CORRELATION BETWEEN LOTTO COVER OR WHEEL GUARANTEES
FOR VARIOUS POOL SIZES
by Colin Fairbrother

Real Lotto players that give some thought to the numbers they play in a rational way realize that a first prize win is highly unlikely but for what should be a modest sum wagered not impossible and for that sum they would like to maximize their wins.

Using the Classic Lotto Pick 6, Pool 49 game to get a 3 integer match or prize the odds are arrived at from the inverse of dividing all the 49c6 ie 13,983,816 CombSix possibilities by the 6c3 ie 20 CombThree possibilities in one line or block multiplied by the 43c3 ie 12,341 CombThree possibilities (the complement to 6c3). The odds are then 1 in 56.66 or rounded up 1 in 57 and one can reasonably expect playing 57 random selection lines per draw that on average 1 win per draw would be achieved.

If you played around 3 times 57 lines ie 171 random selection lines per draw you would reasonably expect to get an average of 3 wins per draw. If for 163 lines played per draw an average of close to 3 wins per draw was guaranteed then this would be relatively attractive (still negative return), if you were bent on wagering as in the UK £163 for £30 or 18% return. However, if the guarantee is only £10 or 6% return and on analysis you find that the structure of the set played is skewed or distorted towards obtaining a single CombThree prize instead of a multiple prize of 3 CombThree wins then the attraction is practically non-existent for most rational players.

There are a few irrational, obdurate players who when confronted with these facts still express a preference for a guaranteed prize no matter what the cost and how little the return. For 1,000 plays or around 6 draws playing 163 random selections the calculated chances of not getting a prize are an insignificant ¼ of 1 percent and

only a 16% chance of a single CombThree prize

with a

83% chance of getting multiple CombThree prizes and a CombFour

for an overall percentage return using UK ticket costs and payouts of 21.6%.

By comparison the 163 line cover

guarantees only 1 CombThree prize

and a

33.3% chance that it will be a single CombThree prize of £10.

There is only a

66.7% chance of getting multiple CombThree prizes and a CombFour

for an overall, lower percentage return of 16.6%. See chart here for various Covers and Pools per line quantity.

Any student of Lotto number set or structure analysis needs to be familiar with Covers or Wheels with Guarantees, without as in my case necessarily endorsing them as a method to play. The minimum number of lines to achieve the guarantee and the structure of the set can be used for reference purposes.

In Lotto Cover or Wheel parlance the Pick is the quantity of numbers filled in for 1 line played and the Pool is the quantity you can choose from. For a 6/49 Lotto game a 3if3 guarantee Cover or Wheel is the minimum number of lines that will ensure when played that of the 49c3 or 18,424 possible CombThrees at least 1 will match with the current record minimum of 1084 lines.

For the various guarantees there are a few facts worth knowing regarding Pool size and a less difficult guarantee. A 3if3 guarantee for say Pool 49 would also be a guarantee for 3if4, 3if5 and 3if6 with some redundancy (ie it can be done in a lesser number of lines). Similarly a 3if4 would guarantee both 3if5 and 3if6 and a 3if5 would guarantee a 3if6. A 3if3 guarantee Cover for Pool 44 would also be a 3if4 guarantee for Pool 45, a 3if5 guarantee for Pool 46 and a 3if6 guarantee for Pool 47. Similarly a 3if4 guarantee for Pool 44 would also be a 3if5 guarantee for Pool 45 and a 3if6 guarantee for Pool 46. Also a 3if5 guarantee for Pool 44 would be a 3if6 guarantee for Pool 45 or a 207 line 3if5 guarantee for Pool 48 is also a 3if6 Cover for Pool 49 with redundancy as it can be done in 163 lines.

The following chart shows there is a correlation between the various guarantees in this case 3if3, 3if4, 3if5 and 3if6 for various Pool sizes and the number of lines required. Silly assertions as by Professor Iliya Bluskov (click for article) that a 3if6 Cover is possible in about 87 lines ie about half the current record of 163, are shown to be absurd. Quite simply as shown in the chart as the Pool is increased for the 3if6 curve there is an increase in the number of lines to make a cover and a dip from around 163 lines to 87 for Pool 49 is just impossible as the entire correlation between the guarantees and for each guarantee curve would defy reconciliation.

An obvious example is the C(44,6,3,5)=154 constructed from two Steiner C(22,6,3,3)=77 each uniquely using 22 of the 44 integers and by definition with maximum compaction ie 154 x 20 = 3080 CombThrees. For Pool 44 the record C(44,6,3,6)=123, which is 31 lines less than the 3if5 and this difference increases to 71 for the current records of C(49,6,3,5)=234 and C(49,6,3,6)=163. How can mathematics Professor Iliya Bluskov specialising in combinatorics, who according to his own description is a "Lottery Expert" seriouly suggest that the difference can be 147 ie much less than the Steiner produced C(44,6,3,5)=154?

