No formula predicts the outcome of Euromillions draw. What mathematics offers is a clear picture of how combinatorial compositions are distributed across the game’s outcome space, and why some appear more often than others over thousands of draws.
From understanding the odds to examining how numbers behave in a random draw, this discussion applies combinatorial mathematics and probability theory to explain how lottery games actually work.
Table of Contents
The odds of winning the Euromillions
Euromillions is a 5/50 lottery. You pick five numbers from 1 to 50, which gives 2,118,760 possible combinations. But matching the jackpot also requires hitting two Lucky Stars from a pool of 12, so the actual odds stretch to about 1 in 139.8 million.1
A ticket that is never purchased has a probability of exactly zero.
The game offers 12 prize tiers beyond the jackpot, ranging from £2.50 to £130,554.30, with an overall chance of winning any prize at about 1 in 13.2
That means a single ticket fails to win any prize about 92.31% of the time. Lose twice in a row and the math looks like this:
P(losing twice) = 0.92312 tickets = 0.8520
Around nine tickets gets you close to a 50/50 shot at any prize. To push that to 99.99%, roughly 115 tickets are required. Even at that coverage, probability theory places the overwhelming majority of wins in the lowest prize tiers, not the jackpot.
P(winning any prize) = 1 – 0.9231115 tickets
No combination of hot numbers, lucky numbers, or quick picks changes the underlying probability. Mathematics is what brings clarity to the game’s structure.
How Euromillions works: what the math actually shows
All numbers have an equal chance of being drawn. All combinations share the same single-draw probability. The odds cannot be beaten and a random process cannot be manipulated. Those facts are settled.
So the real question becomes:
How does combinatorial mathematics describe Euromillions when numbers and combinations have an equal likelihood of winning in a truly random game?
The answer is not a secret formula. It is understanding how combinatorial compositions are distributed across the entire outcome space.
Buying more tickets increases coverage of the outcome space. Even though all combinations have equal single-draw probability, combinatorial compositions are not equally distributed across that outcome space. Their frequency ratios differ, and that difference is measurable over many draws.
In Lotterycodex, I use the term “frequency ratio” rather than “odds” to describe the long-run behavior of combinatorial compositions. The word “odds” is typically interpreted as odds of winning or losing, which is a different framing from what is being measured here. Frequency ratio describes how often a particular combinatorial composition appears relative to others across a large number of draws. It does not imply prediction or control over any single outcome.
A truly random lottery distributes probability evenly across the full number field. It does not favor even numbers over odd, or low over high. For example, the combination 10-12-14-16-18 belongs to a small combinatorial group. Small groups appear less often over time, not because they are unlucky, but because there are fewer of them in the total combination space.
What the numbers actually signal
Look at these combinations:
| 1-2-3-4-5 | 26-27-28-29-30 |
| 6-7-8-9-10 | 31-32-33-34-35 |
| 11-12-13-14-15 | 36-37-38-39-40 |
| 16-17-18-19-20 | 41-42-43-44-45 |
| 21-22-23-24-25 | 46-47-48-49-50 |
Most players will say that any combination can win because the draw does not care. Then, if asked whether they would spend money on those combinations, most would not.3
That hesitation points at something real in the math — something combinatorics can actually quantify.
There is a mathematical explanation for why part of you resists certain combinations.4
Combinatorial mathematics can quantify what that resistance is picking up on.
Why combinatorics and probability are the right tools
A lottery game has a finite structure. That structure gives us enough information to ask and answer probability questions without relying on historical data sampling.
When you ask “What is the likelihood of drawing 10-12-14-16-18?” you are really asking a probability question, not a statistical one. Rephrased:
“What is the probability of a purely low-even combination?”
The answer: 0.0003738035. This composition is expected to appear roughly four times in every 10,000 draws.
In Lotterycodex, one of the prevalent combinatorial groups in Euromillions has 0.0689157809 probability which translates that this group is likely to occur approximately 689 times in 10,000 draws based on the law of large numbers.
The difference between four and 689 occurrences over 10,000 draws is a property of how the combination space is structured, not a prediction of what any draw will produce.
