# Powerball Systems Based on Skips: Pools of Regular Numbers and Power Balls

## By Ion Saliu, Founder of Powerball Mathematics

Worked by Ion Saliu on September 17, 2005; redone on April 29, 2006 for the new Powerball format '5/55 + 1/42'.

## 1. Powerball Strategy Based on Two Pools of Numbers Derived from Skips

Two things were noticeable. My lottery and lotto strategies do not stress groups (pools) of numbers. Secondly, the Powerball, Mega Millions, Thunderball, SuperLotto games fall way behind in attention; I treat other lotto/lottery and gambling games with a whole lot more interest.

My philosophy is that the combination is the fundamental element. The lotto or lottery game draws combinations (or sets in the pick games). The players also play combinations or sets. Therefore, the most natural way for software is to generate lotto, lottery combinations in the native format of the game. I decided the other day to do a test for the Powerball 5/53/42 game. I have the complete results (drawings) file — from the beginning of the game format through September 16, 2005. Of course, I don't do many things without software! I used in–house software similar to the latest version of SKIPS (but with more bells and whistles).

Lotto software, covering many lottery games, is now available to the public: SkipSystem.

Then the test stopped abruptly. I was not aware of the change in the game format. On August 28, 2005 the format changed from (5/53 + 1/42) to (5/55 + 1/42). It takes some time for sufficient data to accumulate. At the time of this writing, the new Powerball game has a database containing 69 drawings.

There are two fundamental elements in this strategy:
1) The most recent skips of every Powerball number
2) The range of analysis.

The favorable skips of the latest data trend are calculated in accordance with the Fundamental Formula of Gambling (FFG). FFG offers a multitude of methods in this regard. I chose one method that offers the highest probability of repeats and also the lowest number of elements in the pool (the lotto numbers to play).

The range of analysis can vary. Any particular range of analysis is called parpaluck for the sake of simplification. For example, parpaluck–1 defines a range of analysis equal to N * 1 ; i.e. a range equal to total numbers in the lottery game (N=55 in the current Powerball game). Parpaluck–3/4 defines a range of analysis equal to N * .75 ; i.e. a range equal to 75% of total numbers in the lottery game (42 in the current Powerball game).

Ion Saliu's Paradox of N Trials is at the foundation of any range of analysis. The parpaluck equal to one (or N) is the showcase of 'Ion Saliu's Paradox of N Trials'. There are cases when common sense is utterly wrong. We were taught in school (or was we?) that if an event has a probability equal to 1/N, we will achieve one success in N trials. NOT! NON! In truth, the degree of certainty has the limit equal to 1 – 1/e, when N tends to infinity. The number e is the seed of the Universe and also the base of the natural logarithm (e = 2.718281828…... The parameter (1 – 1/e) equals approximately 0.63212057….

Let's take a different look at Ion Saliu's Paradox of N Trials applied to this Powerball game format. The lottery commission draws 5 regular numbers out of 55. The individual probability of each regular number is 5/55 = 1/11. Use SuperFormula to calculate the degree of certainty DC for various probabilities p and number of trials (drawings) N.

The degree of certainty is 65% that any regular Powerball number will be drawn in 11 drawings. Equivalently, around 65% of the Powerball numbers will come in 11 trials: 36 numbers.

The degree of certainty is 44% that any regular Powerball number will be drawn in 6 drawings. Equivalently, around 44% of the Powerball numbers will come in 6 trials: 25 numbers.

The degree of certainty is 58% that any regular Powerball number will be drawn in 9 drawings. Equivalently, around 58% of the Powerball numbers will come in 9 trials: 32 numbers.

We can set other parpalucks as well, based on the Fundamental Formula of Gambling. We can set a degree of certainty DC = 99% for the individual probability p = 1/11. The number of draws (trials) for the two parameters is 49. The modulo 49 for a total of 69 drawings is 20. We delete the top 20 draws in the data file and generate a skip report for a parpaluck equal to 49.

I worked with three parpalucks: N, N/2, and N * (.75). The mathematical operator MODULO is applied. If the Powerball data file consists of 207 drawings, parpaluck N has a modulo equal to 42 (207 – 55*3 = 207 – 165 = 42).

In the current Powerball situation, parpaluck N*1 had a modulo equal to 14. I eliminated the top 14 draws from the current Powerball data file. I saved the file under a different name (e.g. PB.1). I ran UtilPB and made the DPB big data file, with PB.1 on top of the SIM–PB simulated data file. I ran SKIPS with the bells and whistles for parpauluck N. It automatically created a file with the best pool of numbers to play: SkipPB. I ran UtilPB again and checked for winners directly in the pool hosted by SkipPB.

