How to Win the Lottery Mathematically

how to win the lottery according to Math

I can almost guarantee 99% of lotto players always pick the wrong combination each draw. And you are probably one of them.

In the lottery, millions of number combinations will never appear in any draw no matter how many times you play them. For example, the 1-2-3-4-5 or 1-2-3-4-5-6 has never appeared in the history of the lottery.

Here are some combinations almost identical to 1-2-3-4-5-6:

  • 2-3-4-5-6-7
  • 3-4-5-6-7-8
  • 4-5-6-7-8-9
  • 1-3-5-7-9
  • 2-4-6-8-9
  • 3-4-6-7-8

And here are some combinations of similar patterns:

  • 12-13-14-17-18-19
  • 20-23-25-27-28-29
  • 33-35-36-37-38-39
  • 40-42-45-48-49

The odds of these number patterns such as the ones listed above are very low that they will not occur in the lottery, never.

Surely, not in your lifetime or your grandsons’ or granddaughters’ lifetime.

Let me tell you the bad news, the lottery has millions of bad number combinations and if you play the lottery, you may have been wasting your money on these bad numbers your whole life.

Why? The best way to explain this is through Mathematical reasoning.

The 1-2-3-4-5-6 and those identical combinations belong to a number pattern that has the least likelihood of appearing.

See the likelihood of their occurrences in 6 ball lottery games:

LotteryPossible Occurrence of 1-2-3-4-5-6 or those with identical number pattern
UK Lotto2 times in 1,000,000 draws
Irish Lottery7x in 1,000,000 draws
Tattslottoonce in 100,000 draws
Any 6/49 lottery game6 times in 1,000,000 draws

And here is the likelihood of them occurring in 5 ball lottery games:

LotteryPossible Occurrence of 1-2-3-4-5 or those with identical number pattern
US Powerballonce in 100,000 draws
Euromillions5 times in 100,000 draws
US Mega Millions7x in 1,000,000 draws
Eurojackpot5 times in 100,000 draws

The tables show that the likelihood of the 1-2-3-4-5 or the 1-2-3-4-5-6 appearing in a lottery draw is almost “impossible”.

How do I know? I use a Mathematical tool called the Probability Formula.

And this is how Science meets the Lottery.

In fact using the same Mathematical tool, you can learn to play the lottery with better odds of winning.

However, the majority of lottery players do not know how to play the lottery.

People keep on playing the wrong pattern.

Below is a graphic representation of why the majority of players do not win the lottery.

A graph showing why majority of lotto players do not win by playing number combinations that never appear in the lottery.

From this point on, let me debunk the popular belief that each number combination has an equal probability of getting drawn.

It’s not true, and I will prove it using the actual results of the lottery.

Firstly, let me start this discussion with a simple example of 9 black marbles and 40 white marbles.

If you blindfold someone and ask to pick a marble, what color do you think will get picked on the first try?

It doesn’t require someone to be more mathematically inclined to know that white is the answer.

It is because the probability is more likely leaning towards the white marbles.

However, we deserve a better explanation than just mere assumption.

Let’s use Mathematics.


In short, you pick black marbles 18.37% of the time while you pick white 81.63% of the time.

In layman’s term:

You get approximately 18 black marbles in every 100 draws

You get approximately 82 white marbles in every 100 draws

Using this simple black and white marbles, you will see how Probability principle works in Mathematics.

Of course, this is far from how the real lottery works.

So let’s take this marble example up a notch.

What are the odds of picking 6 black marbles?
What are the odds of picking 6 white marbles?

Now, the problem gets a little bit complex but the concept is still the same.

Let’s group together black and white marbles into the following sets:

Black marbles = {1,2,3,4,5,6,7,8,9}

White marbles = {10,11,12,13,…,49}

We can come up with many different possibilities:

PatternSample combination
6 Black1-2-3-4-5-6
5 Black + 1 white1-2-4-5-9-38
4 Black + 2 White2-3-7-8-26-42
3 Black + 3 White1-3-9-11-20-48
2 Black + 4 White4-8-14-31-43
1 Black + 5 White6-12-23-39-40-44
6 White10-12-23-39-40-49

Our next objective is to determine the number of favorable combinations for each pattern.

Such objective can be performed using Binomial Coefficient:

nCr = n! / r!(n-r)!

