Wetin Happen for Long Run: Law of Large Numbers for Roulette
How di average dey settle towards −2.70%, and why e dey take longer pass wetin pipo dey think.
Di average loss for every spin dey come close to −1/37 as spins dey add up, and di spread dey shrink like 1/√N. Afta 1,000 red spins, di average dey usually inside ±3% of im expectation.
Law of Large Numbers
Di running average of independent bets dey go towards di expected valuemathsExpected valueDi average result for each bet for di long run. −1/37 of di stake for every standard European bet.Read di whole tin →. Afta N even-money spins with one unitmoniUnitDi base stake wey system or player dey work with. Dem dey describe progressions in units so dat dem go fit work for any table.Read di whole tin →, di average result for every spin get expectation of −1/37 and standard deviationmathsStandard deviationSquare root of variance; di normal distance between one result and di expected value.Read di whole tin → of 1/√N. For N = 100, dat spread na 10% of di stake, almost four times di 2.7% signal. For N = 1,000 e na 3.2%, about di same as di signal. For N = 10,000 e na 1%, and you fit see di edge clear. You no fit see di edge for one session, but for one year e no fit hide.
Wetin Dey Converge and Wetin No Dey
Di average for every spin dey converge. Di total no dey: e dey move far from zero as N dey grow, by about 2.7% of N for expectation, with spread of √N around dat one. Player wey don play 10,000 spins with $10 dey expected to dey $2,700 down, plus or minus $1,000. Di running proportion of redsbetRedEven-money bet on di 18 red numbers. For French: rouge.Read di whole tin → don settle near 48.65% long time; di running balance never settle anywhere.
- p10 −$440
- p25 −$280
- median −$140
- p75 +$0
- p90 +$140
Why Wetin Pipo Feel No Dey Correct
Pipo dey expect di edge to show quick becos 2.70% sound like plenty, and dem dey expect long runs to “balance” becos di law of large numbersmathsLaw of large numbersDi average of results wey no depend on each oda dey move near di expected value as di number of tries dey grow.Read di whole tin → tok sey averages dey converge. Both feelings dey misjudge di scale: di edge small compared to di variancemathsVarianceOne measure of how wide di results dey spread around di expected value. E high for straight up, e low for even-money bets.Read di whole tin → of one single spin (σ = 1 unit against EV = 0.027), so e dey take (1/0.027)² ≈ 1,400 spins before you fit see am for one sigma. And di convergence of di average no dey do anytin to pull di total back; e only dey dilute past fluctuations.
Convergence for Oda Bets
For straight upsbetStraight upOne single number, e dey pay 35:1.Read di whole tin → σ = 5.84, so di same calculation dey give (5.84/0.027)² ≈ 47,000 spins before di edge go reach one sigma. Straight-up player fit play for years and di edge no go different from luck for im own records, while e dey pay exactly di same 2.70% as di red player wey go see am inside weeks.
Wetin E Mean for Testing Systems
Anybody wey tok sey one system dey work suppose fit show am over 10,000 spins or more, with positive average wey dem document. For 1,000 spins, positive average for even money na event wey dey happen one time out of five under di null; for 100 spins e almost be coin flip. Di simulators for dis site dey run 2,000 sessions of di length wey you choose for one second, and dat one na more evidence pass any testimony.
Formula
Questions wey pipo dey ask
Di probabilitymathsProbabilityDi chance of one outcome, from 0 to 1 or from 0% to 100%. One number: 1/37 = 2.70%.Read di whole tin → sey you go ever dey ahead again dey fall towards zero as you dey play. Di total dey go down and e no get bottom.
No. E mean sey past fluctuations dey become smaller part of bigger sample.
For even money, about 10,000 before you fit see di edge clear. For straight ups, about 50,000.
Play with Maths
Try dem by yourself first. Each solution dey folded; open am when you ready.
Two friends play 20 rounds of fair gamemathsFair gameGame where no side get edge: every bet get expected value of zero.Read di whole tin →, coin toss or redbetRedEven-money bet on di 18 red numbers. For French: rouge.Read di whole tin → against blackbetBlackEven-money bet for di 18 black numbers. For French: noir.Read di whole tin → wey dem dey return di zeroswheelZeroDi one green pocket wey no even-money, dozen or column bet dey cover; na im be di source of di house edge. Dem also dey call am: single zero, 0.Read di whole tin →, and dem dey keep running score. Count di rounds wey each one dey ahead (tie dey count for di one wey dey lead just before). Which one likely pass: sey dem share di lead 10 rounds to 10, or sey one friend lead all 20?
Show di solution
Answer One friend wey lead all 20 rounds likely pass well well: 35.2%, if you count any of di two friends, against 6.1% for even 10 to 10.
Di chance sey fair running score go dey level afta 2k rounds na u = C(2k, k)/2²ᵏ: 0.246 afta 10 rounds and 0.176 afta 20. One classic result, di arc-sine law, tok sey di chance sey one friend wey you name go lead for exactly 2k out of 2n rounds na u(2k) × u(2n − 2k).
To lead all 20: u(20) × u(0) = 0.176 for each friend, 35.2% for any of di two. Even share: u(10) × u(10) = 0.246² = 6.1%, di share wey no likely pass all.
So for fair game, di lead no dey change hand often. Player wey don dey ahead di whole evening, or behind di whole evening, never see anytin strange, and e no dey tell anytin about di wheel or about skill. Epstein (2013) give di same law for longer games: half of di time, di side wey get more luck dey lead for pass 85% of di play.
Small laugh
Di average dey converge. Di total just dey waka comot.
Sources
- Bernoulli, J. Ars Conjectandi.
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.