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Covers or Wheels with Guarantees and Unique 3's
 LottoPoster Forums : NUMBER SETS TO PLAY FROM DIFFERENT METHODS : Covers or Wheels with Guarantees and Unique 3's
Message Icon Topic: Cover or Wheel with Guarantee C(49,6,3,6,1)=163. Post Reply Post New Topic
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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: Cover or Wheel with Guarantee C(49,6,3,6,1)=163.
    Posted: January 20 2007 at 8:49pm
For those familiar with Covers as applicable to Lotto the free Covermaster program by John Rawson is no doubt familiar; if not it can be downloaded here.
 
You can easily build a C(49,6,3,6)=163 line cover by merging a C(22,6,3,3)=77 with a C(27,6,3,4)=86. This cover will guarantee a Three win every draw as at least one three in each of the 13,983,816 sixes is covered by the 163 line set.
 
The C(22,6,3,3)=77 is easy to build in CoverMaster - just don't check auto optimize - and has all the 1540 combinations of three that can be formed from a pool of 22 integers. This is a one off bonus and is why for the larger 3if6 covers these 77 lines are always used. In other words it is maximum packed for the number of lines with the possible distinct threes. As each six can only hold 20 threes (77x20=1540) it is impossible to do it in a lesser number of lines. You should note though that only 15 lines are needed for a C(22,6,3,6). Also note that for the 2600 combinations of 6 from a pool of 26 a C(26,6,3,3)=130 can be obtained (130x20=2600).
 
The  C(27,6,3,4)=86 is impossible to the best of my knowledge and attempts to produce in CoverMaster without some manual intervention for the last 275 or so. There are lots of ways you can go about it eg Generate some 77 lines or so with your hits parameter set to 3 and then do a bit of manual scrutiny using whatever you have to assist - Access, Excel, MSQuery etc or you can do it in CoverMaster using techniques that don't seem to be well known in the Lotto crowd.
 
Using Covermaster optimization has only got me to 99.5% so far for a C(27,6,3,4)=86 so you are forced to use other means. You can try building it manually from scratch - just load your parameters - Pool, Pick, Match (Prize or Guarantee) then test and you will have a list of fours appear in a popup. Change the max. entries to a higher number than 25 and test again to get a bigger list of fours. From the menu bar Tools|Search|Best Play will give a list of sixes with coverage. Initially, the 435 means that in the list of uncovered fours using the 20 threes as criteria from that six a coverage of 435 is attainable. The difficulty lies in choosing which six - given that quite a few can have equal coverage. Which six you choose has repercussions on the later makeup of the overall set. After building the 163 line cover in Covermaster choose from the menu 
Tools|Reports|Quick to get the following report.
 
Prize Match
if Hits in Pool
Tested
Covered
% Covered
Uncovered
% Uncovered
2 If 2
1,176
582
49.489800%
594
50.5102
2 If 3
18,424
18,424
100%
0
0
2 If 4
211,876
211,876
100%
0
0
2 If 5
1,906,884
1,906,884
100%
0
0
2 If 6
13,983,816
13,983,816
100%
0
0
2 If 7
85,900,584
85,900,584
100%
0
0
3 If 3
18,424
3,007
16.321100%
15,417
83.6789
3 If 4
211,876
98,719
46.592820%
113,157
53.40718
3 If 5
1,906,884
1,570,086
82.337780%
336,798
17.66222
3 If 6
13,983,816
13,983,816
100%
0
0
3 If 7
85,900,584
85,900,584
100%
0
0
4 If 4
211,876
2,428
1.145950%
209,448
98.85405
4 If 5
1,906,884
102,539
5.377310%
1,804,345
94.62269
4 If 6
13,983,816
2,016,925
14.423280%
11,966,891
85.57672
4 If 7
85,900,584
22,898,862
26.657400%
63,001,722
73.3426
5 If 5
1,906,884
978
0.051290%
1,905,906
99.94871
5 If 6
13,983,816
42,157
0.301470%
13,941,659
99.69853
5 If 7
85,900,584
884,838
1.030070%
85,015,746
98.96993
6 If 7
85,900,584
7,009
0.008160%
85,893,575
99.99184
 
 
Before considering this report in detail let's recap on the basic odds for a 6/49 Lotto game.
 
