Roulette ka Math · Math problem · 10 min

Sawaal: Table Limit

$500 wali table par $10 ka Martingale. Yeh kitni baar toot-ta hai, tootne par kitna nuksaan hota hai, aur kya limit badhaane se madad milti hai?

Chhota jawab

Lagatar chhe haar ise tod deti hain: har cycle 1.8%, lagbhag 60% 100-spin sessions mein. Tootne ki keemat $630 hai. Limit $5,000 karne se tootna kam hota hai (har cycle 0.25%) aur bada hota hai ($5,110). Har bet kiye gaye dollar par expected loss kisi bhi limit par 2.70% hai.

base / limit
$10 / $500
tootne tak haar
6
har cycle p(tootna)
1.8%
tootne par nuksaan
$630

Sawaal

Ek player redbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → par $10 ki bet lagata hai aur har haar ke baad double karta hai, jeet ke baad wapas shuru karta hai. Table maximum $500 hai. (a) Kitni lagatar haar progressionstrategyProgressionKoi bhi rule jo pichhle results ke hisaab se stake badalta hai. Negative progressions haar ke baad ise badhati hain, positive jeet ke baad.Poora padhein → ko tod deti hain? (b) Har cycle probabilityganitProbabilityKisi outcome ka chance, 0 se 1 tak ya 0% se 100% tak. Ek number: 1/37 = 2.70%.Poora padhein → kya hai? (c) Tootne par kitna nuksaan hota hai? (d) $5,000 ki limit jawaabon ko kaise badalti hai? (e) Kya koi bhi limit system ko munafe wala banati hai?

Roulette Table Limits

Hal (a)–(c)

Stakes: 10, 20, 40, 80, 160, 320. Saatvaan 640 > 500 hoga, isliye chhe haar ise tod deti hain. (b) Red par p(haar) = 19/37; (19/37)⁶ = 0.0183, yaani har cycle lagbhag 1.8%. Ek cycle average mein lagbhag do spins chalta hai, isliye 100-spin session mein qareeb 50 cycles hote hain aur p(kam se kam ek baar tootna) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, yaani 63 base unitspaisaUnitBase stake jismein koi system ya player kaam karta hai. Progressions units mein batayi jaati hain taaki kisi bhi table par laagu hon.Poora padhein →, us $10 ke muqable jo cycle jeetne ki koshish kar raha tha.

limittootne tak haarhar cycle p(tootna)tootne par nuksaanp(100 spins mein tootna)wapas paane ke liye zaroori jeet
$100313.5%$70≈ 100%7
$50061.8%$630≈ 60%63
$1,00061.8%$630≈ 60%63
$2,00070.94%$1,270≈ 36%127
$5,00090.25%$5,110≈ 11%511
$20,000110.07%$20,470≈ 3%2,047
Ceiling ko har baar double karne se tootna lagbhag aadha kam hota hai aur do guna mehenga. Dono ka gunanfal nahi badalta.

Hal (d)

$5,000 par: 2,560 tak ke stakes allowed hain; dasvaan stake, 5,120, mana ho jaata hai. Nau haar ise tod deti hain: (19/37)⁹ ≈ har cycle 0.25%, lagbhag 11% 100-spin sessions mein. Tootne par nuksaan: 10 + 20 + … + 2,560 = $5,110. Player ne badi tabaahi ki keemat par kam hone wali tabaahi khareedi hai. Kai sessions mein tootne ki expected keemat p × nuksaan ≈ 0.0183 × 630 = $500 limit par har cycle $11.5 aur $5,000 par 0.0025 × 5,110 = $12.8 hai; rounding ke andar yeh wahi number hai, aur yeh cycle ka expected lossganitExpected lossExpected value guna kul lagayi gayi raqam; ek session ya ek muddat mein ausat nuksaan.Poora padhein → plus woh $10 hai jo jeetne wale cycles kamaate hain.

