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 thesame for each ticket; namely (49c6)*

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

]]>LOTTO WHEELS OR COVERS CON-ARTIST CLAIMS

TOTALLY DEBUNKED IN TABLE

*by Colin Fairbrother*

TOTALLY DEBUNKED IN TABLE

Lesser Pool Covers or Wheels are shown in the following table to overwhelmingly reduce your chances of winning a prize. Using the classic Pick 6, Pool 49 Lotto game (the principle is applicable to any Lotto game) the reduction can be as much as 65% in the reasonably short term, say, 1,000 plays.

Basically the extent of repetition of paying sub-set combinations in the set played such as CombThrees, CombFours and even CombFives along with how much of the Pool is used and good coverage will determine how well a play set structure will perform.

**The guarantee of the prize** when playing a set of Lotto numbers which is a Cover or Wheel with guarantee is only applicable to the Pool of numbers used in that particular Cover or Wheel for the lowest set of lines. For the 6/49 Lotto game the record minimum set of lines with no ifs or buts is 163 for a CombThree prize. Prefixing a guarantee for a lesser Pool Cover or Wheel with a highly improbable 'if' does not make it more likely! If you play a 7 line Full Wheel or all the possible combinations of 6 from a Pool of 7 in a 6/49 Lotto game then, sure, if the 6 main integers drawn are in the 7 integers you have used in your Pool of 7 then you win or share first prize - but the likelihood of this happening is the same as playing any other 7 lines. The principle applies to the sub-prizes and because generally a Cover or Wheel for the minimum set of lines is distorted compared to random selections which are in accord with the odds your chances of success are diminished as the table below makes abundantly clear.

A 3if6 guarantee for a 6/49 Lotto game means that for all the 13,983,816 combinations of six integers at least one CombThree in each combination is in the 163 line set played. While it is possible to have 163 x 20 = 3260 CombThrees the 163 line set actually has only 3007 distinct CombThrees with 142 repeated twice, 11 three times, 18 four times, 5 five times and 3 six times.

Little mathematical ability is needed to work out that if a pool of 12 is used then the possible number of CombThrees is 220 (ie 12c3) but after 4 lines repeats are unavoidable and a minimum number of 15 lines is needed to get them all in. When this 15 line set is applied to a 6/49 Lotto game the coverage of the CombThrees is only 14.8% as shown in the table below. Using the first 15 lines from my 365 line set which uses the full Pool of 49 integers, no repeat CombThrees and coverage progressively calculated to next best play, a coverage some 50% more of 23.5% is obtained.

**Generally, the choice between using the Full Pool or a Lesser Pool for a given Lotto game boils down to wanting overall a better percentage return or yield which is the universal method of measuring ones chances of success in a game of chance. Would you prefer 17 CombThree and 1 CombFour wins over 2 years playing 10 lines once a week in a 6/49 Lotto game or 12 CombThrees and 1 CombFour wins as is the case for a reduced Pool of 10 which has a GUARANTEE OF A COMBTHREE PRIZE FOR THE POOL OF 10 BUT NOT FOR THE POOL OF 49. I prefer the better percentage return or yield of 22.6% compared to 20.6% with more wins more often.**

It is easily seen that using the full Pool for a 6/49 game gives better results than the Covers or Wheels where the Pool used is less than 49. While the guarantee does apply to the lesser Pool **IT DOES NOT APPLY** to the full Pool of 49 and usually because of the repetition of CombThrees, CombFours and even CombFives gives significantly less coverage and consequent yield.

Worth noting is the fact that since the full pool calculations have been done with no duplicate CombThrees then they are all optimized for 3if3 ie each added line covers the maximum 20 CombThrees in a line for the 18,424 possible CombThrees.

Comparing the 3if6 calculations for a given number of lines then the higher the 3if6 figure is the higher the likelihod of more wins more often and the higher the Yield figure the more likely you are to get the expected return over a reasonable number of draw participations. Most constructions will return the expected yield over a thousand or so draws but why wait 20 years when you can get that same return in 2 years?

One can also see that to win 1st prize in a 6/49 Lotto game ie 6if6 then whatever the method used to construct the set no difference is made to the likelihood of winning the jackpot as the only requirement is not to repeat a line.

As the increase in the Yield or Percentage Return is only around the 2 to 4 Percent compared to Random Selections it is only the finicky few, like myself, that bother to deliberately maximize the chances of a better return with the lower prizes and under no dellusions about the chances of winning first prize.

Comment is kept deliberately to a minimim to keep this an accurate reference table. Articles exposing the con-artists that promote these lesser pool Covers or Wheels will refer back to the following table.

Comparison Coverage and Yield for Lesser Pool Covers or Wheels applied to 6/49 Lotto Game | |||||||
---|---|---|---|---|---|---|---|

Copyright © Colin Fairbrother 2012 | |||||||

Lines Played | Guarantee | Pool used for Guarantee | Coverage 6/49 Game for Partial Cover (Nil) using max Pool for lines and Partial Pool Covers | Expected Yield for Draws required to approximate 1000 Plays | |||

3 if 6 | 4 if 6 | 5 if 6 | 6if6 | ||||

1 | 3 if 6 | 6 | 260624 1.86375% | 13804 0.09871% | 259 0.00185% | 1 0.00000007% | 23.6% 1000 draws |

4 if 6 | 6 | ||||||

5 if 6 | 6 | ||||||

6 if 6 | 6 | ||||||

3 if 6 | 7 | ||||||

4 if 6 | 7 | ||||||

5 if 6 | 7 | ||||||

3 if 6 | 8 | ||||||

Fairbrother Partial Cover | 49 | ||||||

2 | 3 if 5 | 9 | 481704 3.44472% | 27203 0.19453% | 518 0.0037% | 2 0.00000014% | 22.6% 500 draws |

3 if 5 | 10 | 504532 3.60797% | 27572 0.19717% | ||||

3 if 5 | 11 | 517148 3.69819% | 27608 0.19743% | ||||

3 if 5 | 12 | 520848 3.72465% | 23.6% 500 draws | ||||

3 if 6 | 13 | 520848 3.72465% | |||||

Fairbrother Partial Cover | 49 | 520848 3.72465% | 27608 0.19743% | 23.6% 500 draws | |||

3 | 3 if 4 | 8 | 518104 3.70503% | 33443 0.23916% | 722 0.00516% | 3 0.00000021% | 20.6% Say 333 draws |

4 if 5 | 8 | 571404 4.08618% | 36764 0.2629% | 765 0.00547% | 15.0% Say 333 draws | ||

4 if 6 | 9 | 575304 4.11407% | 35564 0.25432% | 728 0.00521% | 23.6% Say 333 draws | ||

3 if 4 | 9 | 636924 4.55472% | 38964 0.27864% | 773 0.00553% | 19.6% Say 333 draws | ||

Fairbrother Partial Cover | 49 | 780672 5.58268% | 41412 0.29614% | 777 0.00556% | 23.6% Say 333 draws | ||

4 | 3 if 3 | 7 | 432824 3.09518% | 30163 0.2157% | 763 0.00546% | 4 0.00000029% | 15.0% 250 draws |

3 if 3 | 8 | 656684 4.69603% | 46604 0.33327% | 1012 0.00724% | 21.6% 250 draws | ||

5 if 6 | 8 | 656684 4.69603% | 46604 0.33327% | 1012 0.00724% | 23.6% 250 draws | ||

3 if 6 | 14 | 924448 6.61084% | 53476 0.38241% | 1032 0.00738% | 21.6% 250 draws | ||

3 if 6 | 15 | 896468 6.41075% | 52768 0.37725% | 22.6% 250 draws | |||

Fairbrother Partial Cover | 49 | 1040096 7.46786% | 55216 0.39486% | 1036 0.00741% | 23.6% 250 draws | ||

5 | 4 if 4 | 7 | 432824 3.09518% | 31626 1.65852% | 847 0.00606% | 5 0.00000036% | 18.6% 200 draws |

4 if 6 | 11 | 931215 6.65923% | 59932 0.42858% | 1238 0.00885% | 23.6% 200 draws | ||

Fairbrother Partial Cover | 49 | 1299120 9.29017% | 69020 0.49357% | 1295 0.00926% | |||

6 | 5 if 5 | 7 | 432824 3.09518% | 31024 0.22186% | 889 0.00636% | 6 0.00000043% | 17.0% Say 167 draws |

3 if 4 | 12 | 1199058 8.57461% | 77450 0.55385% | 1542 0.01103% | 21.6% Say 167 draws | ||

4 if 6 | 12 | 1200168 8.58255% | 78846 0.56384% | 1554 0.01111% | |||

Fairbrother Partial Cover | 49 | 1557744 11.13962% | 82824 0.59228% | 23.6% Say 167 draws | |||

7 | 6 if 6 System 7 | 7 | 432824 3.09518% | 31024 0.22186% | 889 0.00636% | 7 0.00000050% | 16.0% Say 143 draws |

5 if 6 | 9 | 874044 6.2504% | 68304 0.48845% | 1604 0.01147% | 17.6% Say 143 draws | ||

3 if 3 | 9 | 933324 6.67432% | 71424 0.51076% | 1640 0.01173% | 20.6% Say 143 draws | ||

4 if 5 | 10 | 1042992 7.45856% | 76728 0.54869% | 1699 0.01215% | 19.6% Say 143 draws | ||

3 if 5 | 15 | 1549064 11.07755% | 95996 0.68648% | 1813 0.01296% | 22.6% Say 143 draws | ||

3 if 6 | 18 | 1457522 10.42292% | 91254 0.65257% | 1801 0.01288% | 20.6% Say 143 draws | ||

Fairbrother Partial Cover | 49 | 1815968 12.98621% | 96628 0.691% | 1813 0.01296% | 23.6% Say 143 draws | ||

8 | 3 if 5 | 16 | 1676136 11.98626% | 105032 0.7511% | 2060 0.01473% | 8 0.00000057% | 21.6% 125 draws |

Fairbrother Partial Cover | 49 | 2073792 14.82994% | 110432 0.78971% | 2072 0.01482% | 23.6% 125 draws | ||

9 | 3 if 4 | 13 | 1583778 11.32579% | 112322 0.80323% | 2308 0.0165% | 9 0.00000064% | 22.6% Say 111 draws |

3 if 6 | 19 | 1652512 11.81732% | 110773 0.79215% | 2299 0.01644% | 20.6% Say 111 draws | ||

Fairbrother Partial Cover | 49 | 2312716 16.53852% | 124236 0.88843% | 2331 0.01667% | 22.6% Say 111 draws | ||

10 | 3 if 3 | 10 | 1262328 9.02706% | 108435 0.77543% | 2510 0.01795% | 10 0.00000072% | 18.6% 100 draws |

4 if 5 | 11 | 1431048 10.2336% | 112220 0.8025% | 2471 0.01767% | |||

4 if 6 | 13 | 1615698 11.55406% | 117330 0.83904% | 2478 0.01772% | 21.6% 100 draws | ||

3 if 6 | 20 | 1849694 13.22739% | 124054 0.88713% | 2554 0.01826% | 20.6% 100 draws | ||

Fairbrother Partial Cover | 49 | 2547860 18.22006% | 138040 0.98714% | 2590 0.01852% | 22.6% 100 draws | ||

11 | 3 if 3 | 11 | 1641948 11.74177% | 133595 0.95535% | 2849 0.02037% | 11 0.00000079% | 21.6% Say 91 draws |

3 if 4 | 14 | 1912218 13.67451% | 139436 0.99712% | 2833 0.02026% | 20.6% Say 91 draws | ||

3 if 5 | 17 | 2198856 15.72429% | 147453 1.05445% | 2845 0.02034% | 21.6% Say 91 draws | ||

Fairbrother Partial Cover | 49 | 2782284 19.89646% | 151844 1.08586% | 2849 0.02037% | 22.6% Say 91 draws | ||

12 | 5 if 5 | 8 | 610148 4.97207% | 56868 0.46341% | 2268 0.01848% | 12 0.00000086% | 18.0% Say 83 draws |

4 if 4 | 9 | 933324 6.67432% | 103404 0.73945% | 2964 0.0212% | 14.0% Say 83 draws | ||

3 if 5 | 18 | 2384112 17.04908% | 163776 1.17118% | 3108 0.02223% | 21.6% Say 83 draws | ||

Fairbrother Partial Cover | 49 | 3016308 21.56999% | 165648 1.18457% | 23.6% Say 83 draws | |||

13 | 3 if 6 | 21 | 2146942 15.35305% | 150314 1.07491% | 3216 0.023% | 13 0.00000093% | 21.6% Say 77 draws |

