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 {X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}, X_{7}, X_{8}, X_{9}, X_{10}}, then the up mean averaging is

U_{1} = X_{1}

U_{2} = (U_{1} + X_{2}) / 2

U_{3} = (U_{2} + X_{3}) / 2

U_{4} = (U_{3} + X_{4}) / 2

U_{5} = (U_{4} + X_{5}) / 2

U_{6} = (U_{5} + X_{6}) / 2

U_{7} = (U_{6} + X_{7}) / 2

U_{8} = (U_{7} + X_{8}) / 2

U_{9} = (U_{8} + X_{9}) / 2

U_{10} = (U_{9} + X_{10}) / 2

The up mean averaging data is {U_{1}, U_{2}, U_{3}, U_{4}, U_{5}, U_{6}, U_{7}, U_{8}, U_{9}, U_{10}}. The down mean averaging is

D_{10} = X_{10}

D_{9} = (D_{10} + X_{9}) / 2

D_{8} = (D_{9} + X_{8}) / 2

D_{7} = (D_{8} + X_{7}) / 2

D_{6} = (D_{7} + X_{6}) / 2

D_{5} = (D_{6} + X_{5}) / 2

D_{4} = (D_{5} + X_{4}) / 2

D_{3} = (D_{4} + X_{3}) / 2

D_{2} = (D_{3} + X_{2}) / 2

D_{1} = (D_{2} + X_{1}) / 2

The down mean averaging data is {D_{1}, D_{2}, D_{3}, D_{4}, D_{5}, D_{6}, D_{7}, D_{8}, D_{9}, D_{10}}. The simplified bidirectional mean averaging is

B_{1} = (U_{1} + D_{1}) / 2

B_{2} = (U_{2} + D_{2}) / 2

B_{3} = (U_{3} + D_{3}) / 2

B_{4} = (U_{4} + D_{4}) / 2

B_{5} = (U_{5} + D_{5}) / 2

B_{6} = (U_{6} + D_{6}) / 2

B_{7} = (U_{7} + D_{7}) / 2

B_{8} = (U_{8} + D_{8}) / 2

B_{9} = (U_{9} + D_{9}) / 2

B_{10} = (U_{10} + D_{10}) / 2

The simplified bidirectional mean averaging data is {B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6}, B_{7}, B_{8}, B_{9}, B_{10}} for this sample set of 10 numbers. A different quantity set of numbers will follow the same basic process where X_{1} is set equal to U_{1} 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 {B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6}, B_{7}, B_{8}, B_{9}, B_{10}}, then the bidirectional mean averaging data is set equal to the original variables {X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}, X_{7}, X_{8}, X_{9}, X_{10}} 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.

W_{1} = D_{i} - 5

W_{2} = D_{i} - 4

W_{3} = D_{i} - 3

W_{4} = D_{i} - 2

W_{5} = D_{i} - 1

W_{6} = D_{i}

W_{7} = D_{i} + 1

W_{8} = D_{i} + 2

W_{9} = D_{i} + 3

W_{10} = D_{i} + 4

W_{11} = D_{i} + 5

W_{12} = D_{i} + 6

The wheel numbers are then {W_{1}, W_{2}, W_{3}, W_{4}, W_{5}, W_{6}, W_{7}, W_{8}, W_{9}, W_{10}, W_{11}, W_{12}}.

The wheel used is as follows:

Index | A | B | C | D | E |

1 | 1 | 2 | 3 | 5 | 7 |

2 | 1 | 2 | 3 | 5 | 12 |

3 | 1 | 2 | 4 | 7 | 8 |

4 | 1 | 2 | 5 | 6 | 12 |

5 | 1 | 2 | 5 | 7 | 10 |

6 | 1 | 2 | 6 | 7 | 11 |

7 | 1 | 2 | 6 | 9 | 10 |

8 | 1 | 3 | 4 | 6 | 11 |

9 | 1 | 3 | 4 | 8 | 9 |

10 | 1 | 3 | 5 | 6 | 12 |

11 | 1 | 3 | 6 | 7 | 8 |

12 | 1 | 4 | 5 | 7 | 11 |

13 | 1 | 4 | 5 | 8 | 9 |

14 | 1 | 4 | 5 | 8 | 10 |

15 | 1 | 4 | 5 | 9 | 12 |

16 | 1 | 4 | 6 | 9 | 11 |

17 | 1 | 4 | 6 | 10 | 12 |

18 | 1 | 4 | 8 | 10 | 12 |

19 | 1 | 5 | 7 | 10 | 11 |

20 | 1 | 5 | 8 | 9 | 11 |

21 | 1 | 5 | 8 | 10 | 12 |

22 | 2 | 3 | 4 | 6 | 9 |

23 | 2 | 3 | 4 | 9 | 10 |

24 | 2 | 3 | 5 | 9 | 11 |

25 | 2 | 3 | 6 | 9 | 10 |

26 | 2 | 3 | 7 | 8 | 10 |

27 | 2 | 3 | 7 | 9 | 11 |

28 | 2 | 3 | 8 | 9 | 12 |

29 | 2 | 4 | 7 | 10 | 11 |

30 | 2 | 5 | 6 | 7 | 11 |

31 | 2 | 6 | 7 | 10 | 12 |

32 | 2 | 6 | 8 | 10 | 11 |

33 | 2 | 6 | 8 | 10 | 12 |

34 | 2 | 7 | 8 | 11 | 12 |

35 | 3 | 4 | 6 | 11 | 12 |

36 | 3 | 4 | 7 | 9 | 12 |

37 | 3 | 4 | 8 | 9 | 10 |

38 | 3 | 5 | 6 | 8 | 9 |

39 | 3 | 5 | 6 | 8 | 11 |

40 | 3 | 5 | 7 | 8 | 12 |

41 | 3 | 6 | 7 | 10 | 12 |

42 | 3 | 8 | 10 | 11 | 12 |

43 | 4 | 5 | 7 | 8 | 9 |

44 | 4 | 5 | 9 | 11 | 12 |

45 | 4 | 6 | 7 | 8 | 10 |

46 | 4 | 7 | 9 | 10 | 11 |

47 | 4 | 7 | 9 | 11 | 12 |

48 | 5 | 6 | 10 | 11 | 12 |

49 | 5 | 8 | 9 | 10 | 12 |

50 | 6 | 7 | 9 | 11 | 12 |

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:

Index | A | B | C | D | E |

1 | 13 | 14 | 15 | 17 | 19 |

2 | 13 | 14 | 15 | 17 | 24 |

3 | 13 | 14 | 16 | 19 | 20 |

4 | 13 | 14 | 17 | 18 | 24 |

5 | 13 | 14 | 17 | 19 | 22 |

6 | 13 | 14 | 18 | 19 | 23 |

7 | 13 | 14 | 18 | 21 | 22 |

8 | 13 | 15 | 16 | 18 | 23 |

9 | 13 | 15 | 16 | 20 | 21 |

10 | 13 | 15 | 17 | 18 | 24 |

11 | 13 | 15 | 18 | 19 | 20 |

12 | 13 | 16 | 17 | 19 | 23 |

13 | 13 | 16 | 17 | 20 | 21 |

14 | 13 | 16 | 17 | 20 | 22 |

15 | 13 | 16 | 17 | 21 | 24 |

16 | 13 | 16 | 18 | 21 | 23 |

17 | 13 | 16 | 18 | 22 | 24 |

18 | 13 | 16 | 20 | 22 | 24 |

19 | 13 | 17 | 19 | 22 | 23 |

20 | 13 | 17 | 20 | 21 | 23 |

21 | 13 | 17 | 20 | 22 | 24 |

22 | 14 | 15 | 16 | 18 | 21 |

23 | 14 | 15 | 16 | 21 | 22 |

24 | 14 | 15 | 17 | 21 | 23 |

25 | 14 | 15 | 18 | 21 | 22 |

26 | 14 | 15 | 19 | 20 | 22 |

27 | 14 | 15 | 19 | 21 | 23 |

28 | 14 | 15 | 20 | 21 | 24 |

29 | 14 | 16 | 19 | 22 | 23 |

30 | 14 | 17 | 18 | 19 | 23 |

31 | 14 | 18 | 19 | 22 | 24 |

32 | 14 | 18 | 20 | 22 | 23 |

33 | 14 | 18 | 20 | 22 | 24 |

34 | 14 | 19 | 20 | 23 | 24 |

35 | 15 | 16 | 18 | 23 | 24 |

36 | 15 | 16 | 19 | 21 | 24 |

37 | 15 | 16 | 20 | 21 | 22 |

38 | 15 | 17 | 18 | 20 | 21 |

39 | 15 | 17 | 18 | 20 | 23 |

40 | 15 | 17 | 19 | 20 | 24 |

41 | 15 | 18 | 19 | 22 | 24 |

42 | 15 | 20 | 22 | 23 | 24 |

43 | 16 | 17 | 19 | 20 | 21 |

44 | 16 | 17 | 21 | 23 | 24 |

45 | 16 | 18 | 19 | 20 | 22 |

46 | 16 | 19 | 21 | 22 | 23 |

47 | 16 | 19 | 21 | 23 | 24 |

48 | 17 | 18 | 22 | 23 | 24 |

49 | 17 | 20 | 21 | 22 | 24 |

50 | 18 | 19 | 21 | 23 | 24 |

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.

Index | Lottery Numbers | Draw Span | Wheel Numbers |

A | B | C | D | E | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

0 | 1 | 13 | 19 | 20 | 24 | - | - | - | - | - | - | - | - | - | - | - | - | - |

1 | 2 | 3 | 5 | 15 | 22 | 20 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

2 | 2 | 10 | 20 | 22 | 26 | 24 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

3 | 1 | 6 | 10 | 21 | 27 | 26 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |

4 | 2 | 3 | 5 | 12 | 20 | 18 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

5 | 2 | 4 | 16 | 17 | 25 | 23 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |

6 | 13 | 17 | 19 | 27 | 30 | 17 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

7 | 3 | 12 | 23 | 29 | 30 | 27 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 |

8 | 16 | 20 | 22 | 25 | 28 | 12 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

9 | 7 | 11 | 18 | 19 | 23 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 |

10 | 4 | 5 | 13 | 23 | 30 | 26 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

11 | 13 | 17 | 20 | 22 | 26 | 13 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

12 | 9 | 22 | 24 | 29 | 30 | 21 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

13 | 22 | 24 | 26 | 27 | 30 | 8 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 |

14 | 5 | 10 | 18 | 26 | 29 | 24 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 |

15 | 3 | 10 | 12 | 16 | 19 | 16 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |

16 | 2 | 8 | 20 | 25 | 28 | 26 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

17 | 2 | 19 | 20 | 25 | 26 | 24 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |

18 | 5 | 6 | 9 | 13 | 21 | 16 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

19 | 5 | 12 | 19 | 21 | 24 | 19 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

20 | 2 | 14 | 22 | 26 | 31 | 29 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |

21 | 4 | 8 | 9 | 13 | 26 | 22 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

22 | 10 | 13 | 17 | 19 | 30 | 20 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

23 | 4 | 6 | 9 | 11 | 12 | 8 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |

24 | 1 | 10 | 21 | 29 | 30 | 29 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |

25 | 1 | 2 | 12 | 21 | 26 | 25 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

26 | 1 | 7 | 15 | 16 | 20 | 19 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

27 | 2 | 7 | 8 | 13 | 25 | 23 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |

28 | 5 | 7 | 17 | 18 | 21 | 16 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |

29 | 1 | 18 | 25 | 27 | 29 | 28 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

30 | 4 | 5 | 15 | 17 | 29 | 25 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 |

31 | 1 | 19 | 20 | 26 | 28 | 27 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

32 | 8 | 9 | 13 | 16 | 24 | 16 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

33 | 4 | 7 | 18 | 20 | 30 | 26 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

34 | 3 | 4 | 10 | 14 | 17 | 14 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

35 | 2 | 3 | 9 | 11 | 24 | 22 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

36 | 3 | 7 | 19 | 20 | 26 | 23 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

37 | 21 | 22 | 23 | 26 | 28 | 7 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

