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Colin F
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Quote Colin F Replybullet Topic: Count of Integers World’s 649 Lotto Games
    Posted: January 18 2005 at 10:41pm

I recently combined all the World's 649 Lotto's main draw numbers to make a table consisting of some 17,511 Draws. See for yourself how close the tally is for each number to the average of 2144.

Regards
Colin Fairbrother 
Lotto Draws have no relationship to one another; the integers serve just as identifiers. Any prediction calculation on one history of draws for a same type game is just as irrelevant as another.
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Quote Bertil Replybullet Posted: April 17 2005 at 1:03pm

 Hi Colin,

I just became aware of your table of frequencies for 17511 draws

and decided to add my two bits of analysis. There is a std.dev.  of

55.44 from the mean 2144.2. Thus the spread covers 4.1 units

of s.d., which covers 95%.

Bertil
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Quote Bertil Replybullet Posted: August 30 2005 at 8:25am

Hi Colin,

Your table of frequencies for the 1-49 integers seem to yield a high

std.dev. of 55.44 if my calculations are correct. There is a formula

for predicting the variance for any lotto and it yields 43.37 as std.dev

Thus the actual value seems to differ too much from theory.

If you know an expert on probability theory you might ask him to

predict the theoretical variance for any mean lotto frequency.

The formula I was given is V=D*n*(N-n)/(N^2), where D= number

of draws and n/N is the lotto matrix. Personally, I think a better

formula would be :D*n(N-2.5-n)/(N-2.5)^2 for any pick6 game.

Please let me know if you find a formula.

Bertil
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Colin F
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Quote Colin F Replybullet Posted: August 30 2005 at 10:30am

Bertil

Professor Iliya Bluskov puts his name to a book of Covers/Wheels which purportedly improve a Lotto players chances of winning prizes. People believed him and still do.

Then along came Colin who proved him wrong and a few other people too. However, the percentage of Lotto players that look for information beyond that of a few numbers they keep track of on a piece of paper, if that, is extremely small. So we have a situation where the people who do a bit more than the latter know they have been proved wrong but just can't bring themselves to acknowledge it - they probably hope this website and me would fade away and then they can pretend it and me never existed.

Bertil the draws are real; the equations are what?

If you toss a coin there is less than 1 in a million chance of getting 20 heads in a row - but it can happen.

On the 18th of August, 1913 in the famous Monte Carlo casino black was coming up repeatedly. Around the 15 in succession mark the casino was in turmoil - everyone wanted to get on red. So they bet millions on red and lost millions on red because black continued coming up 26 times in succession. The chances of this happening were 1 in 136,823,184. If you were using the results from that week to work out a standard deviation ...

Regards
Colin

 
Lotto Draws have no relationship to one another; the integers serve just as identifiers. Any prediction calculation on one history of draws for a same type game is just as irrelevant as another.
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Quote Bertil Replybullet Posted: August 31 2005 at 7:37am

 

 Colin,

 I'm fully aware of short run deviations from ideallity. That is what

 system players try to take advantage of. But I tested 17511 runs of

 6/49 games and found a substantial deviation from a theoretical

 formula. This makes me wonder if the formula is correct. It is based

 on a simple binomial model. If possible, please ask an expert.

 Bertil
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Colin F
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Quote Colin F Replybullet Posted: November 10 2009 at 8:24pm

Here is a table of AllWorld649 Lotto draws increased to 20,682 in 6 columns as requested by Bertil or Stig Holmquist in another forum and as well, similarly all the combinations of 6 integers enumerated lexicographically. The columns P1 to P6 each sum to the number of combinations, respectively 20,682 and 13,983,816.

Stig Holmquist has been obsessed with applying Standard Deviation measurements to Lotto combinations since the late 90's. The problem is he ascribes magnitude to the identifiers which is the stuff of numerology. Whatever calculation you do in Lotto analysis must be applicable if the balls are identified in some other way than numbers eg pictures of animals or hieroglyphic symbols.
 