Fairbrother Partial Cover | 49 | 3249932 23.24067% | 179452 1.28328% | 3367 0.02408% | 23.6% Say 77 draws | ||

14 | 5 if 6 | 10 | 1262328 9.02706% | 131562 0.94082% | 3408 0.02437% | 14 0.00000100% | 17.6% Say 71 draws |

4 if 5 | 12 | 1944978 13.90878% | 168090 1.20203% | 3614 0.02584% | 20.6% Say 71 draws | ||

3 if 4 | 15 | 2315016 16.55497% | 175807 1.25722% | 3602 0.02576% | |||

Fairbrother Partial Cover | 49 | 3483156 24.90848% | 193256 1.382% | 3626 0.02593% | 23.6% Say 71 draws | ||

15 | 3 if 3 | 12 | 2069298 14.79781% | 174528 1.24807% | 3857 0.02758% | 15 0.00000107% | 20.6% Say 67 draws |

3 if 5 | 19 | 2849672 20.37836% | 199060 1.4235% | 3877 0.02772% | 22.6% Say 67 draws | ||

3 if 6 | 22 | 2467416 17.6448% | 179910 1.28656% | 3817 0.0273% | 21.6% Say 67 draws | ||

Fairbrother Partial Cover | 49 | 3705456 26.49817% | 207060 1.48071% | 3885 0.02778% | 23.6% Say 67 draws | ||

16 | 3 if 4 | 16 | 2826296 20.21119% | 214915 1.53688% | 4144 0.02963% | 16 0.00000114% | 22.6% Say 63 draws |

Fairbrother Partial Cover | 49 | 3927356 28.08501% | 220864 1.57943% | 23.6% Say 63 draws | |||

17 | 3 if 6 | 23 | 2823186 20.18895% | 211045 1.50921% | 4391 0.0314% | 17 0.00000122% | 19.6% Say 59 draws |

Fairbrother Partial Cover | 49 | 4145602 29.64571% | 234668 1.67814% | 4403 0.03149% | 23.6% Say 59 draws | ||

18 | 3 if 5 | 20 | 2561412 18.31697% | 204239 1.46054% | 4493 0.03213% | 18 0.00000129% | 17.6% Say 56 draws |

Fairbrother Partial Cover | 49 | 4363448 31.20356% | 248472 1.77685% | 4662 0.03334% | 22.6% Say 56 draws | ||

19 | 4 if 6 | 15 | 2863861 20.47982% | 240047 1.71661% | 4917 0.03516% | 19 0.00000136% | 20.6% Say 53 draws |

Fairbrother Partial Cover | 49 | 4580702 32.75717% | 262276 1.87557% | 4921 0.03519% | 22.6% Say 53 draws | ||

20 | 4 if 4 | 10 | 1262328 9.02706% | 165648 1.18457% | 4880 0.0349% | 20 0.00000143% | 20.6% Say 50 draws |

3 if 4 | 17 | 3280632 23.46021% | 263367 1.88337% | 5176 0.03701% | |||

3 if 6 | 24 | 3112646 22.25892% | 235019 1.68065% | 5002 0.03577% | |||

Fairbrother Partial Cover | 49 | 4797556 34.30792% | 276080 1.97428% | 5180 0.03704% | 22.6% Say 50 draws | ||

21 | 4 if 5 | 13 | 2511978 17.96347% | 242718 1.73571% | 5415 0.03872% | 21 0.00000150% | 20.6% Say 48 draws |

3 if 3 | 13 | 2540538 18.1677% | 237678 1.69966% | 5362 0.03834% | 16.6% Say 48 draws | ||

3 if 5 | 21 | 2884476 20.62725% | 242030 1.73079% | 5359 0.03832% | 21.6% Say 48 draws | ||

Fairbrother Partial Cover | 49 | 5007243 35.80741% | 289884 2.073% | 5439 0.03889% | 22.6% Say 48 draws | ||

22 | 5 if 6 | 11 | 1641948 11.74177% | 211343 1.51134% | 5478 0.03917% | 22 0.00000157% | 18.6% Say 45 draws |

3x2 if 3 | 12 | 2069298 14.79781% | 250008 1.78784% | 5698 0.04075% | 17.6% Say 45 draws | ||

3 if 5 | 22 | 3256671 23.28886% | 267190 1.91071% | 5698 0.04075% | 19.6% Say 45 draws | ||

3 if 6 | 25 | 3446256 24.6446% | 267972 1.9163% | 5658 0.04046% | 21.6% Say 45 draws | ||

Fairbrother Partial Cover | 49 | 5216530 37.30405% | 303688 2.17171% | 5698 0.04075% | 23.6% Say 45 draws | ||

23 | Fairbrother Partial Cover | 49 | 5422099 38.7741% | 317492 2.27042% | 5957 0.04260% | 23 0.00000164% | 23.6% Say 43 draws |

24 | Fairbrother Partial Cover | 49 | 5627268 40.24129% | 331296 2.36914% | 6216 0.04445% | 24 0.00000172% | 23.6% Say 42 draws |

25 | 3 if 3 | 14 | 3051048 21.81842% | 276668 1.97849% | 6339 0.04533% | 25 0.00000179% | 21.6% 40 draws |

4 if 6 | 16 | 3468520 24.80382% | 310618 2.22127% | 6467 0.04625% | 19.6% 40 draws | ||

3 if 4 | 18 | 3761287 26.89743% | 322836 2.30864% | 6467 0.04625% | 20.6% 40 draws | ||

3 if 6 | 26 | 3934656 28.13721% | 311107 2.22476% | 6463 0.04622% | 18.6% 40 draws | ||

Fairbrother Partial Cover | 49 | 5825525 41.65905% | 345100 2.46785% | 6475 0.04630% | 23.6% 40 draws | ||

26 | 3 if 5 | 23 | 3733446 26.69833% | 310091 2.2175% | 6710 0.04798% | 26 0.00000186% | 18.6% Say 38 draws |

Fairbrother Partial Cover | 49 | 6020368 43.0524% | 358904 2.56657% | 6734 0.04816% | 23.6% Say 38 draws | ||

27 | 3 if 6 | 27 | 4416434 31.58247% | 348510 2.49224% | 6993 0.05001% | 27 0.00000193% | 21.6% Say 40 draws |

Fairbrother Partial Cover | 49 | 6208606 44.39851% | 372708 2.66528% | 22.6% Say 40 draws | |||

28 | 6 if 6 System 8 | 8 | 656684 4.69603% | 59724 0.42709% | 2324 0.01662% | 28 0.00000200% | 0 to 20.0% Say 36 draws |

Fairbrother Partial Cover | 49 | 6396444 45.74176% | 386512 2.764% | 7252 0.05186% | 22.6% Say 36 draws | ||

29 | 4 if 5 | 14 | 2893968 20.69512% | 302148 2.1607% | 7319 0.05234% | 29 0.00000207% | 15.6% Say 34 draws |

Fairbrother Partial Cover | 49 | 6583265 47.07774% | 400316 2.86271% | 7511 0.05371% | 22.6% Say 34 draws | ||

30 | 5 if 5 | 9 | 933324 6.67432% | 103404 0.73945% | 5124 0.03664% | 30 0.00000215% | 14.0% Say 33 draws |

3 if 4 | 19 | 4250372 30.39494% | 383203 2.74033% | 7758 0.05548% | 18.6% Say 33 draws | ||

3 if 5 | 24 | 4090196 29.2495% | 349056 2.49614% | 7714 0.05516% | |||

Fairbrother Partial Cover | 49 | 6769686 48.41086% | 414120 2.96142% | 7770 0.05556% | 22.6% Say 33 draws | ||

31 | 3 if 3 | 15 | 3595592 25.71252% | 348099 2.4893% | 7977 0.05704% | 31 0.00000222% | 19.6% Say 32 draws |

3 if 6 | 28 | 4859715 34.75242% | 396962 2.83872% | 8025 0.05739% | 21.6% Say 32 draws | ||

Fairbrother Partial Cover | 49 | 6955463 49.73938% | 427924 3.06014% | 8029 0.05742% | 22.6% Say 32 draws | ||

32 | 4 if 4 | 11 | 1641948 11.74177% | 250008 1.78784% | 7526 0.05382% | 32 0.00000229% | 11.0% Say 31 draws |

Fairbrother Partial Cover | 49 | 7140840 51.06503% | 441728 3.15885% | 8288 0.05927% | 22.6% Say 31 draws | ||

33 | 4 if 6 | 17 | 4077208 29.15662% | 401256 2.86943% | 8515 0.06089% | 33 0.00000236% | 19.6% Say 30 draws |

Fairbrother Partial Cover | 49 | 7314012 52.30341% | 455532 3.25757% | 8547 0.06112% | 23.6% Say 30 draws | ||

34 | 3 if 5 | 25 | 4758516 34.02874% | 405240 2.89792% | 8733 0.06245% | 34 0.00000243% | 20.6% Say 29 draws |

Fairbrother Partial Cover | 49 | 7486784 53.53892% | 469336 3.35628% | 8806 0.06297% | 23.6% Say 29 draws | ||

35 | 3 if 4 | 20 | 4530918 32.40116% | 421917 3.01718% | 8985 0.06425% | 35 0.00000250% | 21.6% Say 29 draws |

3 if 6 | 29 | 5266110 37.6586% | 437895 3.13144% | 9033 0.0646% | 22.6% Say 29 draws | ||

Fairbrother Partial Cover | 49 | 7653657 54.73225% | 483140 3.45499% | 9065 0.06482% | 23.6% Say 29 draws | ||

36 | Fairbrother Partial Cover | 49 | 7820130 55.92272% | 496944 3.55371% | 9324 0.06668% | 36 0.00000257% | 23.6% Say 28 draws |

37 | Fairbrother Partial Cover | 49 | 7982492 57.08379% | 510748 3.65242% | 9583 0.06853% | 37 0.00000265% | 23.6% Say 27 draws |

38 | 5 if 6 | 12 | 2069298 14.79781% | 308616 2.20395% | 8916 0.06376% | 38 0.00000272% | 15.6% Say 26 draws |

3 if 3 | 16 | 4168472 29.80926% | 438126 3.13309% | 9818 0.07021% | 18.6% Say 26 draws | ||

Fairbrother Partial Cover | 49 | 8144454 58.242% | 524552 3.75114% | 9842 0.07038% | 22.6% Say 26 draws | ||

39 | 3 if 5 | 26 | 5253798 37.57056% | 459680 3.28723% | 10045 0.07183% | 39 0.00000279% | 17.6% Say 26 draws |

3 if 6 | 30 | 5906954 42.24136% | 498278 3.56325% | 10089 0.07215% | 22.6% Say 26 draws | ||

Fairbrother Partial Cover | 49 | 8305963 59.39697% | 538356 3.84985% | 10101 0.07223% | |||

40 | 4 if 5 | 15 | 3595592 25.71252% | 412852 2.95236% | 9991 0.07145% | 40 0.00000286% | 14.6% 25 draws |

3 if 4 | 21 | 5107494 36.52432% | 485810 3.47409% | 10288 0.07357% | 17.6% 25 draws | ||

Fairbrother Partial Cover | 49 | 8467072 60.54908% | 552160 3.94856% | 10360 0.07409% | 22.6% 25 draws | ||

41 | 4 if 4 | 12 | 2069298 14.79781% | 359898 2.57368% | 10025 0.07169% | 41 0.00000293% | 11.0% Say 24 draws |

Fairbrother Partial Cover | 49 | 8624873 61.67753% | 565964 4.04728% | 10619 0.07594% | 22.6% Say 24 draws | ||

42 | 4 if 6 | 18 | 5132262 36.70144% | 514347 3.67816% | 10878 0.07779% | 42 0.00000300% | 18.6% Say 24 draws |

Fairbrother Partial Cover | 49 | 8779624 62.78418% | 579768 4.14599% | 10878 0.07779% | 22.6% Say 24 draws | ||

43 | Fairbrother Partial Cover | 49 | 8925529 63.82756% | 593572 4.24471% | 11137 0.07964% | 43 0.00000307% | 22.6% Say 23 draws |

44 | 3 if 3 | 17 | 4763672 34.06561% | 503112 3.59782% | 11336 0.08107% | 44 0.00000315% | 20.6% Say 23 draws |

Fairbrother Partial Cover | 49 | 9071034 64.86809% | 607376 4.34342% | 11396 0.08149% | 22.6% Say 23 draws | ||

45 | 3 if 5 | 27 | 5770688 41.2669% | 527466 3.77197% | 11611 0.08303% | 45 0.00000322% | 17.6% Say 22 draws |

3 if 6 | 31 | 6192996 44.28688% | 544795 3.8959% | 11547 0.08257% | 18.6% Say 22 draws | ||

