Problem: limit latab
Enn Martingale $10 lor enn latab $500. Komie souvan li kase, komie li perdi kan li kase, ek eski ogmant limit-la ede?
Sis pert swivan kas li: 1.8% par sik, apepre 60% bann sesion 100 tour. Kas-la kout $630. Ogmant limit-la ziska $5,000 fer kas-la pli rar (0.25% par sik) ek pli gro ($5,110). Pert atandi par dolar parye se 2.70% kiswa limit-la.
Problem-la
Enn zwer parye $10 lor rouzpariRouzPari peman egal lor 18 nimero rouz. An franse: rouge.Lir tou → ek li doub apre sakenn pert, li rekoumans apre enn gin. Maksimom latab-la se $500. (a) Komie pert swivan kas progresionstrateziProgresionNenport ki regleman ki sanz miz-la dapre bann rezilta avan. Progresion negatif ogmant li apre pert, pozitif apre gin.Lir tou →-la? (b) Ki probabilitematematikProbabiliteSans enn rezilta, depi 0 ziska 1 ouswa depi 0% ziska 100%. Enn nimero: 1/37 = 2.70%.Lir tou → par sik? (c) Komie perdi kan li kase? (d) Kouma enn limit $5,000 sanz bann repons? (e) Eski ena enn limit ki fer sistem-la rapor profi?
Solision (a)–(c)
Bann miz: 10, 20, 40, 80, 160, 320. Setyem-la ti pou 640 > 500, alor sis pert kas li. (b) p(pert) = 19/37 lor rouz; (19/37)⁶ = 0.0183, apepre 1.8% par sik. Enn sik dire apepre de tour an mwayenn, alor enn sesion 100 tour ena apepre 50 sik ek p(omwin enn kas) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, ki se 63 initelarzanIniteMiz baz ki enn sistem ouswa enn zwer servi. Bann progresion dekrir an inite pou ki zot aplik lor nenport ki latab.Lir tou → baz, kont $10 ki sik-la ti pe rod gagne.
Solision (d)
A $5,000: miz ziska 2,560 permet; dizyem miz-la, 5,120, refize. Nef pert kas li: (19/37)⁹ ≈ 0.25% par sik, apepre 11% bann sesion 100 tour. Pert kan li kase: 10 + 20 + … + 2,560 = $5,110. Zwer-la inn aste enn katastrof pli rar o pri enn katastrof pli gro. Lor boukou sesion, kout atandi bann kas se p × pert ≈ 0.0183 × 630 = $11.5 par sik avek limit $500 ek 0.0025 × 5,110 = $12.8 a $5,000; le de pre me pa egal, ek sakenn se pert atandimatematikPert atandiValer atandi miltiplie par total larzan ki ou parye; montan mwayen ki ou perdi lor enn sesion ouswa enn peryod.Lir tou → sik-la plis $10 ki bann sik gagnan raporte.
Solision (e)
Non. Pert atandi par sik = (total miz atandi dan sik-la) / 37, kiswa limit-la, akoz sakenn pari vo −1/37 limem. Limit-la zis deside kouma pert-la distribie ant boukou tipti gin ek enn pe gro pert. San limit ek avek enn kapital zwelarzanKapital zweTotal larzan ki ou met de kote pou zwe. Zesion kapital zwe vedir deside komie ladan ou riske par pari ek par sesion.Lir tou → infini, progresion-la ti pou “marse”, dan sans ki li zame ti pou kase, me li ti pou osi parye enn montan san limit avek enn pert atandi 2.70% sa montan san limit-la. Se pa limit latablatabLimit latabPari minimom ek maksimom ki permet. Maksimom-la se seki aret bann progresion ki doub.Lir tou → ki anpes MartingalestrateziMartingaleProgresion ki doub pari-la apre sak pert. Get seksion Stratezi pou kone seki li fer ek seki li pa fer.Lir tou → marse. Se kalkil-la.
