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Topic: LOTTO YIELDS BY METHOD AND COMBINATIONS PLAYED  
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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: LOTTO YIELDS BY METHOD AND COMBINATIONS PLAYED Posted: October 10 2010 at 10:10pm 
Comparison of Lotto Yields for Various Methods and
Combinations Played in Pick 6, Pool 49 Lotto by Colin Fairbrother Rarely does a layman get the opportunity to show members of academia with high mathematical qualifications to be utterly wrong. Such is the case for me in exposing the utter fallacy of contriving an increased coverage for a lower prize in Lotto to achieve a Cover or Guaranteed Prize for the set played but an overall lower Yield.
While proving the irrelevance of using a particular Lotto game history to construct numbers to play has given me some satisfaction it can rightfully be regarded as overkill because of the Independence of Events rule and certainly is not as sweet as that obtained from proving highly intelligent and qualified people to have been utterly wrong in advocating Lotto players should play various Covers or Wheels with Guarantees.
From the time I became aware of Covers or Wheels around 2000 there was something about them that came across as dubious and contrived  how right I was. For a Pick 6, Pool 49 Lotto game with ticket costs at 50¢ or £1 and the lowest prize at around $4 or £10 in respectively the USA and the UK an outlay of $81.50 or £163 was recommended for a 163 line Cover. Given the odds of getting three integers in a line correct are 1 in 57 for this game how could anyone justify spending more than $28.50 or £57 per draw if the objective was to have at least an average of one win per draw. The answer apparently was in the desire for the guaranteed pittance no matter what the cost.
In the natural order of numbers played in a wagering scenario with multiple prize levels like Lotto one would expect: 
Coverage
Consider a Pick 6, Pool 49 Lotto game. There are 
49c6 or
49!/(43! x 6!) or the inverse of 6/49 x 5/48 x 4/47 x 3/46 x 2/45 x 1/44 or 13,983,816 chances of getting first prize. Similarly, the possible combinations of three integers are the inverse of 3/49 x 2/48 x 1/47 or 18,424. If you played one line such as 01 02 03 04 05 06 then for the first prize you have covered only one line of the possible 13,983,816 combinations of six integers but for the combinations of three integers you have 20 in your one line which covers 260,624 of the 13,983,816 possibilities meaning each contains at least one of those 20 CombThrees.
Next Best Play
From the previous paragraph after say, 01 02 03 04 05 06 the next line with the next best coverage or play of 260224 could be 07 08 09 10 11 12 but there are many more to choose from as long as an integer is not in your first line.
Syntax
There is a convention for describing Covers which if adhered to makes it easy to find any Cover from among maybe hundreds of files.
Combinations minimum(Pool, Pick, Match, Hits, Prize Multiple) = Lines
C(12,6,4,5,3)=22 means for a Pool of 12 and Pick of 6 in this hypothetical Lotto game if you get 6 correct, at a minimum 13 prizes will be obtained and if 5 correct at a minimum 11 prizes. The construction of this set is done such that the 220 combinations of Three integers from a Pool of 12 are repeated twice in the 22 line set meaning all the 924 Combinations of 6 integers from a Pool of 12 have at least two matching CombThrees.
If the context of the Pool and Pick has already been indicated then it may simply be referred to as 4if5. A 4if5 guarantees a 4if6, 3if3, 3if4, 3if5 and 3if6 and the hardest to achieve guarantee is the preferred description.
A special case referred to as a Steiner is where the Cover or minimum number of lines multiplied by the Prize Combination in a line is the same as all the possible Prize Combinations for the Pool. A famous Steiner is C(22,6,3,3,1)=77 where the maximum combinations of three integers from a Pool of 22 ie 1540 is the same as the 77 lines multiplied by the 20 combinations of three integers in a Pick 6 line.