Fairbrother Partial Cover | 49 | 9215858 65.90374% | 621180 4.44214% | 11655 0.08335% | 22.6% Say 22 draws | ||

46 | 3 if 4 | 22 | 5602521 40.06432% | 557053 3.98355% | 11803 0.0844% | 46 0.00000329% | 20.6% Say 22 draws |

Fairbrother Partial Cover | 49 | 9360282 66.93654% | 634984 4.54085% | 11914 0.0852% | 23.6% Say 22 draws | ||

47 | Fairbrother Partial Cover | 49 | 9501569 67.9469% | 648788 4.63956% | 12173 0.08705% | 47 0.00000336% | 21.6% Say 21 draws |

48 | 3 if 3 | 18 | 5374992 38.43723% | 561498 4.01534% | 12364 0.08842% | 48 0.00000343% | 18.6% Say 21 draws |

Fairbrother Partial Cover | 49 | 9642456 68.9544% | 662592 14.73828% | 12432 0.0889% | 23.6% Say 21 draws | ||

49 | 3 if 5 | 28 | 6037010 43.17141% | 556776 3.98157% | 12579 0.08995% | 49 0.00000350% | 21.6% Say 20 draws |

3 if 6 | 32 | 6688504 47.83032% | 604804 4.32503% | 12643 0.09041% | 19.6% Say 20 draws | ||

Fairbrother Partial Cover | 49 | 9772284 69.88281% | 676360 4.83673% | 12691 0.09075% | 22.6% Say 20 draws | ||

50 | 5 if 5 | 10 | 1262328 9.02706% | 165648 1.18457% | 10038 0.07178% | 50 0.00000358% | 16.0% 20 draws |

Fairbrother Partial Cover | 49 | 9901791 70.80893% | 690128 4.93519% | 12950 0.09261% | 22.6% 20 draws | ||

77 | 3 if 3 | 21 | 7243446 51.79878% | 932420 6.66785% | 19943 0.14261% | 77 0.00000551% | 19.6% Say 13 draws |

3 if 3 Steiner | 22 | 7857696 56.19136% | 988988 7.07238% | 19943 0.14261% | 17.6% Say 13 draws | ||

Fairbrother Partial Cover | 49 | 12293995 87.91588% | 1059200 7.57447% | 19943 0.14261% | 21.6% Say 13 draws | ||

84 | 6 if 6 System 9 | 9 | 933124 6.67432% | 103404 0.73945% | 5124 0.03664% | 84 0.00000600% | 20.0% Say 12 draws |

Fairbrother Partial Cover | 49 | 12649151 90.45565% | 1153921 8.25183% | 21756 0.15558% | 21.6% Say 12 draws | ||

85 | Fairbrother Partial Cover | 49 | 12691622 90.75936% | 1167401 8.34823% | 22015 0.15743% | 85 0.00000608% | 23.6% Say 12 draws |

86 | 3 if 4 | 27 | 8385300 59.96432% | 1027937 7.3509% | 22214 0.15886% | 86 0.00000615% | 17.6% Say 12 draws |

Fairbrother Partial Cover | 49 | 12732005 91.04814% | 1180845 8.44437% | 22274 0.15928% | 23.6% Say 12 draws | ||

130 | 3 if 3 Steiner | 26 | 10130120 72.44174% | 1633320 11.68007% | 33670 0.24078% | 130 0.0000092% | 16.6% Say 8 draws |

Fairbrother Partial Cover | 49 | 13679520 97.82394% | 1766101 12.62961% | 33670 0.24078% | 22.6% Say 8 draws | ||

163 | 3 if 6 | 49 | 13983816 100.00000% | 2016925 14.42328% | 42157 0.30147% | 163 0.0000116% | 16.6% Say 6 draws |

Fairbrother Partial Cover | 49 | 13890084 99.32971% | 2197458 15.71429% | 42217 0.30190% | 22.6% Say 6 draws | ||

210 | 6 if 6 System 10 * | 10 | 1262328 9.02700% | 165648 1.18457% | 10038 0.07178% | 210 0.0000150% | 0 to 21% Say 5 draws |

Fairbrother Partial Cover | 49 | 13966433 99.87569% | 2797386 20.00445% | 54390 0.38895% | 22.6% Say 5 draws | ||

* 1000 plays at 210 plays per draw could easily give nothing over 5 draws and even 10 draws; minimum payout if prize is obtained is 35 CombThrees ie £350.00. If you want regular wins then a System at this number of lines is the worst way to play. |

Colin Fairbrother

]]>Probability of Success for Grouped Lotto Numbers

by Colin Fairbrother

by Colin Fairbrother

Classic Lotto has 49 balls or integers from which 6 are randomly picked for the first prize number. Favourite ways of categorizing these numbers by players with a numerology bent are: -

**Odd and Even Integers**

Odds/Evens Debunked

.**Summing the Integers**

Sums in Lotto Debunked Part 1

Sums in Lotto Debunked Part 2

.**High, Middle and Low integers**

Considered in this article.

.

The numerologists, whether they realize that is what they are or not, then go further and say some combinations are less likely to occur than others and so without any rational reason they filter them out.

In reality the integers as used in Jackpot Lotto have no magnitude and even if they did it is irrelevant in the random picking process. The rather glaring mistake that is made by the promoters of these methods is that it is better to play those that occurr more often such as Sums that add to 150.

To illustrate this correlation I will consider some other categorizations and show that the occurrence of the winning numbers in some 1636 draws of the UK Lotto are proportional to the extent the category is represented in the total possibilities.

Consider first the usual scenario where it is stated you are better off playing numbers for a Pick 6 Lotto game where when considered in numerical order the first two lowest integers are from 1 to 16, the middle two are between 17 and 33 and the last two are the highest between 34 to 49. For a Pick 6, Pool 49 Lotto game we have 3,549,600 from the 13,983,816 possibilities that fall within these constraints or 25%. For 1636 draws from the UK Lotto game we have 410 that fit the constraints to give 25% which is little different to considering those with lexicographic ID of < 3,549,601 with 24%, > 10,434,216 with 25% and between IDs 5,217,108 and 8,766,708 with 24%.

We can make the UK 6/49 Lotto game with a first prize chance of winning of 1 in 13,983,816 a game with a chance of winning 1 in 7 by simply dividing the possibilites in either numerical (lexicographic order) or last number order by 7 as shown in the table below with results for the UK 1636 draws:

Category | ID Start | ID Finish | Lex Last | LNO Last | Lex Cat | LNO Cat |
---|---|---|---|---|---|---|

A | 1 | 1997688 | 02 04 20 22 33 49 | 06 07 09 23 25 37 | 224 | 208 |

B | 1997689 | 3995376 | 03 10 16 18 19 35 | 03 13 24 26 31 41 | 229 | 244 |

C | 3995377 | 5993064 | 05 07 11 13 37 38 | 09 32 36 40 41 43 | 244 | 213 |

D | 5993065 | 7990752 | 07 09 10 14 18 40 | 11 13 37 39 43 45 | 227 | 260 |

E | 7990753 | 9988440 | 09 19 24 26 37 46 | 14 31 32 34 40 47 | 233 | 233 |

F | 9988441 | 11986128 | 13 25 27 41 42 49 | 15 16 28 30 46 48 | 247 | 249 |

G | 11986129 | 13983816 | 44 45 46 47 48 49 | 44 45 46 47 48 49 | 232 | 229 |

Another interesting categorization is by the balls or integers and for a 6/49 game 7 equal length sets can be used as in the following; the first in numerical order followed by another with an offset 7 but any order is just as applicable: -

Set 1

Group A: { 1, 2, 3, 4, 5, 6, 7}

Group B: { 8, 9, 10, 11, 12, 13, 14}

Group C: {15, 16, 17, 18, 19, 20, 21}

Group D: {22, 23, 24, 25, 26, 27, 28}

Group E: {29, 30, 31, 32, 33, 34, 35}

Group F: {36, 37, 38, 39, 40, 41, 42}

Group G: {43, 44, 45, 46, 47, 48, 49}

Set 2

Group A: { 1, 8, 15, 22, 29, 36, 43}

Group B: { 2, 9, 16, 23, 30, 37, 44}

Group C: { 3, 10, 17, 24, 31, 38, 45}

Group D: { 4, 11, 18, 25, 32, 39, 46}

Group E: { 5, 12, 19, 26, 33, 40, 47}

Group F: { 6, 13, 20, 27, 34, 41, 48}

Group G: { 7, 14, 21, 28, 35, 42, 49}

A formula may be applied where ! means factorial (Pl + Pk - 1)! / Pk! x (Pl - 1)! ie (7 + 5)! / 6! x 6! or 12! / 6! x 6! or 12 x 11 x 10 x 9 x 8 x 7 / 6 x 5 x 4 x 3 x2 x1 which can be simplified to 11 x 2 x 3 x 2 x 7 = 924 to get the number of combinations

For combinations without repetition of the categories we have the simple formula 7c6 ie 7! / 6! x (7 - 6)! or 7 categories ie

A B C D E F

A B C D E G

A B C D F G

A B C E F G

A B D E F G

A C D E F G

B C D E F G

There are 117,649 possibilities for each of these 7 unmatched categories giving a total of 823,543 which is just less than 6% of all the 13,983,816 possibilities. The table below shows the occurrence for both Group Set 1 and Group Set 2: -

Unmatched UK Lotto | Count Set 1 | Count Set 2 |
---|---|---|

ABCDEF | 15 | 11 |

ABCDEG | 12 | 15 |

ABCDFG | 24 | 12 |

ABCEFG | 11 | 12 |

ABDEFG | 17 | 13 |

ACDEFG | 15 | 13 |

BCDEFG | 14 | 12 |

ABCDFG certainly stands out for Set 1 but is pretty well normal in Set 2. This is something not predictable but not unusual when dealing with random selections.

Considering all the 924 possibilities for groupings of 6 from the 7 groups ie A B C D E F G with repeats allowed, one can aggregate these into 11 groups based on Group Repeats with varying degrees of occurrence as in the table below: -

Group Occurrence for Number Drawn | Possibilities | Percentage of All Possibilities | Expected UK Lotto | Actual UK Lotto |
---|---|---|---|---|

Same Group x 2 | 5423859 | 38.78668 | 635 | 615 |

(Same Group x 2) x 2 | 4418526 | 31.59742 | 517 | 495 |

Same Group x 3 | 1678985 | 12.00662 | 196 | 201 |

Same Group x 2 with Same Group x 3 | 1066730 | 7.62831 | 125 | 158 |

Unmatched or All Different Groups | 823543 | 5.88925 | 96 | 108 |

(Same Group x 2) x 3 | 329280 | 2.35472 | 39 | 32 |

Same Group x 4 | 184485 | 1.31927 | 22 | 20 |

Same x 4 with Same x 2 | 26460 | 0.18921 | 3 | 2 |

Same x 3 with Same x 3 | 25725 | 0.18396 | 3 | 5 |

Same x 5 | 6174 | 0.04415 | 0 | 0 |

All Same Group | 49 | 0.00035 | 0 | 0 |

The last category in the above table has only 49 possibilities and as expected no success for the 1636 UK Lotto draws. Comparing the probability for playing 49 numbers of 0.0000035 or 0.00035% with that for 1 number of 0.0000000715 we see there is a big difference but the latter is the only relevant probability, irrespective of any groupings made, as this is the one that is paid on. Just picture a 6/49 Lotto game as having 13,983,816 relevant groups and all should be clear as 13,983,816 x 0.0000000715=1 or certainty. Multiplying any grouping of distinct lines by 0.0000000715 gives the probability of success.

If money was no object and you were inclined to throw it away, then playing 25,725 lines per draw, where 3 of the integers were from one group and the other 3 from another over the 1,636 draws would have cost £42,086,100. Even though this group configuration did about 70% better than expected with 5 Jackpot wins a quick, rough calculation shows the return including sub prizes is only about the 50% mark.

Comparing theSet 1 and set 2 where the numbers are from the same group as in the table below we see that including 01 02 03 04 05 06 makes no difference to the probability of 0.00035% for the set winning 1st prize. Included for comparison is an optimized set where I deliberately made sure 8 lines with consecutive integers were included and the chances of this set winning first prize are identical to the other two.

The difference for the optimized set is that none of the paying subset CombThrees, CombFours and CombFives are repeated whereas for the other two sets they are excessive. So, with a possible 980 unique CombThrees for 49 lines the optimized set has this and covers 9,795,325 or 70% (as good as it gets) of the 13,983,816 possible combinations of 6 integers from 49. Set 1 and Set 2 have only 245 distinct CombThrees which are repeated 4 times with a consequent coverage of only 21%.

