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The beauty of math is that you can literally predict the outcome of random events and then later you can prove that you are right. Because undeniably so, the numbers don't lie.
So having said that, let's dive deep into the theoretical computations and explain from a mathematical point of view – why lottery results behave in a particular trend. Then we will compare the theoretical computations with actual lottery game results to prove our conclusion.
The Basics of Probability
The traditional school of thought tells us that each lottery combination has the same chances of getting picked in a draw. For example, in a 6/45 lottery format, they say that each 6 number combination has 1 in 8 million chances of getting picked.
This concept is quite flimsy, as it does not explain why 1,2,3,4,5,6 or 40,41,42,43,44,45 has yet to appear in the history of lottery draws.
So if that sort of thinking is questionable, then what is the probability of winning the lottery?
To understand the mathematical concept involved in playing the lottery, let me start this journey with a simple discussion of marbles.
You see from the image above, the white marbles outnumber the five black marbles. If you were to place a wager for $100, what color would you put your money on? Anyone with a sense of simple arithmetic can tell that when you blindly pick a marble from the bunch illustrated above, the likelihood of getting a white marble is extremely high.
In mathematics, there are 2 methods of measuring probability. One deals with independent events and the other one deals with dependent events. We will not discuss these 2 methods comprehensively.
For now, all you need to know is that if we replace the marble in the bag each time we pick, then the probability doesn't change and each event is independent.
So therefore, getting a white marble on the second pick is likely to occur because the probability leans towards the white marble getting picked more often.
On the third pick, you are still more likely to draw a white marble for the same reason.
In mathematics, probability is the likelihood of one or more events happening divided by the number of possible outcomes.
Therefore:
In the case of the black and white marbles above, the chances of drawing a white marble are 45 in 50 whereas the chances of getting a black marble are 5 in 50.
Expressed in percentage, we get:
P(white) = ( 45 / 50 ) x 100 = 90% chances of getting drawn
P(black) = (5 / 50) x 100 = 10% chances of getting drawn
In layman's term, we can say that the white marbles will get picked 90x every 100 draws, and the black marbles will only get picked 10x every 100 draws.
If we were to hold 10 draws, then the probability will be:
P(white) = 9x every 10 draws
P(black) = once every 10 draws
Now, you might ask, what do these marbles have to do with the lottery? Absolutely nothing. But the probabilities have everything to do with it. It's how you can use math to win the lottery.
Let's move on and see how this probability theory can be used to analyze the lottery.
The Best Mix of Odd and Even Lotto Combinations
Which of these combinations do you think have better chances of winning the lottery?
(a) 1-12-20-31-39-42
(b) 3-12-22-35-36-38
(c) 4-15-27-36-39-43
(d) 2-14-24-36-38-42
(e) 5-13-25-37-41-45
For now, you can simply pick an answer and later, we will find out if you are correct. Let's do it just for fun.
It's extremely important that you pick the combination that strongly matches how you would pick a combination in real life. Why? Simple, the combination you pick will answer the question 'what are the chances of winning the lottery?'
If you picked your numbers already, then let's proceed with the probability analysis the same way we analyze the marbles above.
For the sake of illustration, I will use the 6/45 lottery format in this section.
First of all, we need to know how many different combinations of 6 numbers can be generated from 45 numbers.
We use binomial coefficient formula below:
For everyone's sake, I will not explain the formula in detail. As much as possible, I will try to explain everything more simply and to make my point much quicker, I will simply provide the answer.
n = 45 numbers
r = 6 combinations
45C6 = 8,145,060
So in a 6/45 lottery game, there are a total of 8,145,060 possible combinations to play.
Computing for possible combinations of 3 odds and 3 evens, we'll get:
= 2,727,340
We can therefore analyze the probability of this pattern as follows:
P(3 odd & 3 even) = 0.33484590659
If we follow the same formula with the rest of the patterns, we will be able to arrive at the completed table below:
Verdict: The 3 Even & 3 Odd number pattern has the best chance of getting drawn in a 6/45 lottery.