38 | 10 | 12 | 24 | 25 | 27 | 17 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

39 | 4 | 9 | 14 | 18 | 24 | 20 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

40 | 3 | 5 | 17 | 23 | 29 | 26 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

41 | 2 | 3 | 6 | 12 | 28 | 26 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

42 | 8 | 11 | 15 | 18 | 21 | 13 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

43 | 7 | 15 | 16 | 23 | 27 | 20 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

44 | 9 | 15 | 16 | 17 | 28 | 19 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |

45 | 2 | 6 | 7 | 9 | 24 | 22 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |

46 | 1 | 19 | 20 | 25 | 27 | 26 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

47 | 1 | 14 | 15 | 21 | 26 | 25 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

48 | 3 | 5 | 6 | 12 | 13 | 10 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

49 | 2 | 7 | 10 | 12 | 18 | 16 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |

50 | 4 | 8 | 17 | 19 | 28 | 24 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

51 | 3 | 9 | 11 | 14 | 22 | 19 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

52 | 1 | 7 | 10 | 12 | 29 | 28 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |

53 | 7 | 13 | 14 | 30 | 31 | 24 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |

54 | 1 | 4 | 11 | 14 | 17 | 16 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

55 | 1 | 8 | 10 | 22 | 25 | 24 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

56 | 3 | 14 | 16 | 24 | 29 | 26 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

57 | 18 | 19 | 23 | 25 | 26 | 8 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |

58 | 2 | 9 | 21 | 22 | 23 | 21 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 |

59 | 4 | 5 | 8 | 9 | 14 | 10 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

60 | 5 | 8 | 11 | 15 | 20 | 15 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |

The numbers in each column heading of Lottery Numbers, Draw Span and Wheel Numbers can be tallied to find each headings distribution of numbers. For this example, the following tables show their respective distributions.

Number | Frequencies |

Wheel | Draw |

1 | 0 | 11 |

2 | 1 | 14 |

3 | 4 | 12 |

4 | 10 | 11 |

5 | 12 | 12 |

6 | 16 | 6 |

7 | 20 | 11 |

8 | 24 | 9 |

9 | 27 | 12 |

10 | 36 | 11 |

11 | 39 | 7 |

12 | 44 | 11 |

13 | 48 | 10 |

14 | 52 | 9 |

15 | 50 | 8 |

16 | 45 | 8 |

17 | 46 | 11 |

18 | 42 | 9 |

19 | 39 | 11 |

20 | 36 | 12 |

21 | 33 | 10 |

22 | 24 | 11 |

23 | 21 | 8 |

24 | 16 | 9 |

25 | 12 | 10 |

26 | 7 | 13 |

27 | 6 | 7 |

28 | 5 | 7 |

29 | 2 | 9 |

30 | 2 | 9 |

31 | 1 | 2 |

Value | Span Frequency |

1 | 0 |

2 | 0 |

3 | 0 |

4 | 0 |

5 | 0 |

6 | 0 |

7 | 1 |

8 | 3 |

9 | 0 |

10 | 2 |

11 | 0 |

12 | 1 |

13 | 2 |

14 | 1 |

15 | 1 |

16 | 7 |

17 | 2 |

18 | 1 |

19 | 4 |

20 | 4 |

21 | 2 |

22 | 3 |

23 | 3 |

24 | 6 |

25 | 3 |

26 | 8 |

27 | 2 |

28 | 2 |

29 | 2 |

30 | 0 |

The bar graphs of these can be seen next.

The draw span will be referred to later in the post. The wheel and draw distributions will have the Simplified Bidirectional Mean Averaging applied to get the average fluctuations for each data set. The Bidirectional Mean Averaging will be carried out to 8 Iterations for analysis. The next table shows the average fluctuations for each wheel and draw distribution.

