Editor’s Note: This article was contributed by Ricardo Martins, a long-time Lotterycodex community member. While I reviewed and prepared it for publication, the core ideas and analysis are his. I’m sharing it here to make his insights available to a wider audience and to highlight the variety of perspectives within the Lotterycodex community. As always, statistical analysis describes long-run tendencies in random processes, but it does not provide certainty about any specific future draw.
Thank you for coming back for Part 2 of my article “Using Statistics to Play the Lottery”. This time, we will be taking a look at something that is not new to most players, but I hope to put a new spin on it, and I hope we all learn something new about the game with it.
Table of Contents
Understanding Sum Distribution
Very basic idea, you pick the numbers of your lottery, and you add them together. For example, in EuroMillions, you pick 5 balls, between 1 and 50. Some examples would be:
3 + 12 + 26 + 32 + 47 = 120
1 + 2 + 25 + 33 + 50 = 111
In PowerBall you pick 5 balls between 1 and 69.
2 + 23 + 31 + 44 + 61 = 161
10 + 17 + 29 + 33 + 42 = 131
It’s very simple, and it won’t tell you much about which numbers to choose.
But some people actually try to sell the idea that there is a “magical formula” to use sums to predict numbers. Edvin himself wrote about it, and I fully agree that there is nothing that will help you choose your numbers using the sum results.
But if instead of looking for some magical formula, you just do some statistical analysis on the results, you can actually get a deeper understanding of the game, its nature, and how it works over time. So let’s start!
The Totoloto Case Study (5/49 Game)
We will analyze a real-world scenario, Totoloto, a 5/49 game. The lowest and highest sums you can get are as follows.
1 + 2 + 3 + 4 + 5 = 15
49 + 48 + 47 + 46 + 45 = 235
So, any combination you ever pick will always be summed to a value between those two. That means all the 1906884 combinations that are possible in a 5/49 game will always produce a sum between 15 and 235. Now, we have calculated the highest and the lowest points. Let’s find out the middle point, the more balanced result between the two:
235 (highest) – 15 (lowest) = 220 (all possible difference results between highest and lowest)
220 / 2 = 110 (half of the difference)
110 + 15 (lowest possible sum) = 125 (middle point)
It’s important not simply make half of the difference, but also add back the initial 15. That’s because, as we have said, that’s the lowest sum we can get; we will never get a 7 or a 12. So 125 is the middle point.
This is what some lottery tools do online: they suggest combinations that they claim are “mathematically proven to win the lottery”. But they simply calculate the middle point for any given lottery game, and produce random combinations that add up to that value. Thousands of different combinations result in that sum, but it doesn’t help at all! If anything, it gives you unbalanced combinations!
The idea here is not to magically predict the winning numbers, but to gain a deeper understanding of how the game works.
Now, the first thing you’ll notice is that not all sums are equally possible.
1 + 2 + 3 + 4 + 5 = 15
1 + 2 + 3 + 4 + 6 = 16
1 + 2 + 3 + 4 + 7 = 17
1 + 2 + 3 + 5 + 6 = 17
1 + 2 + 3 + 4 + 8 = 18
1 + 2 + 4 + 5 + 6 = 18
1 + 2 + 3 + 5 + 7 = 18
You see, there is only one possible combination to sum 15. Likewise, only one combination will give you 16. But there are two that result in 17, and three result in 18. The same happens the other way around.
49 + 48 + 47 + 46 + 45 = 235
49 + 48 + 47 + 46 + 44 = 234
49 + 48 + 47 + 46 + 43 = 233
49 + 48 + 47 + 45 + 44 = 233
The number of possible combinations that add up to a certain sum value increases as you move away from the extreme points (highest and lowest) and closer to the middle-point, in this case 125.
In fact, if we take the sum results of all the 833 draws that happened between 2012 and 2019 in Totoloto, and we calculate the average of those sums, the result is 124.95, which means 125!
Why Some Sums Are More Common Than Others and Why the Middle Point Is Misused
Seeing as we have many more ways to sum numbers up to 125 than, say, 18, we can say that this result (125) has a higher probability than the other (18). And remember, the average sum result for a large number of plays is actually 125. The natural expectation would be that the observed results would match this probability, with winning combinations that sum to 125 appearing much more often.
However, that is not the case, and in fact, this is where we can learn something new and important about the lottery game. Below I present eight tables that display the sum results over eight years of Totoloto draws.1 Take a few moments to analyze them and try to notice anything important.
There is much to be learned from these, but for now, let’s focus on two things.
First, observed sum results usually start at about 60 and only go up to about 200. You don’t often see results closer to the extreme points, especially the lowest. This is in line with the thought that “all numbers have the same probability, but not all combinations”, something Edvin himself has written about before.
Second, unlike what we would expect, we don’t see sum 125 appearing much more often than others, even though many more possible combinations give us that result, therefore making it more probable to appear. Why?