Frequency ratio: how Euromillions outcome distribution works over time
The underlying probability of the game cannot be changed. But all possible outcomes can be calculated, and combinations can be examined according to which combinatorial groups they belong to. To do that clearly, odds and probability need to stay separate.5
They are not the same thing.
Probability measures the likelihood of a single event. Odds compare favorable outcomes to unfavorable ones. Put simply:
Probability = Chance
Odds = Ratio of favorable to unfavorable shots
The underlying probability cannot be controlled. Combinations can be examined by their frequency ratios, which describe how often different combinatorial groups appear across the outcome space.
Comparing combinations by frequency ratio
At Lotterycodex, odds between combinatorial groups are described using the term frequency ratio rather than “odds in favor.” The reason is practical: “odds in favor” sounds like winning vs. losing, which is not what is being measured here. Frequency ratio describes the relative long-run prevalence of a combinatorial composition across all possible outcomes.
Frequency ratio describes how often a combinatorial composition appears relative to all others over many draws.
Here is the math for two Euromillions compositions:
There are 53,130 ways to combine five even numbers:
Frequency Ratio (5-even) = 53,130/2,065,630 = 1:39
There are 690,000 ways to combine 3-odd and 2-even numbers:
Frequency Ratio (3-odd-2-even) = 690,000/1,428,760 = 1:2
A 0-odd-5-even combination belongs to a group with roughly three favorable occurrences across 100 draws. A 3-odd-2-even combination belongs to a group with about 33 favorable occurrences across those same 100 draws.
A player who consistently picks combinations from low-prevalence frequency ratio groups is selecting from a part of the combination space that, by the math, accounts for fewer occurrences over many draws.
The underlying probability cannot be changed and the lottery’s odds cannot be beaten. Combinatorial mathematics describes how the outcome space is distributed, which is a different kind of information than prediction or forecasting.
How Euromillions draws behave under the law of large numbers
To examine how theoretical probability compares to actual outcomes, I analyzed 1,831 historical Euromillions draws from April 16, 2004 to June 27, 2025, measuring each combinatorial composition’s observed frequency against its expected frequency under probability theory.
Expected frequency for any combinatorial composition is calculated by multiplying the probability by 1,831:
Expected Frequency = Probability × 1,831
For the 3-odd-2-even composition with a probability of 0.3256621797655230:
Expected Frequency (3-odd-2-even) = 0.32566 × 1,831 = 596
Doing the same calculation for each odd-even group produces the full comparison below:
The expected frequencies and actual draw results line up closely. That is the law of large numbers at work, governed by combinatorics6 and probability theory.7
Historical results are not required to perform these calculations. The math lives in the structure of the game itself.
How to choose Euromillions numbers using combinatorial composition
Odd-even analysis alone is not enough. Consider 1-2-3-4-5: it has a balanced 3-odd-2-even mix, but every number is low. That is a problem.
A truly random draw distributes probability across the entire number field. Analyzing only one dimension of a combination gives an incomplete picture. Odd-even and low-high must be combined into a single analysis for the result to be mathematically meaningful.
I developed the Lotterycodex sets in 2017 as a way to study how different combinatorial compositions are distributed under probability theory, not a method for predicting or controlling outcomes. The sets examine the number field across four dimensions simultaneously: LOW-ODD, LOW-EVEN, HIGH-ODD, and HIGH-EVEN. Each lottery format gets its own partition based on the game’s number field.
This is what the Lotterycodex sets accomplish for Euromillions:
From these four sets, all possible combinatorial compositions are extracted and organized into templates. A small number of those templates account for the largest share of occurrences over many draws.
Templates #1, #2, #3, and #4 are the prevalent groups. Templates #1 and #2 carry the highest frequency ratios. Based on combinatorial mathematics, these two groups account for more of the total outcome space than any others in the 5/50 game.
How to use the Lotterycodex calculator for Euromillions
Buying multiple tickets is the only way to increase coverage of the outcome space. A lottery wheel organizes chosen numbers into tickets with less overlap, meaning more of the budget goes toward distinct combinations rather than redundant ones.
