana nemali, kwelidazini noma kwelikholamu; umtfombo wenzuzo yelikhasino. Kuphindze kutsiwe: licandza linye, 0.Fundza konkhe →, lapho kubheja ngakunye kulungile, cishe 60% kuphela bebutawuphetsa buphambili. Busuku lobujwayelekile kuwina lokuncane; silinganiso sikhokhelwa ngubusuku lobuphela ngelicandza.
Sifundvo
Emakhasino abeka imikhawulo kute alawule kugucugucukatibaloKugucugucukaSilinganiso sekutsi imiphumela isabalele kangakanani etungeleta linani lelilindzelekile. Kukhulu ku-straight up, kuncane ekubhejeni lokulingana nemali.Fundza konkhe → kwawo, hhayi kute ehlule tinhlelo. Kwehlula i-Martingale kungumphumela longakahloswa lajabula ngawo, futsi tibalo betingayehlula noma kunjalo. Umdlali locabanga kutsi umkhawulo uyinkinga utfole liwa futsi walitsatsa njengemtsambeko.
Lifomula
Imibuto
Ikuvumela kutsi usinde kuze kube semkhawulweni lophakeme, lapho kuphuka kungavami futsi kukhulu. Kulahlekelwa lokulindzelekile ngeludola ngalinye akugucuki.
Emanani 10, 30, 70, 150, 310: umkhawulo we-$500 uyafinyelelwa ngemuva kwekulahlekelwa kasihlanu, futsi kuphuka kubita $570. 2.70% lofanako.
Kute umdlali munye angabheji ngetulu kwaloko imali yelikhasino lelondvolotiwe lengakukhokha ngaphandle kwenkinga. Kulawula kugucugucuka.
Bumnandi betibalo
Zama loku ngekwakho kucala. Sixazululo ngasinye sifihliwe; sivule nawukulungele.
Luhlelo lwe-MartingalelisuMartingaleKwenyuswa kwelinani lokuphindza kubheja kabili ngemuva kwekulahlekelwa ngakunye. Buka sigaba semasu kute ubone kutsi kwentani nekutsi akwenti ini.Fundza konkhe → lucala nga-$100 kulokubovukubhejaBovuKubheja lokulingana nemali etinombolweni letingu-18 letibovu. NgesiFulentji: rouge.Fundza konkhe → futsi luvumela kubheja lokungu-n kuphela ngemjikeleto: ngemuva kwekulahlekelwa lokungu-n kulandzelana luyayekela bese luphindze lucale. Bhala silinganiso semphumela wemjikeleto munye njengefomula ku-n, bese usibala kubheja lokungu-5, 6 na-7 elisondvweni laseYurophu.
Khombisa sixazululo
Imphendvulo Silinganiso ngu-100 × (1 − (38/37)ⁿ): −$14.26 kubheja lokungu-5, −$17.35 ku-6 na-−$20.52 ku-7. Kuvumela kubheja lokunyenti kwenta umjikeleto wesilinganiso ube mubi kakhulu, akusiko kube ncono.
Umjikeleto uwina $100 ngaphandle kwekutsi konkhe kubheja lokungu-n kulahlekelwe, lokwenteka ngelitfuba la-qⁿ, lapho q = 19/37. Nakulahlekelwa konkhe, emanani labhejwako 100, 200 … 100 × 2ⁿ⁻¹ ahlangana abe ngu-100 × (2ⁿ − 1). Ngako silinganiso ngu-100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Lapha 2q = 38/37, ngetulu kancane kwa-1, ngako (2q)ⁿ ikhula na-n futsi silinganiso siyehla: n = 5 inika 1.1426 na-−$14.26; n = 6 inika 1.1735 na-−$17.35; n = 7 inika 1.2052 na-−$20.52. Situbhu lesidze silahlekelwa kancane, ku-3.6%, 1.8% na-0.9% wemijikeleto, kodvwa kulahlekelwa ngakunye kukhulu kabili, futsi imali lenyenti yeca litafula kukumjikeleto ngamunye, nga-2.70% wako konkhe kubheja.
Elisondvweni laseMelika 2q = 40/38 futsi tibalo tingu-−$29.24, −$36.04 na-−$43.20 (Taylor, 2021). Emdlalweni lote licandzalisondvoLicandzaUmgodzi munye loluhlata longambonjwa kubheja lokulingana nemali, kwelidazini noma kwelikholamu; umtfombo wenzuzo yelikhasino. Kuphindze kutsiwe: licandza linye, 0.Fundza konkhe → 2q = 1, futsi yonkhe imijikeleto inesilinganiso lesingulutfo ncamashi nobe situbhu side kangakanani.
I-“grand Martingale” iphindza kabili linani lelibhejwako ngemuva kwekulahlekelwa bese yengeta liyunithiimaliLiyunithiLinani lelibhejwako lelisisekelo luhlelo noma umdlali lasebenta ngalo. Kwenyuswa kwelinani kuchazwa ngemayunithi kute kusebente kunoma ngumuphi litafula.Fundza konkhe → linye, ngako emanani labhejwako ahamba 1, 3, 7, 15, 31 … Usetulu ngalokungakanani nangabe kuwina kwekucala kufika ekubhejeni kwe-1, kwe-2, kwe-3 nobe kwe-k? Ngeliyunithi le-$10 nemkhawulo we-$500, umjikeleto wephuka kangakhi, futsi umjikeleto ubita malini ngesilinganiso?
