Wheel Bias
A biased wheel favors some pockets because of a physical defect: a loose fret, an uneven rotor, worn pockets. Biased wheels exist and have been exploited. The difficulty is that a perfectly fair wheel produces “hot numbers” constantly, so the question is never whether a number is hot, but whether the whole pattern is more lopsided than chance allows.
1. Sample a fair wheel
Spins drawn from a simulated wheel with no bias at all. Change the sample size and watch the hottest pocket.
2. Observed vs expected
Each bar is a pocket in wheel order. The dashed line is the expected count, n/37. The shaded band is ±2 standard deviations: on a fair wheel about one pocket in twenty will poke outside it by chance alone.
3. Chi-square test
The test adds up how far every pocket is from expectation, squared and scaled. On a fair wheel the sum follows a chi-square distribution with 36 degrees of freedom; a p-value under 0.05 is conventionally called significant.
n = 370 · E = 10.00 · χ² = 32.80 · p = 0.622
4. How many spins you would need
Spins required to detect one pocket hitting more often than 1/37, at a one-sided 5% level with 80% power. A pocket hitting 1 time in 35 instead of 1 in 37 would be a strong edge for a player, and it takes tens of thousands of recorded spins to confirm.
Methodology
Sources
- Pearson, K. (1900). On the criterion that a given system of deviations… Philosophical Magazine, 50(302).
- Thorp, E. O. (1969). Optimal gambling systems for favorable games. Review of the International Statistical Institute, 37(3).
- Ethier, S. N. (2010). The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer. Ch. 13, Roulette.