Roulette ka Math · Core concept · 8 min

Variance

Same expected lossganitExpected lossExpected value guna kul lagayi gayi raqam; ek session ya ek muddat mein ausat nuksaan.Poora padhein → wale straight-up aur red bet alag-alag games jaise kyun lagte hain.

Chhota jawab

Variance naapta hai ki results expected value ke aas-paas kitna faile hain. Har unit stake par straight-up ke liye yeh lagbhag 34 hai aur red ke liye lagbhag 1.

straight-up
σ² ≈ 34.1
red
σ² ≈ 1.00
anupaat
34×
N ke saath scaling
linear

Ek bet ka formula

Jo bet probabilityganitProbabilityKisi outcome ka chance, 0 se 1 tak ya 0% se 100% tak. Ek number: 1/37 = 2.70%.Poora padhein → p ke saath W to 1 pay karta hai, uska har 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 → par result probability p ke saath +W aur probability 1 − p ke saath −1 hai. Expected valueganitExpected valueLambe samay mein har bet ka ausat nateeja. Har standard European bet ke liye stake ka −1/37.Poora padhein → pW − (1 − p) = −1/37 hai. 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 → E[X²] − EV² = pW² + (1 − p) − (1/37)² hai. Straight-upbetStraight-upEk single number, payout 35:1.Poora padhein → ke liye: (1/37)(1225) + 36/37 − 0.0007 ≈ 34.08. RedbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → ke liye: 18/37 + 19/37 − 0.0007 ≈ 0.999.

betpWhar $1 par variancehar $1 par σ
Straight-up1/373534.085.84
Split2/371716.704.09
Street3/371110.753.28
Corner4/3788.052.84
Six line6/3754.972.23
Dozen12/3721.941.39
Even money18/3710.9991.00
Har row par same EV. Coverage badhne par variance ghatta hai.

Session ke liye iska matlab

Independent spinsganitIndependent spinsHistory kuch bhi ho, har spin ki probabilities same rehti hain. Is site ki har calculation ki buniyaad yahi hai.Poora padhein → par variances jud jaate hain, isliye fixed stake par N spins mein total ka variance N × stake² × σ² hai, aur standard deviationganitStandard deviationVariance ka square root; expected value se kisi result ki aam doori.Poora padhein → σ × stake × √N. $10 par 100 spins: straight ups ke liye standard deviation lagbhag $584 aur red ke liye $100 hai, dono ke aas-paas wahi expected −$27. Ek player ke results saikdon mein jhoolte hain; doosra bas halka sa khiskaav dekhta hai. Dono same keemat chuka rahe hain.

100-spin, $10 session ka standard deviation$
Straight-up
$584
Split
$409
Corner
$284
Six line
$223
Dozen
$139
Even money
$100
$
0292$584
Har case mein expected result: −$27.

Independent Roulette Spins

Variance aur systems

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 → har lagaye unit par EV nahi badalti, lekin lagayi gayi raqam aur distribution ki shakal dono badalti hai. MartingalestrategyMartingaleProgression jo har haar ke baad bet double karti hai. Yeh kya karti hai aur kya nahi, iske liye Strategy section dekhiye.Poora padhein → zyadatar sessions ka variance kam karti hai (bahut saari chhoti jeet) aur ek moti tail jodti hai (kabhi-kabhi badi haar); us tail ki wajah se saare sessions ka kul variance flat bettingstrategyFlat bettingHar spin par same raqam lagana. Har progression ko isi baseline se compare kiya jaata hai.Poora padhein → se zyada hai. ParolistrategyParoliPositive progression: jeet ke baad double, haar ke baad ya lagatar teen jeet ke baad reset. Doosra naam: reverse Martingale.Poora padhein → ulta karti hai. System chunna variance profile chunna hai; mean menu mein hai hi nahi.