Hal (e)

Nahi. Har cycle expected loss = (cycle mein expected total stake) / 37, kisi bhi limit par, kyunki har bet apne aap ka −1/37 hai. Limit sirf yeh tay karti hai ki nuksaan kai chhoti jeet aur kuch badi haar ke beech kaise bant-ta hai. Bina limit aur anant bankrollpaisaBankrollKhelne ke liye alag rakhi gayi kul raqam. Bankroll management ka matlab hai tay karna ki har bet aur har session mein iska kitna hissa risk karna hai.Poora padhein → ke progression “kaam” karega, is matlab mein ki woh kabhi nahi tootega, aur woh ek be-hisaab raqam bhi daav par lagayega jiska expected loss us be-hisaab raqam ka 2.70% hoga. Table limittableTable limitAllowed minimum aur maximum bet. Maximum hi doubling progressions ko rokta hai.Poora padhein → woh cheez nahi hai jo MartingalestrategyMartingaleProgression jo har haar ke baad bet double karti hai. Yeh kya karti hai aur kya nahi, iske liye Strategy section dekhiye.Poora padhein → ko kaam karne se rokti hai. Arithmetic rokta hai.

$10 base par Martingale, 100 spins: limit ke hisaab se mean result (simulated)$
$500 ki limit
≈ −$75
$1,000 ki limit
≈ −$75
$2,000 ki limit
≈ −$72
$5,000 ki limit
≈ −$70
flat $10, koi bhi limit
$27
$
04080
Bars nuksaan ka size dikhaate hain. Limit badhaane se mean mushkil se hilta hai; ceiling chahe jo ho, progression har 100 spins mein lagbhag $2,800 daav par lagata hai.

Do asli tables

Thorp ka example $1 ki table hai jiska maximum $1,000 hai, jahan ek player ek-ek dollar karke dinner ke liye $30 jeetne ki ummeed karta hai. Stakes 1, 2, 4 … 512 das bets ki ijaazat dete hain, isliye lagatar das haar progression ko tod deti hain: European wheel par 785 cycles mein ek baar (American par 613 mein ek baar), $1,023 ki keemat par. Tees cycles 96.2% baar sab safal hote hain. Baaki 3.8% mein lagbhag $1,000 jaata hai, aur shaam ka average result −$9.17 hai, jo theek us $339 ka 1/37 hai jo tees cycles average mein daav par lagaate hain. $2,000 maximum wali $5 ki table bhi alag nahi: $1,280 tak ke stakes nau bets dete hain, aur European wheel par 403 cycles mein ek baar (American par 323 mein ek baar) tootna aata hai aur $2,555 ka padta hai.

Double hote stakes table maximum ki taraf badhte hain, jabki dinner agle kamre mein intezaar kar raha hai.

Ek ameer player

Progression ko kam limit ki jagah gehra bankroll dijiye: 1,000 units, ek unit ka base aur red par 1,000 spins, aur jab bhi agla double bache hue paise se zyada ho toh phir se ek unit se shuru. Humne aisi das lakh shaamein simulate keen. European wheel par 52% munafe mein khatam hoti hain, 21% sab kuch haar jaati hain, aur average result −124 units hai. Bina zerowheelZeroAkela green pocket jise koi even-money, dozen ya column bet cover nahi karta; house edge ka source. Doosre naam: single zero, 0.Poora padhein → wale game mein bhi, jahan har bet fair hai, sirf lagbhag 60% munafe mein khatam hongi. Aam shaam ek chhoti jeet hoti hai; average ki keemat woh shaamein chukaati hain jo zero par khatam hoti hain.

wheelmunafe mein khatam1,000 units haar jaanaaverage result
zero nahi (fair game)60%15%0
European wheel52%21%−124 units
American wheel45%30%−250 units
Hamara simulation, har wheel par das lakh shaamein. Averages us raqam ke 1/37 aur 1/19 hain jo progression 1,000 spins mein daav par lagata hai: European wheel par lagbhag 4,600 units aur American par 4,750.

Seekh

Casinos limits apne varianceganitVarianceYeh naap ki results expected value ke aas-paas kitna faile hue hain. Straight-up ke liye zyada, even-money bets ke liye kam.Poora padhein → ko seemit karne ke liye rakhte hain, systems ko haraane ke liye nahi. Martingale ka haarna ek side effect hai jo unhe pasand hai, aur arithmetic use waise bhi hara deta. Jo player sochta hai ki limit hi problem hai, usne khaai ko dhoondh liya hai aur use dhalaan samajh baitha hai.

