Probability Lab · Module

Randomness Analysis

Live statistical tests on billions of samples

Five classical tests run continuously against the simulator's frequency stream. Together they measure whether the underlying RNG is producing genuinely uniform random output, and at what statistical resolution.

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RNG Health Score

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Composite of all 5 tests below

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Uniformity Score

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1.0 = perfectly uniform; lower = bias detected

Statistical Resolution

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Smallest detectable bias, in σ units

Pearson chi-square test (main)

Goodness-of-fit against uniform distribution over 1–50

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χ² statistic

χ² = Σ (O − E)² / E · degrees of freedom = 49 · critical value at p=0.05: 66.34

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Pearson chi-square test (euro)

Goodness-of-fit against uniform distribution over 1–12

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χ² statistic

degrees of freedom = 11 · critical value at p=0.05: 19.68

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Shannon entropy

Information content of the empirical distribution, in bits

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bits (max log₂(50) ≈ 5.644)

H(X) = −Σ p(x) log₂ p(x)

Maximum entropy = perfect uniformity. Any drop indicates the distribution is non-uniform.

Frequency histogram

All 50 main numbers, sorted by frequency

least drawnmedianmost drawn

Variance of frequencies

Spread of observed counts around the mean

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Var / Var_expected ratio

Expected variance under multinomial: σ² = N · p · (1−p) where N = total draws, p = 1/50

A ratio near 1.0 means the simulator's variance matches the theoretical multinomial — exactly what a uniform RNG produces.

What "passing" means

Each test asks a slightly different question of the same data. Chi-square tests compare observed counts against the hypothesis "all numbers are equally likely." A χ² well below the critical value means we can't reject that hypothesis — the simulator looks uniform. Entropy is the maximum if the distribution is uniform; any drop is a signature of bias. Variance ratios near 1.0 confirm the variance matches what an unbiased process would produce.

The simulator passes all five tests because it uses a cryptographic RNG (CSPRNG) that is, for all practical purposes, indistinguishable from true randomness. The tests run continuously to demonstrate the statistical signature of randomness in real time, on real data, at scale.

Caveat: These tests detect bias of certain kinds. No finite suite of tests can prove an RNG is "truly random" — only that it has not yet failed any of them at the resolution provided by the sample size.