Because, as I said at the beginning of Part 1, the lottery game is a “chaotic system”.
Real Draw Data and the Chaotic Nature of the Lottery
A chaotic system of such complexity doesn’t try to follow mathematical predictions; it tries to satisfy its own nature by producing as many different results as it can.
It’s the fact that this system exists inside our Universe, which is made and ruled by mathematical laws, that makes certain patterns appear over time, and in fact, the balance between the chaotic nature of the game and the mathematical laws that rule over the entire Universe is what makes the lottery game so interesting and exciting to play.
The chaotic game won’t obey our natural expectation of having the 125 sum appearing more often, but it cannot escape the mathematical ruling of the Universe that makes the average, our middle point, 125.
But… so what?
You may be asking, how does it help us to win the lottery? It’s all very interesting, the nature of the game vs the nature of the Universe, bla bla bla, how can I make money from it?
Well, the first easy, low-hanging fruit is that the sum of the combination you are going to play should not be too close to the extreme points (15 and 235), since we can see those rarely appeared in the past 8 years, and in fact are extremely hard to appear, probabilistically speaking.
But that’s not much help, because if you build your combinations using the proper mathematical processes, you already avoid those combinations. If you don’t know how, you can use Edvin’s calculator here on the website, which makes use of those mathematical processes and gives you the most probable winning combinations, therefore greatly increasing your chances of winning the lottery.
The problem is that even if you use mathematical processes, you are still left with 10, 20, or 40 possible winning lines. Edvin’s calculator, for example, allows you to choose how many lines you want to produce. If you can afford to play all those lines, good for you! You greatly increase your chances!
But playing those many lines is very expensive, and so we need a way to reduce the number of lines that we are going to play, without affecting our chances of winning. It’s an extremely important part of playing the game, making it sustainable to play over a large period of time until we win.
Editor’s note: In Lotterycodex, the most prevalent combinatorial templates naturally correspond to the most statistically typical sum ranges. These dominant templates are combinatorially optimized through a fair distribution of probability across the entire number field. Therefore, players do not need to compute or chase a ‘most probable’ sum separately, since this information is already embedded in the template structure itself.
The Principle of Elimination
Now, I understand some people advocate not playing for a few weeks, saving money so you can play more lines when you do. Edvin himself has written about it, and I don’t disagree. However, not every player wants to go that route; in fact, some people would lose interest if not playing frequently, for various reasons. So I developed a system to reduce the number of lines (thus reducing the expenses), while still playing frequently.
I call this “The Principle of Elimination”, which is a couple of strategies I have come up with to help us choose which lines NOT to play.
Keep in mind, this is not for every player out there. This is for those who are active players, the ones who want to play every week, knowing they did everything in their power to increase their chances of winning the game. This is for those who want to feel confident and know that once victory comes, they achieved it using their brains and hard work, not simply blind luck.2
There are different strategies which are a part of the Principle of Elimination, but for today’s article, I chose to focus on the sum’s results. So let’s get down to it!
If you made a good reading of the tables I showed you earlier, you will notice that every year (or about 100 draws) about 70% of the sum results appear only once. Around 20% will appear twice. The same sum rarely appears 3 or more times in the same range of draws.
So, if you have produced 20 possible winning combinations, but you can only afford to play 5 or 10, calculate their sums. Now analyze the winning results for the last 6 or 8 months, and calculate their sums. From your possible lines, eliminate those that have a sum already appearing more than once in the past results. That’s simple.
If necessary, you can try changing an odd number for an even number, or the other way around, so your final sum is different. By using this method, you avoid betting on an extremely rare event (that the same sum appears 3 or more times).
Important Clarification
Now hear me out… Let’s make sure we are all on the same page here. It is not impossible for the same sum to appear 3 times, or 4 times. Or even 20 times in a row! Each draw is independent, and one does not affect the other. It is a matter of, statistically speaking, not being the observed behavior of the game, which is in line with its chaotic nature. It all comes down to deeply understanding the game, actively monitoring it, and using that knowledge to our benefit.
You still need to play using a proper combination, built using proper mathematical principles. This is not about “statistics vs combinatorics”. You need to use both if you want to have a shot at winning the game.
If you are not a math wizard, I think you should consider giving a try to Edvin’s Lotterycodex Calculator, he put a lot of work into making it as simple and powerful as possible.
As ever, playing the lottery is a luxury entertainment, and you pay for the right to play.3 Don’t expect to win, just enjoy the game. Having fun is the key part of it. But being an active player gives you the best chances you will ever have to win!
I hope you had fun reading this article and learned something new today. I know I did. If you are an active player, consider giving this method a shot the next time you play, and let us know how it goes for you. As always, I welcome constructive feedback in the comments section!
Also read Part 1: Using Statistics to Play the Lottery (Part 1): Testing the Fairness of the Game
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References