1. Introductory notes: A protomathematics of lottery

A Google search on 'lottery mathematics' yields 1,640,000 results! Sounds like the whole world consists of lottery mathematicians only! BRRRRRRRRAHAHAHAHA......

Probably, lottery mathematics — an attempt is a necessary qualifier — starts with Nurmela and Ostergard, followed by Pak Ching Li and G. H. John Van Rees at University of Manitoba, Canada. They all deal with Lotto Designs and techniques for constructing Lotto designs and determining upper bounds for L(n, k, p, t). Lotto Designs are better known as Lotto Wheels or Abbreviated Lotto Systems.

The would-be lottery mathematicians attempt to discover formulas for constructing the tightest possible lotto wheels. The professors come up with an approximating formula; the margin of error is as large as the area of Canada! They analyze one lotto wheel, a famous one by now: 49 lotto-6 numbers with the '3 of 6' minimum guarantee. In cuckoo speak: LD(49, 6, 6, 3)=163. That is, the guarantee of the lotto wheel for 49-6 numbers is satisfied in 163 lines (or 6-number combinations). The flaw is evident right from the start. It is not one group of 49 numbers, but two separate groups of numbers: 22 and 27, respectively. None of the numbers in the first group meets with any number in the second group.

The task as analyzing a flawed lotto wheel raised to the status of lottery mathematics sounds hilarious. But, hey, we got to start somewhere! The lottery professors even invoke a true mathematician, Euler. They sow the pages with a plethora of theorems and formulae. The plain truth is that only one formula is for real: hypergeometric distribution.

2. Mathematics applied to lotto designs or lotto wheels

The hypergeometric distribution formula calculates the probabilities of getting 'k of m in p from n numbers'. In layperson's words: The probability or odds to hit 3 (k) of 6 (m) winners, when I play a pool of 18 (p) numbers in a lotto game with a total of 49 (n) numbers in the field. The answer in this particular example is '1 in 3.8' (or approximately once every 4 lotto drawings). But if we only play a single combination of 6 numbers instead of an 18-number pool, the '3 in 6' odds are '1 in 56.66' (or approximately once every 57 lottery draws). The calculations are easily available to everybody by visiting the Random Generator, Odds Calculator page at this truly mathematical web site (!)

The wrong assumption here is the construction of a lotto wheel (do you like lotto design better?) consisting of 57 lines (6-number combinations) to satisfy the '3 in 6' minimal guarantee. That will never happen! 57 here is not probability, is not degree of certainty: It is number of trials! These are the three fundamental elements of random phenomena, including lottery and lotto games. The three elements are embodied in the Fundamental Formula of Gambling (FFG). It is the formula of the core of TheEverything. It is all ruled by randomness; The Universe is created-destroyed-and-forever-again by randomness. What differentiates the events is the interaction of the three fundamental elements of randomness: Probability (p), degree of certainty (DC), and number of trials (N).

Most people, especially those who never read the totally truthful (axiomatic) materials at this web site, wrongly assume that certain events, like life, are not random. They assume that their own life reaching tomorrow is a given. Yet, thousands of people will die unexpectedly tomorrow. What we can say correctly is that, for example, all the readers of this page will live to see tomorrow with a high degree of certainty. That's what randomness is all about! Long live my visitors!

"People who believe in gods
Are mighty frightened by the odds."

We don't need a lotto design (sic!) to hit the lottery according to the hypergeometric distribution formula. Just play 57 unique 6/49 lotto combinations. Again, run that incredulously great random number generator by visiting the Random Generator, Odds Calculator page. You'll notice that, for example, the 57-line set of random lotto combinations does not have a '3 in 6' winner in a particular lottery drawing. But, other times, you will get two '3 in 6' winners per draw. If you play the 57-line set in 10 lottery draws, you will record 10 '3 in 6' winners. So, what's the purpose of lotto designs? The huge cost, such as time, effort, extra money for playing, etc.?

No lotto design will come even close to the odds, as far as the number of combinations is concerned. (Exception: the 10-number case for '4 of 6', in 3 combinations.) The aforementioned lottery professors abundantly use the < (less than) and > (greater than) mathematical operators. When things are not sure, those operators come to the rescue. Let's say: The minimal size of LD(49, 6, 6, 3) must be > (greater than) 57 combinations! You betcha! How about LD(18, 6, 6, 4)? It has to be greater than 19. You betcha! But how much, exactly? They don't know… Thus, your best bet is to play a set of random combinations equal in size to the odds calculated by the hypergeometric distribution formula for that particular lotto prize. And you'll be better off financially. I demonstrated that truth with real data in UK National Lottery by comparing the 163-line lotto wheel to a set of random lotto combinations. Read the web pages dealing with lotto wheels at this site.

