Colin Fairbrother

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whether other than through sheer luck the odds can be bettered.

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Recently, I had a bizarre contributor in a Newsgroup who to be honest is not quite compos mentis state the following regarding 5 QuickPicks he purchased for the UK 6/49 Lotto, "Not with so many numbers above 8 TO 10, position 1 surely. 80% Of winning JACKPOT lines have a number between 1 AND 8  in P1"
 
For 1584 draws of the UK Lotto game you do have 67% starting with 1 to 8 when the line is in Numerical Order. To expect a small sample of 5 Random Selections to reflect this distribution is totally unrealistic. Indeed, if we divide the lexicograhic enumeration into thirds then we have 7 draws consecutively in the top third, namely draws 952, 953, 954, 955, 956 and 957. Harry thinks this shouldn't happen!
 
Basically to restrict the numbers played to a grouping that has the highest winnings is to ignore proportionality as is done with Sums, Odds/Evens and other numerology mumbo jumbo. If a grouping is preferred where the wins mostly occur your chances of success have not improved as evidenced by comparing the lines in the group to all possible combinations ratio and wins for that group over 1584 draws of the UK Lotto game. The ratios are pretty much the same as shown in the table below: -
 
From Lex Index To Lex Index First Integer Lex Combinations % All Lex UK Lotto Wins % All UK Wins
1 1712304 1 1712304 12.24 184 11.62
1712305 3246243 2 1533939 10.97 177 11.17
3246244 4616997 3 1370754 9.80 154 9.72
4616998 5838756 4 1221759 8.74 144 9.09
5838757 6924764 5 1086008 7.77 114 7.20
6924765 7887362 6 962598 6.88 113 7.13
7887363 8738030 7 850668 6.08 97 6.12
8738031 9487428 8 749398 5.36 80 5.05
9487429 10145436 9 658008 4.71 81 5.11
10145437 10721193 10 575757 4.12 65 4.10
10721194 11223135 11 501942 3.59 68 4.29
11223136 11659032 12 435897 3.12 49 3.09
11659033 12036024 13 376992 2.70 36 2.27
12036025 12360656 14 324632 2.32 35 2.21
12360657 12638912 15 278256 1.99 32 2.02
12638913 12876248 16 237336 1.70 26 1.64
12876249 13077625 17 201376 1.44 20 1.26
13077625 13247535 18 169911 1.22 26 1.64
13247536 13390041 19 142506 1.02 17 1.07
13390042 13508796 20 118755 0.85 9 0.57
13508797 13607076 21 98280 0.70 7 0.44
13607077 13687806 22 80730 0.58 10 0.63
13687807 13753586 23 65780 0.47 9 0.57
13753587 13806716 24 53130 0.38 10 0.63
13806717 13849220 25 42504 0.30 4 0.25
13849221 13882869 26 33649 0.24 7 0.44
 
Consider the following two samples of 5 lines for a 6/49 Lotto game. The first has lexicograhic indices below or equal to 6,298,682 or in the lower half of the 13,983,816 combinations. The second table has lexicograhic indices higher than or equal to 13,390,042 or the higher 1/24th of the 13,983,816 some 594,000. According to Harry you have a better chance with the first set. The reality is that they have an equal chance at all prize levels both with a Six coverage of 0.00004% and Three coverage of 9.29%
 
Lexicographic Index Comb6
670034 01 06 11 16 21 26
2324197 02 07 12 17 22 27
3803853 03 08 13 18 23 28
5124008 04 09 14 19 24 29
6298682 05 10 15 20 25 30
 
 
Lexicographic Index Comb6
13390042 20 21 22 23 24 25
13849221 26 27 28 29 30 31
13965253 32 33 34 35 36 37
13982893 38 39 40 41 42 43
13983816 44 45 46 47 48 49
  
 
I did point out if you are looking at the results in Numerical Order then for example 1 will always be in Position 1 and 49 always in Position 6. In other words you can't compare integer occurrences fairly when the presentation is in Numerical Order.
 
If the occurrence is compared for the order drawn this is a fair comparison and is shown in the table below for 1584 draws in the UK Lotto game. Integer 23 has the highest occurrence of 48 for the first main ball drawn and integers 20 and 10 have the lowest at 20 for respectively the first and second ball drawn.
 
