Inna di long run: di law a big numba inna roulette
Ow di average settle dung toward −2.70%, an wah mek dat tek longa dan yuh wudda tink.
Di average loss evry spin move closer to −1/37 as di spin dem pile up, an di spread shrink like 1/√N. Afta 1,000 red spin, di average usually deh inside ±3% a weh it fi be.
Law a big numba
Di runnin average a independent bet move toward di expected valuemathsExpected valueDi average result pon evry bet inna di long run. −1/37 a di stake fi evry standard European bet.Read di whole ting →. Afta N even-money spin at one unitmoney mattaUnitDi base stake weh a system or a playa work wid. Dem describe progression inna unit suh dem work at any table.Read di whole ting →, di average result evry spin have expectation −1/37 an standard deviationmathsStandard deviationDi square root a variance; ow far a result usually deh fram di expected value.Read di whole ting → 1/√N. At N = 100 dat spread a 10% a di stake, nearly four time di 2.7% signal. At N = 1,000 it a 3.2%, bout di same as di signal. At N = 10,000 it a 1%, an yuh can see di edge clear clear. Inna one session yuh cyaan see di edge; inna one year yuh cyaan miss it.
Wah settle an wah nuh settle
Di average evry spin settle. Di total nuh settle: it drif further fram zero as N grow, by bout 2.7% a N inna expectation, wid a spread a √N roun dat. A playa weh play 10,000 spin at $10 fi expec fi dung $2,700, give or tek $1,000. Di runnin share a redbet demRedEven-money bet pon di 18 red numba. Inna French: rouge.Read di whole ting → settle near 48.65% long time; di runnin balance nuh settle nowhere.
- p10 −$440
- p25 −$280
- median −$140
- p75 +$0
- p90 +$140
Wah mek yuh gut feelin wrong
People expec di edge fi show up quick cause 2.70% sound like a whole heap, an dem expec long run fi “even out” cause di law a big numbamathsLaw a big numbaDi average a independent result dem come closer an closer to di expected value as di numba a try dem grow (di more spin, di closer to average).Read di whole ting → seh average settle. Both a dem feelin get di scale wrong: di edge small nex to di variancemathsVarianceA measure a ow wide di result dem spread roun di expected value. High fi straight up, low fi even-money bet.Read di whole ting → a one spin (σ = 1 unit gainst EV = 0.027), suh it tek (1/0.027)² ≈ 1,400 spin fi it reach one sigma weh yuh can see. An wen di average settle, dat nuh pull back di total at all; it jus water dung di ups an dung dem weh pass.
Ow it settle fi odda bet
Fi straight upbet demStraight upOne single numba, pay 35:1.Read di whole ting → σ = 5.84, suh di same calculation give (5.84/0.027)² ≈ 47,000 spin fi di edge reach one sigma. A straight-up playa can play fi years an still cyaan tell di edge fram luck inna im own record, while im a pay exactly di same 2.70% as di red playa weh see it inna a few week.
Wah dis mean fi testin system
Anybaddy weh seh a system work fi able fi show it ova 10,000 spin or more, wid a positive average weh write dung. At 1,000 spin, a positive average pon even money a one-inna-five ting unda di null; at 100 spin it nearly a coin toss. Di simulator dem pon dis site run 2,000 session a di length yuh pick inna one second, an dat a more evidence dan any testimony.
Formula
Question dem
Di probabilitymathsProbabilityDi chance fi a result, fram 0 to 1 or 0% to 100%. One numba: 1/37 = 2.70%.Read di whole ting → fi eva come out ahead again drop toward zero di more yuh play. Di total drif dung an nuh stop.
No. It mean seh di ups an dung dem weh pass tun a smalla piece a a bigga sample.
Fi even money, bout 10,000 bifo di edge show clear. Fi straight up, bout 50,000.
Fun wid maths
Try dem yuhself fus. Evry ansa fold up; open it wen yuh ready.
Two fren play 20 roun a a fair gamemathsFair gameA game weh nuh side nuh have no edge: evry bet have a expected value a zero.Read di whole ting →, a coin toss or redbet demRedEven-money bet pon di 18 red numba. Inna French: rouge.Read di whole ting → gainst blackbet demBlackEven-money bet pon di 18 black numba. Inna French: noir.Read di whole ting → wid di zeropon di wheelZeroDi one green pocket weh no even-money, dozen or column bet nuh cover; a weh di house edge come fram. Nex name: single zero, 0.Read di whole ting → dem give back, an keep a runnin score. Count di roun dem weh each one deh ahead (a tie count fi whoever did lead jus bifo). Which one more likely: dat di lead share 10 roun to 10, or dat one fren lead all 20?
Show di ansa
Ansa One fren weh lead all 20 roun far more likely: 35.2%, if yuh count eida fren, gainst 6.1% fi a even 10 to 10.
Di chance seh a fair runnin score level afta 2k roun a u = C(2k, k)/2²ᵏ: 0.246 afta 10 roun an 0.176 afta 20. A classic result, di arc-sine law, seh di chance dat one fren weh yuh name lead fi exactly 2k outta 2n roun a u(2k) × u(2n − 2k).
Fi lead all 20: u(20) × u(0) = 0.176 fi each fren, 35.2% fi eida one. A even share: u(10) × u(10) = 0.246² = 6.1%, di least likely share a all.
Suh inna a fair game di lead hardly change hand. A playa weh deh ahead all evenin, or behind all evenin, nuh see nutten strange, an it nuh seh nutten bout di wheel or bout skill. Epstein (2013) give di same law fi longa game: half di time di luckier side lead fi more dan 85% a di play.
Sumting fi laugh
Di average settle. Di total jus lef.
Source dem
- Bernoulli, J. Ars Conjectandi.
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.