Prize6 | Prize5 | Prize4 | Prize3 | Count | Probability |
---|---|---|---|---|---|

0 | 0 | 15 | 2 | 10 | 0.0476190 |

0 | 2 | 9 | 8 | 60 | 0.2857142 |

0 | 3 | 7 | 9 | 60 | 0.2857142 |

0 | 3 | 8 | 7 | 60 | 0.2857142 |

1 | 0 | 10 | 8 | 15 | 0.0714285 |

1 | 0 | 12 | 4 | 5 | 0.0238095 |

Prize6 | Prize5 | Prize4 | Prize3 | Count | Probability |
---|---|---|---|---|---|

0 | 0 | 0 | 0 | 12721488 | 0.9097293 |

0 | 0 | 0 | 3 | 822510 | 0.0588187 |

0 | 0 | 0 | 4 | 182780 | 0.0130708 |

0 | 0 | 0 | 5 | 91390 | 0.0065354 |

0 | 0 | 1 | 8 | 55575 | 0.0039742 |

0 | 0 | 1 | 9 | 44460 | 0.0031793 |

0 | 0 | 2 | 7 | 44460 | 0.0031793 |

0 | 0 | 3 | 2 | 7410 | 0.0005298 |

0 | 0 | 3 | 4 | 3705 | 0.0002649 |

0 | 0 | 5 | 10 | 468 | 0.0000334 |

0 | 0 | 5 | 11 | 2340 | 0.0001673 |

0 | 0 | 7 | 6 | 2340 | 0.0001673 |

0 | 0 | 15 | 2 | 10 | 0.0000007 |

0 | 1 | 3 | 11 | 2340 | 0.0001673 |

0 | 1 | 4 | 10 | 2340 | 0.0001673 |

0 | 2 | 9 | 8 | 60 | 0.0000042 |

0 | 3 | 7 | 9 | 60 | 0.0000042 |

0 | 3 | 8 | 7 | 60 | 0.0000042 |

1 | 0 | 10 | 8 | 15 | 0.0000010 |

1 | 0 | 12 | 4 | 5 | 0.0000003 |

]]>

Prize6 | Prize5 | Prize4 | Prize3 | Cnt | Probability | Likely 1000 Plays or 36 Draws |
---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 13327132 | 0.95 | 34 |

0 | 0 | 0 | 10 | 596960 | 0.04 | 2 |

0 | 0 | 6 | 16 | 57400 | 0.00 | 0 |

0 | 3 | 15 | 10 | 2296 | 0.00 | 0 |

1 | 12 | 15 | 0 | 28 | 0.00 | 0 |

Prize6 | Prize5 | Prize4 | Prize3 | Combs | Probability | Likely 1000 Plays or 36 Draws |
---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 7571012 | 0.54 | 19 |

0 | 0 | 0 | 1 | 5225888 | 0.37 | 14 |

0 | 0 | 0 | 2 | 759408 | 0.05 | 2 |

0 | 0 | 0 | 3 | 39728 | 0.00 | |

0 | 0 | 0 | 4 | 1268 | 0.00 | |

0 | 0 | 1 | 0 | 337260 | 0.02 | 1 |

0 | 0 | 1 | 1 | 42000 | 0.00 | |

0 | 1 | 0 | 0 | 7224 | 0.00 | |

1 | 0 | 0 | 0 | 28 | 0.00 |

For every draw playing the System 8 you have a 95% chance of getting nothing compared to 54% for the Partial Cover. If you like for the Cover/Wheel 1 in 20 will give you nothing compared to the Partial Cover with only 1 in 2 giving nothing.

Playing the System 8 you can expect to go 24 draws before a win compared to the Partial Cover with a win expected every 2nd draw.

A System 8 or Full Wheel 8 is highly volatile and can go playing the same numbers 175 draws or more without a win. Compare this to the Partial Cover that rarely goes more than 11 draws.

I strongly recommend that you randomize whatever set you are playing before every draw for the game you are playing. Don't fall into the trap of being a slave to Lotto, worrying about missing a draw because that might be the one when your numbers come up.

UK Costs and Payouts for 36 draws (1008 plays).