The All World table Line Total column has an average of 2532.49, a Variance of 2621.80 and Standard Deviation (Average of the Sum of the Squared Differences to Average) of 51.20. Does this provide anything worthwhile in producing a set of numbers to play Lotto? None that I can see.
 
For the AllCombs table the Line Total column sum is
1,712,304 x 49 = 83,902,896 = 13,983,816 x 6. This table is given just for reference and in no way resembles a table that would be produced for the same number of combinations by Random Selections. Why? Because from the second selection you could have a repeat and this very event occurred recently in Bulgaria. The irrelevance of applying Standard Deviation measurements based on attributing magnitude to the identifiers is made abundantly clear in this table with 0 deviation in the relevant line Total column but still irrelevant deviations in the P1 to P6 columns.  You may notice that Column 1 is the inverse of Column 6 and the sequence is given by the combinations of 5 from a pool increasing from 5 to 49.  
 
ID P1 P2 P3 P4 P5 P6 Line Total Difference to Average
01 2517 0 0 0 0 0 2517 -15
02 2316 247 0 0 0 0 2563 31
03 2053 500 24 0 0 0 2577 45
04 1756 644 75 1 0 0 2476 -56
05 1596 794 101 6 0 0 2497 -35
06 1428 939 191 12 1 0 2571 39
07 1247 997 297 24 0 0 2565 33
08 1089 1030 327 69 3 0 2518 -14
09 1003 1073 397 65 7 0 2545 13
10 842 1136 502 99 9 0 2588 56
11 770 1117 568 137 16 0 2608 76
12 671 1042 657 142 14 1 2527 -5
13 559 1022 726 196 25 2 2530 -2
14 455 1006 746 256 37 1 2501 -31
15 387 925 737 287 50 2 2388 -144
16 351 918 857 352 63 7 2548 16
17 292 831 856 399 93 3 2474 -58
18 266 779 873 456 90 13 2477 -55
19 209 717 874 535 133 9 2477 -55
20 178 680 917 549 151 16 2491 -41
21 144 612 909 577 195 19 2456 -76
22 108 562 880 690 240 25 2505 -27
23 108 480 893 725 254 29 2489 -43
24 81 424 848 729 352 45 2479 -53
25 48 364 894 835 375 58 2574 42
26 54 331 770 902 402 82 2541 9
27 32 290 731 924 510 79 2566 34
28 30 246 705 906 544 102 2533 1
29 23 200 643 892 591 126 2475 -57
30 24 170 582 887 706 179 2548 16
31 16 147 519 974 724 217 2597 65
32 6 124 457 937 767 251 2542 10
33 6 97 430 869 868 301 2571 39
34 7 66 371 833 947 338 2562 30
35 7 60 306 836 903 432 2544 12
36 1 42 251 738 1010 464 2506 -26
37 0 16 207 681 1010 560 2474 -58
38 1 18 169 690 1142 654 2674 142
39 0 19 128 581 1058 741 2527 -5
40 1 8 95 460 1107 883 2554 22
41 0 4 69 426 1076 965 2540 8
42 0 3 54 345 1025 1136 2563 31
43 0 2 30 265 1048 1251 2596 64
44 0 0 9 190 903 1421 2523 -9
45 0 0 5 110 805 1539 2459 -73
46 0 0 2 66 674 1774 2516 -16
47 0 0 0 29 502 2079 2610 78
48 0 0 0 0 252 2363 2615 83
49 0 0 0 0 0 2515 2515 -17
 
