How do I prove my state lottery's CGNs are fixed?

How do I prove my state lottery's Computer Generated Numbers are fixed? This is a perplexing and difficult task, but it's not beyond a reasonable plausibility. The task becomes an experiment with a number of steps to follow. Following this opening is a header outline of what can be expected in general; after that, each header is presented in more detail and may have data, graphs, explanations in logic and process, theories, etc. The contents in each header will not get too technical unless it is needed to help support a current or following idea, process, data, graph, computation or theory.

Next is a list of each header and a brief explanation:

• Pretext
      Setup information.
• Correlation
      Simple explanation and graphs in the presentation.
• Computations for Analysis
      Draw mean and span, the Simplified Bidirectional Mean Averaging.
• Wheel Application
      Using the draw mean to derive wheel numbers.
• Distributions and Analysis
      Lottery and Wheel Pool distributions and draw span distribution.
• The Pretest, Experiment 0
      The 60 draw sample data for analysis and correlation from 2007-08-06 to 2007-10-04.
• The Setup and Cost Effective Approach
      If they're watching the Lottery Post Predictions, use it to an advantage.
• The Test, Experiment 1
      Draw and wheel data for the 60 day sample from 2007-10-05 to 2007-12-03.
• The Conclusion, Experiment 1
      Possible proof the Computer Generated Numbers are fixed?
• The Posttest, Experiment 2
      A 90 draw sample and wheel posting to play on current draw number deficiencies started 2007-12-04.

Pretext

Prerequisites: Understanding of lotteries, lottery wheels, analysis and systems; graph reading and table reading skills; math operations, statistics, averages, line equations.

In going through this, I ask that anyone reading this please keep this basic understanding in mind. This is not an attempt to try and predict any particular combination. This has more to do with reasonable expectation. I have stated in my blog and defined the difference between the two. The following is from my blog on 2007-02-23:

"Reasonable Expectation and Prediction are often thought of as being the same; they are not. Prediction has a level of precision that is greater than Reasonable Expectation and Prediction can be derived from Reasonable Expectation. Reasonable Expectation has a level of accuracy that is greater than Prediction, however, Reasonable Expectation can only be derived from many different Predictions."

Understanding this difference between reasonable expectation and prediction is important for this topic. There is a tendency to think this topic is about predicting exact combinations. It is about what would be reasonably expected from a lottery selection and the playing of a wheeled set of numbers in a combinatorial set.

The lottery chosen to work with is the Wisconsin Lottery Badger 5. It is a pick 5 of 31 numbers game and is a completely Computer Generated Number game. It has been a Computer Generated Numbers game since its implementation back on 2003-02-17. The numbers selected will be analyzed based on their individual total frequency for the numbers 1 to 31. This creates a distribution and the distribution can be analyzed using a computational method called the Simplified Bidirectional Mean Averaging to show the average fluctuations in the data throughout the numbers  1 to 31. The span of numbers between the largest and the smallest numbers is also analyzed by distribution.

The selected numbers from the previous day's draw are averaged to get a mean value and that mean value is used to create pool of 12 numbers. Those 12 numbers are then inserted into a set of combinations called a wheel. The wheel is then posted on the Lottery Post's prediction board and recorded for further analysis. The 12 numbers derived are also analyzed based on their individual total frequency relative to the 1 to 31 numbers. Here too, the analysis produces a distribution of numbers 1 to 31 and the Simplified Bidirectional Mean Averaging is applied to find the average fluctuations in the distribution.

Both of these analysis procedures are applied to a set of 60 draws before the posting of the wheels and 60 draws during the posting of the wheels; they are experiment 0 (before) and 1 (during). Both experiments will produce a graph of distributions that can be compared to each other and between the drawn numbers and wheeled numbers. From that, it can be visually inspected to see if there is a correlation between the drawn numbers and the wheeled numbers.

In any means of proving that something is influenced by something else, there needs to be a means of correlating the data in to the data out. The statistical information is provided via a link, however, it's easier to present the data in a graphical form for easy reading. If there are questions about the graphs, refer to the data links provided first. The data can be downloaded to perform additional analysis.