NB ~ Get the latest version of Utility PowerBall. It checks for winning numbers not only combination by combination, but also in groups (pools) of numbers directly. The same is true for other lottery utility software: Util332, Util432, Util532, Util632.

This is the module in SkipSystem applicable to Powerball, Mega Millions, or any other lotto game of the 5+1 type.

In the current Powerball case, parpaluck 1/2 had a modulo equal to 14. I eliminated the top 14 draws from the current Powerball data file. I saved the file under a different name (e.g. PB.2). I ran UtilPB and made the DPB big data file, with PB.2 on top of the SIM–PB simulated data file. I ran SKIPS with the bells and whistles for parpauluck N/2 (range of analysis equal to 28). It automatically created a file with the best pool of numbers to play: SkipPB. I ran UtilPB again and checked for winners directly in the pool hosted by SkipPB.

In the current Powerball situation, parpaluck 3/4 had a modulo equal to 27. I eliminated the top 27 draws from the current Powerball data file. I saved the file under a different name (e.g. PB.3). I ran UtilPB and made the DPB big data file, with PB.3 on top of the SIM–PB simulated data file. I ran SKIPS with the bells and whistles for parpauluck N*.75 (range of analysis equal to 42). It automatically created a file with the best pool of numbers to play: SkipPB. I ran UtilPB again and checked for winners directly in the pool hosted by SkipPB.

The winning cases are shown in the above order. The Powerball data file was checked for winners in its entirety. The future is represented by the top 14 and the top 27 drawings, respectively.

In one type of strategy, the player applies the pools in the SkipPB for N future draws.

In another type of strategy, the player looks for the most recent hit of the strategy. The highest prize is taken into account. For example, in the N*1 case, the most recent big hit was recorded 21 Powerball drawings ago. We can't apply it, because there is an insufficient number of remaining drawings (48 – we need at least 55 for this parpaluck).

In the parpaluck N/2 case, the most recent big hit was recorded 16 drawings ago. We make draw #16 the first line in the DPB data file. Run SKIPS for a range of analysis equal to 28. Play the pool for the next N drawings.

In the parpaluck N*.75 case, the most recent big hit was recorded 34 drawings ago. We can't apply the system, because there is an insufficient number of remaining drawings (35 – we need at least 42 for this parpaluck).

The N/2 does offer the best results, but also the largest pool. Based on FFG, the lotto numbers tend to repeat more often when their current skip is below the FFG median. Other filters must be applied in order to make these strategies playable. Some lottery experts would strongly recommend you wheel the numbers! I strongly advise against wheeling the lotto numbers, including Powerball. Instead, you'd be better off randomly selecting a number of combinations equal to the number of lines in a corresponding Powerball wheel. I showed factual evidence that the lotto wheels diminish the chance to win the highest prizes, especially the jackpot.

At this point in time, MDIEditor Lotto WE is, by far, your best Powerball tool. Honestly, everything else in the field is dust in your eyes. Expand the pool of Powerball numbers in SkipPB to Powerball combinations. Take that result file and feed it to the Purge option in MDIEditor Lotto. Set applicable filters based on the median and the Fundamental Formula of Gambling. Look, for example, for 3 consecutive +/–. Chances are there will be a trend reversal.

This is a raw strategy. For example, the following situations are included in the outcome:
- all lotto numbers can repeat after 1 draw;
- all Powerball numbers can repeat after 2 draws;
- all Powerball numbers can repeat after 3 draws;
etc.

In reality, 0 or 1 or 2 Powerball numbers repeat after 1 draw (we include here both the regular Powerball numbers and the powerballs). 0 or 1 or 2 lotto numbers repeat after 2 draws; 0 or 1 or 2 lotto numbers repeat after 3 draws; … 0 or 1 or 2 or 3 lotto numbers repeat after 10 draws; only 0 or 1 or 2 Powerball numbers have equal SKIPS; etc.

That is, a lot of fine-tuning can be performed. You can analyze the skip reports generated by that phenomenal software of mine, SKIPS. Here is a fragment:

``` Line    Any  Any  Any  Any  Any  Any       Pos  Pos  Pos  Pos  Pos  Pos
no.      1    2    3    4    5    PB        1    2    3    4    5    PB

7     18   17   14   11   10   16        46   38   46   46   10   16
8     7    3    11   33   22   34        46   46   46   33   22   34
9     1    46   13   5    17   3         1    46   18   5    17   3
10     10   1    5    2    10   27        10   46   46   46   10   27
11     46   1    12   2    8    33        46   46   12   2    8    33
```

There are situations when most Powerball lotto numbers have very low skips. Golden opportunities such as this one:

1 4 5 3 25 10... ANY skips for drawing 2 3 12 47 51 10.

