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Lotto Covers Non-Distorted Distribution but repeat subsets
 LottoPoster Forums : WORLD RECORD LOTTO COVERS (WHEELS WITH GUARANTEE) : Best Yield Normal Distribution Lotto Covers : Lotto Covers Non-Distorted Distribution but repeat subsets
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Colin F
Lotto Systems Tester Creator & Analyst
Lotto Systems Tester Creator & Analyst
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Joined: September 30 2004
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Quote Colin F Replybullet Topic: WR 6/49 Lotto Cover non-distorted distribution.
    Posted: January 07 2010 at 9:26pm
Surprisingly, until I pointed out in many posts that mathematical Covers do not have a Normal Distribution of the prize groupings , no one as far as I know, had bothered with publishing record Lotto Covers with either Normal Distribution or non-merged sets but with repeat paying subsets.
 
For a 6/49 Lotto game one can easily generate a 322 line cover in CoverMaster with repeat Threes. This can be reduced to 319 by running the optimizer after deleting 3 lines and to 318 lines by deleting one line and optimizing. I believe anyone who uses Covermaster with some expertise, even if only to check their own work, would have had, like me, these coverings in their files for many years.
 
A thread was started in RGL on January 2, 2010 to see what lower limits could be achieved for a so called "straight" cover ie one without any obvious indication that merged covers had been used, in the least number of lines. The author. a rather strange chap by the name of Gary Dobner who naively believes he is free to libel good people on public newsgroups with impunity, while hiding under some silly pseudonym such as a or gARY and plagiarising others work, gave as his first contribution simply the first optimization from Covermaster without mentioning that source or the programmer, John Rawson. For these covers where duplicates are allowed, when the lines are sorted by dependence (Combinations Uncovered if removed) and viewed in the dependency chart for a 3 if 6 setting in Covermaster there should be a reasonably smooth curve from the highest dependency to the lowest without any extended plateaus which would indicate an undisguised merge cover.
 
For a Normal Distribution cover as the number of lines is increased so there should be a progressive, reasonably smooth increase in the coverage for the measurement of progress towards at least one Three and after less than 56 lines should be in decline with two Threes taking over. So, if you like a cascading, decaying wave formation which I have done, see here. This is not a requirement for the covers produced with the least number of lines and no evidence of merging.
 
In the case of the Stojiljkovic & Belic C(49,6,3,6,1)=163 Cover the dominant grouping is the Threes x 1 at 25.71% which is a distortion with the Threes x 2 at 23.32% not greater than the Threes x 1 and the Threes x 3 at a paltry 7.97%. See Chart example for Pool 45  here. The occurrence of the integers when comparing the two merged sets is close but noticeably different with a range from 18 to 21 and 22 on the 21 mark because of the C(22,6,3,3,1)=77.
 
Source/Author                Lines 3's x 1    >= 3's x 6   Dup 3's  Distr 
Jan 17,2010 Colin Fairbrother 326    6,163    76.33511%     172    36 to 43
Jan 16,2010 Colin Fairbrother 325    6,340    76.03344%     168    37 to 43
Jan 17,2010 Colin Fairbrother 324    6,910    75.25535%     135    37 to 43
Jan 17,2010 Colin Fairbrother 323    6,932    75.03244%     227    37 to 42
Jan 20,2010 Colin Fairbrother 322    9,135    73.85296%     307    37 to 42
Covermaster stand generation  322   10,683    72.61616%     501    37 to 46   
Covermaster 1st optimization  319   11,069    72.01873%     455    37 to 41
Covermaster 2nd optimization  318   11,347    71.88831%     416    36 to 42
Jan 4, 2010  Orion            317   10,843    71.73736%     376    37 to 41
Jan 5, 2010  Lottoarchitect   316   10,983    71.49740%     371    36 to 42
Jan 5, 2010  Gary             315   11,192    71.15073%     356    36 to 41
Jan 5, 2010  Gerry            314   11,219    70.86707%     349    36 to 40
Jan 5, 2010  Peteros          312   12,135    69.83794%     394    37 to 41
Jan 6, 2010  John Rawson      311   12,353    69.53813%     374    35 to 42
Jan 9, 2010  Manfred          308   13,488    68.65323%     373    36 to 40
Source/Author                Lines  3's x 1   >= 3's x 5    Dup 3's  Distr
Jan 21,2010 Colin Fairbrother 307   20,385    80.79488%     297    35 to 39
Source/Author                Lines  3's x 1   >= 3's x 2    Dup 3's  Distr
Jan 25,2010 Colin Fairbrother 210  1,175,936  91.59074%     370    23 to 31
Jan 26,2010 Colin Fairbrother 174  3,259,795  76.68880%     348    15 to 24
 
In considering the best set to play if the cost per line is $1.00 - then for my money I would play the 325 lines if not the 326 or 324 with the 325 giving a 76.03344% chance of getting 6 Threes or more compared to the 308 lines at 68.65323%. It also has a much lower number of duplicate Threes than the lines 322 to 308, which represent wasted chances for a single prize or if you like means your 3if3 chances are not optimized. If you only played 308 lines from my 325 set then your chances of getting one of the 260 combinations of six integers not covered are less than that of getting first prize. 
 
The forth last entry by Manfred was given in the spirit of getting the best result for given criteria demonstrating prowess at calculating coverage limits and not as a good set of Lotto numbers to play. In the same spirit I could have entered a 251 line set that had a much wider distribution range but no limits were set on this, couldn't be reduced and has a dependence chart curve without plateaus that give away a merged set.
 
The third last entry at 307 by myself was mainly entered on the goading of the know all, know nothing Gary and Gerry twosome of RGL who seem to think they know all about making Covers or Wheels but are found wanting when put to the test. The screen shot of the Prize Group analysis below using John Rawson's Covermaster is at the goading of two of the biggest, ignorant assholes in this field of interest, Ian Saliu and Robert Perkis once again through RGL. I have at least two other solutions for the 307 lines. You may note that try as you may 308 lines is the limit for getting the 3's x 6 as the dominant Threes prize group.
 
The second last entry is to put a lower bound on what is achievable. The chances of you getting more than just one Three are 91.59% with my 210 lines and given it is 47 lines more than Stojiljkovic & Belic C(49,6,3,6,1)=163 Cover theirs only gives 74.29%. The cost for playing my 210 lines three times is about the same as playing the S & B four times but your chances of getting more than just one Three are about three times better with my cover. 
 
The last entry at 174 lines which just scraped in on the parameters specified is purely to illustrate that the pursuit of the lowest number of lines by inflating the 3's x 1 is utter folly. The 3's x 2 is just greater at 3,287,977 than the 3's x 1 at 3,259,795. To ram home the point the 3's  x 1 at 3,259,795 has comparable figures in my 365 line Unique 3's™ Cover at about the 14 and 100 combs mark with a peak of 6,484,296 at 53 combs.       
 
What we see in the table above is a gradual worsening of your chances of getting 6 Threes. You may be improving your chances of getting just 1 Three but that is not what Lotto is about. You should have for the set of numbers you play the greatest chance of getting what the odds say you are entitled to.
 
Colin Fairbrother
 
 
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