All Combs 6/49 Lotto (13.983,816)
by Numerical Order Position Lexicographic Enumeration 
ID P1 P2 P3 P4 P5 P6 Line Total
01 1712304 0 0 0 0 0 1712304
02 1533939 178365 0 0 0 0 1712304
03 1370754 326370 15180 0 0 0 1712304
04 1221759 446985 42570 990 0 0 1712304
05 1086008 543004 79464 3784 44 0 1712304
06 962598 617050 123410 9030 215 1 1712304
07 850668 671580 172200 17220 630 6 1712304
08 749398 708890 223860 28700 1435 21 1712304
09 658008 731120 276640 43680 2800 56 1712304
10 575757 740259 329004 62244 4914 126 1712304
11 501942 738150 379620 84360 7980 252 1712304
12 435897 726495 427350 109890 12210 462 1712304
13 376992 706860 471240 138600 17820 792 1712304
14 324632 680680 510510 170170 25025 1287 1712304
15 278256 649264 544544 204204 34034 2002 1712304
16 237336 613800 572880 240240 45045 3003 1712304
17 201376 575360 595200 277760 58240 4368 1712304
18 169911 534905 611320 316200 73780 6188 1712304
19 142506 493290 621180 354960 91800 8568 1712304
20 118755 451269 624834 393414 112404 11628 1712304
21 98280 409500 622440 430920 135660 15504 1712304
22 80730 368550 614250 466830 161595 20349 1712304
23 65780 328900 600600 500500 190190 26334 1712304
24 53130 290950 581900 531300 221375 33649 1712304
25 42504 255024 558624 558624 255024 42504 1712304
26 33649 221375 531300 581900 290950 53130 1712304
27 26334 190190 500500 600600 328900 65780 1712304
28 20349 161595 466830 614250 368550 80730 1712304
29 15504 135660 430920 622440 409500 98280 1712304
30 11628 112404 393414 624834 451269 118755 1712304
31 8568 91800 354960 621180 493290 142506 1712304
32 6188 73780 316200 611320 534905 169911 1712304
33 4368 58240 277760 595200 575360 201376 1712304
34 3003 45045 240240 572880 613800 237336 1712304
35 2002 34034 204204 544544 649264 278256 1712304
36 1287 25025 170170 510510 680680 324632 1712304
37 792 17820 138600 471240 706860 376992 1712304
38 462 12210 109890 427350 726495 435897 1712304
39 252 7980 84360 379620 738150 501942 1712304
40 126 4914 62244 329004 740259 575757 1712304
41 56 2800 43680 276640 731120 658008 1712304
42 21 1435 28700 223860 708890 749398 1712304
43 6 630 17220 172200 671580 850668 1712304
44 1 215 9030 123410 617050 962598 1712304
45 0 44 3784 79464 543004 1086008 1712304
46 0 0 990 42570 446985 1221759 1712304
47 0 0 0 15180 326370 1370754 1712304
48 0 0 0 0 178365 1533939 1712304
49 0 0 0 0 0 1712304 1712304
 
Stig or Bertil thought something wonderful was obtained by a formula that gave the Standard Deviations for the 6 columns for all the 13,983,816 combinations. Well, let's do a comparison between various sets and see if it comes up with anything other than what is expected after random selections displayed in order drawn are presented in numerical order.
 
Most of the tables referred to are too long to show here except firstly a little 8/6/8/5/9 Offset 6/49 gem from my collection with no repeat Threes and close to maximum optimization. Secondly, from December 24, 2004 a 49 line set with equal representation of 6 for each integer, maximized repeat consecutive subsets and poor coverage of only 36.64% compared to an optimum of around 69.88%. This set is isomorphic with each integer having the same occurrence after randomization but as shown all over the place with Standard Deviation based on attributing magnitude to the integers. 
 
Just bear in mind that the Standard Deviation for the sequence 1, 2, 3 ... 47, 48, 49 is 14.29 when erroneously attributed magnitude in a Lotto context ie the identifier 49 is 49 times bigger than the identifier 1 in some way. If we take out either 11 or 39 and do the Standard Deviation on 48 integers we still have 14.29!
 
