Modulo Bias Detector

What is modulo bias?

A provably fair server usually begins with a very large cryptographic integer — commonly 32-bit (4 bytes) or 64-bit (8 bytes). Converting that giant number into something usable, like a roulette pocket (0 to 36) or a card slot (0 to 51), requires scaling it down. Inexperienced developers reach for the modulo operator (%) and call it done.

Done is not correct. Unless the generator’s ceiling ($2^k$) divides evenly by the game’s range ($N$), the values at the start of the range appear more often than those at the end. That residual skew is modulo bias.

The math: Quantifying the bias

The gap between favored and disfavored outcomes comes down to one equation:

Bias = (Floor(2^k / N) + 1) / Floor(2^k / N) - 1

Where:

• $2^k$: The size of the cryptographic input pool (e.g., $2^{32} = 4,294,967,296$ for the standard 4-byte case).
• $N$: The count of game outcomes (e.g., 37 for European Roulette).
• Bias: The relative percentage edge enjoyed by the favored outcomes.

How professional casinos avoid the bias

Quantitative players will grind even a tiny bias into profit across millions of rounds, which is why serious provably fair operators skip plain modulo math. They use **Rejection Sampling** instead — sometimes called cryptographic discarding.

Under this method, any random integer landing in the uneven “leftover” strip at the top of the $2^k$ range gets thrown away entirely: the algorithm advances its cryptographic cursor and draws fresh bytes. The extra processing cost is trivial next to the payoff — every outcome ends up with an identical probability.

Frequently asked questions