There are many systems for choosing winning Lottery numbers. All claim to based on some proven logic using observation, mathematics, computer modelling, statistics, combinatorics, etc. However, few if any, quantify exactly how much their techniques will improve one's odds, and until recently there has been no convenient method for consumers to measure their tickets and check a system's efficacy.
Most of the techniques recommended by Lottery Systems can be distilled down to three categories, Historical Bias, False Groups and Wheeling.
Some systems recommend choosing numbers that were drawn recently or have had higher frequencies recently or overall. They suggest that these "hot" or "active" numbers have a higher probability of occuring in the near future because past observation has "proven" them more likely.
Other systems suggest choosing numbers that have not been drawn in a while or have had lower frequencies recently or overall. They suggest that these numbers have a higher probability of occuring because it is "proven" that number frequencies even out over time.
Both techniques appear to have merit, but neither are actually true. Lotteries are designed to be completely random and immune to historial bias, and they hire experts to monitor and ensure this. Some numbers will over time, have higher frequencies than other numbers. This is expected with randomly drawn numbers, otherwise we could predict with certainty the numbers that were due, which then wouldn't be random.
Some systems attempt to give credibility to their claims with phrases like "it has been shown that one or more balls that were drawn recently (or not) have XYZ percent chance of appearing in the next draw". Usually the percentage is quite high. Sometimes the math is correct, or not, but it doesn't matter as it is presented authoritatively. What they do not tell you, is that the percentage is the same for all numbers, not just last week's. Using MegaMillions' White Balls as an example, there's a 32% (1 - 65C5 / 70C5) chance that next week's draw will contain 1 or more numbers from last week's. But it is also equally likely for any other set of 5 numbers, previously drawn or not.
Systems commonly group numbers into arbitrary categories such as odd/even, high/low and sequential, which have absolutely nothing to do with the lotteries themselves. These grouped numbers use only a portion (usually half) of all available numbers and therefore only match a tiny subset of all possible draws. The systems claim, some using equations and graphs, that their "specially discovered" grouping has a lower probability of occurring - and then incorrectly conclude that individual draws that fall outside of their chosen grouping, must be more likely than individual draws that fall inside.
We're going to prove that premise wrong by using a small lottery game that has only 6 balls, where 3 are drawn. This gives a total of 20 (6C3) possible draws as shown in the table below with all odd and all even highlighted.
If we imagine a 20 sided dice, with each side representing a unique draw from the table above, it should be apparent (as long as you are using a fair dice,) that all of the above draws have an equal probability of occurring.
Looking at the table, we can also see that there are 2 draws (3C3 + 3C3) that are either all odd or all even (1,3,5 and 2,4,6) and 18 draws that are mixed. This does NOT mean draw 3,4,6 is 9 times (18÷2) more likely than draw 1,3,5. If you were throwing the 20 sided dice (as described above,) enough times, you would expect all possible draws to have the same approximate throw frequency. However, if you were to also count the number of times a throw landed on all odd or all even, it would only account for about 10% (20÷2) of all throws, which would be the exact number of 1,3,5s and 2,4,6s thrown. 1,3,5 didn't become less likely because someone arbitrarily grouped some draws that happened to include 1,3,5. Any group's likelihood will be perfectly proportional to the size of the draws it includes. Small groups will have small probability, but the probability of the individual draws they include or exclude remains the same.
Wheeling is where tickets are generated using combinations from a list of chosen numbers. To demonstrate why wheeling is a poor strategy, we're going use a very small lottery with 8 balls, where 3 are drawn. This lottery has two ways to win:
|1||3 of 3|
|2||2 of 3|
We're going to choose 4 numbers (1,2,3,4) to wheel which gives the set:
This number set guarantees a first division win, if any 3 of our 4 chosen numbers are drawn, along with a guaranteed second division win, if any 2 of our 4 numbers are drawn. Here is a table of every possible draw with purple bolded 1st division wins and purpled 2nd division wins.
Looking at the table we can see that the wheeled number set matches half of the possible draws, providing an expected win 50% of the time. This seems positive, until we look at other number sets with the same row count, that provide a win over 89% of the time, such as:
Both sets have the same number of rows, and each row has the same number of winning draws. However, the wheeled set's rows share more of the same draws, averaging more multiple wins for matching draws, but winning less often. While, this may seem to make both sets equally good, the 89% set has two advantages:
- Most lotteries divide the division prizes amongst the winners. This means your expected returns are higher if your wins are spread over more draws. (i.e. winning 1st division 4 times on the same draw generally returns significantly less than winning 1st division 4 times on different draws.)
- Your average wait time before winning is significantly less.
So while wheeling does live up to its promises, it obviously isn't the best way to play.
From the Wheeling example above we can see that not all sets are equal. In fact the false advice above leans tickets towards less than average results. However, even lousy systems will have miracle winners to advertise, given enough followers; it's just that their followers could be doing better. We know tickets exist that not only provide greater expected winnings, but also reduce the expected wait time - it's just a question of finding them. While it's easy to compare short tickets with intentionally small lotteries, assessing longer tickets with lotteries that are millions of times larger is impractical to do by hand.
To aid stamping out false advice and generate genuinely better tickets, we developed a calculator web app. It can measure ticket spread, allowing users to compare tickets and choose the best ones.
If that all seems too hard, you can purchase sets, based on lessons learnt from building the tester, that consistently outperform the expected average set spread. However, unlike other systems we expect you to test the generated numbers to confirm how good they are.