Line #1 in the data file at the start of the test:
12 24 26 48 55 14

Following that drawing, and up to drawing:
14 40 47 54 55 6 (July 22, 2006):

Most significant hits for parpaluck #1:
5 regular & 0 power ball in draw: 14 40 47 54 55 6
4 regular & 0 power ball in draw: 4 14 15 40 48 12

Most significant hits for parpaluck #2:
5 regular & 0 power ball in draw: 14 40 47 54 55 6
4 regular + the power ball in draw: 8 11 30 39 47 23
4 regular & 0 power ball in drawings:
10 20 22 39 48 25
4 14 15 40 48 12
10 20 22 43 49 3
19 28 30 47 54 13

Most significant hits for parpaluck #3:
4 regular & NO power ball in drawings:
18 19 30 31 33 15
4 6 19 38 42 37

* The system hit at least 4 (four) Powerball JACKPOTS as of August 18, 2007 (in a 20-drawing span: draw #3, #8, #9, #20). *

The range of analysis (The Parpaluck) is of the essence. There is an optimal past that influences future the most. Various periods of the past exert various influences on the future.

## 2. How The Powerball Strategy Fared In The Past: Checking For Winners In The Pools Directly

```              POOL-PB Winning Number Checking - With 0 PBall
Group of 22 / 6 Numbers = PBall Combinations:  158004
Data File: PAPB-55 (69)
Skips for N (55 draws)

The Pool:  3  7  10  13  14  15  25  30  35  37  38  39  40  43  46  47  48  51  52  53  54
55 = regular numbers
& PBs:  7  14  20  26  37  41 = Powerball numbers

Line      Drawings                            5+    5&    4+     4&     3+    3&     0&
no.       Checked                             PB    0     PB     0      PB    0      PB

13    7 12 36 48 54  37        in draw #                               13
14   16 35 37 52 54  33        in draw #                        14
16   13 22 27 52 55  26        in draw #                               16
17   10 25 35 38 41  23        in draw #                        17
18    3 37 46 48 51  22        in draw #           18 *
19    9 23 47 53 55  39        in draw #                                     19
20   15 17 43 44 48  29        in draw #                                     20
21    3 13 35 39 40  37        in draw #     21 *
22    7 38 40 51 52  14        in draw #     22 *
23   14 25 32 48 54  10        in draw #                        23
24    4 12 13 30 38  30        in draw #                                     24
25   11 13 30 35 55  37        in draw #                 25
27   10 37 43 50 52  14        in draw #                 27
28    3 18 39 43 54  41        in draw #                 28
29   15 29 38 52 53   7        in draw #                 29
30    6 14 40 46 49  17        in draw #                                     30
33   10 23 42 48 54   3        in draw #                                     33
34    4 30 37 43 46  42        in draw #                        34
35   12 15 30 46 54   1        in draw #                        35
48   17 37 41 48 55  39        in draw #                                     48
50    7 25 29 37 50  17        in draw #                                     50
53    2 10 39 42 43  23        in draw #                                     53
54    6  7 36 51 53  17        in draw #                                     54
59    2 15 25 30 45  17        in draw #                                     59
62    2 10 31 43 55  14        in draw #                               62
68    1  3 30 31 43  36        in draw #                                     68

Total Hits:                              2     1     4      5      3     11     0

POOL-PB Winning Number Checking - With 0 PBall
Group of 30 / 12 Numbers = PBall Combos:  1710072
Data File: PAPB-55 ( 69 )
Skips for N / 2 (28 draws)

The Pool:  1  3  5  7  10  11  13  14  15  22  25  27  28  29  30  35  37  38  39  40  43
46  47  48  49  51  52  53  54  55 = regular numbers
& PBs:  7  10  14  17  20  22  23  26  29  30  37  41 = Powerball numbers

Line      Drawings                            5+    5&    4+     4&     3+    3&     0&
no.       Checked                             PB    0     PB     0      PB    0      PB

3    5 28 32 34 53  10        in draw #                               3
4   13 18 23 28 46  33        in draw #                                     4
7    3  9 24 29 39  42        in draw #                                     7
8    1 10 23 50 55  29        in draw #                               8
9    2 14 24 28 54  13        in draw #                                     9
13    7 12 36 48 54  37        in draw #                               13
14   16 35 37 52 54  33        in draw #                        14
15    1  5  7 21 43  20        in draw #                 15
16   13 22 27 52 55  26        in draw #     16 *
17   10 25 35 38 41  23        in draw #                 17
18    3 37 46 48 51  22        in draw #     18 *
19    9 23 47 53 55  39        in draw #                                     19
