Given a range of drawings, each lotto number shows a clear bias towards being drawn with the rest of the lotto numbers. The newest version of my Super Utilities calculates the frequencies of all lotto pairings for every lotto number. It is a major component of Bright6.exe (2011), definitely the most powerful collection of lotto software applications. The following pairing report was done by function F = Frequency Reports by Lotto Number.
The range of analysis is essential and it depends on the purpose of the report or analysis. This truth became evident the first time when I introduced the wonder grid for lotto-6:
Killer lotto strategy? Pairs, pairings, frequency, wonder grid lottery.
I've seen very different results for different ranges of analysis. The purpose of analysis for the wonder-grid was the number of drawings necessary to count the pairs and to select the top-5 pairs for every lotto-6 number. The wonder grid loses it almost completely, if the range (or parpaluck) is too long.
I discovered also another essential element: The starting point of analysis. That is, what is the best past drawing where I should start tracking the pairs' frequencies? In other words, I should delete all the top drawings in the data file up until the most favorable past lottery draw. The probability is the highest for the draw that HIT LAST for the respective strategy. The last hit (or the highest last winning) represents the PIVOT draw. That drawing becomes line #1 in a data file (e.g. Data-6.2; Data-6 is the original data file; it is up-to-the-date).
I presented a series of lotto wonder-grid reports on this page: Lotto Wonder Grid Revisited: New Pairing Research.
PICK-3 Winning Number Checking
Files: OUT3 ( 19 ) against PA-3 ( 100 )
Line Combination Straight Boxed
no. Checked Winner Winner
1 0 4 9 in draw # 22
1 0 4 9 in draw # 77
2 0 9 4 in draw # 22
2 0 9 4 in draw # 77
4 1 1 6 in draw # 2
4 1 1 6 in draw # 67 *
6 1 6 1 in draw # 2 *
6 1 6 1 in draw # 67
7 1 6 8 in draw # 93 *
9 1 8 6 in draw # 93
10 4 0 9 in draw # 22 *
10 4 0 9 in draw # 77
11 4 9 0 in draw # 22
11 4 9 0 in draw # 77 *
12 6 1 1 in draw # 2
12 6 1 1 in draw # 67
13 6 1 8 in draw # 93
14 6 8 1 in draw # 93
16 8 1 6 in draw # 93
17 8 6 1 in draw # 93
18 9 0 4 in draw # 22
18 9 0 4 in draw # 77
19 9 4 0 in draw # 22
19 9 4 0 in draw # 77
Total Hits: 5 19
I get the same number of straight hits consistently, for either 18 or 19 combinations. No other strategy is involved. Even if playing the same grid every draw for the next 100 draws, the COW (cost of winning) is 1800 (1900). The five wins amount to 2500. The result is profit. Not only does this strategy beat random play, it also beats the monstrous house edge or house advantage (HA) imposed by the lottery (50%)! Random play will yield 1.8 straight wins (1800 / 1000).
Of course, the skips are omnipresent. Nothing can avoid skipping it's mathematical! The strategy above does not hit immediately in most situations. The player can safely sit out 5 drawings between hits. But this pick-3 pair strategy should hit within the next 12-13 draws, based on my FFG calculations. The hit in draw #93 * in the report above indicates a hit after 7 drawings (100 93).
I prefer changing the pairing sets after the first hit. I redo the wonder grid at the winning drawing. This is a very favorable case. I skipped 5 draws; I paid for 2 draws: $38. I won straight. I discard of the lottery strategy. If I continued to play, I would skip 5 drawings again. 93 77 5 = 11. 18 * 11 = $198: Still a good outcome.
The individual probability, p, is the cornerstone of FFG. The probability must be calculated as accurately as possible. The degree of certainty, DC, is also important, but secondary to p. I noticed serious discrepancies for different probabilities.