Fairbrother 6/49 Matrix Offset 8/6/8/5/9
P1 P2 P3 P4 P5 P6
1 9 15 23 28 37
2 10 16 24 29 38
3 11 17 25 30 39
4 12 18 26 31 40
5 13 19 27 32 41
6 14 20 28 33 42
7 15 21 29 34 43
8 16 22 30 35 44
9 17 23 31 36 45
10 18 24 32 37 46
11 19 25 33 38 47
12 20 26 34 39 48
13 21 27 35 40 49
 
 
Equal Representation set 49 lines each integer occurrence 6 
P1 P2 P3 P4 P5 P6
1 2 3 4 5 6
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
31 32 33 34 35 36
37 38 39 40 41 42
43 44 45 46 47 48
1 2 3 4 5 49
6 7 8 9 10 11
12 13 14 15 16 17
18 19 20 21 22 23
24 25 26 27 28 29
30 31 32 33 34 35
36 37 38 39 40 41
42 43 44 45 46 47
1 2 3 4 48 49
5 6 7 8 9 10
11 12 13 14 15 16
17 18 19 20 21 22
23 24 25 26 27 28
29 30 31 32 33 34
35 36 37 38 39 40
41 42 43 44 45 46
1 2 3 47 48 49
4 5 6 7 8 9
10 11 12 13 14 15
16 17 18 19 20 21
22 23 24 25 26 27
28 29 30 31 32 33
34 35 36 37 38 39
40 41 42 43 44 45
1 2 46 47 48 49
3 4 5 6 7 8
9 10 11 12 13 14
15 16 17 18 19 20
21 22 23 24 25 26
27 28 29 30 31 32
33 34 35 36 37 38
39 40 41 42 43 44
1 45 46 47 48 49
2 3 4 5 6 7
8 9 10 11 12 13
14 15 16 17 18 19
20 21 22 23 24 25
26 27 28 29 30 31
32 33 34 35 36 37
38 39 40 41 42 43
44 45 46 47 48 49
 
 
Comparison Standard Deviations per column for various Lotto 6/49 sets
Description Col 1 Col 2 Col 3 Col 4 Col 5 Col 6
Fairbrother Matrix Offset 8/6/8/5/9 in 13 Combs 3.89 3.89 3.89 3.89 3.89 3.89
All 13,983,816 Combs Lexicographic 5.74 7.41 8.11 8.11 7.41 5.74
All World 20,682 Combs 5.74 7.47 8.19 8.14 7.39 5.65
South African 874 Draws in Order Drawn 14.37 14.14 14.24 13.86 14.33 14.32
South African 874 Draws Numerical Order 5.82 7.19 7.91 8.17 7.56 5.62
Biased 874 draws with no 11 identifier 6.03 7.66 8.07 8.23 7.57 5.62
Equal Representation Integers 6x in 49 Combs as in template 13.82 13.94 13.99 13.99 13.94 13.82
Equal Representation Integers 6x in 49 Combs Non Numeric 14.29 14.29 14.29 14.29 14.29 14.29
Equal Representation Integers 6x in 49 Combs Randomization 1 5.68 7.24 7.66 7.53 7.48 4.71
Equal Representation Integers 6x in 49 Combs Randomization 2 7.87 7.30 9.31 9.70 8.73 6.17
Fairbrother Matrix Offset 8 Numeric Order 49 Combs 2.38 2.40 2.41 2.41 2.40 2.38
Fairbrother Matrix Offset 8 Non Numeric Order 49 combs 14.29 14.29 14.29 14.29 14.29 14.29
Maximized Coverage Unique Threes Combs 49 3.26 5.87 7.70 8.15 7.06 5.23
 
 
Summary and conclusions
 
When a Pool 49 Pick 6 Lotto games's draws are presented in 6 columns as drawn as by the South African Lotto operators then over a reasonable number of draws it is expected that each column will have a near equal distribution of the integers.
 