Formula

p(tootna) = (19/37)ᵏ, k = limit dwara allowed haar
tootne par nuksaan = base · (2ᵏ − 1)
E[har cycle nuksaan] = E[cycle mein stake] / 37

Sawaal

Kya bada bankroll ise theek kar dega?

Isse aap oonchi limit tak tik paate hain, jahan tootna kam hota hai aur bada. Har dollar par expected loss wahi rehta hai.

Aur “grand Martingale” ka kya, jo har double par ek unit jodta hai?

Stakes 10, 30, 70, 150, 310: paanch haar ke baad $500 ki limit aa jaati hai, aur tootne ki keemat $570 hai. Wahi 2.70%.

Casinos mein maximum hota hi kyun hai?

Taaki koi ek player casino ke reserve ki aaraam se chukaane layak raqam se zyada bet na laga sake. Variance management.

Maths ke saath mazaa

Pehle khud try kijiye. Har solution chhupa hua hai; jab taiyaar hon tab kholiye.

  1. Ek MartingalestrategyMartingaleProgression jo har haar ke baad bet double karti hai. Yeh kya karti hai aur kya nahi, iske liye Strategy section dekhiye.Poora padhein → redbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → par $100 se shuru hota hai aur ek cycle mein zyada se zyada n bets ki ijaazat deta hai: lagatar n haar ke baad woh chhod kar phir shuru karta hai. Ek cycle ka average result n ke formula ke roop mein likhiye, aur European wheel par 5, 6 aur 7 bets ke liye nikaliye.

    Solution dikhaiye

    Jawab Average 100 × (1 − (38/37)ⁿ) hai: 5 bets ke liye −$14.26, 6 ke liye −$17.35 aur 7 ke liye −$20.52. Zyada bets ki ijaazat average cycle ko behtar nahi, bura banati hai.

    Cycle $100 jeetta hai jab tak sabhi n bets na haarein, jo qⁿ chance se hota hai, jahan q = 19/37. Jab sab haarti hain, stakes 100, 200 … 100 × 2ⁿ⁻¹ jud kar 100 × (2ⁿ − 1) hote hain. Isliye average hai 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).

    Yahan 2q = 38/37, 1 se thoda zyada, isliye (2q)ⁿ n ke saath badhta hai aur average girta hai: n = 5 deta hai 1.1426 aur −$14.26; n = 6 deta hai 1.1735 aur −$17.35; n = 7 deta hai 1.2052 aur −$20.52. Lambi seedhi kam baar haarti hai, 3.6%, 1.8% aur 0.9% cycles mein, lekin har haar dugni badi hoti hai, aur har cycle mein table par zyada paisa jaata hai, har bet ke 2.70% par.

    American wheel par 2q = 40/38 aur aankde −$29.24, −$36.04 aur −$43.20 hain (Taylor, 2021). Bina zerowheelZeroAkela green pocket jise koi even-money, dozen ya column bet cover nahi karta; house edge ka source. Doosre naam: single zero, 0.Poora padhein → wale game mein 2q = 1, aur har cycle ka average theek zero hai, seedhi chahe kitni bhi lambi ho.

  2. “Grand Martingale” haar ke baad stake double karta hai aur ek unitpaisaUnitBase stake jismein koi system ya player kaam karta hai. Progressions units mein batayi jaati hain taaki kisi bhi table par laagu hon.Poora padhein → jodta hai, isliye stakes 1, 3, 7, 15, 31 … chalte hain. Jab pehli jeet 1st, 2nd, 3rd ya kth bet par aaye to aap kitne upar hain? $10 unit aur $500 maximum par cycle kitni baar tootta hai, aur ek cycle ki average keemat kya hai?