De vre latab
Lexanp Thorp se enn latab $1 avek enn maksimom $1,000, kot enn zwer espere gagn $30 pou so dine, enn dolar alafwa. Bann miz 1, 2, 4 … 512 permet dis pari, alor dis pert swivan kas progresion-la: enn fwa lor 785 sik lor enn rou eropeen (enn fwa lor 613 lor enn rou amerikin), pou enn kout $1,023. Trant sik tou reisi 96.2% ditan. Lezot 3.8% perdi apepre $1,000, ek rezilta mwayenn laswaree-la se −$9.17, ki egzakteman 1/37 $339 ki trant sik-la mize an mwayenn. Enn latab $5 avek enn maksimom $2,000 pa diferan: miz ziska $1,280 permet nef pari, ek kas-la vini enn fwa lor 403 sik lor enn rou eropeen (enn fwa lor 323 lor enn rou amerikin) ek li kout $2,555.
Enn zwer ris
Donn progresion-la enn gro kapital zwe olie enn limit ba: 1,000 inite, enn baz enn inite ek 1,000 tour lor rouz, rekoumans a enn inite sakenn fwa ki prosenn doub pli gran ki seki reste. Nou inn simil enn milyon laswaree koumsa. Lor enn rou eropeen 52% fini an avans, 21% perdi tou, ek rezilta mwayenn-la se −124 inite. Mem dan enn zwe san zerorouZeroSel kaz ver ki okenn pari peman egal, douzenn ouswa kolonn pa kouver; sours lavantaz kazino. Osi: enn sel zero, 0.Lir tou →, kot sakenn pari zis, zis apepre 60% ti pou fini an avans. Laswaree ordiner se enn tipti gin; mwayenn-la pey par bann laswaree ki fini a zero.
Leson-la
Bann kazino fix limit pou limit zot prop varyansmatematikVaryansEnn mezir lor kouma bann rezilta dispers otour valer atandi. For pou pari lor enn sel nimero, ba pou pari peman egal.Lir tou →, pa pou bat bann sistem. Bat Martingale se enn efe segonder ki zot kontan gagne, ek kalkil-la ti pou bat li kanmem. Enn zwer ki krwar limit-la se problem-la inn trouv falez-la ek li pran li pou pant-la.
Formil
Kestion
Li les ou tini ziska enn pli gran limit, kot kas-la pli rar ek pli gro. Pert atandi par dolar pa sanze.
Miz 10, 30, 70, 150, 310: limit $500 ariv apre sink pert, ek kas-la kout $570. Mem 2.70%.
Pou ki enn sel zwer pa kapav parye plis ki seki rezerv kazino-la kapav pey san difikilte. Zesion varyans.
Lamizman ar matematik
Esey rezoud zot ou mem avan. Sakenn solision kase; ouver li kan ou pare.
Enn MartingalestrateziMartingaleProgresion ki doub pari-la apre sak pert. Get seksion Stratezi pou kone seki li fer ek seki li pa fer.Lir tou → koumans a $100 lor rouzpariRouzPari peman egal lor 18 nimero rouz. An franse: rouge.Lir tou → ek permet li maksimom n pari par sik: apre n pert ansinn li abandone ek rekoumanse. Ekrir rezilta mwayenn enn sik kouma enn formil an n, ek kalkil li pou 5, 6 ek 7 pari lor enn rou eropeen.
Montre solision-la
Repons Mwayenn-la se 100 × (1 − (38/37)ⁿ): −$14.26 pou 5 pari, −$17.35 pou 6 ek −$20.52 pou 7. Kan ou permet plis pari, sik mwayenn vinn pli mal, pa pli bon.
Enn sik gagn $100 sof si tou n pari perdi, seki arive avek sans qⁿ, kot q = 19/37. Kan zot tou perdi, bann miz 100, 200 … 100 × 2ⁿ⁻¹ adisionn ziska 100 × (2ⁿ − 1). Alor mwayenn-la se 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Isi 2q = 38/37, enpe plis ki 1, alor (2q)ⁿ grandi avek n ek mwayenn-la tonbe: n = 5 donn 1.1426 ek −$14.26; n = 6 donn 1.1735 ek −$17.35; n = 7 donn 1.2052 ek −$20.52. Enn lesel pli long perdi mwins souvan, dan 3.6%, 1.8% ek 0.9% bann sik, me sakenn pert de fwa pli gro, ek plis larzan travers latab sakenn sik, a 2.70% lor sakenn pari.