Going back to the question above, which of these combinations do you think have better chances of getting drawn?
(a) 1-12-20-31-39-42 – (the correct answer)
(b) 3-12-22-35-36-38
(c) 4-15-27-36-39-43
(d) 2-14-24-36-38-42
(e) 5-13-25-37-41-45
If you answered (a), then congratulations are in order because you have 33% more chances of becoming a millionaire in the future – you know how to pick winning lottery numbers.
At this point, you should understand now that what we are talking about here are 'patterns'.
(a) is the correct answer because it is the combination that belongs to the group of patterns that have 33% more chances of winning the lottery.
Now what are the chances of winning the lottery if you picked a combination that isn't (a)?
From a theoretical point of view, the (3 Odd & 3 Even) combinations will theoretically appear more often in a draw as per the same case as the marble concept.
The 3 Even and 3 Odd pattern has 33% more chances of getting picked in a draw, while the All Even number pattern has 0.91 or less than 1% chances in a draw.
In layman's term:
P(3 odd numbers and 3 even numbers)
= occurs approximately 33 times every 100 draws
P(All even numbers)
= occurs approximately once every 100 draws
It is now evident that the number of combinations controls a combination's probability of being drawn.
How to pick winning lottery numbers? The answer is to simply avoid number combinations belonging to a pattern with lower probability, as you would with the black marbles.
In the case of a real lottery game, you should avoid a pattern like All Even or All Odd numbers altogether.
But then again, this is just a theoretical analysis right?
As I have mentioned earlier, the beauty of math is that you can prove it.
So let's prove it and let's use the actual results from actual lottery draws.
The Proof – Theoretical Versus Empirical
In the following tables, I will be comparing Expected frequencies with Observed frequencies. The Expected frequency is the theoretical frequency of each pattern drawn from calculating probabilities multiplied by total number of draws, while the Observed frequency is the actual frequency drawn from actual lottery results a.k.a. 'experimental' or 'empirical probability'.
Therefore:
Expected Frequency = (Probability) X (Number of draws)
Ethiopian airlines reviews. Observed Frequency = The number of occurrences in the actual lottery draws
To prove that our calculation is correct:
Expected Frequency should closely match the Observed Frequency given a sufficiently large sample of draws
Proof From Actual Irish Lottery
922 Draws of Irish Lottery from Nov. 4, 2006 – Sept. 2, 2015
Results taken from Irishlotto.net
Proof From Actual Australian Lottery
2,158 Draws of Australian Lottery from June 2, 1990 to May 28, 2016
Results taken from Osalottos.com
The two aforementioned 6/45 lotteries make it evident that the Law of Large Numbers is at work. The proximity of the expected frequency and the observed frequency proves this.
Accordingly, the results coincide with my previous theoretical computations – proof that applying probability in the lottery can predict how it behaves or how it will behave in the future.
Now that you've breathed all of that in, let me ask you again, what is the probability of winning the lottery? More specifically, what is your probability of winning the lottery?
The answer, if you digested all of that information correctly, will depend on which lottery pattern you will use to play with. If you play with a pattern that has more chances of getting drawn, just like the 3 odd and 3 even pattern, then your probability of winning the lottery is high.
Previously, I mentioned the Law of Large Numbers. But what is it anyway and what does it have to do with the lottery?
The Law Of Large Numbers At Work
The Law of Large Numbers states that when an event is repeated a large number of times, the outcome is always close to the expected value and will tend to become closer as more events are performed to infinity. The events here are the actual lottery draws, and the expected value is the expected frequency after we multiply the probability with the actual number of draws.
The previous tables above states clearly how the Law of Large Numbers intersects with the lottery. As you can see, the patterns we 'expected' to appear more frequently, by probability experimentation and testing, were drawn, in reality, more than other patterns. With this, we can say that the lottery can be represented by Probability Theory and the Law of Large Numbers.
Let's move on and see how this probability theory can be used to analyze the lottery.
