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.
If you are reading this, you probably want to win the lottery and believe statistics will help you predict the winning numbers. Well, let me save you the trouble. Statistics cannot predict the winning numbers.
In fact, nothing can predict the winning numbers.
The lottery game is a chaotic system, with an exceedingly large number of possible outcomes, and far too many variables affecting each result. Some people actually consider the lottery to be a random system instead of a chaotic one, due to its complexity.
Either way, it’s impossible to predict the winning numbers.
Instead what I’m going to teach you is how to become an active player.
I’ll help you to look at the game and see no more randomness and hopelessness, but balance and order made of mathematical laws and observable facts.
That means no more using the quick-pick machine, no more playing the same numbers for over 40 years with no success. You will learn to read the game and decide your next move. You will soon discover that math can be more useful and fun than your high school teacher ever made it to be. When we’re done, you’ll have gained confidence, not just in your lottery ticket, but in yourself.
So let’s begin!
Table of Contents
Understanding the Limits of Statistics in Lottery Games
Statistics is usually defined as the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.1 So the first thing I did was collect the results of the two most popular lotteries in my country, EuroMillions and Totoloto. I studied these results and developed a series of game plans that can be applied to increase our chances of winning, whatever goal we set.
But let’s start with what statistics can’t do, and why they can’t do it. In the popular EuroMillions lottery, 5 numbers are picked from 50 possible (plus two “lucky stars”, picked apart from the main numbers, which we will ignore for now). The game was launched in 2004, and at the time of this writing, 1267 draws have been taken, which means 6335 numbers have been picked over a period of about 16 years.
Below we have the statistics of how many times each number was picked, sorted from the most frequent to the least.
As we can see, the most often picked number is 23 (picked 150 times) and the less often is 46 (picked 97 times).
Given our full sampling rate of 6335 numbers, we can say number 23 represents 2,37% of the entire sampling and number 46 represents 1,53%. Not even 1% difference between the most picked number and the least one.
But let’s say we wanted to use number 23 as our first choice. You don’t win by picking one number right; you have to get all 5. So the next step would be to choose the second most often picked number. In our table above, that would be 44, with 149 picks.
However, statistical analysis quickly reveals that out of the 150 times number 23 was picked, only 17 times was number 44 picked on the same draw. And number 50 (third place in our table above) never got picked together with those 2 other numbers!
That’s why the popular approach of playing the 5 most frequent numbers won’t work.
But maybe you will try to out-smart the odds and check which number gets picked more often, together with 23! That would be number 4, which in all the 150 times number 23 was picked, out of 1267 draws, got picked a grand total of 24 times.
So, 24 out of 150 out of 1267… I would make a joke and say “You have better chances of winning the lottery”, but playing that way… no, you don’t!
Setting Expectations the Right Way
Statistical analysis won’t predict the next winning numbers. But it can help us to better understand the game, set a clear goal, define a strategy, and most importantly, make sure that the mathematical principles we use to play are actually being observed. It will help us have a lot more fun with the game, too!
Speaking of fun, before we dive deeper into the mathematical aspects of the game, I want you to ask yourself this.
“Should I play the lottery?”
No.
Ok, long answer: Depends. The lottery is not an investment or an easy way to make money. The lottery is a luxury entertainment, an expensive path for fun and not profit. When you play the lottery, you actually pay for the right to play it. Don’t expect that money to come back to you. So, if you don’t have a job or can’t pay your bills, you have no reason or right to buy a lottery ticket.
Save that money and choose a low-risk investment or buy some tools and start a new job, don’t waste that money in the dreams of “what if”.
Statistics show more people have played EuroMillions for the last 16 years than the number of possible different combinations, and yet only a few hundred have hit the jackpot.2 Invest your money, don’t waste it.
Now, if you do have the money and time to spare on a luxury entertainment such as the lottery, you might as well play it right!
If you have been following Edvin’s articles, you already know the importance of building a balanced combination of odd/even/high/low numbers.
It’s extremely enlightening, and while Edvin and I don’t necessarily agree on every point, it will get you up to speed on the basic math that you need to start understanding the lottery game. Trust me, read it, and then get back here, you’ll be glad you did.
The basic idea is that you should separate the numbers into two groups, between low (1 – 25) and high (26 – 50), and again between odd and even numbers, getting 4 groups of numbers (odd/low, odd/high, even/low, even/high). The more balanced a combination is among the four groups, the higher the probability it will be picked. Again, read his article!
Checking the Fairness of the Lottery with Statistics
Once you understand how important some mathematical principles are to playing the lottery, you will never again pick numbers at random. But that’s where statistics start playing an important role. That’s because if you are going to trust mathematical principles, you want to make sure, before you play, that those principles are actually observed in the real-world results. Or to say it differently…
“Is my lottery game a fair game?”
