Di Chi-Square Test
Di standard test for “dis 37 counts na wetin fair wheel go produce?”
χ² = Σ (observed − expected)² / expected over all di 37 pockets. Compare am with di chi-square distribution on 36 degrees of freedom to get p-value.
How to Calculate Am
Di count wey you expect for each pocketequipmentPocketOne of di 37 small boxes wey get number for di rotor. Dem also dey call am: slot, cup.Read di whole tin → na n/37. For each pocket, square di difference between wetin you see and wetin you expect, and divide am by wetin you expect; add di 37 terms togeda. Fair wheel dey give values around 36 for average (di degrees of freedomlabDegrees of freedomDi number of values wey no depend on each oda inside one statistic. For chi-square test on 37 pockets na 36, becos di counts must add up to di total.Read di whole tin →), with standard deviationmathsStandard deviationSquare root of variance; di normal distance between one result and di expected value.Read di whole tin → of about 8.5. Above 51.0 na significant for di 5% level; above 58.6 for 1%. Di Analyser for Roulette Wheel Bias dey calculate di statistic and di exact p-valuelabp-valueDi probability to see deviation wey big reach di one wey you see or pass am, if di wheel dey fair.Read di whole tin → from di regularised incomplete gamma function.
One Example Wey We Work
1,000 spins. Thirty-six pockets land between 20 and 34 hits; pocket 8 get 41. Wetin you expect na 27.03. Di pocket-8 term alone na (41 − 27.03)²/27.03 = 7.2. Suppose di oda 36 terms add up to 38 (normal). χ² = 45.2 on 36 df, p ≈ 0.14. E no significant: fair wheel dey do dis one time for seven. Pocket 8 na candidate, no be discovery, and di answer na anoda thousand spins.
Wetin Di Test Assume
Independent spinsmathsIndependent spinsEach spin get di same probabilities no matter wetin happen before. Na di base for every calculation for dis site.Read di whole tin →, and expected counts of at least 5 for each pocket, wey mean n ≥ 185 for di test to even dey valid, and plenty more for am to get use. To mix spins from different wheels dey break di first assumption for spirit (you dey test mixture, no be wheel). Dealer-signature effect go break am for real, and e go push χ² up without any pocket wey dey physically favoured.
Dealer Signature for RouletteRoulette Spins Wey No Dey Depend on One Anoda
Pearson and Monte Carlo
Karl Pearson publish di test for 1900, and one of im first examples na 26,306 Monte Carlo roulette spins wey dem record, wey e find sey dem no uniform at all to be chance, and e declare sey “Monte Carlo roulette… no be game of chance”. Later writers conclude sey di journalists wey record di spins for one newspaper invent many of dem, and dat one na lesson on im own about where data dey come from.
Pearson publish di chi-square goodness-of-fit test, with Monte Carlo spins as example.
Pipo begin question di spin data; e be like sey di recorders invent runs.
Fisher formalise degrees of freedom; di 5% convention take hold.
Statisticians dey use roulette spins as textbook data for uniformity tests.
Di García-Pelayo family apply di method with real data wey dem record, and dem win for Madrid.
Casinos dey run di same test on dem own wheels as normal maintenance.
One Real Test: Automatic Wheels for Las Vegas
Players dey often suspect sey automatic wheel dey steer di ballequipmentBallDi small ball wey dem dey throw for di track against di rotor. Modern balls na ivorine or Teflon, 18–21 mm.Read di whole tin → comot from big bets. One analysis for 2017 test di suspicion on di double-zerowheelZeroDi 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 → wheels of Interblock wey no get dealer for Las Vegas, by betting on arc of five numbers wey dey near each oda, di kind bet wey wheel clocker dey make. For 299 spins, di arc win 29 times against 39.3 wey dem expect, shortfall wey fair wheel dey produce about 4% of di time. Di chi-square over all 38 pockets na 45.29 on 37 degrees of freedom: p = 16.5%. Second sample of 1,204 spins on three wheels give χ² = 34.79, p = 57%, and no arc of any size wey dem try dey significant. Both samples togeda, 1,503 spins, give χ² = 41.75, p = 27%. Di verdict: no evidence of steering. Di first sample alone look suspicious; na di second one settle am.
Beyond Chi-Square
For one single pocket wey you suspect, binomial test get more power. For arc wey you suspect, di sectorwheelSectorPockets wey follow each oda for one arc of di wheel, no be group for di layout.Read di whole tin → z-test for di sector analysis page. Chi-square na di omnibus test: e dey tell you sey sometin no correct without saying wetin. Use am first, den look.
Formula
Questions wey pipo dey ask
Di 37 counts must add up to n, so na only 36 fit change freely.
Wheel wey get one pocket for 1/25 for 10,000 spins go give pocket term of around 60 alone; χ² around 100, p almost zero.
For technical side, yes, with 5.4 expected for each pocket. E no go catch anytin unless na wheel wey dem gum one pocket shut.
Small laugh
Add di squared surprises. Compare am with 51.0.
Sources
- Pearson, K. “On the criterion that a given system of deviations from the probable…”
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Wizard of Odds. “Organic Roulette Experiment.”