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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: ANALYSIS OF LOTTO DRAW HISTORY - THE FINAL WORD
    Posted: July 29 2010 at 6:25pm

Analysis of Lotto Draw History - the Final Word

by Colin Fairbrother

There is no way of telling the start or end of a Lotto history or whether it has been jumbled up without knowing the draw dates or draw IDs which means there is no intrinsic order to the draws - it is simply a bucket of numbers. For a particular time of the day you could set 100 machines or computers to start at exactly the same time and the chances are they will all be different numbers. No draw result is more valid than another and to resolve this scenario a random selection from the 100 results is needed.

The heyday of interest in discussing Lotto in on-line forums and in particular about using the history of draws to determine the numbers to play peaked around 2005. Practically all the many thousands of sites on Lotto number analysis are about assuming a relationship or constraint between the next draw and the draws previous to that, which as I will show is utter bunkum and is really the stuff of numerology with their penchant for errant ersatz science, mathematics and statistics.

You still have freaky people that actually believe that from the history of draws for a particular Lotto game the next jackpot number can be narrowed down. This of course is based on the erroneous assumption mentioned above that there is a relationship between the history and the next draw. Interestingly, give someone a single draw result or a set and a choice between a number of draw histories and they wouldn't be able to pick the relevant history without knowing the answer or looking it up. The point being that whatever is considered positive that one comes up with could be similarly achieved by using any like history.

There are two principles to keep in mind when discussing the history of draws for a Lotto game: -


  • Order is just a convention. Lotto operators normally present the results in numerical order but some show the numbers in the order drawn. For each draw with six main numbers there are 720 ways it can be drawn. Order is about permutations so while in a 6/49 game there are 13,983,816 combinations of 6 integers there are 10,068,347,520 permutations and similarly for a 6/45 game 8,145,060 combinations and 5,864,443,200 permutations. For say a 6/45 game a set of numbers can be randomized by simply varying the order of the integers and transposing eg _10 20 30 40 01 11 21 31 41 02 12 22 32 42 03 13 23 33 43 04 14 24 34 44 05 15 25 35 45 06 16 26 36 07 17 27 37 08 18 28 38 09 19 29 39 .

  • Occurrence is proportional to the representation of the integers in the categorization. The best results are obtained where close to or all the integers are used. Apart from categorizing the integers by their recency and occurrence in the immediate past history others are summing the integers, classifying the numbers as high, low or medium and using the six combination possibilities for Odd or Even in a Pick 6 game. I have shown for all categorizations that the proportionality is maintained and nothing is gained. It should be mentioned as I have previously shown that if you played half the possibilities - 4,072,530 for a 6/45 and 6,991,908 for a 6/49 ie 50% chance of getting the main number - you can still go 9 or 10 draws without success - similar to tossing a coin and getting 9 or 10 of the same side consecutively.

  • In the following table for a Pick 6, Pool 49 Lotto game you can see the chances of success playing all the combinations for the respective Pool. Using only 42 integers from the Pool of 49 you have a 86% chance of success for the 1's but only a 62% chance for the 3's and 38% for the 6's. 

    Integers Used of 496's6's %5's5's %4's 4's %3's3's %2's2's %1's1's %
    770.00210.00350.02350.19 211.79714.29
    1430030.0220020.1010010.473641.98 917.741428.57
    21542640.39203491.0759852.8213307.22 21017.862142.86
    283767402.69982805.15204759.66327617.78 37832.142857.14
    35162316011.6132463217.025236024.71654535.52 59550.603571.43
    42524578637.5185066844.6111193052.831148062.31 86173.214285.71
    4913983816100.001906884100.00211876100.0018424100.00 1176100.0049100.00


    The following table shows 34 draws for a 6/45 Lotto Game with the latest draw either at the top or the bottom of the table. Without looking it up the integers for the last draw could be anywhere and not show up as being untoward.




    There is nothing to stop a Lotto operator from using blank ping pong balls and writing the integers on them prior to shuffling for the draw. Amazingly, on a forum that caters for the Lotto deranged they bemoan the fact that test runs by the Lotto operator are not revealed as if it would make a difference.