Here is what Professor Iliya Bluskov wrote on page 9 in his thesis of 1997 called
New Designs and Coverings-

"If the syndicate chooses to play with tickets that correspond to the blocks of a (14,6,4) covering then they will get
at least one 4-win whenever any 4 of their 14 numbers are drawn. Thus the syndicate will secure a certain garantee. Since any
(14,6,4) covering gives the same guarantee, they should choose the most "economical" covering; that is, the covering with the
smallest known number of blocks, which is currently 80, and hence they purchase the fewest number of tickets."

"Naturally, one can ask: What is the advantage of playing for such a guaranteed win? If we compare playing with 80 random
tickets agaïnst 80 tickets forming a (14,6,4) covering we see that the probability of a 6-win ("hitting the jackpot) is the
same for each ticket; namely (49c6)^{-1}. However, if any 4 of the numbers drawn are among the 14 numbers chosen by
the syndicate, then the 80 tickets of a (14,6,4) covering guarantee àt least one 4-win while 80 random tickets (on the same 14
numbers) guarantee nothing!"

This is pure speil - coming from the likes of snake oil sales people like Gail Howard or Ken Silver it is expected - from a
University Professor in Mathematics and Science in a mathematics thesis it begs belief. Whatever the complexity of any equation Bluskov
came up with I for one would not accept it at face value. Where is the relevance to reality in those two paragraphs?

To secure a certain prize in a hypothetical Pick 6, Pool 14 Lotto game only 4 lines need to be played ie C(14.6,3,6)=4 and
for a 4-win only 14 lines ie C(14.6,4,6)=14. Based on the number of lines (a lot less than 80) surely these are the most
"economical" coverings according to the Professor despite their irrelevance to a 6/49 Lotto game?

The odds for getting a 4-win in a 6/49 Lotto game are 1 in 49c6/((6c4) x (43c2)) = 13983816/(15 x 903) = 1032.39. A 4-win is
expected on average roughly every 1000 plays and for a random seletion 80 line set using the Pool of 49 once every 13 draws. In
fact you don't have to wait that long for a prize as getting any prize is 1 in 54 so a prize would be obtained on average every
draw playing just 54 random selection lines. The odds for getting a 4-win in a Pick 6, Pool 14 hypothetical Lotto game are
1 in 7 so playing 80 lines at least 34 3-wins and 4 4-wins are expected every draw with a 5-win or greater with other prizes 73%
of the draws and bearing no relation to the results in a 6/49 Lotto game.

The chances of getting a 4-win in a hypothetical Pick 6, Pool 14 Lotto game playing the 80 line Cover are certain but not
when applied to a Pick 6, Pool 49 Lotto game. In fact due to the highly distorted structure of the 80 line Cover or Wheel your
chances of getting a 3-Win let alone a 4-Win compared to a random selection using the full Pool of 49 are severely diminished.
The possible number of distinct CombThrees for 80 Pick 6 lines are 80 x 20 = 1600 and distinct CombFours 80 x 15 = 1200 and in
a Random selection of 80 lines close to this is achieved as there are 18,424 distinct CombThrees and 211,876 CombFours available.
However the C(14,6,4,4)= 80 Cover or Wheel has 220 CombThrees repeated 4 times and 144 repeated 5 times with 95 CombFours repeated 2 times, 20
repeated 3 times and 16 repeated 5 times. For a CombThree match instead of 1600 working towards this aim there are only 364 and for the CombFours
instead of a possible 1200 distinct CombFours there are only 870.

The absurdity of qualifying the statement with "... 80 random tickets (on the same 14 numbers) guarantee nothing!" is
blatently obvious. BEFORE THE DRAW THERE IS NO GUARANTEE; IN THE UNLIKELY EVENT THAT ANY FOUR OF THE RESTRICTED POOL NUMBERS CHOSEN ARE ALSO IN THE WINNING SIX NUMBERS DRAWN THEN THE CONSTRUCTION OF THE SET PLAYED GUARANTEES A COMBFOUR PRIZE. THE RUB IS THAT BECAUSE OF THE DISTORTED STRUCTURE WITH SO MANY REPEAT PAYING SUBSETS THIS IS DEFINITELY AND QUANTITAVELY LESS LIKELY TO HAPPEN.

Mostly, major Lottery countries do not offer random selections on anything other than the Pick and Pool applicable to
the game which in this case is 6 and 49 and where they do, like Australia it is restricted, in this case to System Entries or Full Wheels up to 18 integers or 18,564 lines. Even if someone went to the trouble of generating 80 Pick 6 random selection lines with a Pool restricted to 14 then this set would yield over 1,000 plays an 11% return with wins more often compared to 9% for the
C(14,6,4,4)=80 Cover.

Colin Fairbrother

This article first published on the web 16/9/2012 and has been viewed 9616 times. The author may be contacted at fairbros@bigpond.com. To generate sets of Lotto numbers to play for most Lotto games around the world go to LottoToWin ^{®} where for a US$5.00 annual subscription you can store the numbers as well, to check later for wins. The numbers produced have no duplicate subset combinations of two integers (if applicable) or three integers, use all the integers for the particular Lotto game and maximize the coverage of the potential winning main number. The articles on this site have been selected from the many at the forum LottoPoster