Correlation

This is an over simplified description, correlation is the relationship between to different sets of data relative to the same event. Typically, it's thought of as degree of relationship between the Data A (in) to the Data B (out). It's like the heat put into a flask of water and the temperature measured or the electrical current put into a light bulb and the light emitted out. The level of correlation can be expressed as a value from 0 to 1. Later in the topic, this value will be referred to as the R-squared value. The closer to 0 the R-squared value is means there is low correlation between the Data A (in) and the Data B (out). A value closer to 1 means there is a high correlation between the Data A (in) and the Data B (out). Also, there are negative, neutral and positive relationships between the Data A (in) and the Data B (out). The more negative a correlation is means Data A goes in one direction and Data B goes in the opposite direction. The more positive a correlation is means Data A goes in one direction and Data B goes in the same direction. A neutral correlation means Data A changes, but Data B does not change in relationship to Data A. Below are some simple animated graphs to show how this would look.

The animated graphs can be also shown as a line graph to display all the data values at once. The following are the same animated graphs as a non-animated line graphs.

The data can also be graphed as an XY plot with Data A (in) on the horizontal X axis and Data B (out) on the vertical Y axis . The green line on the graph is the general trend line for each. A negative correlation will slant from upper left to lower right and the data will be fairly close to the line, a positive correlation will slant from the lower left to the upper right and the data will be fairly close to the line. A neutral correlation may be slanted or horizontal, but the data will not be very close to the line.

Understanding these last two sets of line graphs is going to be important in showing the relationship between the wheel distributions (Data A) and the draw distributions (Data B). As the distribution data is presented, it will begin as a bar graph and then the Simplified Bidirectional Mean Averaging will be applied to give a line graph showing the average fluctuations in the distributions. The fluctuation distributions will be shown in both formats, line graphs and XY plots.

Computations for Analysis

In order to get the wheel numbers and the average fluctuations in the distributions, a few math operations need to be done. First the mean value of the previously selected draw is needed to derive the wheel numbers. If the previous days draw is represented by these set of values for each column, {A, B, C, D, E}, then the mean is the sum of those numbers divided by the pick size, r. The equation is as follows:

D_m = (A + B + C + D + E) / r

Example, if the previous days draw is {1, 2, 3, 4, 5}, then the mean is D_m = (1 + 2 + 3 + 4 + 5) / 5, D_m  = 15 / 5 or D_m = 3. The mean will be applied to find the wheel numbers later.

Next is the Simplified Bidirectional Mean Averaging and is similar to the posted topic Bidirectional Mean Averaging and the Wave Matrix. As the name suggests, it a simplified version of the bidirectional mean averaging process. It involves a few steps: up mean averaging, down mean averaging, the average between the up an down processes. The process is as follows:

By example, if the sample data is {X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀}, then the up mean averaging is

U₁ = X₁
U₂ = (U₁ + X₂) / 2
U₃ = (U₂ + X₃) / 2
U₄ = (U₃ + X₄) / 2
U₅ = (U₄ + X₅) / 2
U₆ = (U₅ + X₆) / 2
U₇ = (U₆ + X₇) / 2
U₈ = (U₇ + X₈) / 2
U₉ = (U₈ + X₉) / 2
U₁₀ = (U₉ + X₁₀) / 2

The up mean averaging data is {U₁, U₂, U₃, U₄, U₅, U₆, U₇, U₈, U₉, U₁₀}. The down mean averaging is

D₁₀ = X₁₀
D₉ = (D₁₀ + X₉) / 2
D₈ = (D₉ + X₈) / 2
D₇ = (D₈ + X₇) / 2
D₆ = (D₇ + X₆) / 2
D₅ = (D₆ + X₅) / 2
D₄ = (D₅ + X₄) / 2
D₃ = (D₄ + X₃) / 2
D₂ = (D₃ + X₂) / 2
D₁ = (D₂ + X₁) / 2

