Fair dice: 100 million rolls
We ran the app's real dice generator: 1,000,000 new games of 100 rolls, each with a fresh key from the system's cryptographic generator — exactly as in a game.
Results
- Doubles: 16.6620% (expected 16.6667%; deviation −1.26 σ — ordinary chance).
- Pips per roll: 8.1669 (expected 8.1667).
- Longest run of doubles in a row: 9. Runs of 3, 4 and 5 doubles: 315,515, 51,911 and 8,610 — each longer run about 6 times rarer, as it should be.
Chi-square tests
p is the chance that perfect dice deviate as much or more. Anything above 0.01 means no deviation.
| Test | χ² | d.f. | p |
|---|---|---|---|
| 21 roll types | 31.0 | 20 | 0.055 |
| 36 ordered pairs | 39.4 | 35 | 0.279 |
| first die, 6 faces | 9.7 | 5 | 0.084 |
| second die, 6 faces | 1.2 | 5 | 0.946 |
| opening roll of a game | 41.7 | 35 | 0.202 |
| independence of consecutive rolls (21×21) | 406.2 | 400 | 0.404 |
Conclusion: faces, pairs and roll types are uniform, consecutive rolls are independent, and the opening roll is no different from the others. The generator behaves like perfect dice.