    Solution dikhaiye

    Jawab kth bet par jeet aapko k units upar chhodti hai: +1, +2, +3 aur aise hi. $10 aur $500 par cycle paanch haar ke baad tootta hai, 3.57% cycles mein, −$570 ke liye. Average mein ek cycle $85.73 lagaye mein se $2.32 haarta hai: 2.70%.

    kth stake 2ᵏ − 1 units hai: 1, 3, 7, 15, 31. Pehle k − 1 stakes jud kar 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1 hote hain. kth bet par jeet 2ᵏ − 1 deti hai, isliye cycle (2ᵏ − 1) − (2ᵏ − k − 1) = k units upar khatam hota hai: jeet se pehle har haar ke liye ek extra unit.

    $10 per unit par stakes $10, $30, $70, $150 aur $310 hain; agla, $630, $500 maximum se upar hai. Lagatar paanch haar, (19/37)⁵ = 3.57%, ki keemat 10 + 30 + 70 + 150 + 310 = $570 hai.

    Average: k = 1 se 5 ke liye jeet (19/37)ᵏ⁻¹ × 18/37 chance se $10 × k laati hai, aur tootna 3.57% chance se $570 le jaata hai; milakar −$2.32. Lagaya gaya paisa average mein $85.73 per cycle hai, aur 2.32/85.73 = 2.70%. Usi table par normal Martingale chhe haar ke baad tootta hai, 1.83% cycles mein, −$630 ke liye, aur $64.20 lagaye mein se $1.74 haarta hai: phir se 2.70%.

  3. Ek khiladi red par $1 pehli bet ke saath Martingale plan karta hai, aisi table par jiski limit $1,000 hai, isliye das bets mumkin hain: $1, $2, $4 aur aise hi $512 tak. Ek cycle kitni baar sabhi das haarta hai, isme average mein kitna samay lagta hai, aur 200 bets ki shaam kitni baar aise khatam hoti hai?

    Solution dikhaiye

    Jawab Ek cycle lagbhag 784 mein ek baar sabhi das haarta hai, jiski keemat $1,023 hai. Yeh average mein lagbhag 1,600 bets hai, aur 200 bets ki shaam lagbhag 11% baar deewar se takraati hai. Cycle phir bhi average mein $11.31 lagaye mein se $0.31 haarta hai: 2.70%.

    Red par bet 19/37 chance se haarti hai, zero samet. Lagatar das: (19/37)¹⁰ ≈ 0.00128, lagbhag 784 mein 1. Stakes 1 + 2 + 4 + … + 512 jud kar $1,023 hote hain.

    Ek cycle average mein 2.05 bets leta hai, isliye 784 cycles lagbhag 1,600 bets lete hain. Spin dar spin ginein to das haar ki series 200 bets mein kahin na kahin 11.3% baar aati hai.

    Aafaton ke beech khiladi har cycle $1 jeetta hai: buri wali se pehle lagbhag $783, phir −$1,023, poore daur mein lagbhag −$240. Per cycle: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Per cycle lagaya paisa: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Aur −0.31/11.31 = −2.70%.

    Tulna kijiye $500 limit par $10 pehli bet se, chhe bets: deewar 55 cycles mein ek baar aati hai, aur 200 bets ki shaam 84% baar usse milti hai. Oonchi limit crash ko kam baar aur bada banati hai, sasta nahi.

Thodi hansi

Har limit par: kam baar, lekin bada. Gunanfal wahi rehta hai.

Har limit par: kam baar, lekin bada. Gunanfal wahi rehta hai.

Sources

  1. Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010. Chapter 13, roulette.
  2. Thorp, E. O. “Optimal gambling systems for favorable games.” Review of the International Statistical Institute 37(3), 1969.
  3. Thorp, E. O. The Mathematics of Gambling. Gambling Times / Lyle Stuart, 1984.
  4. Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed. Academic Press, 2013.
Roulette Betting StrategiesMartingale Roulette StrategyRoulette ka MathRoulette mein Streaks ki ProbabilityRoulette ToolsRoulette Martingale SimulatorWalkthroughsWalkthrough: Ek Martingale Session
2026-09-28 ko review kiyamodel: European, single zero walamethodology ↗sources ↗