System 8 Prizes £25 x 20 = £500 Yield 24.8%

Partial Cover Prizes £25 x 14 + £50 x 4 + £100 x 1 = £550 Yield 27.28%

OR OPTIMUM STRUCTURED SETS

OR DISTORTED SETS

by Colin Fairbrother

Lotto players can refer to information printed or online from Lotto Operators which give the Odds for winning a prize for one line or ticket and in some cases for the minimum lines playable, which is simply the respective multiple of that applicable to one line.

In this article I will use the Classic Lotto game where 6 numbers or distinct balls are randomly picked from 49 and where a prize is obtained if in any line played there is a match of at least 3, 4, 5 or 6 integers. To simplify matters the bonus ball is ignored and the principles outlined here are applicable to any Pick 5 or Pick 6 Lotto game.

For any given Lotto game matrix the odds or probability, as I shall refer to it from hereon, can be calculated for any given match.

Consider first one line in our 6/49 game, say, 01 02 03 04 05 06, realizing the numbers used are just identifiers with no magnitude. This combination of 6 numbers or integers (CombSix) is just 1 combination to pit against the 1 combination first prize of which there are 13,983,816 distinct possibilities for the draw. Each line or draw also has 6 combinations of 5 integers, 15 combinations of 4 integers and 20 combinations of 3 integers for a possible match. One line is flawless and is no better than any other line.

Any combination of 3 integers will for an enumerated 13,983,816 CombSixes appear 15,180 times. Any combination of 4 integers will appear 990 times and any combination of 5 integers will appear 44 times. Due to overlaps the combinations when considered distinctly are 1,906,884 CombFives, 211,876 CombFours and 18,424 CombThrees.

= 1 / (13983816 / 246820)

= 1 / 56.6559273

= 0.0176504

Multiplying this probability for 1 draw gives 0.0176504 of a match-3.

Multiplying this probability for 29 draws gives 0.5118616 of a match-3.

Multiplying this probability for 57 draws gives 1.0060728 match-3's.

Multiplying this probability for 114 draws gives 2.0121456 match-3's

________________________________________________________

= 1 / (13983816 / 13545)

= 1 / 1032.3968

= 0.0009686

Multiplying this probability for 1 draw gives 0.0009686 of a match-4.

Multiplying this probability for 518 draws gives 0.5017348 of a match-4.

Multiplying this probability for 1033 draws gives 1.0005638 match-4's.

Multiplying this probability for 2065 draws gives 2.000159 match-4's.

________________________________________________________________

= 1 / 54200.837

= 0.0000184

Multiplying this probability for 1 draw gives 0.0000184 of a match-5.

Multiplying this probability for 27180 draws gives 0.500112 of a match-5.

Multiplying this probability for 54380 draws gives 1.000592 match-5's.

Multiplying this probability for 109000 draws gives 2.0056 match-5's.

________________________________________________________________

= 1 / (13983816 / 1)

= 1 / 13983816

= 0.0000001

While highly likely there is no guarantee you will get a 1st prize in 14,000,000 draws.

_________________________________________________________________

As the number of draws increases the likelihood of no prizes decreases.

_________________________________________________________________

By testing against all 13,983,816 possibilities a Prize Table can be produced for one line that gives exactly the same probability as the theoretical caculation. Multiplying the probabilty by the number of plays, which in this case is the same as the draws, is exactly the same as one would do to work out potential winnings using the theoretical calculation..

Combs Probability Likely Likely Likely Likely

6 5 4 3 1 29 518 27180

Draw Draws Draws Draws

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

0 0 0 0 13723192 0.9813625 1 28 508 26673

0 0 0 1 246820 0.0176540 0 1 9 480

0 0 1 0 13,545 0.0009686 0 0 1 26

0 1 0 0 258 0.0000184 0 0 0 1

1 0 0 0 1 0.0000001 0 0 0 0

_______________________________________________________

Note that for two or more CombSixes it is possible to build distortion into the set when compared to Random Selections where for say, a 6/49 Lotto game, it is extremely rare to get high repeat subsets. The System 8 or Full Wheel 8 will be used as an example of an abnormal set as enumerated below -