Number | Frequencies | Fluctuations |

Wheel | Draw | Wheel | Draw |

1 | 0 | 11 | 8.296221 | 11.20978 |

2 | 1 | 14 | 9.455705 | 11.12575 |

3 | 4 | 12 | 11.14963 | 11.00038 |

4 | 10 | 11 | 13.31095 | 10.84371 |

5 | 12 | 12 | 15.85587 | 10.67092 |

6 | 16 | 6 | 18.69422 | 10.498 |

7 | 20 | 11 | 21.72728 | 10.34132 |

8 | 24 | 9 | 24.8474 | 10.20175 |

9 | 27 | 12 | 27.93632 | 10.07813 |

10 | 36 | 11 | 30.86472 | 9.965723 |

11 | 39 | 7 | 33.49274 | 9.866072 |

12 | 44 | 11 | 35.68861 | 9.785218 |

13 | 48 | 10 | 37.33592 | 9.723592 |

14 | 52 | 9 | 38.34711 | 9.684897 |

15 | 50 | 8 | 38.67417 | 9.672521 |

16 | 45 | 8 | 38.31653 | 9.685291 |

17 | 46 | 11 | 37.31168 | 9.713629 |

18 | 42 | 9 | 35.71819 | 9.739883 |

19 | 39 | 11 | 33.61597 | 9.747519 |

20 | 36 | 12 | 31.09881 | 9.718874 |

21 | 33 | 10 | 28.27243 | 9.641936 |

22 | 24 | 11 | 25.25336 | 9.513316 |

23 | 21 | 8 | 22.16709 | 9.332396 |

24 | 16 | 9 | 19.13069 | 9.102002 |

25 | 12 | 10 | 16.25005 | 8.820878 |

26 | 7 | 13 | 13.61314 | 8.488693 |

27 | 6 | 7 | 11.28714 | 8.112503 |

28 | 5 | 7 | 9.31464 | 7.717087 |

29 | 2 | 9 | 7.720732 | 7.331684 |

30 | 2 | 9 | 6.520118 | 6.99004 |

31 | 1 | 2 | 5.718166 | 6.733356 |

Here is a graph of the data.

Now the fluctuations only.

The fluctuation data then can be applied to an XY plot to show the relationship between the wheel and draw average fluctuation. In the XY plot there will be a green line showing the trend line and some additional information about the line and correlation of the data. The R-square value is the degree of correlation between the wheel and draw fluctuations and the equation of the line is given by y = 8.77 + 0.0326x. The R-square value is close to 0, meaning the data in (wheel fluctuation) has a low influence on the data out (draw fluctuation) and can be visually verified by the points and their respective distances from the line.

Next, these basic processes of finding fluctuations and correlation will be applied to some actual draw data in a pretest, Experiment 0.

The Pretest, Experiment 0

For this part, the data can be accessed through the following link for reference, Experiment 0 Data. Experiment 0 never posted any wheels to the Lottery Post's prediction board and was never played in the Badger 5 game. It is the 60 draw data from 2007-08-06 to 2007-10-04. The following graphs are the wheel and draw distributions, their fluctuations and the XY Plot and correlation.

From the graphs, an inference can be made based on the R-square value and the XY Plot. The low R-square value shows there is a low relationship between the wheel numbers (data in) to the draw numbers (data out). Also, from the visual inspection of the plot, the data points are fairly far from the trend line, meaning this tends to suggest there is a low relationship between the wheel and the draw. This is what would be reasonably expected for a random event where no wheel was played or posted. In a truly random setting, it is also reasonable to expect a low correlation if the wheel is played or posted. In the next step, there needs to be a setup for attention grabbing to get those who could possibly be involved to play along. Then Experiment 1 can be tested to see if there is a relationship between data in (wheel) and data out (draw).

The Setup and Cost Effective Approach

Ever had that feeling you're being watched? The feeling goes something like this: You work out a really great system for analyzing and playing numbers in a specific lottery game. You've done all the research, worked all the loose ends and then comes the time to either play or post your find. It works great and then like someone switching off a light, it goes cold.

Well, in the case of posting on the Lottery Post, I think there might be a reason. It's described in one word, discredit. One of the most effective ways of discouraging many people from playing a proven method is to show it doesn't work. I think this is what could be happening at the Lottery Post.

Unfortunately for the ones watching, this can prove to be a very cost effective way of proving the opening sentence in this post. If a $50.00 wheel were played for 60 draws, that would be a $3000.00 cost to play. Far beyond my ability to afford. So, basically I needed a way to attract attention and get the eye of those who would be watching. Loud mouthing, posting wheels and systems seems to have worked. Instead of actually playing the $50.00 wheel, I've used the watchful eye as an advantage.