Set 1 Same Group | Set 2 Same Group | Set Optimized |
---|---|---|

01 02 03 04 05 06 | 01 08 15 22 29 36 | 01 02 03 04 05 06 |

01 02 03 04 05 07 | 01 08 15 22 29 43 | 07 08 09 10 11 12 |

01 02 03 04 06 07 | 01 08 15 22 36 43 | 13 14 15 16 17 18 |

01 02 03 05 06 07 | 01 08 15 29 36 43 | 19 20 21 22 23 24 |

01 02 04 05 06 07 | 01 08 22 29 36 43 | 25 26 27 28 29 30 |

01 03 04 05 06 07 | 01 15 22 29 36 43 | 31 32 33 34 35 36 |

02 03 04 05 06 07 | 08 15 22 29 36 43 | 37 38 39 40 41 42 |

08 09 10 11 12 13 | 02 09 16 23 30 37 | 43 44 45 46 47 48 |

08 09 10 11 12 14 | 02 09 16 23 30 44 | 01 14 20 33 41 46 |

08 09 10 11 13 14 | 02 09 16 23 37 44 | 01 23 25 35 38 44 |

08 09 10 12 13 14 | 02 09 16 30 37 44 | 01 13 36 42 48 49 |

08 09 11 12 13 14 | 02 09 23 30 37 44 | 01 16 19 26 39 45 |

08 10 11 12 13 14 | 02 16 23 30 37 44 | 01 10 17 27 34 40 |

09 10 11 12 13 14 | 09 16 23 30 37 44 | 02 07 13 21 32 45 |

15 16 17 18 19 20 | 03 10 17 24 31 38 | 02 15 20 26 38 47 |

15 16 17 18 19 21 | 03 10 17 24 31 45 | 02 22 31 37 44 49 |

15 16 17 18 20 21 | 03 10 17 24 38 45 | 02 11 25 36 40 43 |

15 16 17 19 20 21 | 03 10 17 31 38 45 | 02 08 17 24 30 33 |

15 16 18 19 20 21 | 03 10 24 31 38 45 | 03 15 23 30 34 46 |

15 17 18 19 20 21 | 03 17 24 31 38 45 | 03 14 21 28 38 43 |

16 17 18 19 20 21 | 10 17 24 31 38 45 | 03 11 18 24 42 44 |

22 23 24 25 26 27 | 04 11 18 25 32 39 | 03 08 29 35 40 47 |

22 23 24 25 26 28 | 04 11 18 25 32 46 | 03 09 16 20 25 48 |

22 23 24 25 27 28 | 04 11 18 25 39 46 | 04 11 17 21 31 48 |

22 23 24 26 27 28 | 04 11 18 32 39 46 | 04 08 23 27 36 39 |

22 23 25 26 27 28 | 04 11 25 32 39 46 | 04 10 15 22 25 32 |

22 24 25 26 27 28 | 04 18 25 32 39 46 | 04 07 26 35 42 46 |

23 24 25 26 27 28 | 11 18 25 32 39 46 | 04 09 24 28 45 49 |

29 30 31 32 33 34 | 05 12 19 26 33 40 | 05 13 24 26 31 43 |

29 30 31 32 33 35 | 05 12 19 26 33 47 | 05 11 15 28 35 39 |

29 30 31 32 34 35 | 05 12 19 26 40 47 | 05 10 16 23 29 42 |

29 30 31 33 34 35 | 05 12 19 33 40 47 | 05 09 18 21 37 47 |

29 30 32 33 34 35 | 05 12 26 33 40 47 | 05 12 22 36 38 45 |

29 31 32 33 34 35 | 05 19 26 33 40 47 | 06 12 17 19 28 46 |

30 31 32 33 34 35 | 12 19 26 33 40 47 | 06 09 22 27 42 43 |

36 37 38 39 40 41 | 06 13 20 27 34 41 | 06 07 24 29 36 37 |

36 37 38 39 40 42 | 06 13 20 27 34 48 | 06 08 15 31 41 45 |

36 37 38 39 41 42 | 06 13 20 27 41 48 | 06 10 21 33 39 49 |

36 37 38 40 41 42 | 06 13 20 34 41 48 | 06 13 20 30 40 44 |

36 37 39 40 41 42 | 06 13 27 34 41 48 | 07 18 23 33 40 48 |

36 38 39 40 41 42 | 06 20 27 34 41 48 | 07 14 19 25 34 49 |

37 38 39 40 41 42 | 13 20 27 34 41 48 | 08 14 26 32 37 48 |

43 44 45 46 47 48 | 07 14 21 28 35 42 | 09 17 29 32 41 44 |

43 44 45 46 47 49 | 07 14 21 28 35 49 | 10 18 19 30 35 41 |

43 44 45 46 48 49 | 07 14 21 28 42 49 | 11 13 19 27 33 37 |

43 44 45 47 48 49 | 07 14 21 35 42 49 | 12 18 20 29 34 43 |

43 44 46 47 48 49 | 07 14 28 35 42 49 | 12 14 30 31 39 47 |

43 45 46 47 48 49 | 07 21 28 35 42 49 | 16 22 28 34 41 47 |

44 45 46 47 48 49 | 14 21 28 35 42 49 | 16 27 32 38 46 49 |

So much for HOW TO WIN THE LOTTERY SCHEMES, however there is some brain stimulation in proving them wrong.

Colin Fairbrother

The possibilities for 1st prize in a Lotto game such as UK Lotto where each line has six distinct integers from a Pool of 49 is 13,983,816. Each line has the same chance of being drawn as any other.

If you played one line your chances of a match for all six integers in one draw is 1/13,983,816 or 0.0000000715112 or 0.00000715112%.

The table below shows your chances of success for one draw with distinct combinations for 1 and multiples of 10 to 11,111,111 lines and the balance of 2,872,705 to make 13,983,816. Adding the probabilities gives 1 or certainty for 13,983,816 distinct combinations.

Lines | Probability |
---|---|

1 | 0.0000000715112 |

10 | 0.0000007151124 |

100 | 0.0000071511238 |

1000 | 0.0000715112384 |

10000 | 0.0007151123842 |

100000 | 0.0071511238420 |

1000000 | 0.0715112384202 |

2872705 | 0.2054306921659 |

10000000 | 0.7151123842018 |

Plays per Draw and Draws can be interchanged so in the UK Lotto playing 1813 plays per draw over 1584 draws from November 1994 gives 2,871,792 plays with a 20% chance of a 1st prize win.

An important concept is proportionality in terms of plays made and chances of success. Time and time again in this field of interest, Lotto number analysis, you will find the facts misconstrued by not taking into account proportionality especially by numerologists whether outed or not.

An example is consecutive ascending Numerical Order numbers of which there are 44 for a Pick 6, Pool 49 Lotto game as in the table below.

Comb6 |
---|

01 02 03 04 05 06 |

02 03 04 05 06 07 |

03 04 05 06 07 08 |

04 05 06 07 08 09 |

05 06 07 08 09 10 |

06 07 08 09 10 11 |

07 08 09 10 11 12 |

08 09 10 11 12 13 |

09 10 11 12 13 14 |

10 11 12 13 14 15 |

11 12 13 14 15 16 |

12 13 14 15 16 17 |

13 14 15 16 17 18 |

14 15 16 17 18 19 |

15 16 17 18 19 20 |

16 17 18 19 20 21 |

17 18 19 20 21 22 |

18 19 20 21 22 23 |

19 20 21 22 23 24 |

20 21 22 23 24 25 |

21 22 23 24 25 26 |

22 23 24 25 26 27 |

23 24 25 26 27 28 |

24 25 26 27 28 29 |

25 26 27 28 29 30 |

26 27 28 29 30 31 |

27 28 29 30 31 32 |

28 29 30 31 32 33 |

29 30 31 32 33 34 |

30 31 32 33 34 35 |

31 32 33 34 35 36 |

32 33 34 35 36 37 |

33 34 35 36 37 38 |

34 35 36 37 38 39 |

35 36 37 38 39 40 |

36 37 38 39 40 41 |

37 38 39 40 41 42 |

38 39 40 41 42 43 |

39 40 41 42 43 44 |

40 41 42 43 44 45 |

41 42 43 44 45 46 |

42 43 44 45 46 47 |

43 44 45 46 47 48 |

44 45 46 47 48 49 |

If playing these 44 consecutives over 1584 draws the chances of a 1st prize match for 69696 plays is only 0.5%. However, for the 14 million random numbers generated using the Mersenne Twister algorithm playing the 44 lines we can expect about 44 successes. We actually get 39 of which 29 are distinct as in the table below with much to the consternation of numerologists 44 45 46 47 48 49 repeating twice. This is not unexpected as the 14 million generated random numbers only has 63% of the distinct combinations so 63% of 44 is 28. See Random Lotto Numbers using the Mersenne Twister

Comb6 | Cnt |
---|---|

05 06 07 08 09 10 | 3 |

13 14 15 16 17 18 | 3 |

35 36 37 38 39 40 | 3 |

15 16 17 18 19 20 | 3 |

26 27 28 29 30 31 | 2 |

09 10 11 12 13 14 | 2 |

36 37 38 39 40 41 | 2 |

14 15 16 17 18 19 | 2 |

30 31 32 33 34 35 | 2 |

44 45 46 47 48 49 | 2 |

24 25 26 27 28 29 | 2 |

03 04 05 06 07 08 | 1 |

04 05 06 07 08 09 | 1 |

11 12 13 14 15 16 | 1 |

02 03 04 05 06 07 | 1 |

29 30 31 32 33 34 | 1 |

33 34 35 36 37 38 | 1 |

37 38 39 40 41 42 | 1 |

38 39 40 41 42 43 | 1 |

39 40 41 42 43 44 | 1 |

41 42 43 44 45 46 | 1 |

42 43 44 45 46 47 | 1 |

43 44 45 46 47 48 | 1 |

27 28 29 30 31 32 | 1 |

Colin Fairbrother

Recently, I had a bizarre contributor in a Newsgroup who to be honest is not quite compos mentis state the following regarding 5 QuickPicks he purchased for the UK 6/49 Lotto, "Not with so many numbers above 8 TO 10, position 1 surely. 80% Of winning JACKPOT lines have a number between 1 AND 8 in P1"

For 1584 draws of the UK Lotto game you do have 67% starting with 1 to 8 when the line is in Numerical Order. To expect a small sample of 5 Random Selections to reflect this distribution is totally unrealistic. Indeed, if we divide the lexicograhic enumeration into thirds then we have 7 draws consecutively in the top third, namely draws 952, 953, 954, 955, 956 and 957. Harry thinks this shouldn't happen!

Basically to restrict the numbers played to a grouping that has the highest winnings is to ignore proportionality as is done with Sums, Odds/Evens and other numerology mumbo jumbo. If a grouping is preferred where the wins mostly occur your chances of success have not improved as evidenced by comparing the lines in the group to all possible combinations ratio and wins for that group over 1584 draws of the UK Lotto game. The ratios are pretty much the same as shown in the table below: -

From Lex Index | To Lex Index | First Integer | Lex Combinations | % All Lex | UK Lotto Wins | % All UK Wins |
---|---|---|---|---|---|---|

1 | 1712304 | 1 | 1712304 | 12.24 | 184 | 11.62 |

1712305 | 3246243 | 2 | 1533939 | 10.97 | 177 | 11.17 |

3246244 | 4616997 | 3 | 1370754 | 9.80 | 154 | 9.72 |

4616998 | 5838756 | 4 | 1221759 | 8.74 | 144 | 9.09 |

5838757 | 6924764 | 5 | 1086008 | 7.77 | 114 | 7.20 |

6924765 | 7887362 | 6 | 962598 | 6.88 | 113 | 7.13 |

7887363 | 8738030 | 7 | 850668 | 6.08 | 97 | 6.12 |

8738031 | 9487428 | 8 | 749398 | 5.36 | 80 | 5.05 |

9487429 | 10145436 | 9 | 658008 | 4.71 | 81 | 5.11 |

10145437 | 10721193 | 10 | 575757 | 4.12 | 65 | 4.10 |

10721194 | 11223135 | 11 | 501942 | 3.59 | 68 | 4.29 |

11223136 | 11659032 | 12 | 435897 | 3.12 | 49 | 3.09 |

11659033 | 12036024 | 13 | 376992 | 2.70 | 36 | 2.27 |

12036025 | 12360656 | 14 | 324632 | 2.32 | 35 | 2.21 |

12360657 | 12638912 | 15 | 278256 | 1.99 | 32 | 2.02 |

12638913 | 12876248 | 16 | 237336 | 1.70 | 26 | 1.64 |

12876249 | 13077625 | 17 | 201376 | 1.44 | 20 | 1.26 |

13077625 | 13247535 | 18 | 169911 | 1.22 | 26 | 1.64 |

13247536 | 13390041 | 19 | 142506 | 1.02 | 17 | 1.07 |

13390042 | 13508796 | 20 | 118755 | 0.85 | 9 | 0.57 |

13508797 | 13607076 | 21 | 98280 | 0.70 | 7 | 0.44 |

13607077 | 13687806 | 22 | 80730 | 0.58 | 10 | 0.63 |

13687807 | 13753586 | 23 | 65780 | 0.47 | 9 | 0.57 |

13753587 | 13806716 | 24 | 53130 | 0.38 | 10 | 0.63 |

13806717 | 13849220 | 25 | 42504 | 0.30 | 4 | 0.25 |

13849221 | 13882869 | 26 | 33649 | 0.24 | 7 | 0.44 |

Consider the following two samples of 5 lines for a 6/49 Lotto game. The first has lexicograhic indices below or equal to 6,298,682 or in the lower half of the 13,983,816 combinations. The second table has lexicograhic indices higher than or equal to 13,390,042 or the higher 1/24th of the 13,983,816 some 594,000. According to Harry you have a better chance with the first set. The reality is that they have an equal chance at all prize levels both with a Six coverage of 0.00004% and Three coverage of 9.29%

