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Prize Tables for Covers or Wheels in 6/49 Lotto Game
 LottoPoster Forums : ANALYSIS OF VARIOUS LOTTO NUMBER SETS : Prize Tables for Covers or Wheels in 6/49 Lotto Game
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Colin F
Lotto Systems Tester Creator & Analyst
Lotto Systems Tester Creator & Analyst
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Quote Colin F Replybullet Topic: Calculating Likely Wins 1 Line or more
    Posted: March 17 2014 at 11:59pm
CALCULATING LIKELY WINS IN LOTTO FOR 1 LINE
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.

Theoretical calculation of odds for one line.

match-3      1 / (49c6 / (6c3 * 43c3)) = 1 / (13983816 / (20 * 12341))
                                                            = 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

________________________________________________________

match-4      1 / (49c6 / (6c4 * 43c2)) = 1 / (13983816 / (15 * 903))
                                                            = 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.

________________________________________________________________

match-5      1 / (49c6 / (6c5 * 43c1)) = 1 / (13983816 / (6 * 43))
                                                            = 1 / (13983816 / 258)
                                                            = 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.

________________________________________________________________

match-6      1 / (49c6 / (6c6 * 43c0)) = 1 / (13983816 / (1 * 1))
                                                            = 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.

_________________________________________________________________

match-0     1 - (0.0176504 + 0.0009686 + 0.0000184 + 0.0000001) = 0.9813625

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

Building a Prize Table for just one line.

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

Building a Prize Table for the 28 Line System 8 or Full Wheel
 
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
Normal Line Partial Cover
 
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












 
Lotto Draws have no relationship to one another; the integers serve just as identifiers. Any prediction calculation on one history of draws for a same type game is just as irrelevant as another.
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