In my recent post https://www.lotterypost.com/thread/148025 , I illustrated how Benford's Law can be shown to apply to the skips found within any Pick 3 game. The graphs shown within that post reference each of the listed states game from a historical perspective...meaning that the entire draw history of each state was used to gather the skip data.
However, the entire draw history does not necessarily have to be used in order to observe the "Law." You can take any states game, that has a good history behind it, and by examining the last 1,000 draws aquire very similar results. Simply put, by looking at how far out (in consecutive games) that each of the last 1,000 numbers drawn was at the time of its last hit, you can see Benford's Law in action.
For example, if you look at the last 1,000 games in Ohio and for each straight that was drawn determine the skip interval of that specific hit to its previous hit, you can observe how the game closely follows the Law .
Graphing out the skips for each of the straights that fell in Ohio for the last 1,000 games produces the following graph:
Of course this same pattern should emerge for any Pick 3 game in any given state. Here is Georgia as a second example:
The reason that the skips will so closely follow this pattern is that because at any givin point in time, there will be very close to 1/3 of the 1,000 possible Pick 3 straights capable of ending their skip at a value that is in accordance to Benford's Law!
THE BENFORD MATRIX
I dubbed this special group of 1/3 the "Benford Matrix" since it encompasses all of the straighs whos skips are out at a value that starts with the leading digit of one. The skip ranges for these straights include the following:
1 (last straight drawn)
10 through 19 games ago
100 through 199 games ago
1,000 through 1,999 games ago
I call any and all straights that have not been drawn for an amount of consecutive games that fall within any of these specified ranges the "Benford Numbers." The numbers that make up this group or matrix are unlike any other. In grouping numbers to be tracked, the most common approach is to group the numbers based on the digits they contain as well as their high/low and even/odd value, their sum or root sum value and also sometimes their frequencies.
The Benford Matrix is quite a bit different. These numbers are added to the group based on how long its been since they each hit. The group is not static either: it slightly changes each draw, but over the course of time, the numbers it contains changes dramatically. I have heard it said (don't remember exactly where) that constantly changing the numbers that you play leads to more wins than playing the same numbers repeatedly. Judging by the performance of the Benford Matrix this might possibly be true.
The Benford Matrix contains an excellent mixture of numbers. As I stated before, there are about 333 numbers in this group at any given time. Around 230 of this amount are the straights that are out between 1000 and 1999 games. These are the "due" to "over-due" collection of numbers within the group. Normally, there are also around 85 numbers in this 1/3 that just hit between 100 to 199 games ago. These 85 numbers, in conjuction with the 9 or 10 that just hit between 10 to 19 games ago...as well as the last combo to be drawn one game ago, comprise the numbers that are set to have short skips or repeats. So, the Benford Matrix is made from a mixture of both recently drawn and long-out numbers.
Think of randomness as the controlling mechanism, or engine, of the game. The exhaust created by the engine directly fuels this group of numbers...its like a turbo charged lottery lol. The same randomness that creates the daily results is the very same randomness that is creating the Benford group of numbers!
I did some testing in Ohio to see how the group would hit over the course of time. Instead of testing it for an exact 100 or even 1,000 consecutive games, I did something a little different. I tested it over several median spans. A median span is a 693 consecutive game span of time, which is the amount of time (games) it takes to have a 50% chance for any straight to be drawn, this also means that 500 unique straights are drawn in this time. Now, since the group is usually around 333 numbers in size, I expected it to win about about 1/3 of all games...and it did. For 693 games, 1/3 equates to about 231 hits. The actual probability for hits from the benford group sum up to be a little over 32% rather than 33.3% or 1/3, so the more precise figure of .32*693=225.77 should be used.
Starting with the last number drawn, I looked back 693 games to see how many hits were from the Benford group. I recorded this number and also how many of them were unique (I wanted to see how many DIFFERENT numbers were hitting from the group, not just how often the group actually hit). Once i did this test (starting with the most recent game) I repeated the same test starting with the second to last game, again examining 693 games backwards. After recording the results, I performed the test a third time...starting with the third most recent game and then a fourth time with the fourth most recent game as so on. In the end, I had tested 7,971 seperate (but connected by overlap) spans of 693 games.
The Benford group hit an average (and a very tight one at that) of 227.49 times within every 693 consecutive game span. That equates to 32.82%, which is right on the money for the probabilities involved. Now for the interesting part. The amount of unique combos that hit from the group in each span was a very tight average of 205.44. This shows the the amount of different numbers hitting from the group is extremely high as a percentage (62%). If you take any static (unchanging) group of numbers and watch their performance over a 693 game span, only about 50% of them should hit. The unique hits from the Benford Matrix hit at a much higher rate because the numbers played are constantly changing.
To me, perhaps the most interesting aspect of all this is the ratio of unique hits to the size of the matrix they come from.
205.44 * 1.618 (Phi) = 332.401
It could be something, or it could be nothing. More to follow...