Lexicographic Index | Comb6 |
---|---|

670034 | 01 06 11 16 21 26 |

2324197 | 02 07 12 17 22 27 |

3803853 | 03 08 13 18 23 28 |

5124008 | 04 09 14 19 24 29 |

6298682 | 05 10 15 20 25 30 |

Lexicographic Index | Comb6 |
---|---|

13390042 | 20 21 22 23 24 25 |

13849221 | 26 27 28 29 30 31 |

13965253 | 32 33 34 35 36 37 |

13982893 | 38 39 40 41 42 43 |

13983816 | 44 45 46 47 48 49 |

I did point out if you are looking at the results in Numerical Order then for example 1 will always be in Position 1 and 49 always in Position 6. In other words you can't compare integer occurrences fairly when the presentation is in Numerical Order.

If the occurrence is compared for the order drawn this is a fair comparison and is shown in the table below for 1584 draws in the UK Lotto game. Integer 23 has the highest occurrence of 48 for the first main ball drawn and integers 20 and 10 have the lowest at 20 for respectively the first and second ball drawn.

Integer | Ball 1 | Ball 2 | Ball 3 | Ball 4 | Ball 5 | Ball 6 |
---|---|---|---|---|---|---|

1 | 35 | 39 | 24 | 27 | 23 | 36 |

2 | 30 | 37 | 37 | 31 | 37 | 27 |

3 | 36 | 30 | 38 | 26 | 27 | 36 |

4 | 30 | 35 | 30 | 30 | 30 | 35 |

5 | 29 | 22 | 31 | 37 | 33 | 31 |

6 | 24 | 38 | 35 | 35 | 37 | 32 |

7 | 22 | 32 | 34 | 21 | 33 | 40 |

8 | 33 | 32 | 26 | 36 | 26 | 32 |

9 | 31 | 35 | 37 | 30 | 36 | 36 |

10 | 36 | 20 | 38 | 37 | 30 | 33 |

11 | 38 | 33 | 39 | 44 | 29 | 32 |

12 | 35 | 37 | 22 | 34 | 30 | 36 |

13 | 25 | 30 | 34 | 28 | 27 | 25 |

14 | 38 | 33 | 30 | 25 | 29 | 25 |

15 | 24 | 36 | 28 | 30 | 28 | 30 |

16 | 38 | 28 | 28 | 25 | 29 | 27 |

17 | 29 | 26 | 38 | 37 | 29 | 31 |

18 | 27 | 43 | 41 | 21 | 32 | 27 |

19 | 30 | 34 | 33 | 36 | 40 | 21 |

20 | 20 | 21 | 26 | 38 | 21 | 34 |

21 | 34 | 28 | 25 | 38 | 24 | 22 |

22 | 33 | 31 | 23 | 33 | 35 | 33 |

23 | 48 | 31 | 30 | 33 | 39 | 33 |

24 | 31 | 32 | 35 | 32 | 37 | 30 |

25 | 33 | 39 | 38 | 32 | 36 | 33 |

26 | 30 | 29 | 32 | 29 | 36 | 33 |

27 | 34 | 27 | 37 | 33 | 43 | 31 |

28 | 40 | 32 | 34 | 32 | 34 | 23 |

29 | 26 | 41 | 30 | 29 | 27 | 33 |

30 | 35 | 31 | 26 | 29 | 39 | 43 |

31 | 34 | 29 | 32 | 39 | 41 | 38 |

32 | 33 | 37 | 39 | 31 | 29 | 36 |

33 | 35 | 34 | 29 | 36 | 36 | 44 |

34 | 36 | 31 | 35 | 24 | 40 | 27 |

35 | 40 | 33 | 35 | 34 | 30 | 30 |

36 | 24 | 29 | 36 | 30 | 33 | 31 |

37 | 29 | 37 | 31 | 30 | 24 | 27 |

38 | 29 | 37 | 38 | 38 | 39 | 39 |

39 | 26 | 31 | 35 | 33 | 44 | 34 |

40 | 37 | 29 | 40 | 35 | 30 | 38 |

41 | 35 | 32 | 27 | 24 | 21 | 26 |

42 | 32 | 31 | 33 | 27 | 33 | 39 |

43 | 44 | 40 | 31 | 34 | 35 | 31 |

44 | 34 | 43 | 22 | 39 | 34 | 41 |

45 | 35 | 23 | 33 | 40 | 22 | 43 |

46 | 33 | 29 | 28 | 33 | 31 | 33 |

47 | 31 | 32 | 32 | 40 | 41 | 31 |

48 | 33 | 33 | 34 | 38 | 30 | 26 |

49 | 30 | 32 | 35 | 31 | 35 | 30 |

-------------------------------------------------

Integer Occurrence For Order Drawn UK Lotto 1584 Draws | ||||||||||

Integer | Position 1 | Position 2 | Position 3 | Position 4 | Position 5 | Position 6 | Max | Min | Avg | SD |

1 | 35 | 39 | 24 | 27 | 23 | 36 | 39 | 23 | 30.67 | 6.83 |

2 | 30 | 37 | 37 | 31 | 37 | 27 | 37 | 27 | 33.17 | 4.40 |

3 | 36 | 30 | 38 | 26 | 27 | 36 | 38 | 26 | 32.17 | 5.15 |

4 | 30 | 35 | 30 | 30 | 30 | 35 | 35 | 30 | 31.67 | 2.58 |

5 | 29 | 22 | 31 | 37 | 33 | 31 | 37 | 22 | 30.50 | 4.97 |

6 | 24 | 38 | 35 | 35 | 37 | 32 | 38 | 24 | 33.50 | 5.09 |

7 | 22 | 32 | 34 | 21 | 33 | 40 | 40 | 21 | 30.33 | 7.39 |

8 | 33 | 32 | 26 | 36 | 26 | 32 | 36 | 26 | 30.83 | 4.02 |

9 | 31 | 35 | 37 | 30 | 36 | 36 | 37 | 30 | 34.17 | 2.93 |

10 | 36 | 20 | 38 | 37 | 30 | 33 | 38 | 20 | 32.33 | 6.71 |

11 | 38 | 33 | 39 | 44 | 29 | 32 | 44 | 29 | 35.83 | 5.49 |

12 | 35 | 37 | 22 | 34 | 30 | 36 | 37 | 22 | 32.33 | 5.61 |

13 | 25 | 30 | 34 | 28 | 27 | 25 | 34 | 25 | 28.17 | 3.43 |

14 | 38 | 33 | 30 | 25 | 29 | 25 | 38 | 25 | 30.00 | 4.98 |

15 | 24 | 36 | 28 | 30 | 28 | 30 | 36 | 24 | 29.33 | 3.93 |

16 | 38 | 28 | 28 | 25 | 29 | 27 | 38 | 25 | 29.17 | 4.54 |

17 | 29 | 26 | 38 | 37 | 29 | 31 | 38 | 26 | 31.67 | 4.80 |

18 | 27 | 43 | 41 | 21 | 32 | 27 | 43 | 21 | 31.83 | 8.64 |

19 | 30 | 34 | 33 | 36 | 40 | 21 | 40 | 21 | 32.33 | 6.47 |

20 | 20 | 21 | 26 | 38 | 21 | 34 | 38 | 20 | 26.67 | 7.63 |

21 | 34 | 28 | 25 | 38 | 24 | 22 | 38 | 22 | 28.50 | 6.25 |

22 | 33 | 31 | 23 | 33 | 35 | 33 | 35 | 23 | 31.33 | 4.27 |

23 | 48 | 31 | 30 | 33 | 39 | 33 | 48 | 30 | 35.67 | 6.80 |

24 | 31 | 32 | 35 | 32 | 37 | 30 | 37 | 30 | 32.83 | 2.64 |

25 | 33 | 39 | 38 | 32 | 36 | 33 | 39 | 32 | 35.17 | 2.93 |

26 | 30 | 29 | 32 | 29 | 36 | 33 | 36 | 29 | 31.50 | 2.74 |

27 | 34 | 27 | 37 | 33 | 43 | 31 | 43 | 27 | 34.17 | 5.46 |

28 | 40 | 32 | 34 | 32 | 34 | 23 | 40 | 23 | 32.50 | 5.50 |

29 | 26 | 41 | 30 | 29 | 27 | 33 | 41 | 26 | 31.00 | 5.48 |

30 | 35 | 31 | 26 | 29 | 39 | 43 | 43 | 26 | 33.83 | 6.40 |

31 | 34 | 29 | 32 | 39 | 41 | 38 | 41 | 29 | 35.50 | 4.59 |

32 | 33 | 37 | 39 | 31 | 29 | 36 | 39 | 29 | 34.17 | 3.82 |

33 | 35 | 34 | 29 | 36 | 36 | 44 | 44 | 29 | 35.67 | 4.84 |

34 | 36 | 31 | 35 | 24 | 40 | 27 | 40 | 24 | 32.17 | 5.98 |

35 | 40 | 33 | 35 | 34 | 30 | 30 | 40 | 30 | 33.67 | 3.72 |

36 | 24 | 29 | 36 | 30 | 33 | 31 | 36 | 24 | 30.50 | 4.04 |

37 | 29 | 37 | 31 | 30 | 24 | 27 | 37 | 24 | 29.67 | 4.37 |

38 | 29 | 37 | 38 | 38 | 39 | 39 | 39 | 29 | 36.67 | 3.83 |

39 | 26 | 31 | 35 | 33 | 44 | 34 | 44 | 26 | 33.83 | 5.91 |

40 | 37 | 29 | 40 | 35 | 30 | 38 | 40 | 29 | 34.83 | 4.45 |

41 | 35 | 32 | 27 | 24 | 21 | 26 | 35 | 21 | 27.50 | 5.17 |

42 | 32 | 31 | 33 | 27 | 33 | 39 | 39 | 27 | 32.50 | 3.89 |

43 | 44 | 40 | 31 | 34 | 35 | 31 | 44 | 31 | 35.83 | 5.19 |

44 | 34 | 43 | 22 | 39 | 34 | 41 | 43 | 22 | 35.50 | 7.56 |

45 | 35 | 23 | 33 | 40 | 22 | 43 | 43 | 22 | 32.67 | 8.64 |

46 | 33 | 29 | 28 | 33 | 31 | 33 | 33 | 28 | 31.17 | 2.23 |

47 | 31 | 32 | 32 | 40 | 41 | 31 | 41 | 31 | 34.50 | 4.68 |

48 | 33 | 33 | 34 | 38 | 30 | 26 | 38 | 26 | 32.33 | 4.03 |

49 | 30 | 32 | 35 | 31 | 35 | 30 | 35 | 30 | 32.17 | 2.32 |

Colin Fairbrother

Using the 1579 results for the UK Lotto, which is Pick 6 from a Pool of 49, and dates from November 19, 1994 to February 9, 2011 the most popular integers can be inferred from the extent that the first prize is shared according to a

The results for first prize and the extent shared are freely available and the calculation is little more than trivial. For a full representation of the 49 integers the 1579 draws is restricted to 80 with the first prize shared by 8 or more. If the more shared the first prize is an indicator of integer popularity then the converse where no first is obtained let alone shared should be an indicator of unpopular integers.

