Findlay, Ohio United States Member #4855 May 28, 2004 400 Posts Offline

Posted: January 6, 2007, 10:03 pm - IP Logged

In my recent posthttp://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...

Findlay, Ohio United States Member #4855 May 28, 2004 400 Posts Offline

Posted: January 14, 2007, 3:10 pm - IP Logged

Using the Benford Matrix

As I stated above, there are usually about 333 straight numbers in the Benford Matrix at any given time. These numbers are the ones that are actually capable of hitting (ending their skips) at value that starts with the digit one. At this moment, there are 341 straights in this matrix for Ohio.

Here is Ohio's current matrix. The combos are listed in ascending order...the number to the right of each is how many consecutive games ago it was last drawn:

Here are some facts to keep in mind while anticipating numbers from the matrix to hit:

When you see a Benford Number hit, there IS about a 55% chance that another hit from the matrix will occur within the next 2 games.

When a Benford Number hits, there is about a 70% chance that another hit from the matrix will occur within the next 3 games!

90% of all Benford hits occur within 6 games of their last hit.

These percentages are based on probability and should apply to the specific matrices of each state. The statistical performance of the probabilites are right on par in Ohio.

In Ohio, the last four consecutive games have not consisted of numbers from the Benford Group, so watch for one to hit here very soon. When the next Benford Number actually does hit, I would anticipate another hit within the next two games afterwards because that 55% chance/event is extremely due!

341 numbers is way too many to play. The idea is to use the group as a pool to choose numbers from. Since digits, root sum and other such measurements should more or less be equally distributed across the matrix, it will help to use other indicators while choosing. For example, in position-two (Ohio), the digit 2 has not been drawn for 59 games now. Banking on a Benford Number with a middle digit 2 will lead you to these 31 numbers from the matrix above:

United States Member #28945 December 25, 2005 1540 Posts Offline

Posted: January 16, 2007, 3:02 pm - IP Logged

Always enjoy your posts Thoth, and the work that goes into them. This one is no exception - very instructive and confirms much.... Thank you.

"In the Beginning was the Point". According to Lottery Lore, these are the only words written in the long-lost ancient Pick 3 Codex - The Scroll of Harmony .Those who came later, The Lottery LoreKeepers, intuited the rest of the riddle as they charted the Point's evolution through TIME by connecting its periods and exposing relationships that predict the next draw(7th): "The Point(1st draw) was born, began to grow and in its bold adolescence became the line(2nd draw), knew young adulthood in the triangle(3rd draw), stable maturity by the square(4th draw), older age as the pentagon(5th draw), and embraced 'death' in the hexagon(6th draw). Here lies the CIrcle of a lottery draw's life and it is Immutable; For as the Point seeds, flowers and fruits, it is changed but essentially unchanged, as it forever rises in ReBirth(7th draw)".

ORLANDO, FLORIDA United States Member #4924 June 3, 2004 6065 Posts Offline

Posted: January 16, 2007, 6:44 pm - IP Logged

Quote: Originally posted by Thoth on January 14, 2007

Using the Benford Matrix

As I stated above, there are usually about 333 straight numbers in the Benford Matrix at any given time. These numbers are the ones that are actually capable of hitting (ending their skips) at value that starts with the digit one. At this moment, there are 341 straights in this matrix for Ohio.

Here is Ohio's current matrix. The combos are listed in ascending order...the number to the right of each is how many consecutive games ago it was last drawn:

Here are some facts to keep in mind while anticipating numbers from the matrix to hit:

When you see a Benford Number hit, there IS about a 55% chance that another hit from the matrix will occur within the next 2 games.

When a Benford Number hits, there is about a 70% chance that another hit from the matrix will occur within the next 3 games!

90% of all Benford hits occur within 6 games of their last hit.

These percentages are based on probability and should apply to the specific matrices of each state. The statistical performance of the probabilites are right on par in Ohio.

In Ohio, the last four consecutive games have not consisted of numbers from the Benford Group, so watch for one to hit here very soon. When the next Benford Number actually does hit, I would anticipate another hit within the next two games afterwards because that 55% chance/event is extremely due!

341 numbers is way too many to play. The idea is to use the group as a pool to choose numbers from. Since digits, root sum and other such measurements should more or less be equally distributed across the matrix, it will help to use other indicators while choosing. For example, in position-two (Ohio), the digit 2 has not been drawn for 59 games now. Banking on a Benford Number with a middle digit 2 will lead you to these 31 numbers from the matrix above:

Findlay, Ohio United States Member #4855 May 28, 2004 400 Posts Offline

Posted: January 22, 2007, 12:03 pm - IP Logged

Quote: Originally posted by cps10 on January 16, 2007

I would be inclined to wait for a skip of 2 to 3 of a Benford number and then move in for the winner. Progressively, this could be done easily.