The only precise action here is generating all possible combinations in a lotto game. The lotto combinations are a particular set of numbers (one of the four types of sets). My freeware PermuteCombine.EXE generates every imaginable set of numbers (words or text as well), both in lexicographical order and in random manner. There is a total of 13983816 combinations in a '6 from 49' lotto game. My free software will generate exactly 13983816 combinations in lexicographic order (from 1,2,3,4,5,6 to 44,45,46,47,48,49). Them lottery professors or lotto wheel aficionados will call those 13983816 combinations a "6 of 6 lotto design" or a "full 6-49 lotto wheel"! The contradictions in terms are blunt. Design or wheel implies reduction; i.e. a reduced set of numbers that satisfy a condition.

I wrote also lottery software to generate them lotto wheels or reduced (abbreviated) lotto systems. The program names read something like WheelCheck.EXE. You can run them for free, if you register as a download member for a nominal and reasonable fee. That formidable lottery software is designed to verify lotto wheels for missing combinations and generate reduced lotto systems (wheels...pardon me...lotto designs).

The lottery wheeling software treats all lotto numbers as fairly as possible. That feature is called balance; i.e. the numbers appear fairly equitably in the reduced system. I label the lotto wheels generated by my software as balanced. For the '1 of 6 in 6 from 49', WheelCheck6.EXE generates 8 and always 8 combinations. Of course, the hypergeometric odds are '1 in 2.42'. That is, some players expect a 3-line lotto wheel! That is not possible. Just look at the 8-line set:

1 2 3 4 5 6
7 8 9 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

The number 49 is missing, but the design still covers '1 of 6 from 49'. The 8-combination system covers up to 53 numbers. The incomplete combination 49,50,51,52,53 will need one more number to make it a 6-number lotto combination. It will be any of the 48 numbers in the 8-line '1 of 6' lotto wheel. The system is balanced in a proportion equal to 98%. 48 of the 49 numbers are equally distributed (one time each). One number (#49) is missing. This lotto system cannot be compressed (reduced) any further. By eliminating just one line, we destroy the '1 of 6 from 49' guarantee.

Compression or reduction is possible for multiple-number guarantees: From '2 of 6 from 49' to '5 of 6 from 49'. Compression or reduction is no longer possible when we reach the '6 of 6 from 49' condition. The 13983816-combination set cannot be reduced by one single line!

The '2 of 6 from 49' lotto design generated by the incredible WheelCheck6 program consists of 48 lines. The numbers are fairly equitably distributed, but there is a bias towards the first 2 numbers. The bias is caused by the fact that the lexicographical lotto generation starts with #1. If we did the generation in descending number (starting at #49), then the bias would favor #s 49 and 48. "Talking about my ge…ge…generation, baby!" (The Who).

The '3 of 6 from 49' lotto design generated by the incredible WheelCheck6 program consists of 514 lines. The numbers are fairly equitably distributed, but there is a bias towards the first 3 numbers. The bias is caused by the fact that the lexicographical lotto generation starts with #1. If we did the generation in descending number (starting at #49), then the bias would favor #s 49, 48 and 47.

I was able to reduce this lotto system down to 416 combinations. The minimal guarantee was preserved, while the balance was improved. The procedure is painstaking, however. First, I generated all 13983816 combinations in lexicographical order. It is a huge file! Next, I ran another great piece of software created by yours truly: Shuffle.EXE. Among other functions, the program shuffles a text file. That is, the lines of a file are distributed in a random manner. The shuffling procedure is easier for 5-number lotto files, for example. After one shuffle, you run another great free program of Parpaluck's: WheelIn6.EXE. That super program creates wheels from text files containing lottery combinations. You can shuffle again, and again, and again…until you realize that you can't shrink the lotto wheel any further.

Another method is to start the generating with the Wheel.EXE programs (as in the Pick.EXE and especially Bright.EXE integrated lottery software packages: Function key F8). You stop the lotto wheeling programs when everything seems to have come to a halt. Use the lottery wheel output file and input it to WheelCheck.EXE. Combine the new output with the output file generated by Wheel.EXE. You will have a complete lotto design assuring the minimum guarantee and a good balance.

The lotto designs or wheels created by these somehow complicated methods are called balanced and randomized. The shuffling procedure assures that no lotto numbers are biased in favor. The lotto wheels are balanced to higher degrees, while maintaining the minimum guarantee or condition. However, there is no formula that can calculate the number of lines (size) of the lotto design. The design sizes differ catastrophically from set to set!

