Masalo: Xadka Miiska
Martingale $10 ah oo miis $500 ah. Intee jeer ayuu jabaa, intee ayuu khasaaraa markuu jabo, kor u qaadista xadkuse ma caawisaa?
Lix khasaare oo isku xiga ayaa jebiya: 1.8% meertadiiba, qiyaastii 60% fadhiyada 100 wareeg ah. Jabku wuxuu ku kacaa $630. Kor u qaadista xadka ilaa $5,000 waxay jabka ka dhigtaa mid naadir ah (0.25% meertadiiba) oo ka weyn ($5,110). Khasaaraha la filayo doollar kasta oo la sharadsaday waa 2.70% xad kasta.
Masalada
Ciyaartoy ayaa $10 ku sharadsada casaankasharadCasaanSharad lacag isle'eg ah oo ku saabsan 18-ka lambar ee casaanka ah. Faransiis: rouge.Akhri dhammaan → oo labanlaaba khasaare kasta kadib, dib u bilaabana guul kadib. Xadka ugu sarreeya ee miisku waa $500. (a) Imisa khasaare oo isku xiga ayaa jebiya kordhintaistaraatiijiKordhinXeer kasta oo beddela sharadka iyadoo lagu salaynayo natiijooyinkii hore. Kordhinnada tabanka ah waxay kor u qaadaan khasaare kadib, kuwa togan guul kadib.Akhri dhammaan →? (b) Waa maxay itimaalkaxisaabItimaalFursadda natiijo, 0 ilaa 1 ama 0% ilaa 100%. Hal lambar: 1/37 = 2.70%.Akhri dhammaan → meertadiiba? (c) Maxaa lagu khasaaraa jabka? (d) Sidee xad $5,000 ah u beddelaa jawaabaha? (e) Xad kasta ma ka dhigaa nidaamka mid faa'iido leh?
Xalalka (a)–(c)
Sharadyada: 10, 20, 40, 80, 160, 320. Kan toddobaad wuxuu noqon lahaa 640 > 500, sidaas darteed lix khasaare ayaa jebiya. (b) p(khasaare) = 19/37 casaanka; (19/37)⁶ = 0.0183, qiyaastii 1.8% meertadiiba. Meertadu waxay socotaa qiyaastii laba wareeg celcelis ahaan, sidaas darteed fadhi 100 wareeg ah wuxuu leeyahay qiyaastii 50 meerto, p(ugu yaraan hal jab) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, taasoo ah 63 unuglacagUnugSharadka saldhigga ah ee nidaam ama ciyaartoy ku shaqeeyo. Kordhinnada waxaa lagu sharxaa unugyo si ay ugu shaqeeyaan miis kasta.Akhri dhammaan → oo sal ah, marka la barbar dhigo $10 ee cycle-ku isku dayayay inuu ku guulaysto.
Xalka (d)
$5,000: sharadyada ilaa 2,560 waa la oggol yahay; sharadka tobnaad, 5,120, waa la diidaa. Sagaal khasaare ayaa jebiya: (19/37)⁹ ≈ 0.25% meertadiiba, qiyaastii 11% fadhiyada 100 wareeg ah. Khasaaraha jabka: 10 + 20 + … + 2,560 = $5,110. Ciyaartoygu wuxuu iibsaday masiibo naadir ah qiimaha mid ka weyn. Fadhiyo badan, qiimaha la filayo ee jabadku waa p × khasaare ≈ 0.0183 × 630 = $11.5 meerto-kiiba xadka $500 iyo 0.0025 × 5,110 = $12.8 xadka $5,000; waa, iyadoo la soo koobayo, isla tirada, waana khasaaraha la filayoxisaabKhasaaraha la filayoQiimaha la filayo oo lagu dhuftay wadarta lacagta la sharadsaday; celceliska lacagta lagu khasaaro fadhi ama muddo.Akhri dhammaan → ee meertada oo lagu daray $10 ee meertooyinka guulaysta ay kasbadaan.
