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Analysis by Grouping Win Results per Possible Winning Number
 LottoPoster Forums : ANALYSIS OF VARIOUS LOTTO NUMBER SETS : Analysis by Grouping Win Results per Possible Winning Number
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Colin F
Lotto Systems Tester Creator & Analyst
Lotto Systems Tester Creator & Analyst
To dream the impossible dream ...

Joined: September 30 2004
Location: Australia
Online Status: Offline
Posts: 678
    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: -
  1. As the number of lines played is increased then so should one's chances of both getting a higher prize and multiples of the lower prizes increase and indeed a point be reached where getting just one of the lowest prize ceased to be a possibility.
  2. Duplicating a paying combination of the integers in Lotto for the set played reduces your chances of getting one of those prizes.
  3. For a Lotto game from which a random selection of five or six integers is made then the best opportunity for success is to play at least a set of numbers which includes all the integers. For a Pool 49 Pick 6 game this is a minimum of 9 lines and for Pick 5 it is a minimum 10 lines.
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
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. 
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.
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 
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.
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.
  • The lowest actual Jackpot Lotto game has a Pick of 6 and Pool of 25 numbers. Leaving out just one of these integers removes a significant 24%  of the possible Jackpot winning numbers from consideration. Leaving out just one integer removes 12.24% from consideration in a Pick 6, Pool 49 Lotto game - however, if you reduce the pool to 10 integers a whopping 99.998% of the possible winning numbers have been removed from consideration. How Ilyya Bluskov, a Professor of Combinatorics, could promote such a system to Lotto players defies comprehension.
  • Since the 1980's and the publication of a list of Cover tables or Wheels by Ivan Dimitrov subsequently popularized by Gail Howard the guarantee of a prize in Lotto if a template is used for a Pool of numbers less than that of the full pool has been promoted. The effect of reducing the Pool which makes the guarantee non-applicable to the game played is glossed over by a glib statement that says if you get so and so correct from the drawn number in the set promoted you will get such and such a prize. An honest approach which I use is to calculate the prize table for the same set using the actual Pool for the Lotto game you are playing. Then we see a dramatic difference - your chances of getting any prize using a Pool of 10 and 20 lines is only 9% compared to double that of 18% when the full Pool of 49 is used. 
  • If you play just a few integers of the Pool and repeat a paying subset then your prize may be higher when you do get it but it will not be as frequent. Real Lotto players playing a reasonable number of lines, say 18 in a 6/45 game and 10 in a 6/49 want the frequency of wins to be as high as possible for the money spent. If you are a regular weekly player then some years playing a small partial Pool you may do quite well but other years get only a a third or less of what is expected. Playing the same set each draw and restricting the Pool to say 12 integers and repeating the subset Threes twice which can be easily done manually in 22 combinations of six but keeping the Fours unique is a tested exception to the rule on repetitions regarding yield but you still have a longer Maximum Draws with no wins (as many as 47 draws - ie over 1022 plays - compared with less than half that - 21 draws ie 453 plays - when using the full pool with Unique 3's - in a 6/49 game). The advantage is lost with a higher partial Pool and repeating subsets.
  • Lotto is a game of chance not certainty. A normal Lotto player acquaints himself with the odds usually given on the back of an entry form for say the lowest prize and any prize and realizes if the latter is 1 in 54 plays as for a 6/49 game then if playing a random selection 18 lines per draw a win every 3 draws on average can be expected. The notion that a player wants a certain lowest prize win every draw at nearly 3 times the cost is absurd, nonsensical and something dreamed up by misguided boffins not Lotto players. If with some tweaking one can get an extra few percentage points in Yield by using all the integers, not repeating paying subsets and maximizing the yield through coverage without distorting the prize distribution then well and good and that is what I recommend.
  • A normal Lotto player is concerned mainly about the size of the First Prize, the odds for getting that prize and how fair the operator is in paying a proportionate prize for the lower prizes relative to the odds. This is best summarized by Yield which is the overall percentage payout for say 100 actual draws repeated say 10 times with different draws or a randomized set. A prize table can also be calculated fairly rounding out the likelihood of each prize group. The notion that an increased coverage for a given number of lines or the making of a Cover for the lowest prize in the least number of lines by distorting the prize distribution has been shown to produce an inferior Yield. 
Colin Fairbrother
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