The best balance for a lotto-6 system was achieved in a lottery newsgroup where I participated. Two other participants collaborated. We reduced an 18-number system created by WheelCheck down to 51 lotto-6 combinations. That lotto wheel offers a highly balanced frequency of both singular numbers and number-pairings. Nothing else out there comes even close to that balancing act.

3. Step One: One lottery number at a time

Everything said and done to this point is only the appetizer. It is not even a stepping-stone. For the true mathematics of lottery starts here and now. We go back to the Fundamental Formula of Gambling (FFG) right away. That's how I started the lottery strategy back in 1997. It was the very beginning of my Internet experience. Very importantly also, it signed the birth certificate of lottery mathematics. The main lottery strategy page started this way:

“…FFG has a column p=1/8 that describes exactly a lotto game drawing 6 winning numbers from a field of 48 numbers. '6 divided by 48' is 1/8 or 0.125. That's how you calculate the probability p, when considering one lotto number at a time!

Evidently, each lotto or lottery combination has an equal probability p as the rest, but the combinations appear with different frequencies. The FFG median is the key factor in the biased appearance. The lotto numbers tend to repeat more often when their running skip is less than or equal to the 'probability median'. The 'probability median' or FFG median can be calculated by the Fundamental Formula of Gambling (FFG) for the degree of certainty DC = 50%. This revolutionary premise constitutes the backbone of the lottery and lotto strategy that follows.”

See above: The '1 of 6 from 49' lotto design required 8 lines. There is more to it. Ion Saliu's Paradox of N Trials demonstrates that if we randomly generate 8 lotto 6-49 combinations, only 63% of the numbers will be unique; the rest, 37%, will be repeats. The FFG median for this case is 6. If we generate 6 random lottery combinations, only half (50%) of the numbers will be unique. Equivalently, half of the 48 (or 49) lotto numbers will have not come out in 6 drawings.

This lottery strategy, as it was the case with the '1 of 6 from 49' lotto design, doesn't offer anything in the way of further reduction. We know now that after 6 drawings only 50% of the lotto numbers came out. We also know that after two more drawings 63% of the lotto 6/48 (let's say 6/49) would have come out. Thus, the drawings #7 and #8 will spit out 13% more lotto numbers. In absolute terms, 13% represents 6 or 7 lotto numbers (out of 48 or 49). There were 24 (or 25 -- the half) numbers that came out in the previous 6 draws. FFG expects 6 new numbers to come out in the next 2 draws. The 6 numbers can be distributed over the next drawings in patterns like 0-6 or 6-0 to 15 or 5-1, but more likely 3-3. If we play only the numbers from the last 6 lottery drawings, we expect to hit 3 lotto winners. But, things could be drastically different at (rare) times. As my lottery strategy page demonstrated, real lottery results showed that all 6 winners came from the previous 6 drawings, even from the last 5, or even 4, lotto draws!

Since no much reduction is possible, this incipient lottery strategy is the weakest.

4. Step Two: A pair has a count of two and is more powerful, too

4.1. I developed a more powerful lottery strategy three or four years after the incipient lottery strategy. I called this second coming 'wonder-grid'. The lottery wonder grid considered pairs of lotto numbers, instead of single lotto numbers. There was plenty of statistical evidence that the lotto numbers showed strong biases in pairing with other lotto numbers. The top 5 pairs for each lotto number came out a lot more frequently than the rest of the pairs: Around 50% of all pairing frequency.

The weakness of the 'lottery wonder-grid' was non-discrimination. It treated equally all lottery numbers. I discovered at a later time the science of positive discrimination. You can read much more right at this website. We found out at Step One that the lottery numbers do not come out with an equal frequency. As a matter of fact, a few lotto numbers come out 6 times in 50 drawings, while other numbers do not show up! On the other hand, the wonder grid treated every lotto number equally. It played each and every number and its top pairs.

I discovered an important zone of the area of lottery pairings I named least pairings. That discovery preceded the Incipient Lottery Strategy described by Step One. The least pairings was one of the earliest filters (or restrictions) in my lottery software (beginning 1988). The term least pairings is relative. We can fully define it by establishing the value of least. Least refers to the minimum threshold of the pair frequency. The default value I established for least in my lottery software is 0 (zero). That is, every lottery pair with a frequency of zero (no show) is considered to belong to the least pairings restriction (filter). But the least pairings filter is validated sometimes by an upper limit higher than zero. There are drawings when the least lottery pair has a frequency higher than 5, even 10! Setting the lottery filter that high absolutely devastates the lotto odds! Even a least pairing equal to 0 can reduce millions of lotto combinations!