Integer | Count |
---|---|

07 | 20 |

23 | 19 |

19 | 18 |

12 | 16 |

29 | 16 |

11 | 16 |

17 | 15 |

03 | 14 |

24 | 14 |

09 | 13 |

14 | 13 |

08 | 13 |

18 | 12 |

05 | 12 |

27 | 12 |

22 | 11 |

10 | 11 |

44 | 11 |

33 | 10 |

32 | 10 |

38 | 10 |

39 | 9 |

06 | 9 |

42 | 9 |

43 | 9 |

04 | 9 |

34 | 9 |

20 | 8 |

02 | 8 |

31 | 8 |

48 | 8 |

25 | 8 |

30 | 8 |

47 | 7 |

16 | 7 |

13 | 7 |

41 | 7 |

40 | 7 |

46 | 6 |

45 | 6 |

36 | 6 |

21 | 6 |

28 | 6 |

01 | 6 |

26 | 5 |

15 | 5 |

49 | 4 |

35 | 4 |

37 | 3 |

The question may arise for some, "Should I restrict the numbers played to those that are less popular on the basis that I am less likely to share First Prize?"

The answer is very simple. If you reduce the integers that form the lines of the set you play then generally you reduce drastically your chances of success. There are many supporting articles for this advice on this site one of the main ones being **ANALYSIS OF 15 LOTTO NUMBER SETS - WORST TO BEST **where a set of around 28 numbers played is drawn from various Wheels or Covers and from the full Pool of numbers or a lesser amount.

A simple comparison between the winning numbers with the two most popular played integers 07 and 23 gives 8 wins whereas the two most unpopular played 35 and 37 give 12 or 37 and 49 give 19. Taking 11 and 23 which are high in the ranking for both wins and popularity gives 18.

The UK Lotto doesn't pay on getting one integer right in a line - the table below shows the top 46 repeating triples, which pay a constant £10, from the 18,424 possible and for 7 in red where all three integers are in the most popular - for 4 in blue with all integers not in the most popular. That is nearly double and certainly not supporting the notion to play unpopular integers.

Triple | Cnt |
---|---|

022340 | 9 |

103448 | 9 |

022847 | 8 |

194445 | 8 |

092742 | 8 |

063045 | 8 |

083343 | 8 |

061219 | 8 |

182444 | 7 |

061019 | 7 |

243334 | 7 |

253745 | 7 |

154248 | 7 |

101331 | 7 |

113343 | 7 |

183343 | 7 |

123945 | 7 |

052225 | 7 |

194449 | 7 |

032231 | 7 |

023435 | 7 |

113846 | 7 |

212438 | 7 |

042440 | 7 |

262846 | 7 |

183236 | 7 |

010738 | 7 |

183239 | 7 |

172940 | 7 |

172238 | 7 |

152327 | 7 |

052438 | 7 |

012745 | 7 |

323345 | 7 |

313341 | 7 |

313240 | 7 |

Winning Sixes with Popular Integers |
---|

03 10 14 17 19 27 |

06 07 14 19 22 27 |

03 06 11 17 23 29 |

08 14 24 27 29 32 |

09 11 23 27 29 33 |

12 18 22 24 32 33 |

05 09 12 14 18 38 |

05 06 18 22 24 38 |

07 11 12 17 23 39 |

07 11 19 24 32 39 |

03 12 17 18 22 42 |

06 10 19 29 32 42 |

07 17 23 32 38 42 |

08 14 18 23 27 44 |

06 07 11 23 32 44 |

08 11 14 18 33 44 |

05 23 24 29 39 44 |

Winning Sixes with Unpopular Integers |
---|

02 13 20 36 37 40 |

15 26 30 35 40 41 |

01 13 26 28 35 45 |

04 25 26 30 37 45 |

02 25 31 37 41 45 |

04 21 25 37 45 46 |

02 04 16 28 37 47 |

28 34 41 45 46 49 |

01 04 16 37 47 49 |

My take on all this - absolute rubbish especially the bit that infers the bonus number can be selected and that this has any relevance to the winning six. See this article from BBC News

From my early days in Lotto number analysis I have wondered about the extent ways of filling in the betting slip are repeated and articles on this subject can be found on the web. The splitting of the jackpot shortly after the UK Lotto started on November 14, 1994 between 133 winners is most likely due to the layout of the numbers being a common "random" selection pattern wise on the betting slip as shown below for 7 17 23 32 38 42: -

01 02 03 04 05 06 **07** 08 09 10

11 12 13 14 15 16 **17** 18 19 20

21 22 **23** 24 25 26 27 28 29 30

31 **32** 33 34 35 36 37 **38** 39 40

41 **42** 43 44 45 46 47 48 49

The pattern for the most popular 24 integers inferred from the winning sixes is shown below with a clear division between the stronger right side and the left.

01 02 **03** 04 **05 06 07 08 09 10**

21 **22 23 24** 25 26 **27** 28 **29** 30

31 32 33 34 35 36 37 38 39 40

41 **42** 43 **44** 45 46 47 48 49

Integer | Count Most Played | Rank Most Played | Count Most Winning | Rank Most Winning |
---|---|---|---|---|

07 | 20 | 1 | 182 | 24 |

23 | 19 | 2 | 214 | 2 |

19 | 18 | 3 | 194 | 15 |

12 | 16 | 4 | 194 | 15 |

29 | 16 | 4 | 184 | 22 |

11 | 16 | 4 | 214 | 2 |

17 | 15 | 5 | 188 | 19 |

03 | 14 | 6 | 193 | 16 |

24 | 14 | 6 | 195 | 14 |

09 | 13 | 7 | 203 | 8 |

14 | 13 | 7 | 178 | 27 |

08 | 13 | 7 | 182 | 24 |

18 | 12 | 8 | 189 | 18 |

05 | 12 | 8 | 181 | 25 |

27 | 12 | 8 | 204 | 7 |

22 | 11 | 9 | 188 | 19 |

10 | 11 | 9 | 194 | 15 |

44 | 11 | 9 | 212 | 4 |

33 | 10 | 10 | 213 | 3 |

32 | 10 | 10 | 203 | 8 |

38 | 10 | 10 | 220 | 1 |

39 | 9 | 11 | 201 | 10 |

06 | 9 | 11 | 198 | 11 |

42 | 9 | 11 | 193 | 16 |

43 | 9 | 11 | 213 | 3 |

04 | 9 | 11 | 189 | 18 |

34 | 9 | 11 | 189 | 18 |

20 | 8 | 12 | 156 | 34 |

02 | 8 | 12 | 197 | 12 |

31 | 8 | 12 | 212 | 4 |

48 | 8 | 12 | 193 | 16 |

25 | 8 | 12 | 210 | 5 |

30 | 8 | 12 | 201 | 10 |

47 | 7 | 13 | 204 | 7 |

16 | 7 | 13 | 173 | 30 |

13 | 7 | 13 | 169 | 32 |

41 | 7 | 13 | 162 | 33 |

40 | 7 | 13 | 208 | 6 |

46 | 6 | 14 | 185 | 21 |

45 | 6 | 14 | 196 | 13 |

36 | 6 | 14 | 179 | 26 |

21 | 6 | 14 | 170 | 31 |

28 | 6 | 14 | 194 | 15 |

01 | 6 | 14 | 183 | 23 |

26 | 5 | 15 | 187 | 20 |

15 | 5 | 15 | 174 | 29 |

49 | 4 | 16 | 192 | 17 |

35 | 4 | 16 | 202 | 9 |

37 | 3 | 17 | 177 | 28 |

More relevant is the table below that clearly shows that as the integer Pool is increased so does the number of wins.

Integer Range | Wins | Cum Wins |
---|---|---|

<= 16 | 1 | 1 |

<= 19 | 2 | 3 |

<= 20 | 1 | 4 |

<= 22 | 7 | 11 |

<= 24 | 3 | 14 |

<= 25 | 1 | 15 |

<= 26 | 7 | 22 |

<= 27 | 10 | 32 |

<= 28 | 7 | 39 |

<= 29 | 11 | 50 |

<= 30 | 11 | 61 |

<= 31 | 12 | 73 |

<= 32 | 25 | 98 |

<= 33 | 21 | 119 |

<= 34 | 24 | 143 |

<= 35 | 27 | 170 |

<= 36 | 27 | 197 |

<= 37 | 32 | 229 |

<= 38 | 45 | 274 |

<= 39 | 60 | 334 |

<= 40 | 89 | 423 |

<= 41 | 62 | 485 |

<= 42 | 82 | 567 |

<= 43 | 90 | 657 |

<= 44 | 122 | 779 |

<= 45 | 135 | 914 |

<= 46 | 121 | 1035 |

<= 47 | 172 | 1207 |

<= 48 | 179 | 1386 |

<= 49 | 193 | 1579 |

Colin Fairbrother

by Colin Fairbrother

There are two main indexes used with enumerated Lotto numbers but many more can be made depending on requirements. An index is simply a mapping of the combination integers to a numerical sequence for quick finding.

The first is Lexicographic or Numerical Order. For a Pick 6, Pool 49 Lotto game -

1 01 02 03 04 05 06

2 01 02 03 04 05 07

3 01 02 03 04 05 08

4 01 02 03 04 05 09

5 01 02 03 04 05 10

6 01 02 03 04 05 11

7 01 02 03 04 05 12

8 01 02 03 04 05 13

etc to

13983816 44 45 46 47 48 49

The 2nd is Colex or Last Number Order (Pool order) with the comb integers in numerical order -

1 01 02 03 04 05 06

2 01 02 03 04 05 07

3 01 02 03 04 06 07

4 01 02 03 05 06 07

5 01 02 04 05 06 07

6 01 03 04 05 06 07

7 02 03 04 05 06 07

8 01 02 03 04 05 08

etc to

13983816 44 45 46 47 48 49

There are only 8 combs where the Lexicographic Index is the same as the Colex: -

Index_Lex | Index_Colex | Comb6 |
---|---|---|

1 | 1 | 01 02 03 04 05 06 |

2 | 2 | 01 02 03 04 05 07 |

463288 | 463288 | 01 04 20 21 28 29 |

1094145 | 1094145 | 01 10 16 28 32 33 |

12889672 | 12889672 | 17 18 22 34 40 49 |

13520529 | 13520529 | 21 22 29 30 46 49 |

13983815 | 13983815 | 43 45 46 47 48 49 |

13983816 | 13983816 | 44 45 46 47 48 49 |

Each integer appears 1,712,304 times for any enumeration of all possibilities and this can be calculated by (Combinations/Pool) * Pick or 49c6/Pool * Pick.

Dividing the 13,983,816 combinations by 1 to 14 or 21 to 24 or 28, 33, 37, 42, 43, 44, 46 or 49 will give equal groupings and in the case of 37 this is 377941.

For an equal partition into 7 groups of the 13,983,816 possibilities in the UK Lotto for 1588 draws we see that the occurrence is pretty even whether the Lexicographic or Colex Index is used as in the table below: -

Range ID | Index Start | Index Finish | Lexicographic Index Occurrence | Colexicographic Index Occurrence |
---|---|---|---|---|

1 | 1 | 1997688 | 215 | 204 |

2 | 1997690 | 3995375 | 225 | 238 |

3 | 3995376 | 5993064 | 235 | 207 |

4 | 5993065 | 7990752 | 217 | 249 |

5 | 7990753 | 9988440 | 228 | 230 |

6 | 9988441 | 11986127 | 241 | 240 |

7 | 11986128 | 13983816 | 227 | 220 |

Let's get a furphy out of the way regarding consecutive draws and the likelihood of an index being close. Each combination of 6 integers is just as likely to be picked as any other and just as likely to be followed by the same number as any other number. The reason it doesn't happen often is because there are 13,983,816 alternatives. Within a group defined by dividing the total combinations by a factor in the previous paragraph, just because we have given the combinations an index order does not mean one number is more or less likely to be picked in a draw - such attributes are delusional and are the stuff of wacky. irrational numerologists.

Without a doubt the most absurd assertion for 2010 in basically an irrelevant, moribund and ignored news group goes to Manfred the numerologist writing on November 28, 2010, "*If we give each of the possible combinations a number from 1 to 13983816 and we watch the lottery drawings, I found that the next drawing is about only 2 000 000 to 400 000 combinations and sometimes only 20 000 combinations below or above the last drawing. Does anyone know a wheel generator where I can load e.g. all 2 000 000 combinations below the last drawing and to generate a small wheel which covers best the loaded lines? With such wheels we could improve our winning chances because we need not to play with all 13 983 816 combinations."* See here -http://groups.google.com/group/rec.gambling.lottery/browse_thread/thread/2d1ec2affe0ca28c#

What utter nonsense. In a Pick 6, Pool 49 Lotto game any of the 13,983,816 combinations of 6 objects has the same chance of being picked and followed by once again any of the same number of combinations. To single out the lexicographic index from multi-million other index possibilities and use the absolute index difference between two consecutive draws as the magic elixir to improving one's chances simply by playing numbers with a lexicographic index between 400,000 and 2,000,000 has to be the corniest notion for 2010.