Optimization
Consider a hypothetical Pick 6, Pool 12 Lotto game in which you have set your objective to play 22 lines. The possible unique combinations of three integers from a Pool of 12 is 220 so for 22 combinations of six integers you need to repeat the CombThrees twice for all to have the same multiple. You could try a time consuming method of manually setting it out making sure you don't repeat a Three more than twice and a Four more than once. (If your Fours are unique then so are your Fives and Sixes.)
A much quicker method for producing this set is to first generate 22 lines next best play 5if6 accepting the first in lexicographic order to give: 
1 01 02 03 04 05 06
2 01 02 03 07 08 09 3 01 02 03 10 11 12 4 01 04 05 07 08 10 5 01 04 05 09 11 12 6 01 06 07 08 11 12 7 02 04 06 07 09 10 8 02 05 06 08 09 11 9 03 04 06 08 09 12 10 03 05 06 07 10 11 11 01 05 06 09 10 12 12 02 03 04 05 07 12 13 02 03 04 08 10 11 14 07 08 09 10 11 12 15 01 03 04 07 09 11 16 02 05 06 08 10 12 17 01 02 04 08 09 12 18 01 02 05 07 10 11 19 01 03 05 08 09 10 20 01 04 06 08 10 11 21 02 03 06 09 11 12 22 01 03 04 07 10 12 If this set set is now optimized by changing an integer up or down to a limit set (in this case 5) and testing for a better coverage the following Cover is obtained C(12,6,4,5,3)=22 with the integers changed in red.
1 01 02 06 04 05 11
2 01 02 03 07 08 06
3 01 02 03 10 11 12
4 06 04 05 07 08 09
5 04 05 09 10 11 12
6 01 05 07 08 11 12
7 02 04 06 03 09 10
8 02 05 03 08 09 11
9 03 04 06 08 11 12
10 03 05 06 07 10 11
11 01 05 03 09 06 12
12 02 03 04 05 07 12
13 02 07 04 08 10 11
14 03 07 08 09 10 12
15 01 03 04 07 09 11
16 02 05 06 08 10 12
17 01 02 04 08 09 12
18 01 02 05 07 09 11
19 01 03 05 08 04 10
20 01 09 06 08 10 11
21 02 07 06 09 11 12
22 01 06 04 07 10 12
Yield
For the amount of money you care to wager over say 100 draws your return for the lower prizes can be reasonably calculated according to probability rules. A marked deviation usually indicates some distortion has been introduced and generally, with few exceptions this applies to most so called wheels or covers that have been touted as beneficial in Lotto since the 1980's where the lowest number of lines has been the main priority.
Optimization or maximization as illustrated above is OK. However, when it is used to artificially skew a design such as favoring lower over higher prizes to produce a lower number of lines with an overall lower Yield that is bordering on fraud.
History
Recreational mathematics goes back to the 1600's so unless you're over 400 years old I doubt that you can lay claim to such a simple design. Let us not forget that while people did not have computers they still had the same or better intelligence and perseverance. Slide Rules were first built in England in 1632 and were still being used by NASA engineers in the space program that put a man on the moon in the 1960's. Lo Shu Magic Squares date from around 2800 BC
Coup de Gras Graphs
The following graph highlights the distortion of optimized sets for a 6/49 Lotto game from around 126 combs onwards especially for the 3'sx1, 3'sx2, 3'sx3 and 3'sx4.
A picture's worth a thousand words. Anyone not able to recognize there is something radically wrong with the above graph compared to the natural curves of the graph below has rocks in their head.
Emphasis must be made that in the chart the sets have used the full Pool of 49 integers  using a partial pool as advocated by a nutty professor can lead to abysmally poor results. A realistic perspective is made by showing the full range for the Percentage Yield indicating that the variation is within a narrow band of some 4 percentage points. Random Selections are below the others up to around the 90 combs mark when the Optimized dips below and again more severely at around the 145 combs and mildly below at the 163 mark. The 3if6 and 4if6 are noticeably superior to the Optimized set staying together to around the 100 combs mark from where the 4if6 gives superior results up to around the 112 mark.
Summary
Colin Fairbrother


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