Thoth - do you have Ohio's max skip of a Benford number? How easy is this to apply to other states/draws?

CPS10,

Sorry for the delayed response...

The max skip in Ohio for the Benford group is 20 draws. Here is the top 30 largest skips of the group:

Top 30

Skips

1

20

2

20

3

19

4

18

5

17

6

17

7

17

8

16

9

16

10

16

11

15

12

15

13

15

14

14

15

14

16

14

17

14

18

14

19

14

20

13

21

13

22

13

23

13

24

13

25

12

26

12

27

12

28

12

29

12

30

12

This table represents the largest skips of the benford group when examining the draw history from game 2,000 through 10,781. If you start the test at game #1 you will have some much larger skips for the group (Top 3 are: 76, 48, 47). I don't count these because it takes about 2000 draws to start to establish the stratetgy to begin with.

Anywhere & Everywhere United States Member #10713 January 23, 2005 290 Posts Offline

Posted: February 3, 2007, 7:43 pm - IP Logged

Thoth,

Tell me what do you think the significance may be from this test. I just recorded a group of 1,000 recent draws in TX with how many had a sum of 10-19, using the benford group first digit #1, the result was the infamous 666 times. Hmmmm!

Also, I understand I am applying a slightly different perspective for review. However, if I am understanding correctly with your application, you are changing the combos every draw accordingly to various #1 digit skip groups?

Also, more food for thought. There has been a lot of studies done on end digits analysis, like end digit, last digit, short sum, root, etc...I wonder if they can be combined at some point, once this part has been ironed out definitively.

Anywhere & Everywhere United States Member #10713 January 23, 2005 290 Posts Offline

Posted: February 3, 2007, 10:41 pm - IP Logged

Well, I dont want to take this study too far ofcourse, however, I think there is a relationship. One reason why there was 66.6% hit rate with sums 10-19, is that there are more combos in this range than 0-9 and 20-27 range. Although, I have no mathematical factual basics, I believe that the end digits 0-5 are most likely. Therefore, Im thinking skips 11-15 are more likely than skips 16-19. Well probably because of something you said before. If something has been out 2-9 draws, then the further out each draw the greater the probability of occurrence. Over the same period sums 11-15 hit 354x and sums 16-19 hit 241x

Greenwich, CT United States Member #4793 May 24, 2004 1822 Posts Offline

Posted: February 5, 2007, 9:53 am - IP Logged

You could also apply the 'probability of probability' to guessing when to play the Benford 1's, the 2's, the 3's etc., adding another layer.

I think that lottery analysis is like slicing a grapefruit. For Pick 3, imagine dividing the grapefruit into one thousand equal pieces. Think of cutting it down the middle, you will have two equal parts, and you can place a bet on which one is more likely to occur. Then slice horizontally, so you have quadrants. Each filter that you apply provides a slice at a different angle, eventually you're left with a very small piece of grapefruit, hopefully the winning number! Benford's Law can provide a couple of excellent cuts through that grapefruit.

Findlay, Ohio United States Member #4855 May 28, 2004 400 Posts Offline

Posted: February 5, 2007, 9:34 pm - IP Logged

Quote: Originally posted by JAG331 on February 5, 2007

You could also apply the 'probability of probability' to guessing when to play the Benford 1's, the 2's, the 3's etc., adding another layer.

I think that lottery analysis is like slicing a grapefruit. For Pick 3, imagine dividing the grapefruit into one thousand equal pieces. Think of cutting it down the middle, you will have two equal parts, and you can place a bet on which one is more likely to occur. Then slice horizontally, so you have quadrants. Each filter that you apply provides a slice at a different angle, eventually you're left with a very small piece of grapefruit, hopefully the winning number! Benford's Law can provide a couple of excellent cuts through that grapefruit.

I like waiting for one of the "Benford numbers" to hit...then gamble on the 55% chance that another one will hit within the next two games. Of course other factors have to be included to narrow the picks in that group down

Using sums is a very effective way to win in the pick3. The most common group of sums are the 12-15 group. To arrive at a sum just add all three digits together. If you take away doubles you have only 10 numbers to work in each of these 4 groups. In the month of Jan. for NY there were 16 hits from the 12-15 group. Not bad considering there only 40 possible non-match numbers from these sums. Go back the past couple weeks in your state and look for the hottest sum and most due sum from the 12-15 group, and play the numbers in it. For example in NY the sum 13 is the most due to show, while the 15 has been the hottest this year. Each time a number hits in the 12-15 group mark it down, this way you can see whats hot, and overdue. Here is a list of the 12-15 sums, minus the doubles.