With one ticket your chances of getting the winning number are 1 in :
49!/(43!*6!) = 13,983,816
or 49c6
If you buy two tickets then bearing in mind there is only one winning ticket, dividing the 13,983,816 possible numbers in no particular order into two sets of numbers each containing 6,991,908 numbers you are guaranteed the winning number will be in one of the sets but it's still just a ratio. If you continued doing this ie divide the 13,983,816 combinations into 163 sets of around 85,790 then you know the winning six is in one of these sets but if you forked out some $42,900 to play one of these sets your chances of getting the winning number are only 1 in 163. Multiplying this by the extreme Monte Carlo factor of 26 (black came up 26 times in succession in 1913) you could spend some $182 million to get a million or so!  
 
Your chances of getting a :
  • five are 1 in 49c6/(6c5 * 43c1) = 54,201
  • four are 1 in 49c6/(6c4 * 43c2) = 104
  • three are 1 in 49c6/(6c3 * 43c3) = 56.6

The chances of getting a three of 1 in 56.6 in relation to the 163 line cover is very relevant. Put another way we are expecting 3 of the three prize wins on average every 170 plays. Now, the 163 line cover may guarantee just 1 of these three prizes but good old Mr Random is relied upon for the other 2. As I have pointed out before there is no appeal in paying some $82 to be guaranteed $2 with no increase in your chances of a higher prize - see Test C(49,6,3,6). If you played a 56 set of numbers with a good spread of the threes then you will average a three win every draw and even if you played just a 28 set of numbers you would average a three win every other draw. 

Firstly, looking at the Covermaster report we see that Covermaster does validate the cover and this is a handy feature as one can use it to verify any Cover. If you delete lines and use Menu|Tools|Search|Best Play to reconstruct the cover the limitations of the algorithm used soon become apparent. Yes, with one or two or even 10 selections deleted it can reconstruct to 163 lines but try deleting the last 10 lines and you end up with a 166 line cover.
 
Secondly, from the 18,424 possible threes only 3,007 are included in the cover ie 16.32%. So, for every draw 83.68% of the threes are excluded ie if we divide our threes into roughly 6 bands then using the cover we are confining ourself to 1 band with a particular mathematical relationship for each random draw where all integers or threes are given an equal chance. 
 
If you look at Menu|Tools|Reports|Detailed this will show you the repetition of threes in the cover. Each six can hold 20 threes. Of the 3,007 threes used to make the cover:
        • 2,828 appear once
        • 142 twice
        • 11 three times
        • 18 four times
        • 5 five times
        • 3 six times
A valid question to ask is if you are really after a few threes wouldn't it be better to do a random selection of 163 lines from my Unique Threes 450 line Cover which has 9000 distinct threes? In other words you would have 163 lines with 3,260 distinct threes.
  
The first line used in constructing any cover for a 6/49 Lotto covers 260, 624 sixes. ie at least 1 three in the six. For example using 01 02 03 04 05 06 we are covering all the 15,180 '01 02 03' threes and for the other 19 threes 3x14,190, 6x 13,244 and 10x12,341. Compare this with Menu|Tools|Reports|Charts|Dependency where we see that 5 of the sixes included are covering only between 3,000 to 4,000 sixes. Does the thought cross your mind that choosing a six made up from the 84% of the threes unused has some merit?
 
Covers may be used as an alternative to random selections when choosing one's lotto numbers but they do not perform over a reasonable period of time any better. There is a challenge in creating some of the larger covers and where there is a will there is a way for Covers come from the discrete side of mathematics -  which means all the different ways it can be done are known and if tested then the optimum result can be obtained.   
 
