Roulette Standard Deviation
Roulette standard deviation, yaani 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 → ka square root, kisi result aur uski expectation ke beech ki aam doori hai.
Har $1 stake par σ straight-up ke liye 5.84 aur red ke liye 1.00 hai. Session ke liye stake aur spins ki ginti ke square root se guna kijiye.
Rule of thumb
Lagbhag do-tihaai sessions expectation se ek σ ke andar khatam hote hain aur 95% do ke andar. $10 par 100 redbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → spins (expected −$27, σ = $100) mein zyadatar sessions −$127 aur +$73 ke beech girte hain, aur lagbhag saare −$227 aur +$173 ke beech. Is range ke bahar ka result kisi cheez ka saboot nahi, siwaay iske ki bees mein se ek session aisa karta hai.
√N kyun
Expected lossganitExpected lossExpected value guna kul lagayi gayi raqam; ek session ya ek muddat mein ausat nuksaan.Poora padhein → N ke anupaat mein badhta hai; failaav √N ke anupaat mein. Chhote N par failaav haavi rehta hai aur sessions halke jhukaav wale coin flips hain. Bade N par loss failaav se aage nikal jaata hai. Crossover, jahan expected loss ek standard deviationganitStandard deviationVariance ka square root; expected value se kisi result ki aam doori.Poora padhein → ke barabar hota hai, N = (σ/edge)² = (1/0.027)² ≈ 1,370 spins par hai even money ke liye, aur straight upsbetStraight-upEk single number, payout 35:1.Poora padhein → ke liye (5.84/0.027)² ≈ 46,800. In numbers se neeche kismat faisla karti hai; upar edge.
Ise use karna
Koi session anokha tha ya nahi, yeh jaanne ke liye (result − EV)/σ nikaliye. $10 ka red bettor jo 100 spins +$150 par khatam karta hai, woh (150 + 27)/100 = 1.77σ par hai, 25 mein ek wala outcome: lucky, khaas nahi. Usi +$150 par straight-up bettor 0.3σ par hai: ek aam raat. Ek hi dollar result alag bets par alag matlab rakhta hai.
Proportion ka standard error
Yahi arithmetic batata hai ki data mein edge dekhne ke liye kitne spins chahiye. Observed win proportion ka standard error √(p(1 − p)/N) hai. Red ke liye N = 1,000 par: √(0.4865 × 0.5135 / 1000) ≈ 1.6%. Fair coin aur red ke beech fark 1.35% hai, isliye 1,000 spins ke baad bhi aap bharose se dono ko alag nahi bata sakte. 10,000 ke baad standard error 0.5% hai aur aap bata sakte hain. Isi liye wheel-bias pakadne ke liye itne spins chahiye.
Skew par ek chetavani
Chhote session mein straight-up results normal distribution se bahut door hain: zyadatar sessions mein har spin par lagatar −$10 hota hai aur kabhi-kabhi +$350 ki chhalaang. ±1σ rule failaav sahi batata hai lekin shakal theek se nahi, jab tak N saikdon mein na ho. Even-money betsbetEven-money betRed/black, odd/even, high/low. 18 numbers, payout 1:1.Poora padhein → ke liye normal approximation lagbhag 30 spins se achha hai.
Formula
Sawaal
Yeh saboot hai ki aapne lagbhag 40 sessions khele, ya chaalees mein us ek lucky mein the. Dobara karke dekhiye.
Yeh kam se kam jhoolta hai: average ka standard error σ/√N hai. N = 10,000 tak yeh stake ka 1% hai.
Maths ke saath mazaa
Pehle khud try kijiye. Har solution chhupa hua hai; jab taiyaar hon tab kholiye.
European wheel ke 100 spins mein, redbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → par 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 → kam se kam 51 jeet par aage hai, aur straight-upbetStraight-upEk single number, payout 35:1.Poora padhein → par ek unit aage hai agar uska number kam se kam 3 baar hit kare. Bell curve se dono chances ka andaaza lagaiye, phir unhe exact jawabon se milaiye.
Solution dikhaiye
Jawab Red: bell curve 35.6% deta hai aur exact jawab 35.5% hai. Straight-up: bell curve 55.0% deta hai aur exact jawab 50.9% hai. Shortcut red ke liye kaam karta hai aur single number ke liye chook jaata hai.
Red: jeet ka average 100 × 18/37 = 48.6 hai, spread √(100 × 18/37 × 19/37) = 5.0 ke saath. Aage hone ka matlab 51 ya zyada; beech ka nishaan, 50.5, average se 0.37 spreads upar hai, aur bell curve uske upar 35.6% rakhta hai. 51, 52 … 100 jeet ke exact chances jodne par 35.5% milta hai.
Straight-up: hits ka average 100/37 = 2.70 hai, spread √(100 × 1/37 × 36/37) = 1.62 ke saath. Teen ya zyada hits ka matlab 2.5 se upar, average se 0.12 spreads neeche: bell curve se 55.0%. Exact mein yeh 1 minus 0, 1 aur 2 hits ke chances hai, 6.5% + 17.9% + 24.7%, jo 50.9% chhodta hai.
Bell curve ko average ke dono taraf bahut saare results chahiye. Red ke paas 100 spins ke baad hain; jo number average mein teen se kam baar hit karta hai uske paas nahi. Curve “koi hit nahi” ko bhi 8.7% deta hai, jahan asli chance 6.5% hai. Taylor (2021) binomial aur bell curve ko isi tarah saath-saath rakhte hain.
Thodi hansi
Bees mein se ek session two sigma ke bahar hota hai. Koi na koi hamesha tail par hota hai.
Sources
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed.