Ugur: talast n ṭṭabla
Martingale s $10 ɣef ṭṭabla n $500. Acḥal i terrẓ, acḥal i texser mi ara terreẓ, yerna asali n talast yettɛawan?
Sḍis n lexsarat yeṭṭafaren rrẓent-tt: 1,8% i tunnḍa, qrib 60% n tɣimiyin n 100 n tuzzyiwin. Aṛeẓ yettkellif $630. Asali n talast ɣer $5 000 yettarra aṛeẓ d qlil ugar (0,25% i tunnḍa) d ameqqran ugar ($5 110). Lexsara yettwaṛǧan i yal adular yettwarehnen d 2,70% s yal talast.
Ugur
Amurar yettrehin $10 ɣef uzeggaɣrrehnAzeggaɣRrehn s lexlaṣ imsawi ɣef 18 n wuṭṭunen izeggaɣen. S tefransist: rouge.Ɣer akk →, yessnernay s sin deffir yal lexsara, yettuɣal ɣer tazwara deffir rrbeḥ. Talast tafellayt n ṭṭabla d $500. (a) Acḥal n lexsarat yeṭṭafaren i yerrẓen anagraw n usnernitastratijitAnagraw n usnerniYal alugen i yettbeddilen rrehn ɣef leḥsab n yigmaḍ yezrin. Asnerni deffir lexsara yessali-t deffir lexsara, asnerni deffir rrbeḥ deffir rrbeḥ.Ɣer akk →? (b) D acu i d taprubabilittusnaktTaprubabilitAzal n yiwen ugmuḍ ad d-yeḍru, seg 0 ɣer 1 neɣ seg 0% ɣer 100%. Yiwen wuṭṭun: 1/37 = 2,70%.Ɣer akk → i tunnḍa? (c) D acu i yexsren deg uṛeẓ? (d) Amek ara tbeddel talast n $5 000 tiririyin? (e) Tella talast ara yerren anagraw d win i d-yettawin rrbeḥ?
Tifrat (a)–(c)
Rrhun: 10, 20, 40, 80, 160, 320. Wis sebɛa ad yili 640 > 500, ihi sḍis n lexsarat rrẓent-t. (b) p(lexsara) = 19/37 ɣef uzeggaɣ; (19/37)⁶ = 0,0183, qrib 1,8% i tunnḍa. Tunnḍa teṭṭul qrib snat n tuzzyiwin s ulemmas, ihi taɣimit n 100 n tuzzyiwin tesɛa qrib 50 n tunnḍiwin, p(ma drus yiwen uṛeẓ) ≈ 1 − 0,9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, d 63 n yiferdisenidrimenAferdisRrehn agejdan s wacu i yettxeddem unagraw neɣ amurar. Inagrawen n usnerni ttwaglamen s yiferdisen akken ad ilin i yal ṭṭabla.Ɣer akk → n tazwara, mgal $10 i tebɣa ad terbeḥ tunnḍa.
Tifrat (d)
S $5 000: rrhun alamma d 2 560 ttwasirgen; rrehn wis mraw, 5 120, yettwagi. Tẓa n lexsarat rrẓent-t: (19/37)⁹ ≈ 0,25% i tunnḍa, qrib 11% n tɣimiyin n 100 n tuzzyiwin. Lexsara deg uṛeẓ: 10 + 20 + … + 2 560 = $5 110. Amurar yuɣ tamsibt yettwafen qlil ugar s ssuma n tin yugaren. Deg waṭas n tɣimiyin, ssuma yettwaṛǧan n yiṛuẓen d p × lexsara ≈ 0,0183 × 630 = $11,5 i tunnḍa s talast n $500 d 0,0025 × 5 110 = $12,8 s $5 000; sin yid-sen qerben maca ur msawan ara, yerna yal yiwen d lexsara yettwaṛǧantusnaktLexsara yettwaṛǧanAzal yettwaṛǧan seg tmerna n yidrimen yettwarhen; d alemmas n wayen yettwaxsar deg yiwet n tɣimit neɣ deg yiwen wakud.Ɣer akk → n tunnḍa rnu $10 i rebḥent tunnḍiwin n rrbeḥ.