Xalka (e)
Maya. Khasaaraha la filayo meertadiiba = (wadarta sharadka la filayo ee meertada) / 37, xad kasta, sababtoo ah sharad kastaa waa −1/37 naftiisa. Xadku wuxuu kaliya go'aamiyaa sida khasaaraha loogu qaybiyo guulo yaryar oo badan iyo khasaarooyin waaweyn oo yar. Xad la'aan iyo raasamaal ciyaareedlacagRaasamaalka ciyaartaWadarta lacagta loo qoondeeyay ciyaarta. Maaraynta raasamaalka ciyaarta waa go'aaminta inta laga khatarinayo sharad kasta iyo fadhi kasta.Akhri dhammaan → aan dhammaad lahayn, kordhintu way “shaqayn lahayd”, macnaha ah inaysan weligeed jabin, waxayna sidoo kale sharadsan lahayd qadar aan xad lahayn oo khasaaraha la filayo uu yahay 2.70% qadarkaas aan xadka lahayn. Xadka miiskamiisXadka miiskaSharadka ugu yar iyo ugu badan ee la oggol yahay. Kan ugu badan ayaa joojiya kordhinnada labanlaabka.Akhri dhammaan → ma aha waxa joojiya inuu MartingaleistaraatiijiMartingaleKordhin labanlaabta sharadka khasaare kasta kadib. Eeg qaybta Istaraatiijiyadda si aad u ogaato waxa ay qabato iyo waxa aysan qaban.Akhri dhammaan → shaqeeyo. Xisaabta ayaa ah.
Laba miis oo dhab ah
Tusaalaha Thorp waa miis $1 ah oo xadkiisa ugu sarreeyaa yahay $1,000, halkaas oo ciyaartoy rajeynayo inuu ku guulaysto $30 casho ah doollar doollar. Sharadyada 1, 2, 4 … 512 waxay oggolaadaan toban sharad, sidaas darteed toban khasaare oo isku xiga ayaa jebiya kordhinta: hal mar 785 meertadiiba giraangir Yurub ah (hal mar 613 kiiba mid Mareykan ah), qiimo $1,023 ah. Soddon meerto dhammaantood way guulaystaan 96.2% waqtiga. 3.8%-ka kale waxay khasaaraan qiyaastii $1,000, celceliska natiijada habeenkuna waa −$9.17, taasoo si sax ah u ah 1/37 $339 ee soddonka meerto ay celcelis ahaan sharadsadaan. Miis $5 ah oo xadkiisa ugu sarreeyaa yahay $2,000 kama duwana: sharadyada ilaa $1,280 waxay oggolaadaan sagaal sharad, jabkuna wuxuu yimaadaa hal mar 403 meertadiiba giraangir Yurub ah (hal mar 323 kiiba mid Mareykan ah) wuxuuna ku kacaa $2,555.
Ciyaartoy hodan ah
Kordhinta sii raasamaal qoto dheer halkii xad hoose: 1,000 unug, sal hal unug ah iyo 1,000 wareeg oo casaan ah, iyadoo mar kale laga bilaabayo hal unug mar kasta oo labanlaabka xigaa ka badan yahay inta hartay. Waxaan jilnay hal milyan habeen oo noocaas ah. Giraangir Yurub ah 52% waxay ku dhammeeyaan faa'iido, 21% waxay khasaaraan wax walba, celceliska natiijaduna waa −124 unug. Xataa ciyaar aan ebergiraangirEberHal godka cagaaran ee aan daboolin sharad lacag isle'eg ah, dersin ama tiir; isha faa'iidada kasiinada. Sidoo kale: single zero, 0.Akhri dhammaan → lahayn, oo sharad kastaa caddaalad yahay, kaliya qiyaastii 60% ayaa ku dhammaan lahaa faa'iido. Habeenka caadiga ahi waa guul yar; celceliska waxaa bixiya habeennada ku dhammaada eber.
Casharka
Kasiinooyinku waxay dejiyaan xadad si ay u xaddidaan kala-duwanaanshahoodaxisaabKala-duwanaanshoCabbir muujinaya sida ballaaran ee natiijooyinku ugu kala firdhsan yihiin qiimaha la filayo agtiisa. Sare hal lambar, hoose sharadyada lacag isle'eg.Akhri dhammaan →, ma aha inay jebiyaan nidaamyada. Jebinta Martingale waa saameyn dhinac ah oo ay ku faraxsan yihiin, xisaabtuna si kastaba way jebin lahayd. Ciyaartoy u maleynaya in xadku yahay dhibaatada wuxuu helay dhagaxa jarka ah wuxuuna u qaatay jiirada.