Therefore, we can complete the following table below:

PatternFavorable Combinations
6 black marbles84
5 black marbles + 1 white5,040
4 black marbles + 2 white98,280
3 black marbles + 3 white marbles829,920
2 black marbles + 4 white marbles3,290,040
1 black marbles + 5 white marbles5,922,072
6 white marbles3,838,380

Now we have enough data to compute the likelihood of an event to happen.

In our case, we are looking for the likelihood of 6 black marbles to appear versus that of the 6 white marbles.

We do this using the Probability Formula.

P(A) = Number of favorable outcomes / total number of possible outcomes


P(6 black marbles) = 84 / 13,983,816

= 0.00000600694

Or 0.000600694%

P(6 white marbles) = 3,838,380 / 13,983,816

= 0.27448730732

Or 27.448730732%

So now, we can answer our previous questions:

What is the probability of picking 6 black marbles?0.000600694%
What is the probability of picking 6 white marbles?27.45%

Or in Layman’s term:

6 black marbles6 times in every 1,000,000 draws
6 white marbles27 times in every 100 draws

From the table above, we can see 1-2-3-4-5-6 which represents one of the 6 black marbles will not likely happen during a draw. In short, almost “impossible”.

Now, here is the question:

Isn’t this the same reason 1,2,3,4,5,6 doesn’t appear yet in the history of the lottery?

It is.

Too sad, according to the report by the National Lottery Camelot, about 10,000 people play this combo every week.

Mathematically, this is just – “waste of money”

But contrary to popular belief, 1-2-3-4-5-6 is not the worst combination in the Lottery.

In a 6/45 game, the worst is 40-41-42-43-44-45

However, whether it’s the worst or just one of the bad combinations, the idea remains, you should never play them. Plain and simple.

Below are some examples of bad number combinations in the lottery:


I can list millions of these and we will run out of space. If you have been playing the lottery, you could have been falling into one of these bad combos and their similar variations.

You get the gist, but we will get to that later.

Let’s talk about Probability analysis as we apply it in a real-world lottery.

Odd and Even Number Combinations in Lottery

To apply Probability in a real-world lottery game, first, we need to know the rules of the game.

For example, in U.S. Powerball, you pick five numbers from 1 to 69.

In UK Lotto, you pick six numbers from 1 to 59.

In some lottery systems, the game requires you to pick six numbers from 1 to 49.

Probability analysis differs from each lottery format.

For the sake of discussion, I will use Euromillions here and then I will use the Actual lottery results to prove my Probability computation. Then, progressively, we will go ahead with other popular lotteries.

In Euromillions, you pick five numbers from 1 to 50.

With this lotto format, we come up with the following number sets:

Odd numbers = {1,3,5,7,9,11,…,49}

Even numbers = {2,4,6,8,10,…,50}

In Euromillions, there are 25 odd and 25 even numbers.

Below are examples of number combinations using the odd and even pattern:

PatternSample Combination
2 odd + 3 even4-12-5-24-17
3 odd + 2 even9-22-31-44-49
all odd numbers9-13-21-43-47
all even numbers4-14-22-36-50

The following table will show you the complete list of possible patterns and the corresponding probability:

3 odd + 2 even0.3256621797655230
3 even + 2 odd0.3256621797655230
1 odd + 4 even0.1492618323925310
1 even + 4 odd0.1492618323925310
5 odd0.0250759878419453
5 even0.0250759878419453

Now with this table, we now have a powerful tool to come up with a possible way to predict how Euromillions behave over time.

In Euromillions actual draws from April 16, 2004, to February 28, 2017, 970 total draws already took place.

Using the probability data, we can determine the estimated frequency for each of the Odd and Even number pattern with the following formula below:

Estimated frequency = 970 X Probability

In the case of 3 odd and 2 even, we multiply 0.3256621797655230 by 970 draws.

Therefore, we get:

Estimated frequency (3 odd and 2 even) = 315.892314373 or 316

So let’s recreate the complete table and this time, I include an additional column for estimated frequency.

PatternsProbabilityEstimated Frequency
3 odd + 2 even0.3256621797655230316
3 even + 2 odd0.3256621797655230316
1 odd + 4 even0.1492618323925310145
1 even + 4 odd0.1492618323925310145
5 odd0.025075987841945324
5 even0.025075987841945324

Now, right before your own eyes, I’ll prove to you that our estimation matches extremely close to the actual data.