Next is the heart of this post, Experiment 1.

The Test, Experiment 1

The Excel Sheet for the data and graphs can be found here, Experiment 1 Data. Experiment 1 posted wheels on the Lottery Post's prediction board during the period 2007-10-05 to 2007-12-03, a total of 60 draws. The following are the distributions, fluctuations and correlation graphs for the data.

In the fluctuations graph, there is an obvious visual correlation between the data in (wheel) and the data out (draw). As the wheel number fluctuation increases, the draw number fluctuation decreases and the same is true in the opposite, as the wheel number fluctuation decreases, the draw number fluctuation increases. This would tend to suggest there may be a negative correlation between the two. Following this graph is the XY plot that can help in determining the correlation.

Next is the XY plot of the fluctuations and shows the plotted data points and tend line. In the lower right are the R-square value and line equation based on 31 data points.

From the graphs, it visually shows there is a possible correlation between the data in (wheel) to the data out (draw), the points are very close to the line. From the the R-square value of 0.957, it's very close to 1 and suggests there is a close correlation between the wheel and draw data, numerically. The -0.0526 value in the line equation suggests there is a negative correlation between the data in (wheel) and the data out (draw). As a comparison between the before and after, examine the XY plots of both Experiment 0 and 1. There is a dramatic difference between the two. Below is the XY plot for Experiment 0.

Going back to the distributions graph, looking at just the wheel data, there appears to be something like a normal distribution curve. Below is a bar graph of just the wheel distribution data; this data can be analyzed to find a mean wheel number and standard deviation. The mean and standard deviation can then be used to find what are the 50% of the highest frequently posted wheel numbers. In other words, where does the bulk of the wheel numbers lay? The mean value is just the average of all the wheel numbers that are to be inserted into the combinations of the wheel and the standard deviation is the value related to those numbers, also.

To find the wheel number mean, add up all the numbers that are to be inserted in the wheel combinations and divided by the total set of numbers. The wheel number mean is then equal to 16.11667. The standard deviation value for the wheel numbers is 5.24639. This operation can be found in the Experiment 1 Data Excel file by click on the bottom tab, Wheel # Analysis. These values can then be used to create a normal distribution curve and find the 50% bulk of the wheel numbers. Below is a graph of the wheel distribution and normal distribution.

The graph shows there is a peak at the number 16; this is consistent with the mean value of 16.11667. Also, looking at the graph there appears to be a bulk of the data centered around the mean. To find the 50% bulk of the data that is closest to the mean and is 25% above and 25% below the mean can be determined by multiplying the standard deviation by 0.674489524681121 and then adding and subtracting it from the mean. This will produce a lower limit and upper limit of where the 50% bulk of the data is falling. The lower limit is

L_{b} = 16.11667 - (0.674489524681121 * 5.24639)

L_{b} = 16.11667 - 3.53864

L_{b} = 12.57803

L_{u} = 16.11667 + (0.674489524681121 * 5.24639)

L_{u} = 16.11667 + 3.53864

L_{u} = 19.65530

The L_{b} and L_{u} values can now establish a set of high work bulk numbers that represent the highest frequency numbers where the work is being done by the wheel. The set of numbers are going to be the integer values that are greater than L_{b} and less than L_{u}; these are {13, 14, 15, 16, 17, 18, 19}. In addition to the high work bulk of numbers, there is a counter part that shows the low work bulk of numbers where the least amount of work is being done. It is also 50% of the numbers and is the remaining set of numbers not covered by the high work bulk numbers of {13, 14, 15, 16, 17, 18, 19}; they are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}.

It's important to understand these as what numbers are doing the most work and how this relates to the numbers being drawn by the lottery. Below is a distribution table of the drawn lottery numbers. From this table it can be shown what are the highest frequency numbers and lowest frequency numbers by finding the mean or average frequency of the table.