Usually draw results are presented in Numerical Order rather than order drawn and if an add on game applies such as the Australian Lotto Strike where the order drawn of the first four random selections is guessed then the results are given separately.
 
Statistics on integers in the history of draws for a Lotto game are usually given as absence (recency) or occurrence (frequency). Attributing magnitude to the integers is unnecessary and is usually the stuff of numerologists.
 
If magnitude is attributed then from the sequence 1, 2, 3, 4 .... 47, 48, 49 a Standard Deviation is derived of 14.288690166. A calculation of the Standard Deviation per column on a sample 874 draws presented in drawn order from the South African 6/49 Lotto game gives close to the 14.29 for each column. If the draws are now converted to numerical order and a calculation done on the Standard Deviation per column then the results obtained may also be obtained using the following conversion factors: -
Col 1 and 6:   0.4017
Col 2 and 5:   0.5185
Col 3 and 4:   0.5675
 
The usefulness of this presupposes a desire to want to know a Standard Deviation measurement per column for the draws when presented in Numeric order. From my long experience in dealing with Lotto numbers and how pseudo analysts work the only use I see for this is an added "tool" for those that predict the numbers. Tests already exist based on occurrence of the integers that adequately measure the fairness of the Random Selections.
 
I think it comes down to understanding what the rows and columns represent. A Lotto draw recorded in a row usually has an identifying number and date. From the draw you can record a hit in the appropriate 6 columns from 49 and then get some meaningful data by counting the hits in each column. Aggregating Lotto draws by column when presented in numerical order leads to nowhere meaningful. You can't get back to how the integers were drawn as for each draw there are 720 possibilities.
 
Colin Fairbrother
Lotto Draws have no relationship to one another; the integers serve just as identifiers. Any prediction calculation on one history of draws for a same type game is just as irrelevant as another.
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Colin F
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Quote Colin F Replybullet Posted: November 22 2009 at 1:13am
Stig Holmquist aka Bertil proposed that the Standard Deviation of the columns when presented in numerical order may be useful in determining the randomness of a Lotto game's draws.
 
In alt.math.recreational November 9,2009 he wrote, "Colin was on a fishing expedition hoping to disprove my data. You might as well have told him to wait until Hell freezes over before you can come up with any data to disprove mine. I know there is a formula for predicting the means and s.d., but I can't find it. Some time ago it was used to predict the values for the 5/55 Powerball game."
 
 
The South African 6/49 Lotto game presents the draws with the integers shown as drawn. Placing the integers in numerical order for 874 draws the following Standard Deviations are obtained for the columns: -
 
Col 1 5.82
Col 2 7.19
Col 3 7.91
Col 4 8.17
Col 5 7.56
Col 6 5.62
 
South African 6/49 Lotto 874 draws
ID P1 P2 P3 P4 P5 P6 Line Total
01 100 0 0 0 0 0 100
02 110 17 0 0 0 0 127
03 91 15 2 0 0 0 108
04 65 25 4 0 0 0 94
05 52 35 3 1 0 0 91
06 72 32 5 0 0 0 109
07 55 49 14 2 0 0 120
08 54 42 17 1 0 0 114
09 38 51 17 3 0 0 109
10 24 47 18 6 1 0 96
11 29 52 25 6 1 0 113
12 21 49 19 5 3 0 97
13 27 42 29 4 0 0 102
14 25 45 37 9 1 1 118
15 25 35 33 12 2 0 107
16 14 54 41 19 1 0 129
17 11 34 40 27 6 0 118
18 14 29 32 25 3 1 104
19 8 31 42 17 7 0 105
20 10 29 40 21 11 0 111
21 6 22 50 17 8 1 104
22 4 25 30 28 12 1 100
23 6 24 39 39 9 1 118
24 0 15 49 30 11 2 107
25 3 8 30 33 15 2 91
26 3 12 31 42 13 2 103
27 1 6 30 42 19 5 103
28 2 8 28 24 27 5 94
29 3 12 24 39 28 3 109
30 1 8 34 43 30 15 131
31 0 5 18 44 26 7 100
32 0 5 21 42 26 5 99
33 0 4 10 36 50 17 117
34 0 3 21 37 36 11 108
35 0 1 11 41 45 15 113
36 0 2 5 29 42 16 94
37 0 0 4 23 43 20 90
38 0 0 5 19 44 24 92
39 0 0 4 21 38 30 93
40 0 1 7 12 38 44 102
41 0 0 2 26 51 38 117
42 0 0 3 21 33 49 106
43 0 0 0 13 46 45 104
44 0 0 0 8 41 69 118
45 0 0 0 4 41 58 103
46 0 0 0 2 32 86 120
47 0 0 0 1 25 86 112
48 0 0 0 0 9 112 121
49 0 0 0 0 0 103 103
 