The Actual Euromillions 5/50 Results
From April 16, 2004 to February 28, 2017

PatternsEstimated Frequency in 970 DrawsActual Frequency in 970 Draws
3 odd + 2 even316354
3 even + 2 odd316291
1 odd + 4 even145140
1 even + 4 odd145140
5 even2425
5 odd2420

Probability estimation compared to actual Euromillions 970 draws

Looking at the probability analysis table above, we can come up with some conclusions, but perhaps the most important takeaway is this:

Playing with the wrong pattern is just a “waste of money”

From the words of the late Gail Howard:

That which is MOST POSSIBLE happens MOST OFTEN.
That which is LEAST POSSIBLE happens LEAST OFTEN.

But hang on, how many Euromillions players are mindful of the number patterns when they pick numbers?

Very few.

Let’s take a look at my probability study with other popular lotteries:

The Actual TattsLotto 6/45 Lottery Results
Total of 582 draws from January 7, 2006 to March 04, 2017

PatternEstimated Frequency in 582 DrawsActual Frequency in 582 Draws
3 odd and 3 even194.88187
4 odd and 2 even146.16154
2 odd and 4 even132.24143
5 odd and 1 even52.9050
1 odd and 5 even43.2839
6 odd and 0 even7.215
6 even and 0 odd5.334

Probability estimation compared to actual results of the TattsLotto 6/47

The Actual EuroJackpot 5/50 Results
Total of 258 draws from March 23, 2012 to March 03, 2017

PatternEstimated Frequency in 258 DrawsActual Frequency in 258 Draws
3 odd, 2 even8497
2 odd, 3 even8465
4 odd, 1 even3947
1 odd, 4 even3936
0 odd, 5 even611
5 odd, 0 even62

Probability estimation compared to actual lottery results of the Eurojackpot 5/50

The Actual Irish Lottery 6/47 Results
Total of 155 draws from September 5, 2015 to March 01, 2017

PatternsEstimated frequency in 155 drawsActual frequency in 155 draws
3 odd and 3 even51.743361372353
4 odd and 2 even38.807521029242
2 odd and 4 even35.27956457237
1 odd and 5 even11.657595249910
5 odd and 1 even14.11182582889
6 odd and 0 even1.942932541651
6 even and 0 odd1.457199406233

Probability estimation compared to actual Irish Lottery 6-47 results

The Actual U.S. Mega Millions 5/75 Results
Total of 353 draws from October 22, 2013 to March 07, 2017

PatternsEstimated frequency in 353 drawsActual frequency in 353 draws
3 odd and 2 even115128
2 odd and 3 even11298
4 odd and 1 even5658
1 odd and 4 even5153
5 odd and 0 even108
0 odd and 5 even98

Probability estimation compared to actual U.S. Megamillions 5-75 results

The Actual U.S. Powerball 5/69 Results
A total of 146 Draws from October 7, 2015 to March 04, 2017.

PatternsEstimated frequency in 146 drawsActual frequency in 146 draws
3 odd and 2 even4842
2 odd and 3 even4643
4 odd and 1 even2327
1 odd and 4 even2130
5 odd and 0 even42
0 odd and 5 even42

Probability estimation compared to actual results of the U.S. Powerball

The Actual UK Lotto 6/59 Results
Total of 147 Draws from October 10, 2015 to March 08, 2017

PatternsEstimated frequency in 147 drawsActual frequency in 147 draws
3 odd and 3 even4844
4 odd and 2 even3642
2 odd and 4 even3436
5 odd and 1 even1310
1 odd and 5 even1210
6 odd and 0 even24
6 even and 0 odd21

Probability estimation compared to the actual results of the UK lotto 6/59

See the complete list of my lottery analysis

The Lottery can be Predicted to an Extent

As the evidence unfolds from our study of the odd and even numbers in the Lottery, we can say that Lottery behaves in a predictable pattern.

Therefore, the lottery could be predicted to an extent, according to Math.

But the lottery is not all about odd and even numbers.

There is more to the lottery than meets the eye. If you study the winning numbers of the lottery very deeply, you will discover winning patterns that could be the key to your lottery success.

Now, let’s talk about a better strategy.

The Geometry of Chance and How It Works To Improve Your Chances of Winning the Lottery

One school of thought tells us that a surefire way to win the lottery is to buy all the tickets. Right?

Correct. However, it’s not practically feasible, and no one with the right mind will do it.

If you remember, at the beginning of the article I made mention of number combinations that are destined not to appear in any draw.