The Best Mix of Odd and Even Lotto Combinations
Which of these combinations do you think have better chances of winning the lottery?
(a) 1-12-20-31-39-42
(b) 3-12-22-35-36-38
(c) 4-15-27-36-39-43
(d) 2-14-24-36-38-42
(e) 5-13-25-37-41-45
For now, you can simply pick an answer and later, we will find out if you are correct. Let's do it just for fun.
It's extremely important that you pick the combination that strongly matches how you would pick a combination in real life. Why? Simple, the combination you pick will answer the question 'what are the chances of winning the lottery?'
If you picked your numbers already, then let's proceed with the probability analysis the same way we analyze the marbles above.
For the sake of illustration, I will use the 6/45 lottery format in this section.
First of all, we need to know how many different combinations of 6 numbers can be generated from 45 numbers.
We use binomial coefficient formula below:
For everyone's sake, I will not explain the formula in detail. As much as possible, I will try to explain everything more simply and to make my point much quicker, I will simply provide the answer.
n = 45 numbers
r = 6 combinations
45C6 = 8,145,060
So in a 6/45 lottery game, there are a total of 8,145,060 possible combinations to play.
Computing for possible combinations of 3 odds and 3 evens, we'll get:
= 2,727,340
We can therefore analyze the probability of this pattern as follows:
P(3 odd & 3 even) = 0.33484590659
If we follow the same formula with the rest of the patterns, we will be able to arrive at the completed table below:
Verdict: The 3 Even & 3 Odd number pattern has the best chance of getting drawn in a 6/45 lottery.
Going back to the question above, which of these combinations do you think have better chances of getting drawn?
(a) 1-12-20-31-39-42 – (the correct answer)
(b) 3-12-22-35-36-38
(c) 4-15-27-36-39-43
(d) 2-14-24-36-38-42
(e) 5-13-25-37-41-45
If you answered (a), then congratulations are in order because you have 33% more chances of becoming a millionaire in the future – you know how to pick winning lottery numbers.
At this point, you should understand now that what we are talking about here are 'patterns'.
(a) is the correct answer because it is the combination that belongs to the group of patterns that have 33% more chances of winning the lottery.
Now what are the chances of winning the lottery if you picked a combination that isn't (a)?
From a theoretical point of view, the (3 Odd & 3 Even) combinations will theoretically appear more often in a draw as per the same case as the marble concept.
The 3 Even and 3 Odd pattern has 33% more chances of getting picked in a draw, while the All Even number pattern has 0.91 or less than 1% chances in a draw.
In layman's term:
P(3 odd numbers and 3 even numbers)
= occurs approximately 33 times every 100 draws
P(All even numbers)
= occurs approximately once every 100 draws
It is now evident that the number of combinations controls a combination's probability of being drawn.
How to pick winning lottery numbers? The answer is to simply avoid number combinations belonging to a pattern with lower probability, as you would with the black marbles.
In the case of a real lottery game, you should avoid a pattern like All Even or All Odd numbers altogether.
But then again, this is just a theoretical analysis right?
As I have mentioned earlier, the beauty of math is that you can prove it.
So let's prove it and let's use the actual results from actual lottery draws.
The Proof – Theoretical Versus Empirical
In the following tables, I will be comparing Expected frequencies with Observed frequencies. The Expected frequency is the theoretical frequency of each pattern drawn from calculating probabilities multiplied by total number of draws, while the Observed frequency is the actual frequency drawn from actual lottery results a.k.a. 'experimental' or 'empirical probability'.
Therefore:
Expected Frequency = (Probability) X (Number of draws)
Ethiopian airlines reviews. Observed Frequency = The number of occurrences in the actual lottery draws
To prove that our calculation is correct:
Expected Frequency should closely match the Observed Frequency given a sufficiently large sample of draws
Proof From Actual Irish Lottery
922 Draws of Irish Lottery from Nov. 4, 2006 – Sept. 2, 2015
Results taken from Irishlotto.net
Proof From Actual Australian Lottery
2,158 Draws of Australian Lottery from June 2, 1990 to May 28, 2016
Results taken from Osalottos.com
The two aforementioned 6/45 lotteries make it evident that the Law of Large Numbers is at work. The proximity of the expected frequency and the observed frequency proves this.