4.2. Let's do some non-cuckoo mathematics regarding Step Two and the Least Pairings in Lottery. How many pairs or pairings are there in a lotto game, say, '6 from 49'? Two easy methods to calculate. The 49 numbers can be paired in C(49, 2) = [(49 * 48) / (1 * 2)] = 1176. The lotto game draws 6 numbers, therefore total pairs in the draw is C(6, 2) = 15 pairs. In final analysis, the 6-49 loto game yields 1176 / 15 = 78.4 or approximately 79 integer elements.

The other method does it in one step, very much like calculating the lotto combinations. [(49/6) * (48/5)] = 78.4.

Those are NOT real elements. They are derived elements to help us perform probability calculations. We did it easier at Step One, when we calculated the probability for singular lotto numbers (e.g. 1/8 or '1 in 8'). Probability is what we need first and foremost. The formula above can be reworked to calculate pair probabilities directly. [(6/49) * (5/48)] = 0.012755 or '1 in 78.4'.

We shall run at this point the most superb probability and statistics software: SuperFormula.EXE. We need to calculate the FFG median for lottery pairs. We should choose 'Option 2: The program calculates p'. I suggest we use 10 for the first element of p and 784 for the second element. The result for FFG median (DC = 50%) is something like 54. So, half of the lotto 649 pairs will come out in 54 drawings. Hold up! The WheelCheck program generated a lotto wheel that covers '2 of 6 from 49' in 48 combinations! Actually, that wheel can be reduced to 30-something lotto combinations while preserving the minimum guarantee! Well, that's the power of reduction!

We can generate the lotto pairing file by running another piece of great lottery software: Util.EXE. The 'Stats' function generates a plethora of frequency reports, including for pairs. The program automatically creates a distinct file dedicated to the 'least pairings'. As per above: Just the least pairings with a frequency equal to zero eliminate millions of lotto combinations, when set as a filter (or restriction the generating software). A least pairing set to minimum frequency = 2 can eliminate ALL lottery combinations sometimes. When it hits, it can generate very few lotto combinations! That's the power of positive discrimination. There are good lotto numbers and lottery pairings, as far as frequency goes. But just one bad pairing destroys the fun! By avoiding such pairing, we can eliminate a large number of lotto combinations that have a very low probability to hit the jackpot.

We generate the pairs for a range of analysis (named parpaluck for the sake of simplification) equal to 54. We expect, based on Ion Saliu Paradox, 13% new pairs in the next 79 - 54 = 25 lottery drawings. We can do one or two things. We enable the same file of least pairings and play the output for the next 25 draws. We can also recreate the least pairings lottery file for each drawing separately. We play the output only for the next lottery drawing. Of course, we can combine the two and then we purge the duplicate combinations (another function in Util.EXE).

We are free to select other parpalucks as well. FFG offers a wide range of possibilities. Set parpaluck to N, for example (79 in the case of lotto 649 pairings). We know that 63% of the lotto pairs will come out in that range. We calculate now the degree of certainty for a case equal to N * 1.5 (one and a half of N); it is something like 120 in this case. Or, we can just set parpaluck = 100 draws. The degrees of certainty are 78.6% and 72.3%, respectively. Seems to me, the parpaluck equal to 100 is a more efficient method of applying the least pairings as a lottery software filter. We can play the next 21 drawings and expect just under 10% of new lotto pairs to come out. We have here a 90% degree of certainty that all lotto pairs will repeat in that range of 21 drawings (from 79 to 100). I don't know about you, kokodrilo (big-time gambler), but it looks like a good bet to me!

The advantage of Step One is the availability of complete lottery software to apply the strategy. Step Two has lottery software to generate the least pairings files and load them as filters (restrictions). This step, however, does NOT have yet software to generate the winning reports, like the W files available via function key F5 in my integrated lottery software packages. If such software becomes available, you will be notified via the New Writings and/or What's New pages. Do yourself a favor and visit them from time to time.

5. Step Three: A triplet has a count of three and is more powerful, too…three

Next to the pairs are the triplets on the lottery stepladder. Every lotto number comes out most with two other numbers and thus forming a triple or triplet (or any similar name).

There is a caveat. The lotto triplets require a much larger lottery data file. That is, a much larger number of past drawings. Few lottery commissions have comparable lotto game histories.

