Advanced Mathematical Analysis of Random Number Distributions - Educational Tool
Benford's Law states that in many naturally occurring datasets, the first digit is more likely to be small. Specifically, the digit "1" appears as the first digit about 30% of the time.
We compare the actual first-digit distribution in our dataset against this theoretical distribution. Large deviations might indicate non-random patterns.
Chi-Square Test: We use the χ² statistic to quantify the deviation:
Shannon Entropy measures the randomness or unpredictability in a dataset. Higher entropy indicates more randomness.
Where p(x) is the probability of observing value x, and N is the number of unique values.
Interpretation:
A Markov Chain models the probability of transitioning from one state to another. We analyze which numbers tend to appear after specific numbers.
We calculate the conditional probability that number j appears given that number i appeared in the previous draw.
Heat Map: The chart shows transition probabilities where darker colors indicate stronger associations.
Monte Carlo simulation uses repeated random sampling to estimate probabilities and outcomes. We run thousands of simulated draws to understand match distributions.
For each simulation iteration, we:
Results show: The probability distribution of getting 0, 1, 2, 3, 4, 5 matches, plus bonus ball.
The Fourier Transform decomposes a time series into its constituent frequencies, revealing periodic patterns that might not be obvious.
We apply DFT to the sum of numbers in each draw over time. High-magnitude frequency components indicate periodic patterns.
Interpretation:
In a truly random lottery, all numbers should appear with approximately equal frequency over a large sample size.
Where O_i is the observed frequency of number i, and E is the expected frequency.
Null Hypothesis: The numbers are uniformly distributed (random).
If χ² is very large and p-value < 0.05, we reject the null hypothesis, suggesting non-random patterns.
"Hot" numbers have appeared frequently in recent draws, while "Cold" numbers haven't appeared for a long time.
Important Note: In truly random systems, past results don't influence future outcomes (no "memory"). However, analyzing these patterns helps understand variance and psychological biases.
This analysis is useful for educational purposes to demonstrate how gamblers often fall prey to the "hot hand fallacy" or "gambler's fallacy."