Tifrat (e)
Ala. Lexsara yettwaṛǧan i tunnḍa = (adrim akk yettwaṛǧan deg tunnḍa) / 37, s yal talast, acku yal rrehn d −1/37 n yiman-is. Talast tettfru kan amek ara tettwabḍu lexsara gar waṭas n yirbaḥen imecṭaḥ d kra n lexsarat timeqqranin. War talast d ugerrujidrimenAgerrujAkk idrimen yettwaǧǧan i wurar. Asefrek n ugerruj d tagrut: acḥal seg-s ara tesrem i yal rrehn d yal taɣimit.Ɣer akk → war taggara, anagraw ad «iteddu», s unamek belli werǧin ad yerreẓ, ad yerhen daɣen azal war talast s lexsara yettwaṛǧan n 2,70% n wazal-nni war talast. Mačči d talast n ṭṭablaṭṭablaTalast n ṭṭablaRrehn adday d rrehn afellay i yettwasirgen. Talast tafellayt d ayen i yeḥebsen inagrawen n usnegmu.Ɣer akk → i yesseḥbasen MartingaletastratijitMartingaleAnagraw n usnerni i yessneggmen rrehn deffir yal lexsara. Walit taggayt n tstratijit i wayen i yettxemmem d wayen ur yettxemmim ara.Ɣer akk →. D leḥsab.
Snat n ṭṭwabel n tidet
Amedya n Thorp d ṭṭabla n $1 s talast tafellayt n $1 000, anda amurar yessaram ad yerbeḥ $30 i imensi, adular s adular. Rrhun 1, 2, 4 … 512 ttaǧǧan mraw n rrhun, ihi mraw n lexsarat yeṭṭafaren rrẓent anagraw: yiwet n tikkelt deg 785 n tunnḍiwin ɣef uḍebsi awrupi (yiwet deg 613 ɣef umarikani), s ssuma n $1 023. Tlatin n tunnḍiwin akk rnent 96,2% n wakud. 3,8% nniḍen xesrent qrib $1 000, yerna agmuḍ alemmas n tmeddit d −$9,17, d 1/37 s timmad-is n $339 i ttrehinent tlatin n tunnḍiwin s ulemmas. Ṭṭabla n $5 s talast tafellayt n $2 000 ur temgarad ara: rrhun alamma d $1 280 ttaǧǧan tẓa n rrhun, aṛeẓ yettas-d yiwet n tikkelt deg 403 n tunnḍiwin ɣef uḍebsi awrupi (yiwet deg 323 ɣef umarikani), yettkellif $2 555.
Amurar ameṛkanti
Efket i unagraw agerruj ameqqran deg umkan n talast tadday: 1 000 n yiferdisen, rrehn n tazwara n yiwen uferdis d 1 000 n tuzzyiwin ɣef uzeggaɣ, s tuɣalin ɣer yiwen uferdis yal tikkelt asnerni s sin d-iteddun yugar ayen d-yeqqimen. Nexdem tasimulasyunt n umelyun n tmeddiyin am tigi. Ɣef uḍebsi awrupi 52% fukkent s rrbeḥ, 21% xesrent kullec, yerna agmuḍ alemmas d −124 n yiferdisen. Ula deg wurar war ilemaḍebsiIlemTaxxamt tazegzawt tasuft ur yeṭṭif ula d yiwen n rrehn s lexlaṣ imsawi, n ddzina neɣ n tgejdit; d aɣbalu n lmaziyya n lkazinu. Daɣen: yiwen yilem, 0.Ɣer akk →, anda yal rrehn d n lḥeqq, qrib 60% kan i yettfakken s rrbeḥ. Tameddit n yal ass d rrbeḥ amecṭuḥ; alemmas yettwaxelleṣ s tmeddiyin i yettfakken ɣef yilem.
Tasirt
Lkazinu sbaddent tilisa akken ad ḥedden tavaryanstusnaktTavaryansAktazal n wamek i d-ttwazzaren igmaḍ i tama n wazal yettwaṛǧan. Meqqret i rrehn ɣef yiwen wuṭṭun, meẓẓiyet i rrhun s lexlaṣ imsawi.Ɣer akk →-nsent, mačči akken ad ɣelbent inagrawen. Taɣelbt n Martingale d ayen d-yettasen ɣer tama i ten-iɛeǧben, yerna leḥsab ad tt-iɣleb akken yebɣu yili. Amurar i iɣilen belli talast d ugur yufa-d amkan n uɣelluy, yeɣleḍ-it d tasawent.
