Modulo Bias Detector

What is modulo bias?

When a casino’s server generates a random outcome, it starts with a massive cryptographic integer—typically a 32-bit (4-byte) or 64-bit (8-byte) number. To scale this massive number down to a specific game range, such as a roulette wheel (0 to 36) or a card deck (0 to 51), amateur developers often use the modulo operator (%).

This creates a mathematical bias. If the maximum possible value of the random generator ($2^k$) is not perfectly divisible by the game’s range ($N$), the numbers at the beginning of the range will have a slightly higher probability of appearing than the numbers at the end of the range.

The math: Quantifying the bias

To calculate the exact probability difference between the favored and disfavored outcomes, we use the following 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 a 4-byte float standard).
• $N$: The number of outcomes in the game (e.g., 37 for European Roulette).
• Bias: The relative percentage increase in probability for the favored outcomes.

How professional casinos avoid the bias

Because even a minuscule bias can be exploited by card counters and quantitative gamblers over millions of rounds, professional provably fair casinos do not use simple modulo math. Instead, they implement **Rejection Sampling** (also called cryptographic discarding).

If the generated random integer falls within the uneven “leftover” region at the very top of the $2^k$ range, the algorithm discards that number entirely, moves the cryptographic cursor forward, and draws a new set of bytes. While this slightly increases the processing cost, it guarantees that every single outcome has an identical, mathematically perfect probability of occurring.

Frequently asked questions