Integer Ball 1 Ball 2 Ball 3 Ball 4 Ball 5 Ball 6
1 35 39 24 27 23 36
2 30 37 37 31 37 27
3 36 30 38 26 27 36
4 30 35 30 30 30 35
5 29 22 31 37 33 31
6 24 38 35 35 37 32
7 22 32 34 21 33 40
8 33 32 26 36 26 32
9 31 35 37 30 36 36
10 36 20 38 37 30 33
11 38 33 39 44 29 32
12 35 37 22 34 30 36
13 25 30 34 28 27 25
14 38 33 30 25 29 25
15 24 36 28 30 28 30
16 38 28 28 25 29 27
17 29 26 38 37 29 31
18 27 43 41 21 32 27
19 30 34 33 36 40 21
20 20 21 26 38 21 34
21 34 28 25 38 24 22
22 33 31 23 33 35 33
23 48 31 30 33 39 33
24 31 32 35 32 37 30
25 33 39 38 32 36 33
26 30 29 32 29 36 33
27 34 27 37 33 43 31
28 40 32 34 32 34 23
29 26 41 30 29 27 33
30 35 31 26 29 39 43
31 34 29 32 39 41 38
32 33 37 39 31 29 36
33 35 34 29 36 36 44
34 36 31 35 24 40 27
35 40 33 35 34 30 30
36 24 29 36 30 33 31
37 29 37 31 30 24 27
38 29 37 38 38 39 39
39 26 31 35 33 44 34
40 37 29 40 35 30 38
41 35 32 27 24 21 26
42 32 31 33 27 33 39
43 44 40 31 34 35 31
44 34 43 22 39 34 41
45 35 23 33 40 22 43
46 33 29 28 33 31 33
47 31 32 32 40 41 31
48 33 33 34 38 30 26
49 30 32 35 31 35 30
 
-------------------------------------------------
 
Integer Occurrence For Order Drawn UK Lotto 1584 Draws
Integer Position 1 Position 2 Position 3 Position 4 Position 5 Position 6

Max

Min

Avg

SD

1 35 39 24 27 23 36 39 23 30.67 6.83
2 30 37 37 31 37 27 37 27 33.17 4.40
3 36 30 38 26 27 36 38 26 32.17 5.15
4 30 35 30 30 30 35 35 30 31.67 2.58
5 29 22 31 37 33 31 37 22 30.50 4.97
6 24 38 35 35 37 32 38 24 33.50 5.09
7 22 32 34 21 33 40 40 21 30.33 7.39
8 33 32 26 36 26 32 36 26 30.83 4.02
9 31 35 37 30 36 36 37 30 34.17 2.93
10 36 20 38 37 30 33 38 20 32.33 6.71
11 38 33 39 44 29 32 44 29 35.83 5.49
12 35 37 22 34 30 36 37 22 32.33 5.61
13 25 30 34 28 27 25 34 25 28.17 3.43
14 38 33 30 25 29 25 38 25 30.00 4.98
15 24 36 28 30 28 30 36 24 29.33 3.93
16 38 28 28 25 29 27 38 25 29.17 4.54
17 29 26 38 37 29 31 38 26 31.67 4.80
18 27 43 41 21 32 27 43 21 31.83 8.64
19 30 34 33 36 40 21 40 21 32.33 6.47
20 20 21 26 38 21 34 38 20 26.67 7.63
21 34 28 25 38 24 22 38 22 28.50 6.25
22 33 31 23 33 35 33 35 23 31.33 4.27
23 48 31 30 33 39 33 48 30 35.67 6.80
24 31 32 35 32 37 30 37 30 32.83 2.64
25 33 39 38 32 36 33 39 32 35.17 2.93
26 30 29 32 29 36 33 36 29 31.50 2.74
27 34 27 37 33 43 31 43 27 34.17 5.46
28 40 32 34 32 34 23 40 23 32.50 5.50
29 26 41 30 29 27 33 41 26 31.00 5.48
30 35 31 26 29 39 43 43 26 33.83 6.40
31 34 29 32 39 41 38 41 29 35.50 4.59
32 33 37 39 31 29 36 39 29 34.17 3.82
33 35 34 29 36 36 44 44 29 35.67 4.84
34 36 31 35 24 40 27 40 24 32.17 5.98
35 40 33 35 34 30 30 40 30 33.67 3.72
36 24 29 36 30 33 31 36 24 30.50 4.04
37 29 37 31 30 24 27 37 24 29.67 4.37
38 29 37 38 38 39 39 39 29 36.67 3.83
39 26 31 35 33 44 34 44 26 33.83 5.91
40 37 29 40 35 30 38 40 29 34.83 4.45
41 35 32 27 24 21 26 35 21 27.50 5.17
42 32 31 33 27 33 39 39 27 32.50 3.89
43 44 40 31 34 35 31 44 31 35.83 5.19
44 34 43 22 39 34 41 43 22 35.50 7.56
45 35 23 33 40 22 43 43 22 32.67 8.64
46 33 29 28 33 31 33 33 28 31.17 2.23
47 31 32 32 40 41 31 41 31 34.50 4.68
48 33 33 34 38 30 26 38 26 32.33 4.03
49 30 32 35 31 35 30 35 30 32.17 2.32
               
 
 
 
Colin Fairbrother
 


This article first published on the web 26/2/2011 and has been viewed 5082 times.
The author may be contacted at fairbros@bigpond.com. To generate sets of Lotto numbers to play for most Lotto games around the world go to LottoToWin ® where for a US$5.00 annual subscription you can store the numbers as well, to check later for wins. The numbers produced have no duplicate subset combinations of two integers (if applicable) or three integers, use all the integers for the particular Lotto game and maximize the coverage of the potential winning main number. The articles on this site have been selected from the many at the forum LottoPoster
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