The down mean averaging data is {D₁, D₂, D₃, D₄, D₅, D₆, D₇, D₈, D₉, D₁₀}. The simplified bidirectional mean averaging is

B₁ = (U₁ + D₁) / 2
B₂ = (U₂ + D₂) / 2
B₃ = (U₃ + D₃) / 2
B₄ = (U₄ + D₄) / 2
B₅ = (U₅ + D₅) / 2
B₆ = (U₆ + D₆) / 2
B₇ = (U₇ + D₇) / 2
B₈ = (U₈ + D₈) / 2
B₉ = (U₉ + D₉) / 2
B₁₀ = (U₁₀ + D₁₀) / 2

The simplified bidirectional mean averaging data is {B₁, B₂, B₃, B₄, B₅, B₆, B₇, B₈, B₉, B₁₀} for this sample set of 10 numbers. A different quantity set of numbers will follow the same basic process where X₁ is set equal to U₁ and then the following mean averaging; also, X_n is set equal to D_n and then the following mean averaging, where n is the quantity of numbers in the set.

Another step is needed to make the averaging results more smooth; to work out any big differences in the data. This process is called iteration and it is simply the reapplication of the bidirectional mean averaging method to the data derived by the bidirectional mean averaging. If the data from the bidirectional mean averaging is {B₁, B₂, B₃, B₄, B₅, B₆, B₇, B₈, B₉, B₁₀}, then the bidirectional mean averaging data is set equal to the original variables {X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀} and the bidirectional mean averaging process is repeated. The new data is said to have been reiterated in the bidirectional mean averaging process, hence the term, iteration. The first iteration is the first application of the method, the second iteration is the reapplication of the method and so on. An example of the method and the following iterations are shown in the graph below. From the graph, Iteration 8 shows the average fluctuations in the distribution. Below the example are a few more graphs.

This graph shows Iteration 8 plotted against a second axis on the right, Average Fluctuation.

An additional calculation is needed for a distribution discussed later in the post. It is the draw span and it's the difference between the largest number and the smallest number in an individual draw. If the draw data is {A, B, C, D, E} and the draw data is in ascending order, then the draw span is

D_s = E - A, where E > A

Example, if the draw is {1, 2, 3, 4, 5}, then the draw span is D_s = 5 - 1 or D_s = 4.

Wheel Application

The draw mean is used to derive the 12 numbers that will be inserted in a wheel for posting on the Lottery Post's prediction board. First, the Integer part of the draw mean is found; it is D_i = Int(D_m). The D_i value is then used to add and subtract incremental values above and below the D_i value to get a set of 12 numbers. Below are the equations for finding W_n values 1 to 12, where n is 1 to 12.

W₁ = D_i - 5
W₂ = D_i - 4
W₃ = D_i - 3
W₄ = D_i - 2
W₅ = D_i - 1
W₆ = D_i
W₇ = D_i + 1
W₈ = D_i + 2
W₉ = D_i + 3
W₁₀ = D_i + 4
W₁₁ = D_i + 5
W₁₂ = D_i + 6

The wheel numbers are then {W₁, W₂, W₃, W₄, W₅, W₆, W₇, W₈, W₉, W₁₀, W₁₁, W₁₂}.

The wheel used is as follows:

If the previous day's lottery numbers are {10, 13, 20, 22, 28}, then the draw mean is D_m = (10 + 13 + 20 + 22 + 28) / 5, D_m = 93 / 5 or D_m = 18.6 and the Integer value is then D_i = Int(18.6) or D_i = 18. The the wheel numbers then become {18 - 5, 18 - 4, 18 - 3, 18 - 2, 18 - 1, 18, 18 + 1, 18 + 2, 18 + 3, 18 + 4, 18 + 5, 18 + 6} or {13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} and when applied to the list of combinations, the wheel becomes the play or prediction post lines as follows:

This process is done for each days drawing and posting.

Distributions and Analysis

There are three distributions that will be looked at and analyzed. They are the lottery and wheel numbers distributions and the draw span distribution. The table below shows a 60 draw example of lottery numbers, wheel numbers and draw span for each draw.