Accordingly, the results coincide with my previous theoretical computations – proof that applying probability in the lottery can predict how it behaves or how it will behave in the future.
Now that you've breathed all of that in, let me ask you again, what is the probability of winning the lottery? More specifically, what is your probability of winning the lottery?
The answer, if you digested all of that information correctly, will depend on which lottery pattern you will use to play with. If you play with a pattern that has more chances of getting drawn, just like the 3 odd and 3 even pattern, then your probability of winning the lottery is high.
Previously, I mentioned the Law of Large Numbers. But what is it anyway and what does it have to do with the lottery?
The Law Of Large Numbers At Work
The Law of Large Numbers states that when an event is repeated a large number of times, the outcome is always close to the expected value and will tend to become closer as more events are performed to infinity. The events here are the actual lottery draws, and the expected value is the expected frequency after we multiply the probability with the actual number of draws.
The previous tables above states clearly how the Law of Large Numbers intersects with the lottery. As you can see, the patterns we 'expected' to appear more frequently, by probability experimentation and testing, were drawn, in reality, more than other patterns. With this, we can say that the lottery can be represented by Probability Theory and the Law of Large Numbers.
There's More To The Lottery Than Meets The Eye
Now, we're only scratching the surface.
Now you know that Probability is very useful especially for lottery players, let's get more granular than just odd and even patterns.
Let's discover those lottery patterns that have the highest probability of winning.
Understanding Lottery Patterns
According to a report published in The Guardian, about 10,000 people play 1,2,3,4,5,6 every week. If ever this combination does get drawn in a lottery, they would be bringing home approximately £400 each since most jackpots are around the £4M mark. These people won't be coming to the bank laughing for sure.
Of course, there's one surefire way to win the lottery. Buy all the possible number combinations. Right?
Wrong.
In 2014, Renato Gianella, a Brazilian mathematician explained that not all combinations have equal chances of being drawn.
Therefore, in this article I will use his study, The Geometry Of Chance, to lucidly explain why 1,2,3,4,5,6 will never get drawn in any of the world's lotteries. Eventually, this will also explain why not all number combinations have equal chances of getting picked in a draw.
In Gianella's study, numbers are grouped together into different probability patterns or template as Gianella calls it.
When you extract all the patterns for a 6/45 lottery game, we will be able to come up with 210 different combination templates.
We will not be able to cover all the patterns here. Fortunately, you can view the complete list when you sign up as Lottometrix member. It's free to become member.
Now, here's the most important thing to remember; each pattern has a different probability of getting drawn, depending on the number of possible combinations.
As with the marble example, when a pattern is outnumbered by other patterns, it would be hard for that pattern to be picked in a draw.
Let's take some pattern samples from a game with a 6/45 format and show their corresponding theoretical probabilities and estimated occurrences.
From the table above, theoretically, you can see that patterns 162, 203 and 205 almost never occur in a draw.
In layman's term, combinations belonging to pattern 205 occur only once in every 100,000 draws. This incurs the notion that playing with this combination is a total waste of time and money.
Unfortunately, many lottery players blindingly choose numbers based on this pattern. You are probably one of them.
And truly, it has never occurred in the history ever since the lottery started.
Again, all these are theoretical analysis. Right?
Let's see how our theoretical computations fare with actual lottery draws.