For my AllWorld database of various 6/49 games from around the world with 20,682 draws the results are shown in the table below with Index Count being the actual occurrence of the indexes and **LexDiffCount being the ridiculous numerology calculation that says there is fictitiously more or less**: -

Range Start | Range Finish | LexDiffCount | IndexCount |
---|---|---|---|

1 | 400000 | 1169 | 593 |

400001 | 800000 | 1167 | 591 |

800001 | 1200000 | 1100 | 574 |

1200001 | 1600000 | 1073 | 586 |

1600001 | 2000000 | 1108 | 562 |

2000001 | 2400000 | 1009 | 597 |

2400001 | 2800000 | 963 | 604 |

2800001 | 3200000 | 948 | 645 |

3200001 | 3600000 | 829 | 613 |

3600001 | 4000000 | 850 | 595 |

4000001 | 4400000 | 823 | 603 |

4400001 | 4800000 | 841 | 560 |

4800001 | 5200000 | 830 | 559 |

5200001 | 5600000 | 739 | 607 |

5600001 | 6000000 | 716 | 594 |

6000001 | 6400000 | 650 | 594 |

6400001 | 6800000 | 601 | 535 |

6800001 | 7200000 | 592 | 598 |

7200001 | 7600000 | 574 | 594 |

7600001 | 8000000 | 534 | 585 |

8000001 | 8400000 | 494 | 610 |

8400001 | 8800000 | 427 | 575 |

8800001 | 9200000 | 403 | 586 |

9200001 | 9600000 | 375 | 603 |

9600001 | 10000000 | 303 | 585 |

10000001 | 10400000 | 289 | 610 |

10400001 | 10800000 | 221 | 570 |

10800001 | 11200000 | 236 | 623 |

11200001 | 11600000 | 196 | 591 |

11600001 | 12000000 | 191 | 607 |

12000001 | 12400000 | 154 | 579 |

12400001 | 12800000 | 147 | 564 |

12800001 | 13200000 | 118 | 592 |

13200001 | 13600000 | 85 | 594 |

13600001 | 13983816 | 24 | 566 |

For a given Pick and Pool what numerologist Manfred is doing is producing a calculated index from 13983816c2 or the combinations of two indexes from 13983816 of which there are 97,773,547,969,020. Simplifying things by using Pick 6, Pool 7 the distortion becomes obvious from the table below where the lower calculated indexes are favored: -

Comb1Index | Comb1 | Comb2Index | Comb2 | Diff |
---|---|---|---|---|

1 | 01 02 03 04 05 06 | 2 | 01 02 03 04 05 07 | 1 |

2 | 01 02 03 04 05 07 | 3 | 01 02 03 04 06 07 | 1 |

3 | 01 02 03 04 06 07 | 4 | 01 02 03 05 06 07 | 1 |

4 | 01 02 03 05 06 07 | 5 | 01 02 04 05 06 07 | 1 |

5 | 01 02 04 05 06 07 | 6 | 01 03 04 05 06 07 | 1 |

6 | 01 03 04 05 06 07 | 7 | 02 03 04 05 06 07 | 1 |

1 | 01 02 03 04 05 06 | 3 | 01 02 03 04 06 07 | 2 |

2 | 01 02 03 04 05 07 | 4 | 01 02 03 05 06 07 | 2 |

3 | 01 02 03 04 06 07 | 5 | 01 02 04 05 06 07 | 2 |

4 | 01 02 03 05 06 07 | 6 | 01 03 04 05 06 07 | 2 |

5 | 01 02 04 05 06 07 | 7 | 02 03 04 05 06 07 | 2 |

1 | 01 02 03 04 05 06 | 4 | 01 02 03 05 06 07 | 3 |

2 | 01 02 03 04 05 07 | 5 | 01 02 04 05 06 07 | 3 |

3 | 01 02 03 04 06 07 | 6 | 01 03 04 05 06 07 | 3 |

4 | 01 02 03 05 06 07 | 7 | 02 03 04 05 06 07 | 3 |

1 | 01 02 03 04 05 06 | 5 | 01 02 04 05 06 07 | 4 |

2 | 01 02 03 04 05 07 | 6 | 01 03 04 05 06 07 | 4 |

3 | 01 02 03 04 06 07 | 7 | 02 03 04 05 06 07 | 4 |

1 | 01 02 03 04 05 06 | 6 | 01 03 04 05 06 07 | 5 |

2 | 01 02 03 04 05 07 | 7 | 02 03 04 05 06 07 | 5 |

1 | 01 02 03 04 05 06 | 7 | 02 03 04 05 06 07 | 6 |

The first absurdity from numerologist Manfred is that the highest occurring index difference is 0 which doesn't exist and happens when one index is followed by the same index as occurred recently in the Bulgarian 6/42 game.

The second absurdity is that numerologist Manfred favors the highest occurring calculated indexes differences as being more likely but excludes the indexes from 1 to 399,000 which occur most often according to his calculation.

The third absurdity as shown in the table above for the AllWorld 6/49 games is that the distribution does not concur with the facts. **An actual analysis of index distribution shows as expected a reasonably uniform distribution of winning indexes and definitely no concentration of wins in the lower one seventh of the indexes.**

The fourth absurdity is that in the face of such overwhelming evidence that his contention is not supported at all he still continues to promote it but this is not unusual behavior for an irrational occultist.

If we divide the 13,983,816 combinations into 37 groups each with 377941 combinations from Lexicographic, Colex or any of multi-millions of other orders then a draw that falls in one of our groups is unlikely to be followed by another in the same group simply because there are 36 alternatives. Using my 20,682 database of 6/49 draws and testing consecutive draws we would expect 20,682/37 = 559 to be in each group and that indeed is pretty much the case.

Which index has the better distribution of the integers? At first glance you may favor the lexicographic over the colex but on a closer look you will find that they are very similar. **Neither has a uniform distribution of the integers over the whole range but that is of no consequence for the winning number as all are equal and unique and any equal partition of the possibilities is just as good as another.**

Looking first at a lexicographic index we see for the first 1,712,304 each combination has the identifier 01 and from index 1712305 the 01 is dropped. Similarly from index 3246244 the 02 is also dropped. The last 377941 combinations do not have the integers from 1 to 20. At index 13983810 a total of 42 integers have been dropped albeit with only 7 combinations to go.

With the colex index we see the last 1,712,304 combinations have the integer 49. The first 377941 combinations do not have the integers from 30 to 49. At index 7 a total of 42 integers have not been used albeit with 13983809 combinations to go. Basically the order is the reversal of the other.

It is possible to create a set of 49 lines with each integer appearing 6 times as I posted in 2004. See http://lottoposter.com/forum_posts.asp?TID=68. Making 285,384 repeats of this set by using a different order of the integers and without repeating a line would give the 13,983,816 combinations with Uniform Distribution of the integers.

Here is an example for 98 lines with each integer appearing 12 times but with the recommendation you do not play it, due to the high number of repeat paying subsets -

Comb6 |
---|

01 10 24 38 42 44 |

08 17 22 26 33 35 |

06 11 25 45 48 49 |

13 29 31 34 39 43 |

03 09 12 28 30 41 |

04 05 15 16 20 32 |

02 07 14 21 23 40 |

18 19 36 37 46 47 |

01 10 24 27 38 44 |

08 17 22 33 35 42 |

06 25 26 45 48 49 |

11 29 31 34 39 43 |

09 12 13 28 30 41 |

03 04 05 16 20 32 |

02 07 14 15 21 40 |

19 23 36 37 46 47 |

01 10 18 24 27 44 |

08 22 33 35 38 42 |

17 25 26 45 48 49 |

06 11 29 31 39 43 |

09 12 13 30 34 41 |

03 04 16 20 28 32 |

02 05 14 15 21 40 |

07 23 36 37 46 47 |

01 10 18 19 24 27 |

08 33 35 38 42 44 |

17 22 26 45 48 49 |

06 11 25 29 31 39 |

09 13 30 34 41 43 |

03 04 12 16 28 32 |

05 14 15 20 21 40 |

02 07 23 37 46 47 |

01 18 19 24 27 36 |

10 33 35 38 42 44 |

08 17 22 26 48 49 |

06 11 25 29 39 45 |

09 13 30 31 34 43 |

03 04 12 28 32 41 |

05 14 15 16 20 21 |

02 07 23 40 46 47 |

01 18 19 27 36 37 |

10 24 33 38 42 44 |

08 17 22 26 35 49 |

06 11 25 39 45 48 |

09 13 29 31 34 43 |

03 12 28 30 32 41 |

04 05 15 16 20 21 |

02 07 14 23 40 46 |

18 19 27 36 37 47 |

01 02 03 04 05 06 |

07 08 09 10 11 12 |

13 14 15 16 17 18 |

19 20 21 22 23 24 |

25 26 27 28 29 30 |

31 32 33 34 35 36 |

37 38 39 40 41 42 |

43 44 45 46 47 48 |

01 02 03 04 05 49 |

06 07 08 09 10 11 |

12 13 14 15 16 17 |

18 19 20 21 22 23 |

24 25 26 27 28 29 |

30 31 32 33 34 35 |

36 37 38 39 40 41 |

42 43 44 45 46 47 |

01 02 03 04 48 49 |

05 06 07 08 09 10 |

11 12 13 14 15 16 |

17 18 19 20 21 22 |

23 24 25 26 27 28 |

29 30 31 32 33 34 |

35 36 37 38 39 40 |

41 42 43 44 45 46 |

01 02 03 47 48 49 |

04 05 06 07 08 09 |

10 11 12 13 14 15 |

16 17 18 19 20 21 |

22 23 24 25 26 27 |

28 29 30 31 32 33 |

34 35 36 37 38 39 |

40 41 42 43 44 45 |

01 02 46 47 48 49 |

03 04 05 06 07 08 |

09 10 11 12 13 14 |

15 16 17 18 19 20 |

21 22 23 24 25 26 |

27 28 29 30 31 32 |

33 34 35 36 37 38 |

39 40 41 42 43 44 |

01 45 46 47 48 49 |

02 03 04 05 06 07 |

08 09 10 11 12 13 |

14 15 16 17 18 19 |

20 21 22 23 24 25 |

26 27 28 29 30 31 |

32 33 34 35 36 37 |

38 39 40 41 42 43 |

44 45 46 47 48 49 |

Colin Fairbrother

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Where does one draw the line in considering some of the ill formed, hare brained ideas in Lotto? One that belongs in the same group as Sums, Odds & Evens and High and Low numbers is Deltas.

Generally, if in the game of Lotto you are considering, there is no payout for whatever category, then it is pretty much irrelevant. Consider that the bare bones requirements of a Lotto game such as the classic Pick 6, Pool 49 are: -

- 49 objects to randomly choose 6 from.

. - A jumbling or randomizing method for those 49 objects.

. - A random picking method where each object has equal opportunity.

A strong and large brown paper bag in which 49 ping pong balls are placed after each ball has a unique symbol written on it, then agitated up and down and round and round would suffice, with the six balls extracted by hand sight unseen.

There is no requirement to use numbers and all the theory of probability still applies. However, for most Lotto games numbers are used to conveniently distinguish one object from another and in that sense only, without the number carrying any magnitude. Once magnitude is attributed to the identifiers and calculations made where the Lotto operator does not pay on, then the realm of numerology has been entered.

More often than not where magnitude is attributed to the identifiers you will not find numerology mentioned and indeed the shysters will deny any connection. They know that to most people numerology, astrology, palmistry etc are amusing but not to be taken seriously. If you are so inclined to take these subjects seriously there is a web site that despite an increasing membership is well in decline as it concentrates more and more on Pick 3 freaks and those who pick their numbers by their nightmares at Freaky Lotto Site for the mentally impaired

Deltas are a strange concoction that despite their easy debunking keep cropping up with outrageous claims and little more to justify them than the proponent's claim to having studied them for 20 years or so.

To arrive at the Delta for a number such as 3 1 2 5 6 4 you must first put it in numerical order ie 1 2 3 4 5 6 and then starting from the left deduct the integer to the left with no integer treated as 0. So, 1 - 0 = 1, 2 - 1 = 1, 3 - 2 = 1, 4 - 3 = 1, 5 - 4 = 1 and 6 - 5 = 1 giving the Delta as 1 1 1 1 1 1. If you think that is a rather useless result then we have something in common. An example of an outrageous claim is that 90% of the Lotto numbers drawn will have Deltas where the maximum integer is 15 or less.