Many here remember that I did not have an answer for the length of the report for the lottery wonder-grid. I tried all kinds of ranges, such as N/2, N*2, N*3, up to N*5 (N represents the largest number in the lotto game; N=10 in the pick 3, 4 lotteries).
There are ways to calculate p very precisely. The probabilities are different if one lotto number (digit) or lottery pairs are considered. I did not make such a distinction early in the (lotto pairing) game.
1) Probability for one lotto number (lottery digit)
This is the easiest case. We can calculate easily total number of combinations (straight pick-3, 4 sets) for the game. Then, we need to calculate how many of the combinations (sets) contain one particular number.
For pick-3: each digit appears in 271 straight sets. p = 271/1000
For pick-4: each digit appears in 3439 straight sets. p = 3439/10000
For lotto games: I will exemplify for 6/49 lotto games.
One lotto number is combined with the rest of the numbers (N-1) taken 5 at a time. C(48, 5) = 1712304 combinations. That's how many times one particular 6/49 lotto number appears in the total amount of combinations. P = 1712304 / 13983816 = 1 in 8.17.
That is also the result of 49/6. In the lotto cases, p for one number is more simply calculated as M/N (e.g. 6/49; the reverse, 49/6 is the probability expressed as 1 in something).
2) Pair probability
Calculations are trickier here.
For the pick games, the number of pairs is equal to C(10, 2) + 10 = 45 + 10 = 55 pairs; p = 1/55
For a lotto 6/49 game. Each 2-number group (lotto pair) is combined with the rest of the lotto numbers (N-2) taken 4 at a time. C(47, 4) = 178365 combinations. (Incidentally, that's how many combinations are eliminated by the TWO filter in Power6.EXE for min_TWO = 1).
p = 178365 / 13983816 = 1 in 79.
3) Choosing the degree of certainty (DC)
I'll give more details for pick-3.
I want to work with integers regarding total number of digits. I convert DC in FFG to number of digits that show up within various numbers of draws N.
I do the calculations using SuperFormula.EXE, option F FFG). I select case 2 = The program calculates p. Type 271 for the 1st element, 1000 for the 2nd element. Type the degree of certainty as an integer, not percentage.
p = 271/1000
DC = 30 = N = 2 : 3 digits
DC = 50 = N = 3 : 5 digits
DC = 70 = N = 4 : 7 digits
DC = 80 = N = 6 : 8 digits
I take the pivot draw the one with the last hit regarding playing the most frequent numbers.
The DC=30% generated 4 digits with frequencies above 0. Two of the digits repeated the next drawing (DC=50, N=3). The same three digits repeated a total of 4 times in the next 4 drawings, including a boxed combination (N=6, DC=80). The probability p for pick-3 is calculated as regardless of position, so only the boxed play is valid.
Pick-3 pair analysis:
p = 1/55
DC = 25 = N = 16 : 14 pairs
DC = 51 = N = 39 : 28 pairs
DC = 76 = N = 78 : 42 pairs
From N=16 to N=39 (a balance of 13 drawings) the pair strategy should hit. It's important to do the grid for the pivot draw, as in the pick-3 report at I.
Actually, in the report above I generated the top-3 pairs, while eliminating the worst-7 pairs. That is a tough strategy: it is budget-oriented.
Lotto 6/49 pair analysis:
p = 1/79
DC = 26 = N = 24 : 20 pairs
DC = 50 = N = 54 : 39 pairs
DC = 74 = N = 105 : 58 pairs
So, I believe the optimal range for the lotto-6 wonder grid should be done for 24 drawings back, starting at the pivot draw. The pairs should register a major hit within the next 54 24 = 30 drawings, or thereabouts.
It would very interesting to see other people's results. Super Utilities allow generation of lotto combinations based on the best pairings, while eliminating the worst lotto pairs. Option M = Make/Break/Position, then Break 5/6).