Theoretical Probability Analysis Versus Actual Lottery Results
Australian Lottery
From June 2, 1990 to May 21, 2016
2,158 draws
| Combination Pattern | Probability | Estimated Occurrence | Expected Frequency in 2,158 Draws | Observed Frequency in 2,158 Draws |
| Pattern 1 | 0.0298340344 | 3x in 100 draws | 64 | 62 |
| Pattern 31 | 0.0089502103 | 9x in 1000 draws | 19 | 19 |
| Pattern 48 | 0.0074585086 | 7x in 1000 draws | 16 | 16 |
| Pattern 83 | 0.0039778712 | 4x in 1000 draws | 9 | 10 |
| Pattern 162 | 0.0008839714 | 9x in 10000 draws | 2 | 2 |
| Pattern 203 | 0.0000257825 | 2x in 100,000 draws | 0 | 0 |
| Pattern 205 | 0.0000103130 | Once in 100,000 draws | 0 | 0 |
Table conclusion:
To win, play pattern 1.
To lose and waste money for the rest of your life, choose patterns 162, 203, 205.
The problem is that you do not know if your numbers are based on bad patterns unless you are a member of Lottometrix team. Accordingly, patterns 203 and 205 not once occurred in a single draw from the 2,158 draws of the Australian lottery.
The results are yet again, clear about the behavior of the lottery being predicted to an extent. Evidently, all these theoretical probability works we're doing guide us on how to predict the lottery mathematically.
Due to the extensive list of patterns, it wouldn't be wise to put all of them here.
However, the complete list of patterns for the world's most popular lotteries along with their comparison to actual lottery results are freely accessible inside Lottometrix exclusive members area.
Old Irish Lottery
From November 4, 2006 to September 02, 2015
922 draws
| Combination Pattern | Probability | Estimated Occurrence | Expected Frequency in 922 draws | Observed Frequency in 922 draws |
| Pattern 1 | 0.0298340343 | 3x in 100 draws | 28 | 34 |
| Pattern 31 | 0.0089502103 | 9x in 1000 draws | 8 | 9 |
| Pattern 48 | 0.0074585086 | 7x in 1000 draws | 7 | 7 |
| Pattern 83 | 0.0039778712 | 4x in 1000 draws | 4 | 3 |
| Pattern 162 | 0.0008839714 | 9x in 10000 draws | 1 | 1 |
| Pattern 203 | 0.0000257825 | 2x in 100,000 draws | 0 | 0 |
| Pattern 205 | 0.0000103130 | Once in 100,000 draws | 0 | 0 |
Table conclusion:
In the table above, the results and draws used were taken from 922 previous draws of the Irish Lottery from before it changed its matrix to 6/47. This is so we can be consistent with our analysis since we derived our patterns from the old 6/45 Irish Lotto format.
Again, the probability coincides with reality as expected frequencies match close to observed frequencies.
If you want to win, play pattern 1.
Otherwise, play patterns 162, 203 or 205 and waste money forever.
Old 5/59 Powerball
From January 7, 2009 to October 03, 2015
704 draws
| Combination Pattern | Probability | Estimated Occurrence | Expected Frequency in 704 draws | Observed Frequency in 704 draws |
| Pattern 1 | 0.0199744886 | 2x in 100 draws | 14 | 15 |
| Pattern 12 | 0.0089885199 | 9x in 1000 draws | 6 | 6 |
| Pattern 63 | 0.0071908159 | 7x in 1000 draws | 5 | 5 |
| Pattern 92 | 0.0040448339 | 4x in 1000 draws | 3 | 4 |
| Pattern 126 | 0.0032358672 | 3x in 1000 draws | 2 | 1 |
| Pattern 181 | 0.0016778570 | 2x in 1000 draws | 1 | 1 |
| Pattern 247 | 0.0000503357 | 5x in 10,000 draws | 0 | 0 |
| Pattern 252 | 0.0000251679 | 2x in 10,000 draws | 0 | 0 |
Table conclusion:
Patterns 247 and 252 proved their inadequacy of helping you win. No results in both the Expected Frequency and Observed Frequency column, taken from 704 draws of the old 5/59 US Powerball.
In the new U.S. Powerball only 6 patterns are the best. And there are 456 patterns to avoid.
The question is how do you know your numbers are based on the best patterns. Fortunately many people fall into the bad patterns when they choose their numbers.
If you are a member of Lottometrix, you can view these patterns when you log inside the members area.