1 2 3 4 5 6

1 2 3 4 5 7

1 2 3 4 5 8

1 2 3 4 6 7

1 2 3 4 6 8

1 2 3 4 7 8

1 2 3 5 6 7

1 2 3 5 6 8

1 2 3 5 7 8

1 2 3 6 7 8

1 2 4 5 6 7

1 2 4 5 6 8

1 2 4 5 7 8

1 2 4 6 7 8

1 2 5 6 7 8

1 3 4 5 6 7

1 3 4 5 6 8

1 3 4 5 7 8

1 3 4 6 7 8

1 3 5 6 7 8

1 4 5 6 7 8

2 3 4 5 6 7

2 3 4 5 6 8

2 3 4 5 7 8

2 3 4 6 7 8

2 3 5 6 7 8

2 4 5 6 7 8

3 4 5 6 7 8

By testing against all 13,983,816 possibilities a Prize Table can be produced which gives the groupings and by dividing by 13983816 the probability for each group based on the structure of the set. If the same set is used for more simulated draws then simply multiplying the probabilty for each group by the number of simulated draws required will give proportionally the likely wins make-up.

Likely Wins From 28 Lines 6/49 Lotto with System 8 | |||||||||||||

6 | 5 | 4 | 3 | Combs | Probability | Likely 1 Draw | Likely 2 Draws | Likely 12 Draws | Likely 37 Draws | Likely 120 Draws | Likely 1936 Draws | Likely 2985 Draws | |

- | - | - | - | 13327132 | 0.9530397 | 1 | 2 | 11 | 35 | 114 | 1845 | 2845 | |

- | - | - | 10 | 596960 | 0.0426893 | 0 | 0 | 1 | 2 | 5 | 83 | 127 | |

- | - | 6 | 16 | 57400 | 0.0041047 | 0 | 0 | 0 | 0 | 1 | 8 | 12 | |

- | 3 | 15 | 10 | 2296 | 0.0001642 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | |

1 | 12 | 15 | 0 | 28 | 0.0000020 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

13983816 | 1 | 2 | 12 | 37 | 120 | 1936 | 2985 | ||||||

These figures are a guide. For the more exotic like the 3 Fives with 15 Fours and 10 Threes ie each line of 28 line set has a hit then flexibility is required; 0.5 is only 2985 draws whereas closer to 1 is more like 6090 draws. Initially one can use the Ceiling function but if the total does not equal the draws a manual check may be required. The allocation between no prizes and prizes is the first priority and within prizes some can be consolidated to a lower prize but always guided by probability.

Building a Prize Table for the Twenty Eight

By testing against all 13,983,816 possibilities a Prize Table can be produced which gives the groupings and by dividing by 13983816 the probability for each group based on the structure of the set. If the same set is used for more simulated draws then simply multiplying the probabilty for each group by the number of simulated draws required will give proportionally the likely wins make-up.

Likely Wins 28 Lines Partial Cover, Unique Threes, Full Pool Optimized 6/49 Lotto | |||||||||

6 | 5 | 4 | 3 | Combs | Probability | Likely 1 Draw | Likely 2 Draws | Likely 37 Draws | Likely 1936 Draws |

- | - | - | - | 7571012 | 0.5414124 | 1 | 1 | 20 | 1048 |

- | - | - | 1 | 5225888 | 0.3737097 | 0 | 1 | 14 | 723 |

- | - | - | 2 | 759408 | 0.0543062 | 0 | 0 | 2 | 105 |

- | - | - | 3 | 39728 | 0.0028410 | 0 | 0 | 0 | 6 |

- | - | - | 4 | 1268 | 0.0000907 | 0 | 0 | 0 | 0 |

- | - | 1 | 0-1 | 379260 | 0.0271214 | 0 | 0 | 1 | 53 |

- | 1 | - | - | 7224 | 0.0005166 | 0 | 0 | 0 | 1 |

1 | - | - | - | 28 | 0.0000020 | 0 | 0 | 0 | 0 |

13983816 | 1 | 29 | 37 | 1936 | |||||

]]>