Long-Run Convergence: Roulette mein Law of Large Numbers
Average kaise −2.70% ki taraf baith jaata hai, aur isme intuition ke andaaze se zyada samay kyun lagta hai.
Spins badhne ke saath har spin ka average loss −1/37 ki taraf converge hota hai, aur spread 1/√N ki tarah sikudta hai. 1,000 red spins ke baad average aam taur par apni expectation ke ±3% ke andar hota hai.
Law of large numbers (bade numbers ka niyam)
Independent bets ka chalta average expected valueganitExpected valueLambe samay mein har bet ka ausat nateeja. Har standard European bet ke liye stake ka −1/37.Poora padhein → ki taraf jaata hai. 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 → par N even-money spins ke baad, har spin ke average result ki expectation −1/37 hai aur standard deviationganitStandard deviationVariance ka square root; expected value se kisi result ki aam doori.Poora padhein → 1/√N. N = 100 par yeh spread stake ka 10% hai, 2.7% signal se lagbhag chaar guna. N = 1,000 par yeh 3.2% hai, lagbhag signal jitna. N = 10,000 par yeh 1% hai, aur edge saaf dikhta hai. Ek session mein edge dikhta nahi, ek saal mein chhupta nahi.
Kya converge hota hai aur kya nahi
Har spin ka average converge hota hai. Total nahi hota: N badhne ke saath woh zero se aur door khiskta hai, expectation mein N ka lagbhag 2.7%, aur uske aas-paas √N ka spread. Jisne $10 par 10,000 spins khele hain, uska expected nuksaan $2,700 hai, $1,000 kam-zyada. RedsbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → ka chalta anupaat kab ka 48.65% ke paas baith chuka hai; chalta balance kahin nahi baitha.
- p10 −$440
- p25 −$280
- median −$140
- p75 +$0
- p90 +$140
Intuition kyun fail hota hai
Log ummeed karte hain ki edge jaldi dikhega kyunki 2.70% bahut lagta hai, aur ummeed karte hain ki lambe silsile “barabar” ho jaayenge kyunki law of large numbersganitLaw of large numbersTrials ki ginti badhne par independent results ka average expected value ki taraf jaata hai.Poora padhein → kehta hai ki averages converge hote hain. Dono intuitions scale ko galat aankte hain: ek spin ke 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 → ke muqable edge chhota hai (σ = 1 unit, jabki EV = 0.027), isliye one-sigma dikhne mein (1/0.027)² ≈ 1,400 spins lagte hain. Aur average ka converge hona total ko wapas nahi kheenchta; woh bas pichhle utaar-chadhaav ko patla kar deta hai.
Doosri bets ke liye convergence
Straight-upsbetStraight-upEk single number, payout 35:1.Poora padhein → ke liye σ = 5.84, isliye wahi calculation edge ke one sigma tak pahunchne ke liye (5.84/0.027)² ≈ 47,000 spins deta hai. Straight-up player saalon khel sakta hai aur uske apne records mein edge luck se alag nahi dikhega, jabki woh bilkul wahi 2.70% de raha hai jo red player deta hai, jise yeh hafton mein dikh jaata hai.
Systems test karne ke liye matlab
Jo bhi kehta hai ki koi system kaam karta hai, use 10,000 ya usse zyada spins par documented positive average dikhana chahiye. 1,000 spins par even money par positive average null ke tahat paanch mein ek baar hone wali baat hai; 100 spins par yeh lagbhag sikka uchhaalna hai. Is site ke simulators aapki chuni lambai ke 2,000 sessions ek second mein chalaate hain, jo kisi bhi testimonial se zyada saboot hai.
Formula
Sawaal
Kabhi dobara aage hone ki probabilityganitProbabilityKisi outcome ka chance, 0 se 1 tak ya 0% se 100% tak. Ek number: 1/37 = 2.70%.Poora padhein → khelte-khelte zero ki taraf girti hai. Total bina kisi seema ke neeche khiskta hai.
Nahi. Iska matlab hai ki pichhle utaar-chadhaav ek bade sample ka chhota hissa ban jaate hain.
Even money ke liye, edge saaf dikhne se pehle lagbhag 10,000. Straight-ups ke liye, lagbhag 50,000.
Maths ke saath mazaa
Pehle khud try kijiye. Har solution chhupa hua hai; jab taiyaar hon tab kholiye.
Do dost ek fair gameganitFair gameAisa game jismein kisi bhi side ka edge nahi: har bet ki expected value zero hai.Poora padhein → ke 20 rounds khelte hain, coin toss ya zeroswheelZeroAkela green pocket jise koi even-money, dozen ya column bet cover nahi karta; house edge ka source. Doosre naam: single zero, 0.Poora padhein → lautaye jaane ke saath redbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → ke khilaaf blackbetBlack18 black numbers par even-money bet. French mein: noir.Poora padhein →, aur chalta score rakhte hain. Gin-iye ki har ek kitne rounds aage raha (tie us ke hisaab mein jo theek pehle aage tha). Kya zyada sambhav hai: ki badhat 10 aur 10 rounds mein bati ho, ya ek dost sabhi 20 mein aage rahe?
Solution dikhaiye
Jawab Ek dost ka sabhi 20 rounds mein aage rehna kahin zyada sambhav hai: kisi bhi dost ko ginkar 35.2%, jabki barabar 10 aur 10 ke liye 6.1%.
2k rounds ke baad fair chalta score barabar hone ka chance u = C(2k, k)/2²ᵏ hai: 10 rounds ke baad 0.246 aur 20 ke baad 0.176. Ek classic result, arc-sine law, kehta hai ki ek tay dost ke 2n mein se theek 2k rounds aage rehne ka chance u(2k) × u(2n − 2k) hai.
Sabhi 20 mein aage: har dost ke liye u(20) × u(0) = 0.176, kisi bhi ke liye 35.2%. Barabar hissa: u(10) × u(10) = 0.246² = 6.1%, sabse kam sambhav hissa.
Isliye fair game mein badhat kam hi haath badalti hai. Jo khiladi poori shaam aage raha, ya poori shaam peeche, usne kuch anokha nahi dekha, aur isse wheel ya skill ke baare mein kuch pata nahi chalta. Epstein (2013) lambe games ke liye yahi law deta hai: aadhe samay lucky taraf khel ke 85% se zyada hisse mein aage rehti hai.
Thodi hansi
Average converge hota hai. Total bas nikal jaata hai.
Sources
- Bernoulli, J. Ars Conjectandi.
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.