    What is a Lotto history? It is made up of independent random number selection events that occur at a predetermined date and time by a Lottery Operator after selling tickets where people have nominated a set of numbers which they hope will match as much as is possible with the draw result. The time and date have no effect on the number drawn and serve to identify it only.

    Often I have come across the shysters claiming such and such supports their spiel when in fact it it just the natural distribution. Consider a Pick 6, Pool 45 Lotto game. It is possible for all the integers to occur within 8 draws but this is highly unlikely. Here is a more likely distribution (in fact actual): -


    Draw Integer Occurrence Repeats Integer Non-Occurrence
    16039
    212033
    315330
    419526
    522823
    6241221
    7271518
    8291916
    9302415
    10322813
    11353110
    1236369
    1338407
    1438407
    1540505
    1640565
    1740625
    1841674
    1941734
    2041794
    2141854
    2241914
    2341974
    24431012
    25431072
    26431132
    27431192
    28431252
    29431312
    30431372
    31431432
    32441481
    33441541
    34451590

    Now, you may be tempted to think that the prior six draws has shrunk the number of integers by more than 50% to 24 from the Pool of 45. The important point to remember is that this is applicable to practically all six randomly selected lines not just the prior six draws.

    There is a big question about the logic of making an association with an as yet non existent next draw. If you make the association it is made by you and then you must ask yourself the question, "Am I receiving anything extra from that of using any other random selection of six lines from the 8,145,060 possibilities?"

    Lotto operators in the main still provide frequency or occurrence and absence or recency data and charts on their websites. The UK National Lottery gives for the 6/49 game the ball-set and machine used as if it mattered. Disclaimers such as this one from Tattersalls in Australia are not uncommon: -

    "Please Note: All Tatts lotto games are entertaining games of chance where all numbers are drawn randomly. Therefore, each number has an equal chance of being drawn, regardless of how frequently it has been drawn previously."

    If you're not familiar with my articles on Signatures see below: -
     

    Introducing Signatures
    Top 31 Signatures
    Absence or Recency

    From the articles there are two overwhelming facts - lack of repetition and inconsistency between ostensibly that which should be giving the best results if there was some correlation between history and the next draw.

    The whole scenario of Lotto history analysis thus becomes farcical and results in the whole process being nothing other than a quaint way of jumbling the numbers to produce a set of numbers to play, which is more than likely inferior to random selections.

    All the thousands of websites touting history analysis as being some guide to future draws are basing their "analysis" on a false assumption.

    The good thing about signatures is that it handles both absence and occurrence or each considered separately.

    Consider the oft used method of 5 categories (Repeat, Hot, Warm, Luke, Cold) for each integer with the following 5 signature examples: -

    Repeat: in previous draw eg HHMMMMMMMM
    Hot: 3 or more hits past 10 draws eg MMHMMHMHMM
    Warm: 2 hits in past 10 draws eg MMMMHMMMHM
    Luke: 1 hit excluding repeat in past 10 draws eg MMHMMMMMMM
    Cold: No hit in past 10 draws eg MMMMMMMMMMM

    A less coloured method is to use P for previous, S for a single occurrence in say the past 10 draws and a number indicating absence but not necessarily ranking. Similarly, use D for a double occurrence, T for a triple or higher occurrence and N for a non-occurrence with a number indicating absence. So, you could end up with a recent 10 draw 6/45 history with the absence number ordered being categorized as _
     
    Previous OccurrenceSingle OccurrenceDouble OccurrenceTriple OccurrenceNo Occurrence
    P00:{02,12,20,27,30,31}




    S01:{09}D01:{7,45}T01:{06,32}


    D02:{15,33,34}


    S03:{10,11,28}D03:{39}


    S04:{01,40}D04:{3,21,22}


    S05:{24}D05:{18,41,43}


    S06:{4,35}D06:{38}


    S07:23,25,37}








    S09:{05,13,26,29}






    N10:{08,14,16,17,19,36,42,44}

    Alternatively, you could use just four categories and have the previous draw results in H, W, L and C with absence eg


    AbsenceHotWarmLukeCold
    0{30}{12,20,31}{02,27}
    1{6,32}{7,45}{09}
    2
    {15,33,34}