Number | Draw |

1 | 11 |

2 | 11 |

3 | 9 |

4 | 10 |

5 | 8 |

6 | 13 |

7 | 12 |

8 | 12 |

9 | 10 |

10 | 11 |

11 | 11 |

12 | 6 |

13 | 6 |

14 | 9 |

15 | 9 |

16 | 7 |

17 | 10 |

18 | 8 |

19 | 4 |

20 | 14 |

21 | 5 |

22 | 14 |

23 | 10 |

24 | 6 |

25 | 14 |

26 | 6 |

27 | 10 |

28 | 13 |

29 | 7 |

30 | 12 |

31 | 12 |

From this an average frequency can be found by summing the frequencies and dividing by the total numbers in the set. The average frequency is 9.67742 and can be used to setup a table that shows the numbers below average and above average.

Experiment 1 |

Draw Number Frequencies |

Below average | Above average |

3 | 1 |

5 | 2 |

12 | 4 |

13 | 6 |

14 | 7 |

15 | 8 |

16 | 9 |

18 | 10 |

19 | 11 |

21 | 17 |

24 | 20 |

26 | 22 |

29 | 23 |

| 25 |

| 27 |

| 28 |

| 30 |

| 31 |

The table shows the high work bulk wheel numbers in gray. It also shows that almost half of the below average draw number frequencies are the high work bulk wheel numbers. The same was applied to Experiment 0. The data can be accessed in this link, Experiment 0 Data. The following is the table for Experiment 0 showing the below average and above average draw number frequencies. As it turns out, the high work bulk wheel numbers have the same set of numbers in Experiment 0, {13, 14, 15, 16, 17, 18, 19}.

Experiment 0 |

Draw Number Frequencies |

Below average | Above average |

2 | 1 |

6 | 3 |

7 | 4 |

8 | 5 |

9 | 10 |

11 | 12 |

15 | 13 |

16 | 14 |

18 | 17 |

21 | 19 |

25 | 20 |

26 | 22 |

28 | 23 |

30 | 24 |

27 |

29 |

31 |

Notice the high work bulk numbers are fairly evenly distributed between the below average and above average draw number frequencies and there is actually more by count in the above average column.

The draw span also changed slightly between Experiment 0 and 1. Below are some graphs showing the change. The shift can be seen where the peek of each fluctuation line is. In Experiment 0 the peek is at 23. The peek in Experiment 1 is at 25, meaning there was a slight shift in the span of drawn numbers. This shows the draw span increased slightly during Experiment 1 as compared to Experiment 0.

All this data will play an important role in the next part, the conclusion.

The Conclusion, Experiment 1

Is this possible proof the Wisconsin Lottery Badger 5 Computer Generated Numbers are fixed? Well, looking at just small slice of the possible data that could be derived for this topic, it appears to be fixed. There is the direct relationship between the data in (wheel) and data out (draw) in Experiment 1. Also, the high work bulk wheel numbers are clustered in the low frequency drawn numbers. In addition, the draw span shifted slightly to higher span values.

Some might say, "Well, this is a small sample and the data can interpreted in may ways. Don't think much of it." This is true if it were not for Experiment 2 running currently. The next part will address the implications of the posttest, Experiment 2.

The Posttest, Experiment 2

Experiment 2 is designed to pick up on the low frequency or deficient draw numbers established in Experiment 1. The deficient draw numbers contain almost all the high work bulk wheel numbers established in Experiment 1. To play on these deficient draw number, a wheel with a fixed set of numbers is used to be inserted in a wheel for posting. The numbers are {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} and is a total set of 13 numbers. This also contains the high work bulk wheel numbers from Experiment 1, {13, 14, 15, 16, 17, 18, 19}.

This setup creates a paradox for the would be wheel watchers. On one hand, if they stop trying to discredit the wheel by returning to a truly random state, the wheel begins to look good. However, on the other hand, if the continue to try and discredit the wheel by randomly manipulating the draw numbers, they will continue the current deficiencies and possibly increase the deficiencies which would go against the very notion of a truly random setting.

There is one other possible choice, but that would lead to something of an admission of guilt, change the play matrix or discontinue the game. The basis is covered, all we have to is watch them watching us.

To be continued... with Experiment 2...