Tasemselt
Isteqsiyen
Yettaǧǧa-ken ad teqqimem alamma d talast meqqren ugar, anda aṛeẓ d qlil ugar d ameqqran ugar. Lexsara yettwaṛǧan i yal adular ur tbeddel ara.
Rrhun 10, 30, 70, 150, 310: talast n $500 tettwaweḍ deffir semmus n lexsarat, aṛeẓ yettkellif $570. Yiwet n 2,70%.
Akken yiwen umurar ur yezmir ad yerhen ugar n wayen tezmer ad txelleṣ s talwit ugerruj n lkazinu. D asefrek n tavaryans.
Urar s tusnakt
Ɛerḍet-tent s yiman-nwen qbel. Yal tifrat tettwaffer; ldit-tt mi ara tilim twejdem.
MartingaletastratijitMartingaleAnagraw n usnerni i yessneggmen rrehn deffir yal lexsara. Walit taggayt n tstratijit i wayen i yettxemmem d wayen ur yettxemmim ara.Ɣer akk → yebda s $100 ɣef uzeggaɣrrehnAzeggaɣRrehn s lexlaṣ imsawi ɣef 18 n wuṭṭunen izeggaɣen. S tefransist: rouge.Ɣer akk → u yettaǧǧa i yiman-is n n rrhun ma yugar i yal tunnḍa: deffir n n lexsarat ṭṭafarent yeḥbes u yebda tikkelt nniḍen. Arut agmuḍ alemmas n yiwet n tunnḍa s tsemselt n n, u ḥesbet-t i 5, 6 d 7 n rrhun ɣef uḍebsi awrupi.
Sken-d tifrat
Tiririt Alemmas d 100 × (1 − (38/37)ⁿ): −$14,26 i 5 n rrhun, −$17,35 i 6 d −$20,52 i 7. Asirag n ugar n rrhun yessexṣar tunnḍa talemmast, ur tt-yesselhay ara.
Tunnḍa terbeḥ $100 ala ma xesren akk n-nni n rrhun, ayen yeḍran s tprubabilit n qⁿ, anda q = 19/37. Mi ara xesren akk, rrhun 100, 200 … 100 × 2ⁿ⁻¹ ttemrennun d 100 × (2ⁿ − 1). Ihi alemmas d 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Dagi 2q = 38/37, cwiṭ ugar n 1, ihi (2q)ⁿ yettimɣur s n u alemmas yettɣelli: n = 5 yettak-d 1,1426 d −$14,26; n = 6 yettak-d 1,1735 d −$17,35; n = 7 yettak-d 1,2052 d −$20,52. Asellum aɣezfan ugar yettexsar drus n tikkal, deg 3,6%, 1,8% d 0,9% n tunnḍiwin, maca yal lexsara tugar s snat, u ugar n yedrimen ttɛeddin ɣef ṭṭabla yal tunnḍa, s 2,70% n yal rrehn.
Ɣef uḍebsi amarikani 2q = 40/38 u imḍanen d −$29,24, −$36,04 d −$43,20 (Taylor, 2021). Deg wurar war ilemaḍebsiIlemTaxxamt tazegzawt tasuft ur yeṭṭif ula d yiwen n rrehn s lexlaṣ imsawi, n ddzina neɣ n tgejdit; d aɣbalu n lmaziyya n lkazinu. Daɣen: yiwen yilem, 0.Ɣer akk → 2q = 1, u yal tunnḍa s ulemmas d ilem s tidet akken yebɣu yili teɣzi n usellum.
«Grand Martingale» yessnernay rrehn s snat deffir lexsara u yerna yiwen uferdisidrimenAferdisRrehn agejdan s wacu i yettxeddem unagraw neɣ amurar. Inagrawen n usnerni ttwaglamen s yiferdisen akken ad ilin i yal ṭṭabla.Ɣer akk →, ihi rrhun d 1, 3, 7, 15, 31 … Acḥal i terbḥem mi ara d-yas rrbeḥ amezwaru ɣef rrehn wis 1, wis 2, wis 3 neɣ wis k? S uferdis n $10 d talast tafellayt n $500, acḥal n tikkal i yerreẓ tunnḍa, d acḥal i tetteklif tunnḍa s ulemmas?