    3
    {39}{10,11,28}
    4
    {03,21,22}{01,40}
    5
    {18,41,43}{24}
    6
    {38}{04,35}
    7

    {23,25,37}
    8



    9

    {05,13,26,29}
    10


    {08,17}
    11


    {36}
    12



    13


    {44}
    14


    {19}
    15


    {42}





    26


    {16}
    30


    {14}

    The important point is that the combinations have not been reduced from 8,145,060 simply the convention used to represent the integers have extra complexity by carrying absence and occurrence data. As I will show the added complexity means the repetition of the subset combinations is practically non-existent, which is not unexpected but is not what we want. The gross inferiority of the subset repetitions when compared to integers alone is proof enough that no beneficial relationship exists in using history.

    Adding complexity where it is not needed to get a win is illogical. Consider tossing a coin where you win if you get heads. You could add complexity by adding the quadrant compass direction the top of the head side or tails side is pointing towards when it lands. So, instead of two possibilities we now have 8 but the win is paid on only two possibilities and nothing is gained.

    For the 10 draw previous history shown above we have 6 integers in P, 1 in H, 13 in W, 16 in L and 8 in C. The only consistent figure is P which is always 6 but the rest are random.

    Category Averages
    Cat   Avg Occurrence
    P    6
    H    4
    W    9
    L   15
    C   11


    Reasonable Possibilities:

    P W L L C C
    P W L L L C
    P H W L L C
    P W W L L C
    W W L L L C
    H W L L L C
    W L L L C C
    H W W L L C
    W W L L C C

    The average for H is 4 but it varies from nothing to 9.
    The average for W is 9 but it varies from 2 to 17.
    The average for L is 15 but it varies from 5 to 26.
    The average for C is 11 but it varies from 4 to 18.

    Top 7 of 472 from sample of 2095 draws:
     
    PfxsPHWLC PfxsCnt
    06 04 10 15 10 30
    06 04 09 16 10 28
    06 04 10 14 11 27
    06 04 11 13 11 22
    06 05 08 15 11 22
    06 03 11 16 09 22
    06 03 11 15 10 22

    One is excused for thinking that surely this is just a very poor method for randomizing the integers by their previous occurrence without any regard to the coverage obtained, their repetition and with little possibility of having all integers included for a reasonable number of plays.

    With 5 categories (Previous, Hot, Warm, Luke and Cold) and a Pick 6 Lotto game we have precisely 210 possible combinations with repetition allowed as shown in the table below.


    However, this does not diminish the number of possibilities which for a Pick 6, Pool 45 Lotto game is 8,145,060.  For the PWWLLC category alone there are 449,280 possibilities and for WWLLCC 262,080 for this particular history, too many to play thus requiring a random selection. This of course begs the question why bother as you could just as easily do a small random selection from all possibilities?

    If out of the many thousands of Signature Combinations you went for D02 D04 S06 S06 N10 N10 (just one of 262,080 in category WWLLCC) you would have done alright. You have 16 integers you can use and 252 combinations to play as below: -


    You probably understand my reluctance years ago when writing about signatures to release details of this system with its potential to be misconstrued. The problem is of the 210 Signature Combinations there is no basis for favoring one over the other and when this is combined with the varying number of integers in each signature all 8,145,060 combinations of six integers are possible as it should be, so it's back to randomness and a good template.

    If you had 01 only in P00, 2 only in S01, 3 only in S02, 4 only in S03, 5 only in S05 and 6 only in S06 and you played P00 S01 S02 S03 S04 S05 you would have won if the winning number was 01 02 03 04 05 06.

    To halve the possibilities in a 6/49 game remove just 6 integers and for a 6/45 game remove 4 or 5. Halving the Pool means just a very small fraction of all the possibilities are considered and this has a drastic effect on the yield.  See Analysis of 15 Lotto Number Sets.

