1. Introduction & Philosophy (v3.1)

The Lucky Picks Fairness Score is an automated statistical auditing system designed to evaluate the randomness of lottery draw outcomes. It serves as an Exploratory Data Analysis (EDA) tool rather than a forensic verdict.

1.1 Risk Surface vs. Hypothesis Test

While we utilize standard statistical tools (like Chi-Square tests and p-values), the Fairness Score is not a binary hypothesis test. In academic statistics, a p-value < 0.05 strictly means “Reject the Null Hypothesis.”

However, in the real world of physical systems, “perfect” randomness is rare. Minor mechanical imperfections can create statistically significant but practically irrelevant deviations.

Therefore, the Fairness Score calculates a Risk Surface. It answers the question: “To what extent does this lottery exhibit systemic bias that a player could exploit (or be harmed by)?”

A low score is an Early Warning of systemic issues, not necessarily proof of fraud.

2. Statistical Framework

We employ a multi-dimensional analysis using Scan Statistics, Permutation Tests, and Dynamic Effect Size scaling.

2.1 The Scan Statistic (Detection)

Instead of just testing the “Overall” history, we use a Scan Statistic approach. We analyze multiple overlapping time windows (e.g., Last 100, Last 500, Overall) to detect localized anomalies that might be washed out in the long run.

For every dimension, we calculate the Chi-Square statistic:

Where:

• O_i = Observed frequency
• E_i = Expected frequency (under uniform distribution)
• k = Number of categories

2.2 Permutation Tests (Validation)

To distinguish “lucky streaks” from “structural bias,” we run Monte Carlo Permutation Tests. We generate hundreds of synthetic, perfectly random lotteries and run the same Scan Statistic on them.

If the observed anomaly in the real lottery is more extreme than 95% of the synthetic lotteries, it is flagged as statistically significant.

2.3 Dynamic Effect Size (Cramer’s V)

Statistical significance (p < 0.05) is not enough. In large datasets (like Powerball), trivial deviations can look significant. We calculate Cramer’s V to measure the magnitude of the effect.

New in v3.1: The threshold for Cramer’s V now scales dynamically with the pool size (Degrees of Freedom). This allows us to be strictly sensitive to small biases in large games (69+ balls) while remaining lenient with noise in small games (Pick 3).

2.4 The Scoring Algorithm

We translate statistical outputs into a scalar Fairness Score (S ∈ [0, 100]) using a continuous, non-linear penalty function.

Where:

• P_sig is the Significance Penalty (Confidence).
• W_effect is the Effect Weight (Magnitude).

3. Component Isolation & Integrity Checks

A unique feature of our methodology is the Component-Based Integrity Check.

3.1 The Collision Test (The “Tipton” Defense)

Before running complex statistics, we check for Duplicate Draws. In large lotteries, repeating an exact combination is mathematically virtually impossible. If we detect collisions beyond a tiny noise threshold, we trigger a Critical Failure immediately, as this indicates a broken or tampered RNG (similar to the Eddie Tipton fraud).

3.2 The “Weakest Link” Logic

Standard averaging can mask a compromised system. If a lottery has perfectly random Main numbers (Score: 100) but a heavily biased Bonus drum (Score: 10), a simple average might yield a “Passing” score of 55-80.

We reject this approach. Our algorithm enforces a strict cap:

If any critical component (Frequency, Temporal, Pattern, or Bonus) fails (Score < 50), the entire lottery is flagged as compromised.

4. Analysis Dimensions

The final Fairness Score is a weighted composite of three distinct analyses:

Dimension	Weight	Description
Frequency	50%	Tests if every number is drawn with equal probability using Scan Statistics.
Temporal	25%	Tests for Seasonality (Day/Month bias) and Dependence (Hot/Cold persistence) using Permutation Tests.
Pattern	25%	Tests for anomalies in combinatorial patterns (Even/Odd, High/Low, Sums).

5. Limitations

• Post-Hoc Analysis: This is a retrospective audit. Past randomness does not guarantee future performance.
• Sample Size: New lotteries with fewer than 100 draws may yield inconclusive results (“Insufficient Data”).
• Physical vs. Digital: This methodology is optimized for physical ball machines. While applicable to RNG (Digital) draws, it cannot inspect the underlying code.

6. Validation

We don’t just assume our math works—we prove it. We stress-tested the v3.1 algorithm against thousands of “rigged” simulations to ensure it correctly identifies bias without triggering false alarms.

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This methodology is subject to continuous peer review and improvement. Last updated: December 2025 (v3.1).