How many triples or triplets are there in a lotto game, say, '6 from 49'? Two easy methods to calculate. The 49 numbers can have C(49, 3) = [(49 * 48 * 47) / (1 * 2 *3)] = 18424 triples. The lotto games draws 6 numbers, therefore total triplets in the draw is C(6, 3) = 20. In final analysis, the 6-49 lotto game yields 18424 / 20 = 921.2 or approximately 922 integer elements.

The other method does it in one step, very much like calculating the lotto combinations. [(49/6) * (48/5) * (47/4)] = 921.2.

Those are NOT real elements. They are derived elements to help us perform probability calculations. We did it easier at Step One, when we calculated the probability for singular lotto numbers (e.g. 1/8 or '1 in 8'). Probability is what we need first and foremost. The formula above can be reworked to calculate triplet probabilities directly. [(6/49) * (5/48) * (4/47)] = 0.00108554 or '1 in 921.2'.

We shall run at this point the most impressive probability and statistics software: SuperFormula.EXE. We need to calculate the FFG median for lottery triples. We should choose 'Option 2: The program calculates p'. I suggest we use 10 for the first element of p and 9212 for the second element. The result for FFG median (DC = 50%) is something like 638. So, half of the lotto 649 triplets will come out in 638 drawings. Wait a minute! The WheelCheck program generated a lotto wheel that covers '3 of 6 from 49' in 514 combinations! Actually, that wheel can be reduced to 400-something lotto combinations while preserving the minimum guarantee! Well, that's the power of reduction!

Step Three is very poor when it comes to lotto software availability. There is nothing to generate a file like the least triplets. Only the very old 16-bit Tools.EXE (1995!) lottery software calculates the frequencies of triplets for 5-, 6-, and 7-number lotto games. There is no least file, however. If such lotto software becomes available…you know what to do. A least triplets lotto filter would be even more devastating to the odds than the least pairings. P(r)ay that such software comes to life soon!

Hold on! There is some good news! I just created (7/15/08) ToolsLotto6.EXE, an upgrade to Util632. ToolsLotto6 has two new functions:
~ Triplets Rundown (3-number lotto groups);
~ Quadruplets Rundown (4-number lotto groups).

The Triplets Rundown has some powerful capabilities:
~ calculates the frequency of every triplet in a 6-number lotto game;
~ sorts the lotto triplets by frequency in descending order;
~ creates a 'least triplet file' the same way as 'least pairings' for a 6-number lotto game.

ToolsLotto6.EXE is available as totally free software. Paid membership is NOT required to download the program. First-time visitors have asked me to provide a totally free piece of lottery software, so that they can see what they are dealing with. It is a convincing proof that lottery mathematics does not require special (or scary!) brains; just some attention and effort are needed.

Read the presentation and download ToolsLotto6.EXE from here: "Special upgrade to the lottery utility software for 6-number lotto games".

Further steps — Four or Five — are absolutely impractical. Human life is too short to accommodate such long lotto mathematics requirements. Besides, no lotto game will last that long. The lotto game formats change way too fast already! For example, the quadruplets amount to 211876 4-number groups, or 14125 derived elements ('4 in 6' lotto 6/49 groups). The FFG median is 9790 drawings... I don't think there is or will be a lotto game history that long: 97 years with two lotto draws a week!

This has been a quite comprehensive introduction to lottery mathematics. The treatise can be expanded a lot by analyzing the multitude of filters already available in my lottery and lotto software. I am talking now about the derived filters or second degree filters, as opposed to the three primary filters analyzed in this treatise. The derived filters aren't much different, if different at all. They also have FFG medians and degrees of certainty. Analyzing all those filters would probably require 100 times more publishing space than this introduction.

Let's not call this treatise an introduction anymore. Let's address it properly: Lottery mathematics in a nutshell.

Best of luck!

6. Lottery links and resources at this web site

Resources in Lotto, Lottery Software, Wheeling.
See a comprehensive directory of the pages and materials on the subject of lottery, lotto, software, systems, and wheels. The link is displayed in a new window. If the page content is of interest to you, copy-and-paste to that lottery page of yours: SaliuLottery.txt or something similar. Copy as much as you can in that text file. By editing that page in your own words will result in a very effective lottery manual.

Download lottery software, lotto software.

Barnes & Noble.com: Shop the Internet's Largest Bookstore for books and textbooks.

Watch the latest videos on YouTube.com

Socrates Home   Search Site   Random Generator, Odds Calculator   New Writings, Pages   Download Lottery Software   Sitemap, Content