    When considering past draws to bring into a calculation for Pick6 Lotto games with 45 or 49 integers the Pool you need to consider a minimum of 8 or 9 lines and realistically no less than 10 for "due" in the inner range and something like 12 or more to "grade" th "cold" objects An analysis of considerable draws can produce a reference list that rates each signature by the count in the sample for each following winning integer. Then one can count the groups of say, 6 signatures that have preceded a draw and there we find the problem. The more you try to narrow it down by using a longer signature string the less likelihood of a repeat. The more you reduce the signature length the greater the number of possibilities for your slots. Restricting the signature length to about 10 you may get a couple of groupings that repeat twice from a 470 draw sample which tell you to use something like: S06 S06 D02 D04 N10 N10 or W W L L C C
    S00 S02 S08 D00 N10 N10 or W L L L C C

    A minimum of two repeats is necessary to be of any future use and if used the probability is that a new single occurrence is more likely and even a double before a triple occurrence occurs. Bear in mind that we have at least 6 for H* from the previous draw, and most probably 6 of MH*, MMH*, MMMH*, MMMMH*, MMMMMH* and those alone need up to 36 integers with nearly 2 million combinations! The insurmountable problem is that for say, signature length 12, you have in a Pool 49, Pick 6 game, six possibilities for P if only one appearance in the past 10 draws and then you need to select 2 from say 16 for W, then 2 from say 16 for L then 1 from 11 for C with little difference between them for ranking - this would give 12 very arbitrary combs. The results after many innovations and exhaustive testing using my LottoTester™ program give at or about thesame yield as using random selections.

    In other words you could have just as easily taken whatever number of combinations you wanted to play from the immediate previous draws and simply randomized it for the pool applicable to the game.

    If intent on using history the likelihood or probability of one of the integers being picked by absence or recency in the past ten draws in a Pick 6 game is given by the following table:-

    SignatureCountProbabilityRank
    MMMMMMMMMM*100.241
    H*00.132
    MH*10.123
    MMH*20.114
    MMMH*30.095
    MMMMH*40.076
    MMMMMH*50.067
    MMMMMMH*60.068
    MMMMMMMH*70.059
    MMMMMMMMH*80.0410
    MMMMMMMMMH*90.0311

    The likelihood of an integer repeating in the next draw main numbers can easily be miscalculated. In a 2094 draw sample for a Pick 6, Pool 45 Lotto game there are 2 occasions where 4 integers repeat (.1%), 49 where 3 integers repeat (2.3%), 296 where 2 integers repeat (14.1%) and 869 with just 1 repeat (41.5%). The likelihood of 1 or more integers repeating is then 58%, less than the usual figure given of around 74.5% (double counting?).

    The subject has been exhaustively studied by myself using sophisticated programming techniques and modeling involving many hundreds of hours to the point where I am not being pretentious by saying I am an expert in this area. Of course I realize that rationally it can be dismissed but this is seen as a capitulation by the shysters so more work needs to be done to sway those that have been sucked in by the spiel. Programming is a hobby interest of mine and Lotto is the focus I use with plenty of interesting challenges over the years from around 1999. It means no matter what the dubious claim when using Lotto history I have the wherewithal to shoot it down in flames. There is nothing to be gained over random selections using history no matter what waffle the pseudo-analysts go on with and that concurs with probability theory on independent events. For an example of a Lotto game where the draw is not independent see: PerfectLotto™.

    For a marginal improvement over random selections you can do no better than my online program LottoToWin  available for a token $5.00 per year subscription.

    My program ignores history and concentrates on producing a set of numbers to play by using all integers, not repeating paying subsets and maximizing without "optimizing" the coverage.

    Recall that H for Hot is 3 or more occurrences of the respective integer in the past 10 draws, W for Warm is 2 occurrences, L for Luke is 1 occurrence, C for Cold is 0 occurrence and the 0 based number after the letter represents absence as in L0 in the previous draw or L9 one occurrence in the previous 10 draws C15 no Occurrences past 10 draws and last occurrence 16 draws ago. To get these wins using absence and occurrence in draw history you would have needed at least two lines and most probably more since only H2W6C13C18 gave 02 03 04 05, only L2L4L6L7 gave 02 03 04 06 and only W1L3L6L6 gave 03 04 05 06.

    It could have been obtained by using the template above with this number order which is just one of 5,864,443,200 permutations of 45 from 45 integers: -

    The reality is at every drawing everything is new
    and each integer has the same likelihood of being picked
    as any other integer.