20   15 17 43 44 48  29        in draw #                               20
21    3 13 35 39 40  37        in draw #     21 *
22    7 38 40 51 52  14        in draw #     22 *
23   14 25 32 48 54  10        in draw #                 23
24    4 12 13 30 38  30        in draw #                               24
25   11 13 30 35 55  37        in draw #     25 *
26   14 15 26 28 36  36        in draw #                                     26
27   10 37 43 50 52  14        in draw #                 27
28    3 18 39 43 54  41        in draw #                 28
29   15 29 38 52 53   7        in draw #     29 *
30    6 14 40 46 49  17        in draw #                 30
33   10 23 42 48 54   3        in draw #                                     33
34    4 30 37 43 46  42        in draw #                        34
35   12 15 30 46 54   1        in draw #                        35
44   28 34 37 45 53  32        in draw #                                     44
48   17 37 41 48 55  39        in draw #                                     48
49    5 13 14 31 49  30        in draw #                 49
50    7 25 29 37 50  17        in draw #                 50
51    6 11 14 27 41  38        in draw #                                     51
53    2 10 39 42 43  23        in draw #                               53
54    6  7 36 51 53  17        in draw #                               54
55    7 21 43 44 49  29        in draw #                               55
59    2 15 25 30 45  17        in draw #                               59
61    7 22 27 48 50  17        in draw #                 61
62    2 10 31 43 55  14        in draw #                               62
63    3 28 34 35 50   8        in draw #                                     63
64    5  9 27 49 52   6        in draw #                        64
66    8 10 11 48 50  16        in draw #                                     66
68    1  3 30 31 43  36        in draw #                        68

Total Hits:                              6     0     9      5      10    11     0

POOL-PB Winning Number Checking - With 0 PBall
Group of 24 / 8 Numbers = PBall Combinations:  340032
Data File: PAPB-55 ( 69 )
Skips for N * .75 (42 draws)

The Pool:  2  3  4  6  10  12  14  15  16  18  19  21  23  30  33  34  37  42  43  45  46
52  53  54 = regular numbers
& PBs:  1  3  7  11  14  18  41  42 = Powerball numbers

Line      Drawings                            5+    5&    4+     4&     3+    3&     0&
no.       Checked                             PB    0     PB     0      PB    0      PB

2    4  5  6 18 27  24        in draw #                                     2
4   13 18 23 28 46  33        in draw #                                     4
6    4 23 35 43 44  35        in draw #                                     6
9    2 14 24 28 54  13        in draw #                                     9
10   20 34 37 39 45  32        in draw #                                     10
11   18 22 23 38 45  31        in draw #                                     11
14   16 35 37 52 54  33        in draw #                        14
18    3 37 46 48 51  22        in draw #                                     18
24    4 12 13 30 38  30        in draw #                                     24
27   10 37 43 50 52  14        in draw #                 27
28    3 18 39 43 54  41        in draw #                 28
29   15 29 38 52 53   7        in draw #                               29
30    6 14 40 46 49  17        in draw #                                     30
31    3  6 14 17 18  14        in draw #                 31
32    2 16 21 32 43   6        in draw #                        32
33   10 23 42 48 54   3        in draw #                 33
34    4 30 37 43 46  42        in draw #     34 *
35   12 15 30 46 54   1        in draw #     35 *
36    4 10 33 34 50  14        in draw #                 36
37    9 12 16 23 43  37        in draw #                        37
39    8 15 16 45 51  11        in draw #                               39
40    3 21 34 42 45  39        in draw #           40 *
41   19 30 41 45 54  14        in draw #                 41
42    2 15 26 37 42  36        in draw #                        42
43   19 23 27 42 52   1        in draw #                 43
44   28 34 37 45 53  32        in draw #                        44
53    2 10 39 42 43  23        in draw #                        53
57    4  9 18 51 54  10        in draw #                                     57
59    2 15 25 30 45  17        in draw #                        59
60    8 18 38 45 54  23        in draw #                                     60
62    2 10 31 43 55  14        in draw #                               62
68    1  3 30 31 43  36        in draw #                                     68

Total Hits:                              2     1     7      7      3     12     0
```

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