The Lotterycodex calculator functions as an advanced wheeling system that classifies combinations by frequency ratio. Templates #1 and #2 carry the highest long-run prevalence based on combinatorial mathematics. Under the law of large numbers, those two templates are expected to keep dominating as more draws occur.8
Template #55 is expected to appear about twice in every 5,000 draws. Combinations in that group belong to a compositional category with very low long-run prevalence in the 5/50 outcome space.
Mathematical analysis does not change the probability of winning any single draw. It does not make one specific combination more likely than another. What it describes is how the game distributes outcomes across all possible combinations over time, which is a different kind of information than prediction or forecasting.
The Lotterycodex calculator handles the combinatorial calculations and classifies combinations by template and frequency ratio.
The lottery is entertainment. Resources on lottery addiction are available for anyone who finds participation becoming more than recreational. Information on responsible play is available through this page and this one.
Share your thoughts on how to win euromillions
Have thoughts on the math behind Euromillions? Questions about combinatorial compositions or frequency ratios? Join the conversation below.
Thank you for reading 🙂
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Frequently Asked Questions
There is no guaranteed way to win Euromillions. The only theoretical method that would guarantee a jackpot is purchasing every possible combination — all 139.8 million of them when the Lucky Stars are factored in. That is neither practical nor economical. Playing in a group, using a lottery wheel, or examining combinations by combinatorial composition does not change the fact that outcomes are random and no win is assured.
No selection method can predict or influence the outcome of a random draw. Combinatorial mathematics describes which compositions are more prevalent across the total outcome space. Frequency ratios, derived from combinatorics and probability theory, show which groups of combinations account for more of the outcome space and appear more often over many draws under the law of large numbers. This is a descriptive measure, not a prediction tool, and it does not change the probability of any single draw.
Euromillions, like all lottery games, is built with a negative expected value. The total prize pool paid out is less than total ticket revenue. That gap does not close with different number selection or more tickets. Lottery participation is entertainment, not a financial instrument or income source.
Hot and cold numbers have no mathematical basis in a random lottery. Each draw is independent — what came up last week has no bearing on what comes up this week. The Law of Large Numbers describes that, over enough draws, each number’s frequency tends toward its theoretical probability. Chasing streaks or avoiding numbers based on recent draw history is a form of the gambler’s fallacy.
A frequency ratio describes how often a particular combinatorial composition appears relative to all others across a large number of draws. It is not a prediction tool and does not indicate what the next draw will produce. What it shows is how different groups of combinations are distributed across the total outcome space. A composition with a frequency ratio of 1:2 occupies a much larger share of that space than one with a ratio of 1:39 — and under the law of large numbers, larger groups appear more often over time. This is a descriptive, long-run statistical measure derived from combinatorial mathematics.
Each draw is an independent random event. Skipping one has no effect on any future draw. Missing a draw does not make the next one statistically overdue in any way. That belief is the gambler’s fallacy. Each draw begins fresh, with the same probability distribution as every other.
References
Clever but its still random
I agree. The lottery is truly random and you should be thankful that it is. Because of its randomness, any calculation you make can be precise and accurate within the perspective of the law of large numbers. If you want solid proof, read this: https://lotterycodex.com/truly-random-lottery/
Can we make an app for calculate the 3odd 2 even? That it just gives the user only 3odd 2 even conbinations?
We have a better calculator for what you need but we handle not only 3-odd-2-even but also low and high numbers altogether in one combinatorial analysis. Please read https://lotterycodex.com/lottery-formula/
Really helpful and explained in a simple way .very interesting way it was put to the reader .
Useful and well explained.
I would like to join a syndicate to play euromillions and improve the probabilities of winning.
Thank you
Why is it when I do the euro million,which ever number I pick Camelot pick the next number ie no 7 camelot picks number 8 if I pick 3 Camelot picks 4.this is definitely a fixed game.
It’s a fair game. Balls are drawn randomly. You cannot expect the game to favor you to the detriment of others.
Hahaha same as me.runs around my numbers
My own concern is “ Why is it that it’s the older people that mostly win the lottery
2) Why must it be Up North always or far outside London
3) Have seen people who have been playing for 10 years and more that has never win nothing
4) is lottery a scam or is it real ?
