Colin Fairbrother

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Probability of Success for Grouped Lotto Numbers
by Colin Fairbrother

Classic Lotto has 49 balls or integers from which 6 are randomly picked for the first prize number. Favourite ways of categorizing these numbers by players with a numerology bent are: -
All these categorizations have been considered in articles on the LottoPoster website.

The numerologists, whether they realize that is what they are or not, then go further and say some combinations are less likely to occur than others and so without any rational reason they filter them out.

In reality the integers as used in Jackpot Lotto have no magnitude and even if they did it is irrelevant in the random picking process. The rather glaring mistake that is made by the promoters of these methods is that it is better to play those that occurr more often such as Sums that add to 150. What is not mentioned is the occurrence is proportional to the extent they are represented in all the possibilities which is really applying the simplest notion of probability.

To illustrate this correlation I will consider some other categorizations and show that the occurrence of the winning numbers in some 1636 draws of the UK Lotto are proportional to the extent the category is represented in the total possibilities.

Consider first the usual scenario where it is stated you are better off playing numbers for a Pick 6 Lotto game where when considered in numerical order the first two lowest integers are from 1 to 16, the middle two are between 17 and 33 and the last two are the highest between 34 to 49. For a Pick 6, Pool 49 Lotto game we have 3,549,600 from the 13,983,816 possibilities that fall within these constraints or 25%. For 1636 draws from the UK Lotto game we have 410 that fit the constraints to give 25% which is little different to considering those with lexicographic ID of < 3,549,601 with 24%, > 10,434,216 with 25% and between IDs 5,217,108 and 8,766,708 with 24%.  

We can make the UK 6/49 Lotto game with a first prize chance of winning of 1 in 13,983,816 a game with a chance of winning 1 in 7 by simply dividing the possibilites in either numerical (lexicographic order) or last number order by 7 as shown in the table below with results for the UK 1636 draws:


Category ID Start ID Finish Lex Last LNO Last Lex Cat LNO Cat
A 1 1997688 02 04 20 22 33 49 06 07 09 23 25 37 224 208
B 1997689 3995376 03 10 16 18 19 35 03 13 24 26 31 41 229 244
C 3995377 5993064 05 07 11 13 37 38 09 32 36 40 41 43 244 213
D 5993065 7990752 07 09 10 14 18 40 11 13 37 39 43 45 227 260
E 7990753 9988440 09 19 24 26 37 46 14 31 32 34 40 47 233 233
F 9988441 11986128 13 25 27 41 42 49 15 16 28 30 46 48 247 249
G 11986129 13983816 44 45 46 47 48 49 44 45 46 47 48 49 232 229

Another interesting categorization is by the balls or integers and for a 6/49 game 7 equal length sets can be used as in the following; the first in numerical order followed by another with an offset 7 but any order is just as applicable: -

Set 1
Group A:  {  1,  2,   3,  4,   5,   6,   7}
Group B:  {  8,  9, 10, 11, 12, 13, 14}
Group C:  {15, 16, 17, 18, 19, 20, 21}
Group D:  {22, 23, 24, 25, 26, 27, 28}
Group E:  {29, 30, 31, 32, 33, 34, 35}
Group F:  {36, 37, 38, 39, 40, 41, 42}
Group G:  {43, 44, 45, 46, 47, 48, 49}

Set 2
Group A:  {  1,   8, 15, 22, 29, 36, 43}
Group B:  {  2,   9, 16, 23, 30, 37, 44}
Group C:  {  3, 10, 17, 24, 31, 38, 45}
Group D:  {  4, 11, 18, 25, 32, 39, 46}
Group E:  {  5, 12, 19, 26, 33, 40, 47}
Group F:  {  6, 13, 20, 27, 34, 41, 48}
Group G:  { 7, 14, 21, 28, 35, 42, 49}

A formula may be applied where ! means factorial (Pl + Pk - 1)! / Pk! x (Pl - 1)! ie (7 + 5)! / 6! x 6! or 12! / 6! x 6! or 12 x 11 x 10 x 9 x 8 x 7 / 6 x 5 x 4 x 3 x2 x1 which can be simplified to 11 x 2 x 3 x 2 x 7 = 924 to get the number of combinations with repeats allowed of the categories.