Das straight-up sessions, $10 flat: variance saamne dikhta huaresult vs shuruaat
+$3,360$0−$500+$1,4400125250375500spinsresult
1,000 sessions × 500 spins · violet p10/p25/p75/p90, light blue median, 83 halke samples · kisi spin ke quantiles dekhne ke liye chart par cursor le jaiye
  • p10 −$500
  • p25 −$500
  • median −$500
  • p75 +$40
  • p90 +$1,120
1,000 sessions × 500 spins · y-axis 1st–99th percentile tak kata hua (extremes −$500 se +$5,120) · 500 spins · $10 base · straight-up

Flat Betting Roulette StrategyParoli Roulette Strategy (Reverse Martingale)

Player ke liye yeh kyun matter karta hai

Variance hi chhote sessions ko unpredictable aur lambe sessions ko predictable banata hai. 100 spins mein σ = $100 aur EV = −$27 ke saath, edge noise ka ek-chauthai hai aur jeetne wala session aam baat hai. 10,000 spins mein EV −$2,700 hai aur σ $1,000; ab edge noise ka lagbhag teen guna hai aur jeet ka silsila three-sigma event hai. Chhote samay mein variance player ka dost hai aur lambe samay mein house ka.

Combination ka variance

Ek hi spin par kai bets ke outcomes aapas mein jude hote hain (17 par hit blackbetBlack18 black numbers par even-money bet. French mein: noir.Poora padhein → aur doosre dozenbetDozen1–12, 13–24 ya 25–36, payout 2:1.Poora padhein → par bhi hit hai), isliye variances seedhe nahi judte. Bet Builder tool kisi bhi chipspaisaChipEk fixed raqam ka token, jise bet lagane ke liye layout par rakha jaata hai. Doosra naam: check.Poora padhein → set ka exact distribution saare 37 outcomes gin kar nikalta hai, aur yahi ise karne ka akela sahi tareeka hai.

Bet Builder

Formula

Var = p·W² + (1 − p) − EV²
σ = √Var
σ_session = σ · stake · √N

Sawaal

Kya zyada variance ka matlab bure odds hai?

Nahi. Har bet ki expected value same hai. Zyada variance ka matlab uske aas-paas bade jhoole.

Kis bet ka variance sabse kam hai?

Even-money betsbetEven-money betRed/black, odd/even, high/low. 18 numbers, payout 1:1.Poora padhein → ka, har unit par σ ≈ 1. La partageniyamLa partageFrench rule: zero aane par even-money stake ka aadha wapas milta hai. In bets par house edge aadha hokar 1.35% ho jaata hai.Poora padhein → ke saath aur bhi kam, jo mean bhi sudhaarta hai.

Casinos high-variance bets kyun allow karte hain?

Har dollar par expected profit same hai; variance sirf yeh badalta hai ki casino ki kamaai kitni oonchi-neechi hai, aur use smooth karne ke liye casino ke paas hazaron players hain.

Maths ke saath mazaa

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

  1. Do khiladi European wheel par 100 spins tak har spin $1 lagate hain, ek hamesha redbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → par, doosra hamesha 17 par. Aakhir mein kiske aage hone ki sambhavna zyada hai?

    Solution dikhaiye

    Jawab 17 wala khiladi: 50.9%, jabki red ke liye 35.5%. Dono average mein $2.70 haarte hain.

    Red: khiladi 100 mein se 51 ya zyada jeet par aage hai, har ek probabilityganitProbabilityKisi outcome ka chance, 0 se 1 tak ya 0% se 100% tak. Ek number: 1/37 = 2.70%.Poora padhein → 18/37 ke saath. Binomial distribution 35.5% deta hai, aur theek 50 jeet, yaani break even, aur 7.7%.

    Number 17: har jeet $36 lautati hai, isliye do jeet (−$28) kaafi nahi aur teen (+$8) kaafi hain. 1/37 par 100 spins mein teen ya zyada hits ka chance 50.9% hai.

    Straight-up khiladi zyada baar aage rehta hai, lekin results zyada ubad-khabad hain: ek hit kam ya zyada result ko $36 se hila deta hai, aur jo khiladi peeche hain woh aam taur par zyada peeche hote hain. Bollman (2024) American aankde deta hai, 26.5% aur 49.2%.

Thodi hansi

Same expected loss. Inmein se ek aapko zor se batata hai.

Same expected loss. Inmein se ek aapko zor se batata hai.

Sources

  1. Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010. Chapter 13, roulette.
  2. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1. Wiley, 3rd ed., 1968. Gambler’s ruin, chapter XIV.
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2026-09-28 ko review kiyamodel: European, single zero walamethodology ↗sources ↗