Using these numbers to play the lottery is a waste of money.

So the way to increase your chances of winning a big jackpot is to play the number pattern that carries the best probability.

So the question now is how we know the best and the worst number patterns in the lottery.

We use the method proposed by Renato Gianella from his study called The Geometry of Chance. I am going to talk about the Geometry Of Chance in a separate article.

In this article, I’ll show you how I use The Geometry of Chance to help lottery players pick the best numbers so that your chances are more leaning on the winning side.

For example, there are 2 million playable number combinations in Euromillions 5/50. Using Gianella’s method, we can reduce this number into 196 manageable templates.

From these templates, I use Probability formula to determine the best, the fair and the worst number patterns to play.

This is how I divide the patterns in Euromillions 5/50.

The Best PatternsThe Fair PatternsThe Worst Patterns
Pattern #1Pattern #2,

Pattern #3,

… up to Pattern #86

Pattern #87

Pattern #88

… up to pattern #196

There are 196 patterns in Euromillions and only one is the best.

In Mathematics, once we get the probability we have the power to predict how likely an event will perform over time.

So let’s pick some of these patterns from The Euromillions 5/50.


I know the numbers sound dull. Doesn’t it?

Let me convert those probability numbers in layman’s term.

#14x in every 100 draws
#202x in every 100 draws
#892x in every 1,000 draws
#1192x in every 1,000 draws
#1965x in every 100,000 draws

So obviously it is pattern #1 that stands out.

If you always pick numbers based on patterns #89,#119, and #196, you will never win the lottery. Guaranteed!

Let me prove it.

Since we can measure the Probability of each pattern, we can use this to determine the expected frequency of any pattern over a period.

I will use the formula below:

Expected Frequency = Probability X The number of draws

This time, let’s take things up a notch. Let’s use the Actual Euromillions Results to compare our computations with the real draws.

983 draws already took place from April 16, 2004, to April 14, 2017.


PatternApproximate Expected Occurrence in 983 Draws
#142 times
#2017 times
#893 times

And to prove that Math works, here is the table to show the comparison:

Euromillions 5/50 Results
From April 16, 2004 to April 14, 2017
Total Draws: 983 draws

PatternExpected Frequency in 983 DrawsActual Frequency in 983 Draws
Pattern #14242
Pattern #201716
Pattern #8932
Pattern #11922
Pattern #19600

Expected frequency versus actual results of the Euromillions 5/50

Once again, Mathematics is a useful tool to measure how the Euromillions behaves.

The same Probability principle applies in all the lottery systems in the world.

My Fearless Predictions of the Lottery

Like I have told you earlier, we can predict the lottery to an extent.

The following formula is what we use:

Estimated Frequency = Probability X The number of draws

Therefore, for Euromillions 5/50, if we want to predict how many times pattern #1 and pattern #196 will appear in 2000 draws, this is what we get:

Estimated Frequency (pattern #1)

= 0.0424776756 x 2000

= 84.9553512  (or approximately 85 times in 2000 draws)

Estimated Frequency (pattern #196)

= 0.0000594687 x 2000

= 0.1189374 (or no appearance at all)

Notice the huge difference between the two patterns.

My recommendations for all players of Euromillions 5/50

  • Focus on Pattern #1.
  • Avoid the rest of the patterns.

Below are my recommendations for other lotteries:

LotteryRecommended PatternsPatterns to Avoid
U.S. Powerball 5/69Patterns #1,#2,#3,#4,#5,#6Patterns #7 to #462
Australian TattsLotto 6/45Patterns #1,#2,#3Patterns #4 to #210
Irish Lotto 6/47Patterns #1,#2,#3Patterns #4 to #210
U.S. Mega Millions 5/75Patterns #1,#2,#3,#4,#5,#6Patterns #7 to #792
EuroJackpot 5/50Pattern #1Pattern #2 to #196
UK Lotto 6/59Pattern #1Patterns #2 to 462

See the complete list of my lottery analysis

Conclusion: How to Win the Lottery

True, it’s not easy to win the lottery.

I’ll be honest with you, no one can reverse engineer the random nature of the lottery.  Surely, I cannot.

But you can play better. With the right strategy in place, you can increase your chances of winning the lottery.

That’s where Mathematics comes to help.

Here are my recommendations:

  1. Avoid the bad odd-even patterns
  2. Focus on number patterns that have the best probability. (see the complete list of my probability analysis)