For combinations without repetition of the categories we have the simple formula 7c6 ie 7! / 6! x (7 - 6)! or 7 categories ie

A B C D E F
A B C D E G
A B C D F G
A B C E F G
A B D E F G
A C D E F G
B C D E F G

There are 117,649 possibilities for each of these 7  unmatched categories giving a total of 823,543 which is just less than 6% of all the 13,983,816 possibilities. The table below shows the occurrence for both Group Set 1 and Group Set 2: -


Unmatched UK Lotto Count Set 1 Count Set 2
ABCDEF 15 11
ABCDEG 12 15
ABCDFG 24 12
ABCEFG 11 12
ABDEFG 17 13
ACDEFG 15 13
BCDEFG 14 12
 
ABCDFG certainly stands out for Set 1 but is pretty well normal in Set 2. This is something not predictable but not unusual when dealing with random selections.

Considering all the 924 possibilities for groupings of 6 from the 7 groups ie A B C D E F G with repeats allowed, one can aggregate these into 11 groups based on Group Repeats with varying degrees of occurrence as in the table below: -


Group Occurrence for Number Drawn Possibilities Percentage of All Possibilities Expected UK Lotto Actual UK Lotto
Same Group x 2 5423859 38.78668 635 615
(Same Group x 2) x 2 4418526 31.59742 517 495
Same Group x 3 1678985 12.00662 196 201
Same Group x 2 with Same Group x 3 1066730 7.62831 125 158
Unmatched or All Different Groups 823543 5.88925 96 108
(Same Group x 2) x 3 329280 2.35472 39 32
Same Group x 4 184485 1.31927 22 20
Same x 4 with Same x 2 26460 0.18921 3 2
Same x 3 with Same x 3 25725 0.18396 3 5
Same x 5 6174 0.04415 0 0
All Same Group
49 0.00035 0 0

The table shows that proportionality applies however you categorize the integers.

The last category in the above table has only 49 possibilities and as expected no success for the 1636 UK Lotto draws. Comparing the probability for playing 49 numbers of 0.0000035 or 0.00035% with that for 1 number of 0.0000000715 we see there is a big difference but the latter is the only relevant probability, irrespective of any groupings made, as this is the one that is paid on. Just picture a 6/49 Lotto game as having 13,983,816 relevant groups and all should be clear as 13,983,816 x 0.0000000715=1 or certainty. Multiplying any grouping of distinct lines by 0.0000000715 gives the probability of success.

If money was no object and you were inclined to throw it away, then playing 25,725 lines per draw, where 3 of the integers were from one group and the other 3 from another over the 1,636 draws would have cost £42,086,100. Even though this group configuration did about 70% better than expected with 5 Jackpot wins a quick, rough calculation shows the return including sub prizes is only about the 50% mark.

Comparing theSet 1 and set 2 where the numbers are from the same group as in the table below we see that including 01 02 03 04 05 06 makes no difference to the probability of 0.00035% for the set winning 1st prize. Included for comparison is an optimized set where I deliberately made sure 8 lines with consecutive integers were included and the chances of this set winning first prize are identical to the other two.

The difference for the optimized set is that none of the paying subset CombThrees, CombFours and CombFives are repeated whereas for the other two sets they are excessive. So, with a possible 980 unique CombThrees for 49 lines the optimized set has this and covers 9,795,325 or 70% (as good as it gets) of the 13,983,816 possible combinations of 6 integers from 49. Set 1 and Set 2 have only 245 distinct CombThrees which are repeated 4 times with a consequent coverage of only 21%.