A game is considered “fair” when the observed results over a large number of plays are a close match to the expected frequency of those results. For example, when flipping a fair coin, you have 50% chance for both heads and tails, plus a small deviation for either side.3
In 1000 flips you would expect 500 results for each side. Add deviation, and it wouldn’t be uncommon to find between 400 and 600 results for each side.
But if you took 1000 flips and found an 850-150 result, you would know that the coin was biased, unbalanced, and should not be trusted.
In EuroMillions, the most probable results are between the 3-odd-2-even and 2-odd-3-even combinations. The same goes for 3-high-2-low and 2-high-3-low combinations.
In Edvin’s article I mentioned earlier, he put it to the test and found a close match between the expected results and the actual observed results. See below.
Therefore, we have a basis for accepting EuroMillions as a “fair game”.
We could also test EuroMillions as a fair game using our statistics from the earlier example. We have already calculated that the most often picked number represents 2,37% of the entire sampling of numbers (1-50). The least often picked number represents 1,53% of that same sampling.
Given that the 50 numbers represent 100% of the sampling, and each ball has a 1/50 probability of being picked, we conclude each number should represent 2% of the entire sampling rate.
The actual results for these 16 years actually match that pretty closely, with less than 0,5% deviation in either direction from the expected frequency!
You can see it in a graphical representation below.
Let us now have a look at “Totoloto”. This is the Portuguese national lottery, a 5/49 game (plus one lucky number, which we will ignore for now, just as we did with EuroMillions). I collected the results from 2012 to 2019, 8 full years, 833 draws total. I wanted to use only full-year results in my analysis, so no data from the current year. In 2011, some rules were changed, so I decided to start in 2012. Below are the results from the same analysis.
Keep in mind that being a 5/49 instead of a 5/50 could lead to a higher degree of deviation in the balance between different results. But in fact, Totoloto proved to be even more balanced than EuroMillions!
You may ask why this all matters, but it’s the most important first step in analyzing your favorite lottery game. Because if you find a clear bias pointing to either of the less probable combinations, you know the results are not being drawn correctly, maybe the game system was manipulated at some point, somehow. You either choose to play that knowledge to your own profit, or you alert the authorities so they can stop the game. I would do the second… After I had done the first a couple of times 😉 Just kidding.
But all jokes aside, even though I don’t expect to find any proven manipulation on national lotteries these days, it’s very important to start by analyzing if the game you will be playing is a fair game. Only then can you trust Combinatorics to help you create a good combination and pick the winning numbers.
When Strategy Gets Complicated, Let the Calculator Help
Speaking of combinatorics, we have been talking about it as if it were an easy process, a clear act of picking numbers from different sets.4 But it’s not that easy.
Even when splitting numbers into 4 sets, you still have many possible choices, and it’s easy to lose focus on what makes a good combination. It happens to most players, so if you are just starting to learn about strategy in the lottery game, or if you have been trying to do it on your own without much positive results, I would like to encourage you to try out the Lotterycodex Calculator. It’s an amazing piece of software, extremely easy to use, that guides you on the number selection process and gives a small number of possible winning lines that you can use within your own strategy.
I have run a few simulations on it already, and I was amazed at how precise it was, eliminating any low probability combinations and targeting a few winning scenarios even at a small number of draws. Even when I tried to make deliberately bad choices and force the software to make mistakes, it would correct and warn me.
FULL DISCLOSURE! I don’t receive any commission or fee from the Calculator sales; this is simply my unbiased opinion as a user of the software!
Like I said, winning the lottery is not easy, but if you are going to play, play it right, so you don’t spend your entire life searching for a victory that never comes.
That’s all for now. In Part 2, we will answer questions that players have been asking for years, we will start defining goals and strategies, and we will even take a small trip at… Monte Carlo?
Yes! I have some great stuff I want to share with everyone here. The next parts will have more mathematics involved, but I wanted to start with the basics, and I hope everyone had the opportunity to learn something new today.
Thanks for reading, and I hope to see you in the comments section. Constructive feedback is welcome!
Also, a special word for Portuguese players, I would love to hear from you, my own country, and I will keep sharing some interesting and useful analysis about both Totoloto and EuroMillions, so I hope to have you here next time!
A final word of gratitude for Edvin, he was the one who encouraged me to actually share some of my work here on the website. Thank you so much for the opportunity and encouragement!
Please read part 2: Using Statistics to Play the Lottery (Part 2): Understanding Sum Distributions and the Principle of Elimination
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References