Hot and Cold Roulette Numbers: A Chi-Square Test

Some players call the numbers that came up most often in recent spins "hot" and the least frequent ones "cold". Here we check whether that split holds up in our data. Past results do not change the next round.

Over 52 hours we recorded 40,779 spins across 12 single-zero roulette tables. For each table we asked one question: could these counts plausibly come from a wheel where all 37 pockets are equally likely? On every table the answer was yes. The most interesting detail is a single number, 18 on Auto Roulette, which landed far more often than expected, and the reason why that still proves very little.

The setup in one paragraph

A single-zero wheel has 37 pockets: 1 to 36 plus a zero. If it is fair, each pocket has a 1 in 37 chance on every spin, so 4,320 spins should give each number about 116.8 hits. Real counts always scatter around that figure. The chi-square goodness-of-fit test measures how much scatter there is in total by adding up (observed − expected)² / expected across all 37 pockets. With 36 degrees of freedom, a total above roughly 51.0 would be unusual at the 5% level.

Calculation: Source: our rounds table, about 52 hours, 12 single-zero tables, 40,779 spins. The collector ran without interruption over this window (58 connection sessions, longest gap 4 seconds). 76 spins missed by the live feed were filled from the source's history list. 623 of the 624 table-hours (99.8%) have records; the one empty hour falls in a period of about an hour and a half when Fortune Roulette reported no results. Method: chi-square goodness-of-fit per table (36 degrees of freedom), p-values from the incomplete gamma function; per-number z-scores with a Bonferroni correction for 444 comparisons; a binomial expectation for the count of |z| ≥ 2; the sum of the 12 chi-square values (432 degrees of freedom); 2,000 simulated fair-wheel runs of 4,320 spins with a fixed seed; the same simulation for all 12 tables at their own spin counts (2,000 runs each). Code: docs/makaleler/hesap/rulet-sayi-dagilimi-ki-kare.py.

What the 12 tables showed

The chi-square statistic ranged from 26.2 (Fireball Roulette) to 48.1 (Monopoly Roulette). None crossed the 5% threshold. The lowest p-value was 0.086. Added together, the 12 chi-square values come to 475.4 against an expected 432 (12 × 36), with a standard deviation of about 29.4: a little high, but within the range chance produces (about 1.5 standard deviations).

If you ran this test on 12 perfectly fair wheels, you would expect a spread of p-values much like this one, including a couple below 0.1. We tested each table on its own because each is a separate physical wheel; pooling them would mix different wheels together. On every one of these tables we saw all 37 results, 0 to 36.

The number that stood out

On Auto Roulette, number 18 came up 155 times against an expected 116.8. That is about 3.6 standard deviations above the mean, which on its own would happen roughly once in 3,000 tries. At the other end, 34 came up only 87 times.

Here is why we do not read much into it:

• We looked in many places. 12 tables × 37 numbers = 444 separate checks. The more places you look, the more likely the most extreme one is extreme. With a Bonferroni correction, the chance of seeing at least one deviation this large somewhere is about 15%.
• The overall count is at the upper edge of ordinary. Across the 444 pairs, 28 had a z-score beyond ±2. Pure chance would give about 20; 28 is a little high (about 1.8 standard deviations). That is not enough to point at any one wheel.
• The spread is rare for one table, not for twelve. In our 2,000 simulated fair wheels, the gap between the most and least frequent number in 4,320 spins was usually between 35 and 59. Auto Roulette's gap of 68 appeared in only 0.5% of simulations. But when we simulated all 12 tables at their own spin counts (2,000 runs each), at least one table showed a gap this extreme about 5% of the time: at the edge of unusual, but not proof of a fault on its own.

So the honest summary is: Auto Roulette's table passes the overall test, while one number is an outlier worth re-checking over a longer period. Fifty-two hours is not enough to call a wheel biased, and we do not.

How to read number counts on a stats page

When you look at a table's history, some number will always lead and another will always trail. That is how randomness looks over a few thousand spins. A leading number is not on a streak and a trailing number is not catching up; the next spin is still 1 in 37 for each. We tested that idea directly in “Can the Last 50 Results Pick the Next Spin? A Backtest”, and looked at colour runs in “Red or Black Streaks in Roulette: How Long Do They Run?”.

If you are curious how betting systems try to exploit patterns like these, “Does the Martingale System Work?” walks through the arithmetic.

Limits of this data

Fifty-two hours gives about 2,000 to 4,400 spins per table. That is enough to catch a badly broken wheel but not a subtle one. We store results exactly as the source reports them; a stray missed spin is possible, but the collector did not stop during this window. Current counts for every table, including Lightning Roulette, are on the games page.

Play roulette for fun, and set your budget before you start.