You can go through all the 13,983,816 Combinations of six integers working out the Deltas and arrive at the summary given in the table below: -

MaxInteger | Cnt |
---|---|

1 | 1 |

2 | 63 |

3 | 665 |

4 | 3367 |

5 | 11529 |

6 | 31031 |

7 | 70993 |

8 | 144495 |

9 | 269087 |

10 | 460767 |

11 | 707722 |

12 | 957572 |

13 | 1149777 |

14 | 1251607 |

15 | 1262696 |

16 | 1202676 |

17 | 1097691 |

18 | 970131 |

19 | 836226 |

20 | 706860 |

21 | 588525 |

22 | 484275 |

23 | 394680 |

24 | 318780 |

25 | 255024 |

26 | 201894 |

27 | 158004 |

28 | 122094 |

29 | 93024 |

30 | 69768 |

31 | 51408 |

32 | 37128 |

33 | 26208 |

34 | 18018 |

35 | 12012 |

36 | 7722 |

37 | 4752 |

38 | 2772 |

39 | 1512 |

40 | 756 |

41 | 336 |

42 | 126 |

43 | 36 |

44 | 6 |

The highest occurrence group at 1,262,696 is for Deltas with a maximum integer of 15 but 16 is not far behind at 1,202,676. If you add the group quantities where the maximum integer is less than or equal to 15 you get 6,321,372 or 45.21%. This is a long way from 90%.

Just as in Sums where the proponents claim you are better off playing numbers that sum to 150 ( see http://www.lottoposter.com/forum_posts.asp?TID=143) so the Delta dolts claim you are better off playing Delta numbers with a maximum integer of 15. What they don't tell you is that proportionality is maintained.

So, examining a reasonable size 6/49 Lotto history like my AllWorld 6/49 with 20,682 draws you will find as in the table below 45% or so (actually 44.31%) are numbers with a maximum integer of 15. Nothing is gained - despite all the shenanigans proportionality is maintained.

DeltaMax | Cnt |
---|---|

3 | 1 |

4 | 2 |

5 | 16 |

6 | 35 |

7 | 101 |

8 | 209 |

9 | 376 |

10 | 618 |

11 | 1029 |

12 | 1428 |

13 | 1702 |

14 | 1810 |

15 | 1836 |

16 | 1848 |

17 | 1599 |

18 | 1463 |

19 | 1292 |

20 | 1014 |

21 | 922 |

22 | 704 |

23 | 598 |

24 | 476 |

25 | 386 |

26 | 310 |

27 | 240 |

28 | 170 |

29 | 127 |

30 | 117 |

31 | 67 |

32 | 58 |

33 | 39 |

34 | 26 |

35 | 25 |

36 | 16 |

37 | 10 |

38 | 6 |

39 | 4 |

40 | 2 |

Colin Fairbrother

**Manually Constructing a Best Lotto Wheel or Cover**

by Colin Fairbrother

An understanding of what will deliver the best results on average for a set of numbers played in Lotto can be obtained by manually constructing the set. Let's use the Classic Lotto game with a Pool of 49 integers and a Pick of 6 integers.

Most Lotto games do not pay on getting 1 integer correct in the set you play but if they did **and the Pool was 48** with a Pick of 6 then the following 8 lines cannot be improved on ie the coverage of the Ones is 100% and it is not possible to achieve this with a lesser number of lines: -

01 02 03 04 05 06

07 08 09 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

It is also isomorphic which means we can randomize the integers and the coverage of the Ones at 100% is the same so it can be used as a template.

For a Pool of 49 there are 13,983,816 possible combinations of 6 integers compared to 12,271,512 for a Pool of 48. See Table of various Combinations obtainable for a given Pool. Just one extra integer has increased the number of combinations by 1,712,304, see Table of extra Combinations for an increase in Pool size.

To extend our Lotto Design beyond 8 lines there are 1,712,304 possibilities to cover the remaining One (ie 49) but if we restrict our selection to not duplicating a combination of two integers it is considerably less. We see that {1,2}, {1,3}, {1,4}, {1,5} and {1,6} already have an appearance but {1,7} does not. Choosing {1,7} means we now have the integers 1 and 7 appearing twice. Looking at the first available integer greater than that paired with 7 we see that {13,14}, {13,15}, {13,16} {13,17} and {13,18} have an appearance so let's opt for {13,19}. Looking at our set we see that both {1,7} and {13,19} appear in the first column and the next integer is 25 which we can pair with the as yet unused integer 49.

The 9th line could then be 01 07 13 19 25 49 but there are plenty of other possibilities that give the same coverage of the Ones. **However, all Pick 6, Pool 49 Lotto games do not pay on getting one integer correct so let's go to the first paying subset which is a combination of Three integers and let the One's and Two's fall as they may with no regard for bunkum "balancing" which is the stuff of numerology rather than that of best yield.** So far we have 2,312,716 CombSixes covered or 16.53852% coverage with 180 distinct Threes in the 9 lines.

The thought may have occurred to you that rather than repeating 5 integers on one line we spread them over 2 lines. Locking our first 7 lines and initializing the 8th line with something like 10 42 43 44 45 46 and the 9th line with something like 10 45 46 47 48 49 the last 2 lines are easily optimized in CoverMaster. An example to use as a template for randomization is given below with 1 7 42 43 44 46 and 41 45 46 47 48 49 for the last two lines, which now covers an extra 45 CombSixes to give a Coverage of 16.53943%. Yes, I agree the probability of those 45 occurring is about the same as winning first prize!

01 02 03 04 05 06

07 08 09 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

01 07 42 43 44 46

41 45 46 47 48 49

For the 10th line let's consider which one will give the best extra coverage of the Sixes for a Three prize without repeating a Three and using a transparent manual method.

The thought may have occurred to you that rather than repeating 5 integers on one line we spread them over 2 lines. Locking our first 7 lines and initializing the 8th line with something like 10 42 43 44 45 46 and the 9th line with something like 10 45 46 47 48 49 the last 2 lines are easily optimized in CoverMaster. An example to use as a template for randomization is given below with 1 7 42 43 44 46 and 41 45 46 47 48 49 for the last two lines, which now covers an extra 45 CombSixes to give a Coverage of 16.53943%. Yes, I agree the probability of those 45 occurring is about the same as winning first prize!

01 02 03 04 05 06

07 08 09 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

01 07 42 43 44 46

41 45 46 47 48 49

For the 10th line let's consider which one will give the best extra coverage of the Sixes for a Three prize without repeating a Three and using a transparent manual method.

The best known record coverage for 9 lines is 16.5394% attributed to Adolf Muehl and for 10 lines it is 18.22231%, attributed to Normand Veilleux each being the first known person to achieve this. See Record Partial Coverings.This is not as big a deal as it used to be with the current speed and RAM memory of current computers as for around this number of line sets it is quite easy to evaluate all possibilities. The 10 line record can be easily replicated using John Rawson's CoverMaster program optimizer - (Download here and send request for zip unlock to honest.John@ntlworld.com) - after seeding with say, the 10 lines below using similar to the 9th line, a 1 and 5 integers in the 2nd column which gives 18.22006% coverage: -

02 03 04 47 48 49

05 06 07 44 45 46

08 09 10 41 42 43

11 12 13 38 39 40

14 15 16 35 36 37

17 18 19 32 33 34

20 21 22 29 30 31

23 24 25 26 27 28

01 02 05 08 11 14

01 03 06 09 12 15

After optimization in CoverMaster we get from the above set the following set with a coverage of 18.22231% of the Sixes: -

04 06 07 44 45 49

17 18 19 32 34 46

01 02 05 07 08 14

11 13 33 38 39 40

23 24 25 26 27 28

20 21 22 29 30 31

02 03 10 35 36 48

08 10 16 41 42 43

14 15 16 36 37 47

01 03 09 12 15 43

17 18 19 32 34 46

01 02 05 07 08 14

11 13 33 38 39 40

23 24 25 26 27 28

20 21 22 29 30 31

02 03 10 35 36 48

08 10 16 41 42 43

14 15 16 36 37 47

01 03 09 12 15 43

... but is it better in yield than the original 10 lines? Over 100 draws both sets are most likely to give: -

No Wins 82

Three Wins 17

Four Win 1

If you transpose using CoverMaster you can rearrange this set with the same coverage to: -

01 02 03 04 05 06

07 08 09 10 11 12

07 08 09 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

01 31 33 34 40 43

05 32 38 41 42 43

06 31 32 36 37 47

02 34 35 36 38 48

04 39 44 45 46 49

19 20 21 22 23 24

25 26 27 28 29 30

01 31 33 34 40 43

05 32 38 41 42 43

06 31 32 36 37 47

02 34 35 36 38 48

04 39 44 45 46 49

Has replacing the lines -

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

- really achieved anything as far as Lotto yields are concerned apart from some jumbling of the numbers for an imagined gain?

Is this telling us something?

Can this set be arrived at in a transparent manual way?

Can we use this to work out a way of testing a sample of the possibilities for the next line?

**Is there any point to optimization beyond next best play if it doesn't improve the yield in Lotto? See a comparison ****here****?**

Colin Fairbrother

Probably an even worse result would be obtained by the following wheel:-

01 45 46 47 48 49

02 45 46 47 48 49

03 45 46 47 48 49

04 45 46 47 48 49

05 45 46 47 48 49

06 45 46 47 48 49

07 45 46 47 48 49

08 45 46 47 48 49

09 45 46 47 48 49

10 45 46 47 48 49

11 45 46 47 48 49

12 45 46 47 48 49

13 45 46 47 48 49

14 45 46 47 48 49

15 45 46 47 48 49

16 45 46 47 48 49

17 45 46 47 48 49

18 45 46 47 48 49

19 45 46 47 48 49

20 45 46 47 48 49

21 45 46 47 48 49

22 45 46 47 48 49

23 45 46 47 48 49

24 45 46 47 48 49

25 45 46 47 48 49

26 45 46 47 48 49

27 45 46 47 48 49

28 45 46 47 48 49

29 45 46 47 48 49

30 45 46 47 48 49

31 45 46 47 48 49

32 45 46 47 48 49

33 45 46 47 48 49

34 45 46 47 48 49

35 45 46 47 48 49

36 45 46 47 48 49

37 45 46 47 48 49

38 45 46 47 48 49

39 45 46 47 48 49

40 45 46 47 48 49

41 45 46 47 48 49

42 45 46 47 48 49

43 45 46 47 48 49

44 45 46 47 48 49

01 44 45 46 47 48

02 44 45 46 47 48

03 44 45 46 47 48

04 44 45 46 47 48

05 44 45 46 47 48

AS far as first prize is concerned this set is just as good as any other as all the lines are different. As a set to maximize your winnings in the short or long term it is extremely inefficient with one Five repeated 44 times and another 5 times.

The prize groupings appear in the table below: -

Six | Five | Four | Three | Total | % | Cum % |
---|---|---|---|---|---|---|

0 | 12,290,712 | 87.89 | 87.89 | |||

1 | 168,720 | 1.21 | 89.10 | |||

2 | 28,120 | 0.20 | 89.30 | |||

3 | 1,520 | 0.01 | 89.31 | |||

4 | 970,294 | 6.94 | 96.25 | |||

5 | 267,140 | 1.91 | 98.16 | |||

6 | 43,700 | 0.31 | 98.47 | |||

7 | 2,320 | 0.02 | 98.49 | |||

8 | 30 | 0.00 | 98.49 | |||

9 | 50,616 | 0.36 | 98.85 | |||

1 | 8 | 21,090 | 0.15 | 99.00 | ||

2 | 7 | 2,280 | 0.02 | 99.02 | ||

3 | 6-46 | 112,068 | 0.80 | 99.82 | ||

4 | 45 | 15,200 | 0.11 | 99.93 | ||

5 | 44 | 1,580 | 0.01 | 99.94 | ||

6 | 43 | 40 | 0.00 | 99.94 | ||

8 | 41 | 2,812 | 0.02 | 99.96 | ||

1 | 7 | 41 | 760 | 0.01 | 99.97 | |

2 | 6-47 | 0-41 | 4,507 | 0.03 | 100.00 | |

3 | 46 | 0 | 210 | 0.00 | 100.00 | |

4 | 45 | 0 | 10 | 0.00 | 100.00 | |

7 | 42 | 0 | 38 | 0.00 | 100.00 | |

1 | 6-48 | 0-42 | 0 | 49 | 0.00 | 100.00 |

We see that for 12,290,712 or 87.89% of the 13,983,816 possibilities no prize is obtained. A good set of numbers from LottoToWin would for the same number of lines give a normal distribution of the prizes and not deliver a prize for only 30% of the possibilities. A prize on average for 70% of the draws as in my 49 line set with no diminution of getting a higher prize or only 12% of the draws as in the "Joe Roberts" variety?

Colin Fairbrother

]]>