Qaacido
Su'aalo
Wuxuu kuu oggolaanayaa inaad u badbaaddo xad ka sarreeya, halkaas oo jabku naadir yahay oo ka weyn yahay. Khasaaraha la filayo doollarkiiba isma beddelo.
Sharadyada 10, 30, 70, 150, 310: xadka $500 waxaa la gaaraa shan khasaare kadib, jabkuna wuxuu ku kacaa $570. Isla 2.70%.
Si aanu hal ciyaartoy u sharadsan karin wax ka badan inta kaydka kasiinadu si raaxo leh u bixin karo. Maaraynta kala-duwanaanshaha.
Madadaalo Xisaabeed
Marka hore adigu isku day. Xal kasta waa qarsoon yahay; fur marka aad diyaar u tahay.
MartingaleistaraatiijiMartingaleKordhin labanlaabta sharadka khasaare kasta kadib. Eeg qaybta Istaraatiijiyadda si aad u ogaato waxa ay qabato iyo waxa aysan qaban.Akhri dhammaan → wuxuu ka bilaabmaa $100 casaansharadCasaanSharad lacag isle'eg ah oo ku saabsan 18-ka lambar ee casaanka ah. Faransiis: rouge.Akhri dhammaan → wuxuuna u oggol yahay naftiisa ugu badnaan n sharad meertadiiba: n khasaare oo isku xiga kadib wuu iska daayaa oo dib ayuu u bilaabaa. Celceliska natiijada hal meerto u qor qaacido n ah, kuna xisaabi 5, 6 iyo 7 sharad giraangir Yurub ah.
Muuji xalka
Jawaab Celcelisku waa 100 × (1 − (38/37)ⁿ): −$14.26 5 sharad, −$17.35 6 iyo −$20.52 7. U oggolaanshaha sharadyo badan celceliska meertada ayuu ka sii xumeeyaa, kama wanaajiyo.
Meertadu waxay keentaa $100 haddii aysan dhammaan n sharad lumin, taas oo dhacda fursad qⁿ ah, halkaas oo q = 19/37. Markay dhammaan lumaan, sharadyada 100, 200 … 100 × 2ⁿ⁻¹ waxay isku noqdaan 100 × (2ⁿ − 1). Sidaa darteed celcelisku waa 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Halkan 2q = 38/37, wax yar ka badan 1, sidaa darteed (2q)ⁿ wuu la koraa n celceliskuna wuu dhacaa: n = 5 wuxuu bixiyaa 1.1426 iyo −$14.26; n = 6 wuxuu bixiyaa 1.1735 iyo −$17.35; n = 7 wuxuu bixiyaa 1.2052 iyo −$20.52. Jaranjaro dheer ayaa dhif ku lumisa, 3.6%, 1.8% iyo 0.9% meertooyinka, laakiin khasaare kastaa waa laba jeer ka weyn yahay, lacag badanina waxay miiska dhex martaa meerto kasta, 2.70% sharad kasta.
Giraangir Mareykan ah 2q = 40/38 tirooyinkuna waa −$29.24, −$36.04 iyo −$43.20 (Taylor, 2021). Ciyaar aan ebergiraangirEberHal godka cagaaran ee aan daboolin sharad lacag isle'eg ah, dersin ama tiir; isha faa'iidada kasiinada. Sidoo kale: single zero, 0.Akhri dhammaan → lahayn 2q = 1, meerto kastaana celcelis ahaan waa si sax ah eber si kasta oo jaranjaradu u dheer tahay.
“Grand Martingale” wuxuu sharadka labanlaabaa khasaare kadib wuxuuna ku daraa hal unuglacagUnugSharadka saldhigga ah ee nidaam ama ciyaartoy ku shaqeeyo. Kordhinnada waxaa lagu sharxaa unugyo si ay ugu shaqeeyaan miis kasta.Akhri dhammaan →, sidaa darteed sharadyadu waxay u socdaan 1, 3, 7, 15, 31 … Imisa ayaad faa'iido ku jirtaa marka guusha koowaad timaado sharadka 1aad, 2aad, 3aad ama kaad? Unug $10 ah iyo ugu badnaan $500, intee jeer ayay meertadu jabtaa, maxayse meertadu ku kacdaa celcelis ahaan?
