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: -

High, Middle and Low integers Considered in this article. .

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