Set 1 Same Group Set 2 Same Group Set Optimized
01 02 03 04 05 06 01 08 15 22 29 36 01 02 03 04 05 06
01 02 03 04 05 07 01 08 15 22 29 43 07 08 09 10 11 12
01 02 03 04 06 07 01 08 15 22 36 43 13 14 15 16 17 18
01 02 03 05 06 07 01 08 15 29 36 43 19 20 21 22 23 24
01 02 04 05 06 07 01 08 22 29 36 43 25 26 27 28 29 30
01 03 04 05 06 07 01 15 22 29 36 43 31 32 33 34 35 36
02 03 04 05 06 07 08 15 22 29 36 43 37 38 39 40 41 42
08 09 10 11 12 13 02 09 16 23 30 37 43 44 45 46 47 48
08 09 10 11 12 14 02 09 16 23 30 44 01 14 20 33 41 46
08 09 10 11 13 14 02 09 16 23 37 44 01 23 25 35 38 44
08 09 10 12 13 14 02 09 16 30 37 44 01 13 36 42 48 49
08 09 11 12 13 14 02 09 23 30 37 44 01 16 19 26 39 45
08 10 11 12 13 14 02 16 23 30 37 44 01 10 17 27 34 40
09 10 11 12 13 14 09 16 23 30 37 44 02 07 13 21 32 45
15 16 17 18 19 20 03 10 17 24 31 38 02 15 20 26 38 47
15 16 17 18 19 21 03 10 17 24 31 45 02 22 31 37 44 49
15 16 17 18 20 21 03 10 17 24 38 45 02 11 25 36 40 43
15 16 17 19 20 21 03 10 17 31 38 45 02 08 17 24 30 33
15 16 18 19 20 21 03 10 24 31 38 45 03 15 23 30 34 46
15 17 18 19 20 21 03 17 24 31 38 45 03 14 21 28 38 43
16 17 18 19 20 21 10 17 24 31 38 45 03 11 18 24 42 44
22 23 24 25 26 27 04 11 18 25 32 39 03 08 29 35 40 47
22 23 24 25 26 28 04 11 18 25 32 46 03 09 16 20 25 48
22 23 24 25 27 28 04 11 18 25 39 46 04 11 17 21 31 48
22 23 24 26 27 28 04 11 18 32 39 46 04 08 23 27 36 39
22 23 25 26 27 28 04 11 25 32 39 46 04 10 15 22 25 32
22 24 25 26 27 28 04 18 25 32 39 46 04 07 26 35 42 46
23 24 25 26 27 28 11 18 25 32 39 46 04 09 24 28 45 49
29 30 31 32 33 34 05 12 19 26 33 40 05 13 24 26 31 43
29 30 31 32 33 35 05 12 19 26 33 47 05 11 15 28 35 39
29 30 31 32 34 35 05 12 19 26 40 47 05 10 16 23 29 42
29 30 31 33 34 35 05 12 19 33 40 47 05 09 18 21 37 47
29 30 32 33 34 35 05 12 26 33 40 47 05 12 22 36 38 45
29 31 32 33 34 35 05 19 26 33 40 47 06 12 17 19 28 46
30 31 32 33 34 35 12 19 26 33 40 47 06 09 22 27 42 43
36 37 38 39 40 41 06 13 20 27 34 41 06 07 24 29 36 37
36 37 38 39 40 42 06 13 20 27 34 48 06 08 15 31 41 45
36 37 38 39 41 42 06 13 20 27 41 48 06 10 21 33 39 49
36 37 38 40 41 42 06 13 20 34 41 48 06 13 20 30 40 44
36 37 39 40 41 42 06 13 27 34 41 48 07 18 23 33 40 48
36 38 39 40 41 42 06 20 27 34 41 48 07 14 19 25 34 49
37 38 39 40 41 42 13 20 27 34 41 48 08 14 26 32 37 48
43 44 45 46 47 48 07 14 21 28 35 42 09 17 29 32 41 44
43 44 45 46 47 49 07 14 21 28 35 49 10 18 19 30 35 41
43 44 45 46 48 49 07 14 21 28 42 49 11 13 19 27 33 37
43 44 45 47 48 49 07 14 21 35 42 49 12 18 20 29 34 43
43 44 46 47 48 49 07 14 28 35 42 49 12 14 30 31 39 47
43 45 46 47 48 49 07 21 28 35 42 49 16 22 28 34 41 47
44 45 46 47 48 49 14 21 28 35 42 49 16 27 32 38 46 49
 

Randomizing the optimized set gives the best chance for success for the lower prizes but as all the lines are different the chances of success for first prize are just the same as any other set of 49 lines where the lines are different.

So much for HOW TO WIN THE LOTTERY SCHEMES, however there is some brain stimulation in proving them wrong.


Colin Fairbrother
 
 
 




This article first published on the web 3/9/2011 and has been viewed 7043 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
© Colin Fairbrother 2004 to 2012 All rights reserved.