Muuji xalka
Jawaab Guul sharadka kaad waxay kaa dhigtaa k unug faa'iido: +1, +2, +3 iyo wixii la mid ah. $10 iyo $500 meertadu waxay jabtaa shan khasaare kadib, 3.57% meertooyinka, −$570. Celcelis ahaan meertadu waxay lumisaa $2.32 $85.73 la sharadsaday: 2.70%.
Sharadka kaad waa 2ᵏ − 1 unug: 1, 3, 7, 15, 31. k − 1 sharad ee hore waxay isku noqdaan 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Guul sharadka kaad waxay bixisaa 2ᵏ − 1, sidaa darteed meertadu waxay ku dhammaataa (2ᵏ − 1) − (2ᵏ − k − 1) = k unug faa'iido: hal unug oo dheeraad ah khasaare kasta oo guusha ka horreeya.
$10 unuggiiba sharadyadu waa $10, $30, $70, $150 iyo $310; kan xiga, $630, wuu ka badan yahay ugu badnaanta $500. Shan khasaare oo isku xiga, (19/37)⁵ = 3.57%, waxay ku kacaan 10 + 30 + 70 + 150 + 310 = $570.
Celcelis: guulaha waxay keenaan $10 × k fursad (19/37)ᵏ⁻¹ × 18/37 ah k = 1 ilaa 5, jabkuna wuxuu qaataa $570 fursad 3.57% ah; wada jir −$2.32. Lacagta la sharadsaday celcelis ahaan waa $85.73 meertadiiba, 2.32/85.73 = 2.70%. Martingale-ka caadiga ah isla miiskaas wuxuu jabaa lix khasaare kadib, 1.83% meertooyinka, −$630, wuxuuna lumiyaa $1.74 $64.20 la sharadsaday: mar kale 2.70%.
Ciyaartoy ayaa qorsheeya Martingale casaan ah oo sharadka koowaad yahay $1 miis xadkiisu yahay $1,000, sidaa darteed toban sharad ayaa suurtagal ah: $1, $2, $4 iyo wixii la mid ah ilaa $512. Intee jeer ayay meertadu lumisaa dhammaan tobanka, intee in le'eg ayay taasi qaadataa celcelis ahaan, intee jeerse ayuu fiid 200 sharad ahi sidaas ku dhammaadaa?
Muuji xalka
Jawaab Meertadu waxay lumisaa dhammaan tobanka qiyaastii hal mar 784-kiiba, taas oo ku kacda $1,023. Taasi waa qiyaastii 1,600 sharad celcelis ahaan, fiid 200 sharad ahna wuxuu darbiga ku dhacaa qiyaastii 11% waqtiga. Meertadu weli waxay lumisaa $0.31 celcelis ahaan $11.31 la sharadsaday: 2.70%.
Sharad casaan ah wuxuu lumaa fursad 19/37 ah, eberka lagu daray. Toban isku xiga: (19/37)¹⁰ ≈ 0.00128, qiyaastii 1 784-kiiba. Sharadyada 1 + 2 + 4 + … + 512 waxay isku noqdaan $1,023.
Meertadu waxay qaadataa 2.05 sharad celcelis ahaan, sidaa darteed 784 meerto waxay qaadataa qiyaastii 1,600 sharad. Haddii wareeg wareeg loo tiriyo, taxane toban khasaare ah wuxuu meel uun ka soo baxaa 200 sharad 11.3% waqtiga.
Musiibooyinka dhexdooda ciyaartoygu wuxuu helaa $1 meertadiiba: qiyaastii $783 ka hor tan xun, kadib −$1,023, qiyaastii −$240 muddada oo dhan. Meertadiiba: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Lacagta la sharadsaday meertadiiba: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Iyo −0.31/11.31 = −2.70%.
Is barbar dhig sharad koowaad oo $10 ah xad $500 ah, lix sharad: darbigu wuxuu yimaadaa hal mar 55 meertoba, fiid 200 sharad ahna wuxuu la kulmaa 84% waqtiga. Xad sare ayaa burburka ka dhiga mid dhif ah oo weyn, kama dhigo mid ka jaban.
Kaftan yar
Naadir oo ka weyn, xad kasta. Taranku waa joogto.
Ilaha
- 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.