    When you stand back after applying all the sophisticated programming to extract something from history you may realize as I did that you're really just maximizing the best results obtainable from a small number of lines. The same template result for 9 lines in a 6/45 Lotto game can be obtained in a few seconds using pen and paper by randomizing the following 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
    09 10 11 43 44 45

    Using the 6/45 sample history above you could have decided to use: -

    P01 W01 W02 L01 L02 C01
    P02 W03 W04 L03 L04 C02
    P03 W05 W06 L05 L06 C03
    P04 W07 L07 L08 L09 C04
    P05 W08 L10 L12 L13 C05
    P06 W09 W10 W11 L13 C06
    H01 W12 L14 L15 L16 C07
    H02 W13 L01 L02 L03 C08

    This gives in numerical order: -

    02 07 08 09 10 45
    11 12 14 15 28 33
    01 16 20 34 39 40
    03 04 17 24 27 35
    19 21 23 25 30 37
    05 18 22 31 36 41
    06 13 26 29 42 43
    09 10 11 32 38 44

    The order used for the 45 integers is: -
    02 07 45 09 10 08 12 15 33 11 28 14 20 34 39 01 40 16 27 03 24 04 35 17 30 21 23 25 37 19 31 22 18 41 05 36 06 43 13 26 29 42 32 38 44

    Consider the sobering fact that for a Pick 6 Pool 45 lotto game where getting four integers correct has odds of 1 in 733 more than two thirds of the integers are in the previous 10 draws but this only gives less than one third of the winning Fours. In other words two thirds of the winning Fours require near the full complement of the 45 integer Pool.

    For the sample of 2094 draws in a Pick 6 Pool 45 Lotto game probability formula calculates that playing one line should give 2094/733 = 2.86 or 3 wins for a combination of four integers ie a Comb-Four. Each line has 15 CombFours and there are 148,995 possibilities of which only 31,890 occurred. Obviously, with 117,105 having no appearance you are better off randomizing the line played. If you were lucky and five of your plays had 06 24 26 36 you could have won the maximum repeat of five CombFours: -


    Comb-Fours Count in 2094 Draws

    15
    104
    1933
    27762
    257141
    1171050

    SO, IN 2094 DRAWS OR RANDOM SELECTIONS 2980 COMB-FOURS REPEATED BY CONTRAST NOT ONE COMB-FOUR THAT WAS RELATED TO ABSENCE OR RECENCY AND OCCURRENCE OR FREQUENCY REPEATED.

    BUT THERE'S STILL MORE!

    For the sample of 2094 draws in a Pick 6 Pool 45 Lotto game probability formula calculates that playing one line should give 2094/45 = 47 wins (if paid on) for a combination of three integers ie a CombThree. Each line has 20 CombThrees and there are 14190 possibilities of which 13450 occurred. If you were lucky and twelve of your plays had 01 18 37 or 05 34 42 you could have won the maximum repeat of twelve CombThrees.

    CombThrees   Count in 2094 Draws


    212
    311
    1010
    399
    1248
    3017
    7516
    14045
    23584
    31593
    31792
    21201
    7400

    So, for a Pick 6, Pool 45 Lotto with 14,190 Comb-Three possibilities we have in 2094 draws 11,330 CombThrees that repeated, 2120 that occurred only once and 740 with no appearance.


    BY CONTRAST JUST 2 COMB-THREES REPEATED ONLY TWICE THAT WERE RELATED TO ABSENCE OR RECENCY AND OCCURRENCE OR FREQUENCY.


    Consider a simple example where you played the line 01 02 03 04 05 06 for the 2094 draws. Three combinations of four integers wins are expected and that is what you would have got 02 03 04 05,  02 03 04 06 and 03 04 05 06.



    THIS ARTICLE IS UNDOUBTEDLY THE COUP DE GRAS
    FOR USING LOTTO HISTORY ANALYSIS TO PRODUCE NUMBERS TO PLAY.


    You're to be congratulated for thinking Lotto history analysis is as a way to beneficially produce numbers to play in Lotto, irrelevant and is a gigantic con played out by closet or brazen numerologists or despicable opportunists prostituting their integrity for a few lousy bucks and all using the usual trick of assuming something false is true by pandering to something intuitive but incorrect and then constructing a dung heap on thin air.



    Colin Fairbrother
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