Below is the detailed report from CoverMaster: -
 
 Win 6      Win 5      Win 4      Win 3     Comb 6's    Comb 6's %     Cum %  
----------------------------------------------------------------------------
     -          -          -          1      3595746    25.71362    25.71362
     -          -          -          2      3261643    23.32441    49.03804
     -          -          -          3      1114905     7.97282    57.01086
     -          -          -          4      2858548    20.44183    77.45269
     -          -          -          5       476971     3.41088    80.86357
     -          -          -          6       223556     1.59868    82.46225
     -          -          -          7        93980     0.67206    83.13431
     -          -          -          8        64089     0.45831    83.59262
     -          -          -          9        37278     0.26658    83.85920
     -          -          -         10       220944     1.58000    85.43920
     -          -          -         11        11270     0.08059    85.51979
     -          -          -         12         3665     0.02621    85.54600
     -          -          -         13         1290     0.00922    85.55522
     -          -          -         14          348     0.00249    85.55771
     -          -          -         15          139     0.00099    85.55871
     -          -          -         16           29     0.00021    85.55891
     -          -          -         17            6     0.00004    85.55896
     -          -          -         18            8     0.00006    85.55901
     -          -          -         20         2476     0.01771    85.57672
     -          -          1         0-17    1781368    12.73878    98.31550
     -          -          2         1-15     165076     1.18048    99.49598
     -          -          3         1-13      25917     0.18534    99.68132
     -          -          4         0-10       2138     0.01529    99.69661
     -          -          5         2-7         207     0.00148    99.69809
     -          -          6         0-7          53     0.00038    99.69847
     -          -          7         3-5           8     0.00006    99.69852
     -          -          9          2            1     0.00001    99.69853
     -          1         0-4        0-15      41942     0.29993    99.99846
     -          2         1-2        3-8          44     0.00031    99.99878
     -          3          0         4-7           8     0.00006    99.99883
     1          0         0-3        0-13        163     0.00117   100.00000
                                          13,983,816 

 
Summary
    • A Cover or Wheel with Guarantee is a set of numbers with each line having the same quantity of integers as the main integers picked in a draw such that it will give a nominated minimum number of prizes in each draw according to the condition that must be met of having so many correct in the pool of integers nominated. For C(49,6,3,6,1)=163 -
      .
        • The Pool is 49 integers. Each draw 6 main integers are picked.

 

  • A Three win is the minimum prize.
 
  • Six integers must be correct in the Pool which in this case is sufficient to form a winning number so all the 13,983,816 combinations that could be the winning Six must have a Three in the 163 line set.
 
  • For each draw at least 1 combination from the set must give at least the minimum prize of a Three. More prizes may be achieved but in every case the guarantee must be met.
 
  • There are 163 numbers, lines, blocks or plays in the set. 
 
  • A typical random selection of 163 lines would for about 720,000 of the possible 13,983,816 combinations of 6 integers not give a prize in each draw ie 5%. The C(49,6,3,6,1)=163 is about forcing all the combinations to give a prize but is it worthwhile?
    .
  • From the probability calculation a randomly selected 163 line set would average 2.9 threes per draw in a 6/49 Lotto game. In fact I took the first 163 draws in my All World's 6/49 History Database and found for all the 13,983,816 combinations where Threes only were prizes that the average was 2.86. Comparing this with a 163 line Unique 3's™ number set and the 163 line Cover shows in both cases the Threes returned for 85% of the 13,963,616 combinations the same average of 2.7 per draw. However, for 85% of the possible combinations with Three only prizes the 163 Line Random Selection returned 2.85. In other words the guarantee makes no enhancement to the overall Three prizes and removes any reason for playing such a high number of combinations.
    .
  •  A 163 Line set with maximum repeat of the 5's gives 11,142,652 Combinations where no prize is obtained 79.68% and for all the 3's only winnings this is done in 2,497,070 Combinations or 18%. The average then for the 3's hits with no other prizes is 1.22 well down on the other yields. Excluding the 1st prize there is 1 combination where you could walk away with 45 5's and 118 4's for about $73,220. Definitely not a recommended method of play.
     .
  • In a 6/49 Lotto game a Five prize should be worth around 957 times that paid for a Three prize and that for a Four prize around 18 times that paid for a Three prize. A check of the payouts from the Lottery operators will probably find $2.00 for a Three, $40.00 for a Four and a short changed $1500 or so for a Five. In other words getting a Four is worth 20 Threes and getting a Five is worth around 750 Threes. You can lose a few threes but never throw away your chances with the Fours and Fives by duplicating them in the number set you play.
  • To guarantee the Three prize win every draw the 163 line cover has been structured such that apart from the 3's duplications (142x2;11x3;18x4;5x5;3x6) there is an unavoidable duplication of 17 of the 4's subsets. This is the important difference between a 163 line Unique 3's™ number set and the 163 line Cover. In other words as I have been saying since I first posted on the web it comes down to maximizing your chances with the Fives and Fours.  
Regards
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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