Standard deviation inna roulette
Roulette standard deviation, di square root a 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 di normal distance between a result an wah yuh expec.
Fi evry $1 yuh stake, σ a 5.84 fi a straight up an 1.00 fi red. Fi a session, multiply by di stake an by di square root a di count a spin.
Rule a tumb
Bout two third a session land inside one σ a wah yuh expec, an 95% inside two. Fi 100 redbet demRedEven-money bet pon di 18 red numba. Inna French: rouge.Read di whole ting → spin at $10 (expected −$27, σ = $100) most session fall between −$127 an +$73, an almost all a dem between −$227 an +$173. A result outside dat range nuh prove nutten, only seh one session outta twenty do it.
Wah mek √N
Expected lossmathsExpected lossExpected value times all di money yuh bet; di average amount yuh lose ova a session or a period (di loss yuh fi expec pon average).Read di whole ting → grow in step wid N; di spread grow in step wid √N. Fi small N di spread run tings an session a like coin flip wid a likkle lean. Fi big N di loss run weh fram di spread. Di crossover, weh expected loss equal one standard deviationmathsStandard deviationDi square root a variance; ow far a result usually deh fram di expected value.Read di whole ting →, deh at N = (σ/edge)² = (1/0.027)² ≈ 1,370 spin fi even money an (5.84/0.027)² ≈ 46,800 fi straight upbet demStraight upOne single numba, pay 35:1.Read di whole ting →. Unda dem numba deh, luck decide; ova dem, a di edge decide.
Ow fi use it
Fi judge if a session did unusual, work out (result − EV)/σ. A $10 red betta weh finish 100 spin at +$150 deh at (150 + 27)/100 = 1.77σ, a one-outta-25 result: lucky, but nutten special. A straight-up betta at di same +$150 deh at 0.3σ: a normal night. Di same dollar result mean different ting pon different bet.
Standard error a a proportion
Di same arithmetic tell yuh ow much spin it tek fi see di edge inna di data. Di standard error a a win proportion weh yuh observe a √(p(1 − p)/N). Fi red wid N = 1,000: √(0.4865 × 0.5135 / 1000) ≈ 1.6%. Di gap between a fair coin an red a 1.35%, suh afta 1,000 spin yuh still cyaan tell dem apart fi sure. Afta 10,000 di standard error a 0.5% an yuh can. A dat too mek yuh need suh much spin fi spot a bias wheel.
Watch out fi skew
Straight-up result far fram normal distribution inna a short session: most session a steady −$10 evry spin wid a +$350 jump now an den. Di ±1σ rule describe di spread right, but di shape bad, till N reach di hundreds. Fi even-money betbet demEven-money betRed/black, odd/even, high/low. 18 numba, pay 1:1.Read di whole ting →, di normal approximation good fram bout 30 spin.
Formula
Question dem
It prove seh yuh play bout 40 session, or yuh did inna di lucky one outta forty. Do it again.
It swing less an less: di standard error a di average a σ/√N. By N = 10,000 it a 1% a di stake.
Fun wid maths
Try dem yuhself fus. Evry ansa fold up; open it wen yuh ready.
Ova 100 spin pon a European wheel, 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 → pon redbet demRedEven-money bet pon di 18 red numba. Inna French: rouge.Read di whole ting → deh ahead wid at least 51 win, an one unit pon a straight upbet demStraight upOne single numba, pay 35:1.Read di whole ting → deh ahead if it numba hit at least 3 time. Use di bell curve fi estimate both chance, den compare dem wid di exact ansa dem.
Show di ansa
Ansa Red: di bell curve give 35.6% an di exact ansa a 35.5%. Di straight up: di bell curve give 55.0% an di exact ansa a 50.9%. Di shortcut work fi red an miss fi di single numba.
Red: win average 100 × 18/37 = 48.6, wid a spread a √(100 × 18/37 × 19/37) = 5.0. Ahead mean 51 or more; di halfway mark, 50.5, deh 0.37 spread ova di average, an di bell curve put 35.6% ova dat. If yuh add di exact chance a 51, 52 … 100 win it give 35.5%.
Straight up: hit average 100/37 = 2.70, wid a spread a √(100 × 1/37 × 36/37) = 1.62. Tree or more hit mean ova 2.5, 0.12 spread unda di average: 55.0% by di bell curve. Exactly, it a 1 minus di chance a 0, 1 an 2 hit, 6.5% + 17.9% + 24.7%, an dat lef 50.9%.
Di bell curve need nuff result pon both side a di average. Red have dem afta 100 spin; a numba weh hit less dan tree time pon average nuh have dem. Di curve even give 8.7% to “no hit”, weh di true chance a 6.5%. Taylor (2021) set di binomial an di bell curve side by side di same way.
Sumting fi laugh
One session outta twenty deh outside two sigma. Smaddy always deh pon di tail.
Source dem
- 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.