Fortunately, it's free to become member Lottometrix.
Euro Millions
From April 16, 2004 to May 27, 2016
893 draws
| Combination Pattern | Probability | Estimated Occurrence | Expected Frequency in 893 draws | Observed Frequency in 893 draws |
| Pattern 1 | 0.0424776756 | 4x in 100 draws | 37 | 37 |
| Pattern 43 | 0.0076459816 | 8x in 1000 draws | 7 | 5 |
| Pattern 61 | 0.0056636901 | 5x in 1000 draws | 5 | 5 |
| Pattern 109 | 0.0021238838 | 2x in 1000 draws | 2 | 2 |
| Pattern 189 | 0.0001189375 | 1x in 10,000 draws | 0 | 0 |
| Pattern 196 | 0.0000594687 | 5x in 100,000 draws | 0 | 0 |
Table Conclusion:
By now it should be clear to you that in lottery, pattern is king.
If you want to win the Euro Millions, you should choose pattern 1.
But if you want to lose and waste your money for the rest of your life, go ahead and choose patterns 189 or 196.
Euro Jackpot
From March 23, 2012 to May 27, 2016
217 draws
| Combination Pattern | Probability | Estimated Occurrence | Expected Frequency in 217 draws | Observed Frequency in 217 draws |
| Pattern 1 | 0.0424776756 | 4x in 100 draws | 9 | 10 |
| Pattern 43 | 0.0086017293 | 9x in 1000 draws | 2 | 3 |
| Pattern 61 | 0.0042477676 | 4x in 1000 draws | 1 | 1 |
| Pattern 109 | 0.0039645831 | 4x in 1000 draws | 1 | 1 |
| Pattern 189 | 0.0001189375 | 1x in 10,000 draws | 0 | 0 |
| Pattern 196 | 0.0000594687 | 5x in 100,000 draws | 0 | 0 |
Table conclusion:
To win: Choose pattern 1
To waste money: Choose the rest of the patterns.
How To Pick Winning Lottery Numbers
To win the lottery, simply avoid the bad patterns at all cost and choose to pick numbers based on patterns that will give you the best chance of winning.
Truth is, there are bad combination patterns in the lottery. Believe it or not, there's actually a lot of them.
Contrary to popular belief, 1,2,3,4,5,6 is not the worst.
The table below shows you the list of popular lotteries and their corresponding patterns to play and to avoid.
| LOTTERY | BEST PATTERNS | FAIR PATTERNS | BAD PATTERNS |
| US Powerball | Patterns 1,2,3,4,5 and 6 | Patterns 7 to 155 | Patterns 156 to 462 |
| Irish Lottery | Patterns 1,2 and 3 | Patterns 4 to 46 | Patterns 47 to 210 |
| Euro Jackpot | Only pattern 1 | Patterns 2 to 85 | Patterns 86 to 196 |
| Euro Millions | Only pattern 1 | Patterns 2 to 85 | Patterns 86 to 196 |
| Australian Saturday Lotto | Patterns 1,2 and 3 | Patterns 4 to 85 | Patterns 86 to 210 |
| UK Lottery | Only pattern 1 | Patterns 2 to 30 | Patterns 31 to 431 |
| US Megamillions | Patterns 1,2,3,4,5 and 6 | Patterns 7 to 190 | Patterns 191 to 596 |
If you want to win, you don't want to pick your numbers based on those bad patterns. Do you?
The best way to choose lottery numbers should be based on the best patterns. Thanks to Probability.
In other words, you can now play with combinations formed around the patterns that will give you the best chances of winning the lottery. The numbers say it all, and the numbers don't lie.
The analysis results to a detailed selection, which makes the vast differences between the most appearing and the least appearing patterns more evident.
However, the more detailed the pattern selections get, the more complex the computations.
Daily Lotto Numbers 11 June 2019
Good news is you don't need a math degree to understand the complexities involved.
Past Winning Lottery Numbers
Lottometrix has made all these patterns accessible to all for free. It's free to become member of Lottometrix. Register here