The Standard Deviation for the Line Totals is 10.37
 
If we now massage the data such that identifier 11 has no occurrence then the following Standard Deviations are obtained for the column totals which are not much different to those with identifier 11 having an occurrence of 113: -
 
Col 1 6.03
Col 2 7.66
Col 3 8.07
Col 4 8.23
Col 5 7.57
Col 6 5.62
 
Lotto 6/49 example 874 draws with no-occurrence identifier 11.
ID P1 P2 P3 P4 P5 P6 Line Total
01 199 0 0 0 0 0 199
02 90 50 0 0 0 0 140
03 82 24 3 0 0 0 109
04 57 30 6 1 0 0 94
05 47 36 6 2 0 0 91
06 64 38 7 0 0 0 109
07 46 53 18 2 1 0 120
08 49 43 21 1 0 0 114
09 34 48 21 6 0 0 109
10 22 43 23 7 1 0 96
11 0 0 0 0 0 0 0
12 21 49 19 5 3 0 97
13 27 42 29 4 0 0 102
14 25 45 37 9 1 1 118
15 25 35 33 12 2 0 107
16 14 54 41 19 1 0 129
17 11 34 40 27 6 0 118
18 14 29 32 25 3 1 104
19 8 31 42 17 7 0 105
20 10 29 40 21 11 0 111
21 6 22 50 17 8 1 104
22 4 25 30 28 12 1 100
23 6 24 39 39 9 1 118
24 0 15 49 30 11 2 107
25 3 8 30 33 15 2 91
26 3 12 31 42 13 2 103
27 1 6 30 42 19 5 103
28 2 8 28 24 27 5 94
29 3 12 24 39 28 3 109
30 1 8 34 43 30 15 131
31 0 5 18 44 26 7 100
32 0 5 21 42 26 5 99
33 0 4 10 36 50 17 117
34 0 3 21 37 36 11 108
35 0 1 11 41 45 15 113
36 0 2 5 29 42 16 94
37 0 0 4 23 43 20 90
38 0 0 5 19 44 24 92
39 0 0 4 21 38 30 93
40 0 1 7 12 38 44 102
41 0 0 2 26 51 38 117
42 0 0 3 21 33 49 106
43 0 0 0 13 46 45 104
44 0 0 0 8 41 69 118
45 0 0 0 4 41 58 103
46 0 0 0 2 32 86 120
47 0 0 0 1 25 86 112
48 0 0 0 0 9 112 121
49 0 0 0 0 0 103 103
 
However, for the normal occurrence calculation irrespective of the column there are three sore thumb indications of something being wrong -
firstly the line total is 0 for identifier 11 and 199 for identifier 1 whereas the others are around the average 107 mark. Thirdly, the Standard Deviation for the line totals has more than doubled to 23.13.
 
I guess we can now take this as proof that hell can freeze over! 
 
Colin Fairbrother
Lotto Draws have no relationship to one another; the integers serve just as identifiers. Any prediction calculation on one history of draws for a same type game is just as irrelevant as another.
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