Lor enn rou amerikin 2q = 40/38 ek bann sif se −$29.24, −$36.04 ek −$43.20 (Taylor, 2021). Dan enn zwe san zerorouZeroSel kaz ver ki okenn pari peman egal, douzenn ouswa kolonn pa kouver; sours lavantaz kazino. Osi: enn sel zero, 0.Lir tou → 2q = 1, ek sakenn sik ena enn mwayenn egzakteman zero, nimport komie lesel-la long.
“Grand Martingale” doub miz-la apre enn pert ek azout enn initelarzanIniteMiz baz ki enn sistem ouswa enn zwer servi. Bann progresion dekrir an inite pou ki zot aplik lor nenport ki latab.Lir tou →, alor bann miz al 1, 3, 7, 15, 31 … Komie ou devan kan premie gin vini lor 1e, 2em, 3em ouswa kyem pari? Avek enn inite $10 ek enn maksimom $500, komie souvan enn sik kase, ek komie enn sik kout an mwayenn?
Montre solision-la
Repons Enn gin lor kyem pari les ou k inite devan: +1, +2, +3 ek koumsa deswit. A $10 ek $500 sik-la kase apre sink pert, 3.57% bann sik, pou −$570. An mwayenn enn sik perdi $2.32 lor $85.73 mize: 2.70%.
Kyem miz se 2ᵏ − 1 inite: 1, 3, 7, 15, 31. Premie k − 1 miz adisionn ziska 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Enn gin lor kyem pari peye 2ᵏ − 1, alor sik-la fini (2ᵏ − 1) − (2ᵏ − k − 1) = k inite devan: enn inite an plis pou sakenn pert avan gin-la.
A $10 par inite bann miz se $10, $30, $70, $150 ek $310; prosin-la, $630, depas maksimom $500. Sink pert ansinn, (19/37)⁵ = 3.57%, kout 10 + 30 + 70 + 150 + 310 = $570.
Mwayenn: bann gin ramenn $10 × k avek sans (19/37)ᵏ⁻¹ × 18/37 pou k = 1 ziska 5, ek kasaz-la pran $570 avek sans 3.57%; ansam −$2.32. Larzan mize fer enn mwayenn $85.73 par sik, ek 2.32/85.73 = 2.70%. Martingale ordiner lor mem latab kase apre sis pert, 1.83% bann sik, pou −$630, ek perdi $1.74 lor $64.20 mize: 2.70% ankor.
Enn zwer prevwar enn Martingale lor rouz avek enn premie pari $1 lor enn latab kot limit-la $1,000, alor dis pari posib: $1, $2, $4 ek koumsa deswit ziska $512. Komie souvan enn sik perdi tou le dis, komie letan sa pran an mwayenn, ek komie souvan enn laswaret 200 pari fini koumsa?
Montre solision-la
Repons Enn sik perdi tou le dis apepre enn fwa lor 784, ki kout $1,023. Sa fer apepre 1,600 pari an mwayenn, ek enn laswaret 200 pari tap dan miray apepre 11% di tan. Enn sik touzour perdi $0.31 an mwayenn lor $11.31 mize: 2.70%.
Enn pari lor rouz perdi avek sans 19/37, zero ladan. Dis ansinn: (19/37)¹⁰ ≈ 0.00128, apepre 1 lor 784. Bann miz 1 + 2 + 4 + … + 512 adisionn ziska $1,023.
Enn sik pran 2.05 pari an mwayenn, alor 784 sik pran apepre 1,600 pari. Si ou kont tour par tour, enn seri dis pert paret kik part dan 200 pari 11.3% di tan.
Ant bann dezas zwer-la gagn $1 par sik: apepre $783 avan move-la, apre −$1,023, apepre −$240 lor tou sa peryod-la. Par sik: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Larzan mize par sik: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Ek −0.31/11.31 = −2.70%.
Kompar enn premie pari $10 a enn limit $500, sis pari: miray-la vini enn fwa lor 55 sik, ek enn laswaret 200 pari zwenn li 84% di tan. Enn limit pli ot fer kras-la pli rar ek pli gro, pa pli bon marse.
Enpe riye
Pli rar ek pli gro, kiswa limit-la. Prodwi-la pa sanze.
Sours
- 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.