Khombisa sixazululo
Imphendvulo Kuwina ekubhejeni kwe-k kukushiya usetulu ngemayunithi langu-k: +1, +2, +3 njalo njalo. Nga-$10 na-$500 umjikeleto wephuka ngemuva kwekulahlekelwa lokusihlanu, ku-3.57% wemijikeleto, nga-−$570. Ngesilinganiso umjikeleto ulahlekelwa $2.32 ku-$85.73 lobhejiwe: 2.70%.
Linani lelibhejwako le-k ngu-2ᵏ − 1 emayunithi: 1, 3, 7, 15, 31. Emanani langu-k − 1 ekucala ahlangana abe ngu-2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Kuwina ekubhejeni kwe-k kukhokha 2ᵏ − 1, ngako umjikeleto uphela usetulu nga-(2ᵏ − 1) − (2ᵏ − k − 1) = k emayunithi: liyunithi linye lelengetiwe kukho konkhe kulahlekelwa ngaphambi kwekuwina.
Nga-$10 liyunithi emanani labhejwako angu-$10, $30, $70, $150 na-$310; lelilandzelako, $630, lingetulu kwemkhawulo we-$500. Kulahlekelwa lokusihlanu kulandzelana, (19/37)⁵ = 3.57%, kubita 10 + 30 + 70 + 150 + 310 = $570.
Silinganiso: kuwina kuletsa $10 × k ngelitfuba la-(19/37)ᵏ⁻¹ × 18/37 ku-k = 1 kuya ku-5, kantsi kwephuka kutsatsa $570 ngelitfuba la-3.57%; kanyekanye −$2.32. Imali lebhejiwe inesilinganiso le-$85.73 ngemjikeleto, futsi 2.32/85.73 = 2.70%. I-Martingale levamile etafuleni lelifanako yephuka ngemuva kwekulahlekelwa lokusitfupha, ku-1.83% wemijikeleto, nga-−$630, futsi ilahlekelwa $1.74 ku-$64.20 lobhejiwe: 2.70% futsi.
Umdlali uhlela i-Martingale kulokubovu ngekubheja kwekucala kwe-$1 etafuleni lelinemkhawulo we-$1,000, ngako kubheja lokulishumi kuyenteka: $1, $2, $4 njalo njalo kuze kufike ku-$512. Umjikeleto ulahlekelwa konkhe lokulishumi kangakhi, loko kutsatsa sikhatsi lesingakanani ngesilinganiso, futsi kusihlwa kwekubheja lokungu-200 kuphela ngalendlela kangakhi?
Khombisa sixazululo
Imphendvulo Umjikeleto ulahlekelwa konkhe lokulishumi cishe kanye ku-784, lokubita $1,023. Loko kucishe kube kubheja lokungu-1,600 ngesilinganiso, futsi kusihlwa kwekubheja lokungu-200 kushayisa lubondza cishe ku-11% wesikhatsi. Umjikeleto usalahlekelwa $0.31 ngesilinganiso ku-$11.31 lobhejiwe: 2.70%.
Kubheja kulokubovu kulahlekelwa ngelitfuba la-19/37, licandza lifakiwe. Lokulishumi kulandzelana: (19/37)¹⁰ ≈ 0.00128, cishe 1 ku-784. Emanani labhejwako 1 + 2 + 4 + … + 512 ahlangana abe ngu-$1,023.
Umjikeleto utsatsa kubheja lokungu-2.05 ngesilinganiso, ngako imijikeleto lengu-784 itsatsa cishe kubheja lokungu-1,600. Nakubalwa kujikeleta ngekujikeleta, luchungechunge lwekulahlekelwa lokulishumi luvela kulenye indzawo ekubhejeni lokungu-200 ku-11.3% wesikhatsi.
Emkhatsini wetinhlekelele umdlali uwina $1 ngemjikeleto: cishe $783 ngaphambi kwalomubi, bese −$1,023, cishe −$240 esikhatsini sonkhe. Ngemjikeleto: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Imali lebhejiwe ngemjikeleto: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Futsi −0.31/11.31 = −2.70%.
Catsanisa nekubheja kwekucala kwe-$10 emkhawulweni we-$500, kubheja lokusitfupha: lubondza lufika kanye emijikeletweni lengu-55, futsi kusihlwa kwekubheja lokungu-200 kuluhlangabetana nalo ku-84% wesikhatsi. Umkhawulo lomkhulu wenta kuwa kungavami futsi kube kukhulu, akusiko kube kushibhile.
Lokuhlekisako
Kungavami futsi kukhulu, kunobe ngumuphi umkhawulo. Sibalo lesiphindzaphindziwe asigucuki.
Imitfombo
- 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.