Sken-d tifrat
Tiririt Rrbeḥ ɣef rrehn wis k yeǧǧa-kun s k n yiferdisen n rrbeḥ: +1, +2, +3 akka akka. S $10 d $500, tunnḍa terreẓ deffir semmus n lexsarat, 3,57% n tunnḍiwin, s −$570. S ulemmas, tunnḍa texser $2,32 seg $85,73 yettwarehnen: 2,70%.
Rrehn wis k d 2ᵏ − 1 n yiferdisen: 1, 3, 7, 15, 31. Rrhun imezwura n k − 1 ttemrennun d 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Rrbeḥ ɣef rrehn wis k ixelleṣ 2ᵏ − 1, ihi tunnḍa tekfa s (2ᵏ − 1) − (2ᵏ − k − 1) = k n yiferdisen n rrbeḥ: yiwen n uferdis nniḍen i yal lexsara qbel rrbeḥ.
S $10 i uferdis, rrhun d $10, $30, $70, $150 d $310; win d-iteddun, $630, yugar talast tafellayt n $500. Semmus n lexsarat ṭṭafarent, (19/37)⁵ = 3,57%, tteklifent 10 + 30 + 70 + 150 + 310 = $570.
Alemmas: rrbeḥ yettawi-d $10 × k s tprubabilit n (19/37)ᵏ⁻¹ × 18/37 i k = 1 ar 5, u aɣelluy yettawi $570 s tprubabilit n 3,57%; akk −$2,32. Idrimen yettwarehnen s ulemmas d $85,73 i yal tunnḍa, u 2,32/85,73 = 2,70%. Martingale amagnu ɣef ṭṭabla-nni yerreẓ deffir sḍis n lexsarat, 1,83% n tunnḍiwin, s −$630, u yexser $1,74 seg $64,20 yettwarehnen: 2,70% tikkelt nniḍen.
Amurar yesbadu Martingale ɣef uzeggaɣ s rrehn amezwaru n $1 ɣef ṭṭabla s talast n $1 000, ihi mraw n rrhun zemren ad ilin: $1, $2, $4 akka ar $512. Acḥal n tikkal i texser tunnḍa mraw-nni meṛṛa, acḥal n wakud i yesra waya s ulemmas, d acḥal n tikkal i tekfa akka tmeddit n 200 n rrhun?
Sken-d tifrat
Tiririt Tunnḍa texser mraw-nni meṛṛa qrib yiwet n tikkelt deg 784, u ayagi yetteklif $1 023. D qrib 1 600 n rrhun s ulemmas, u tameddit n 200 n rrhun tewweḍ ɣer talast qrib 11% n wakud. Tunnḍa mazal texser $0,31 s ulemmas seg $11,31 yettwarehnen: 2,70%.
Rrehn ɣef uzeggaɣ yexser s tprubabilit n 19/37, s yilem. Mraw ṭṭafaren: (19/37)¹⁰ ≈ 0,00128, qrib 1 seg 784. Rrhun 1 + 2 + 4 + … + 512 ttemrennun d $1 023.
Tunnḍa tesra 2,05 n rrhun s ulemmas, ihi 784 n tunnḍiwin sran qrib 1 600 n rrhun. Ma nḥesb tuzzya s tuzzya, azrar n mraw n lexsarat yettban-d deg kra n wadeg deg 200 n rrhun 11,3% n wakud.
Gar twaɣiwin, amurar yerbeḥ $1 i yal tunnḍa: qrib $783 qbel tin n diri, syin −$1 023, qrib −$240 deg ubrid meṛṛa. I yal tunnḍa: 1 × (1 − 0,00128) − 1 023 × 0,00128 ≈ −$0,31. Idrimen yettwarehnen i yal tunnḍa: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11,31. U −0,31/11,31 = −2,70%.
Serwset s rrehn amezwaru n $10 s talast n $500, sḍis n rrhun: talast tettas-d yiwet n tikkelt deg 55 n tunnḍiwin, u tameddit n 200 n rrhun temlal-itt 84% n wakud. Talast yugaren terra aɣelluy ur nettili ara aṭas u d ameqqran ugar, ur t-terri ara yerxes ugar.
Taḍsa i usgunfu
Qlil ugar, meqqer ugar, s yal talast. Asgut ur ibeddel ara.
Iɣbula
- 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.