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Topic: Probability of Success for Grouped Lotto Numbers  
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Colin F
Lotto Systems Tester Creator & Analyst To dream the impossible dream ... Joined: September 30 2004 Location: Australia Online Status: Offline Posts: 678 
Topic: Probability of Success for Grouped Lotto Numbers Posted: September 03 2011 at 5:59pm 

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: 
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:
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: 
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: 
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%.
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 

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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