Hopefully, other people will contribute here. There will be NO copyright infringement in this case!
p = 271/1000
DC = 30: N = 2 : 3 digits
DC = 50 : N = 3 : 5 digits
DC = 70 : N = 4 : 7 digits
DC = 80 : N = 6 : 8 digits
There is a strong correlation between the degree of certainty and number of elements drawn. The probability is presented in the Ion Saliu's Paradox of N Trials. If there are N elements in a set and the individual probability is expressed as p = 1/N, then the degree of certainty for one element to appear tends to 1-(1/e) or approximately 63.2%. At the same time, we will notice that approximately 63.2% of the elements come out in N trials. That's because a number of elements repeat, while other elements do not come out. My program OccupancySaliuParadox.EXE proves that correlation between DC and number of elements drawn.
There are 10 pick-3 digits. 30% means 3 digits. I listed the number of digits to show how many digits to expect to be drawn in various numbers of draws. Usually those values do not match in reality, when the number of drawings is too small. But if you do for 10 pick-3 drawings, you'll notice that 6 or 7 digits come out, while 4 or 3 digits don't show up. DC and number of digits come close to each other.
If you generate a TOP5 grid file, it usually starts at pair #1. You can start at another position; e.g. start at pos. #4, with 12 pairs. Pair #4, 5, #15.
I go now with position #1. That's how I did it for pick-3. Digit, plus pos. 1, 2, 3. Then, I eliminated pairs 4, 5, 6, 7, 8, 9, 10 (WORST7). I saw also some GREAT results for lotto-5.
In Break of Super Utilities I use the option that has one fixed number (digit): The first one in each line.
For pick lottery, I usually start at draw #50; also, at drawing #100.
The starting point makes a huge difference. I started at line #1 in my pick-3 data file (November 2006). I recorded 22 straight wins for parpaluck = 16. I deleted the top 100 draws in Pa-3 and saved the file as Pa-3.2. I chose the draw for the oldest hit in the winning report. I recreated the BREAK3.2 output file for the same parpaluck. This time, I recorded 39 straight hits. I chose the next parpaluck for the same PA-3.2 data file. I also applied the worst-7 pair elimination. The results were much, much better:
PICK-3 Winning Number Checking (Parpaluck = 39)
Files: BREAK3.2 (6) against PA-3 (100)
Line Combination Straight Boxed
no. Checked Winner Winner
1 1 5 3 in draw # 87
1 1 5 3 in draw # 93
2 1 3 5 in draw # 87 *
2 1 3 5 in draw # 93
3 5 1 3 in draw # 87
3 5 1 3 in draw # 93 *
4 3 1 5 in draw # 87
4 3 1 5 in draw # 93
5 5 3 1 in draw # 87
5 5 3 1 in draw # 93
6 3 5 1 in draw # 87
6 3 5 1 in draw # 93
Total Hits: 2 10
I think this method offers the best results. Of course, people will try other positions and other ranges. The key fact is mathematical, though: Some numbers repeat in the near future, while other numbers don't show up. The repetition and non-appearance are different from range to range. It is the relation between p, DC and N in the Fundamental Formula of Gambling (FFG).
I created a special page with the parameters calculated with SuperFormula.EXE. The tables cover the following games: pick-3, pick-4, lotto 5/39, and lotto 6/49. For other lottery formats, you need to edit the source file (the HTML code). Insert your calculations in the corresponding table cells. Print the tables for handy reference.
FFG: Numbers (Digits), Degree of Certainty, Number of Trials (Drawings).
I am not saying that it's me who has the grasp of lottery pairing. I am still shocked how well the pairings can perform at some points in the history file. As the example above, I go back 100 lottery drawings. The pairs perform very well even for the previous 100 real drawings. Those previous 100 were out of the range of analysis (parpaluck).
There is another method of working with the parpaluck. I presented it with the Powerball skip system:
Powerball, Mega-Millions strategy, system, based on pools of numbers derived from skips.
But for that you need a complete history file: From the beginning of the lottery game, without interruption.