I too belive it is a scam. We checked online and found most winners were white and not many people of colour who also play the lottery regularly and in the Uk its mainly people up north and not Londoners. Do the research yourself if you dont believe me.
I live up north in between Sunderland and Newcastle and I’ve never won a bean
Hi OLA and Barry; all these lottery things you observed are dictated by probability. Most winners are old people because most lotto players are older people. The same holds that if more white people are playing, then by the law of large numbers, more white players win. It is that simple.
Thank you for a most fantastic read, I was very impressed by your mathematics and equations.
I used to pick the most overdue numbers , but never won .but reading your article I am now going to try the three odd two even number system.
I do also try the hotpick (select three numbers ) to win.
Does playing hot pick where you select few number make winning harder ? I would be fascinated to know.
Very interesting, for all the years I played, I have been a total failure, so opted for lucky dips no joy there either.
Do these probability theory and combinatorics patterns also apply in 6-digit game? Here in the Philippines, we have 5 6-digit lottery games namely 6/42, 6/45, 6/49, 6/55 and 6/58. Over two months ago, a 6/55 lotto draw had 433 winners with a jackpot of more than 200 million pesos (more than 4 million dollars) which to many like myself find it unbelievable or doubtful but turned out to be true or legit. It was a multiple of 9. The winning number was 9, 18, 24, 30, 36 and 42. It begun exactly with the lucky 9 number. How was that probable or indeed superstitious? Thanks.
433 winners hitting the same winning numbers seem unbelievable and questionable, but mathematically speaking, it’s possible. In math, we call this the law truly large numbers. I have an article where I explain why coincidences happen, read this: https://lotterycodex.com/lottery-superstitions/
There are millions of people in the uk and all the other countries that are in the euro why don’t they make the prizes smaller it would be better with prizes of £500,000 if you can’t survive on that you must stupid or greedy most people who win many millions end up getting divorced or unhappy or on drug’s because they don’t know how to handle large amounts of money £500,000 is enough for anyone in the world to survive on and why not put the price of a ticket down to £1 it’s just a cash cow thanks
What calculator should I select for the euromillions? thaks
5/50 calculator.
This is very interesting, Edvin. I always realised intuitively that it must be right to have your 5 numbers spread evenly across the low & high number sets, and across odds & evens. I’m not sure why this is, but I can definitely see that it makes sense in a gut-wise way. That’s why we never see combinations like 2,4,6,8,10, or 27, 29, 31, 33, 35.
I’m a UK citizen and can play Euromillions quite easily but if I was to be abroad in Europe I cannot play the Euromillions as for some reason the rules layed down by the National lottery say I can’t because I don’t live in that European country I am in , yet Europeans can freely play it.
Is that fair ?
Also the prize money increases on a rollover but when you look at the amount of players both in the UK and Europe it doesn’t add up and somebody is making a vast amount of money.
I don’t know why it is but I can tell my wife that it is going to rollover and I can also tell her when it’s going to be won and also the winners are mainly European people and not the UK.
It’s the same with lotto , prize money for winning the jackpot on a Saturday night if you are a single winner is about 3.8 million pounds.
If this rolls over to the following Wednesday the jackpot will pay to a single winner 4 million pounds , that’s an increase of 200k yet if it didn’t roll over to the Wednesday the prize money for a single winner is 2 million pounds , so where has the 1.8 million pounds disappeared to.
It seems like the whole country is being conned.
I play the lottery every week picking random numbers but i have never won an significant amount usually less than a fiver at very long intervals , i have tried different ways of picking numbers even going back to the fist draw ever on euro millions and trying to work out possibilities. So now i will try your lottery codex for euromillions and also work out my own from the information you have given me thank you Les.
If lottery is completely random why haven’t 10 or 50 or 100 or a thousand people shared the jackpot in all the years it’s been running?
The odds of winning the EuroMillions jackpot are approximately 1 in 139.8 million. Every player has the same probability of winning. Because lottery draws are truly random, it is possible to have multiple winners or no winner at all. As the number of players increases, the likelihood of the jackpot being shared also increases. While scenarios where thousands of players share the jackpot are unlikely, they are still statistically possible.