Lotto Numbers and Combinations: Understanding Your Competitive Advantage Using Probability Theory
When most people think of a winning lottery system, they think of a magic button powered by tables of numbers, a shiny crystal ball, or even a psychic extraordinaire that can magically spit out the winning numbers.
But if we objectively look at things, none of those superstitious methods work. You can’t pick the right numbers because no one, not even the lottery organizers, knows the winning numbers. That’s because the lottery is truly random.
So does that mean there’s no way to hack the lottery? Not in the traditional sense. You can’t guarantee winnability. But with the right mathematical strategies, grit, and perseverance, you can get the best shot possible. (See The Lottery and the Winning Formula According to Math)
Before buying lottery tickets, you first need to understand the difference between numbers and combinations and why that difference means everything.
So let’s start with that.
Understanding Lotto Numbers and Combinations
Let’s start with numbers. A lottery number is any of the individual lotto balls. For instance, in a 5/24 system, you have a set of 24 individual balls as follows:
Balls = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}
Each of those individual balls represents a lotto number. In a 5/24 lotto system, you have to pick 5 of those individual numbers and hope those exact balls are drawn to win.
But lotto games come in different formats. For example, in California Lottery Fantasy 5, players can choose from 39 numbers. While in Hoosier Lottery’s Ca$h 5, there are 45 numbers to choose from. And, of course, let’s not forget the most popular 6/49 lotto game, where you have to choose from 49 numbers.
But that’s not all, and sometimes, people get to choose from two sets of numbers. For example, in Tennessee Cash of the Tennessee Lottery, you must choose from 35 plus an extra number from 1 to 5.
A combination is the set of individual numbers you get to pick. For instance, in Louisiana Lotto 6/42, you can pick six individual numbers to create a legit combination that can win a jackpot. Examples include 7–12–17–23–36–42.
The most important thing is that you cannot win a lotto jackpot by just selecting a number. You must combine 5 or 6 numbers to play a lotto game and win if you are lucky.
Now that you know what numbers and combinations are, let’s discuss how to pick the right numbers.
Picking the Right Lotto Numbers
There are many ways people try to pick numbers. Some bet on lucky numbers only. For instance, it could be a number based on numerology, astrology, frequency of occurrence in their life, or even just a favorite number with no apparent reason.
Others take a different approach by avoiding unlucky numbers. For instance, 13 is commonly considered an unlucky number across many cultures.
Can you notice a pattern here? All of those choices are based on nothing but superstition. In the world of random games, superstition has no place as far as winning is concerned. It’s a good source of entertainment, but sadly nothing more.
Now that begs the question:
Couldn’t you look at all the past lottery draws and determine which numbers win the most?
You could do that, as countless others do each year. Sadly though, that’s a strategy as good as mere superstition. Here’s why.
If we look at the first 30 draws of the Canada Lotto 6/49 from 1982 to 2018, the frequency of occurrence for a set of 10 chosen numbers looks like this:
Based on that graph, we should pick the numbers 11, 18, 28, and 35, as they win way more than the others. Right?
Wrong. According to the law of large numbers, the frequency of individual balls can be biased initially, but it always evens out in the long run.
To see that in action, let’s see the frequency of those same ten numbers based on 3,688 draws of the same Canadian lotto this time:
Can you see how everything has eventually balanced out? Just as the law of large numbers promised!
In short, there are no lucky or unlucky numbers. There are no winning or losing numbers. That’s because all numbers have the same probability in the long run.
Picking the Right Lotto Combinations
So far, we’ve established no way to predict the winning numbers as they are random. But the thing is, individual combinations are just as random. Despite what your gut feeling may tell you.
For instance, what do you think of the combination 1–2–3–4–5? Would you bet your money on that?
If you’re like most people, your gut is probably telling you to run away from that combination as fast as you can. There’s no way that combination has the same probability as any other. Or does it?
Here’s the thing. It has the same probability as any other combination. That’s because the winning combination is as random as the individual numbers.
I know what you’re thinking now. If a lottery game is random, surely there’s no way to get the best shot. Not quite.
Unlike numbers and individual combinations, there’s another factor you can use to improve your odds. And that’s understanding the composition and how the composition relates to your ratio of success to failure. And the best way to describe composition is through combinatorial patterns.
In simple words, a combinatorial pattern is how you compose a combination. For example, 5–7–9–10–12–14 has a combinatorial pattern of 3 odd and 3 even numbers.
Now that you know a combinatorial pattern, let’s look at how some patterns offer far better odds than others.
Combinatorial Patterns — Combinations are Not Created Equally
In a 6/49 lotto system, there are a little under 14 million unique combinations (13,983,816, to be precise). But there’s only one winning combination. So you have a winning chance of 1 in almost 14 million.
Where things get interesting is when we compare the probability of patterns. To keep things short and simple, let’s compare a 3-odd-3-even pattern with a 6-even pattern.
Some of you might think that the 6-even combination is good since it only has 134,596 combinations compared to the more than 4 million combinations of the 3-odd-3-even pattern.
But that’s quite interesting, as that pattern only shows up nine times out of 1000 draws. By picking 3 odd and 3 even numbers, you get more favorable shots with 333 draws out of every 1000 attempts.
Therefore, you get a higher ratio of success to failure when you choose the 3-odd-3-even because, according to probability theory, most of the winning combinations must come from this combinatorial group.
Your job as a lotto player is to make sure that you get closer to that drawing behavior to get the best shot possible.
Mathematically, all combinations have the same probability because there’s only one way to win the jackpot. But to get the best shot possible, you probably avoid purely 6-even or 6-odd numbers altogether because you get only one opportunity to match the winning combination out of 100 attempts. You don’t want to wait that long to win a jackpot.
In short, I don’t have a recipe for you to predict the next winning combinations, but I can show you what to avoid.
Let’s compare the expected frequency and actual frequency of all odd and even patterns based on real lottery results to prove how powerful that is.
And to show you that probability theory applies to all lottery systems, we will apply probability calculation in a different format.
This time, let’s use the 6/45 system.
For this example, let’s go with the 734 draws of the Australian Saturday Lotto between January 7, 2006, and February 01, 2020:
Below is a similar probability study conducted for low and high numbers.
Our expected frequency of each pattern is almost identical to the actual frequency in the 734 draws of the Australian Saturday Lotto. This is the beauty of math’s precision and accuracy.
Just don’t forget that the best combinatorial patterns vary from one lotto system to another. You can use a calculator to figure out which ones are the best for you quickly. (see Knowing the Best Lotto Combinations Without a Math Degree)
Probability Theory Applies to All Lottery Systems
As I said, probability theory applies to all lottery systems regardless of the format. Below you will see similar results from other lottery systems I have studied.
UK Lotto 6/59 (Odd-Even Analysis)
UK Lotto 6/59 (Low-High Analysis)
US Powerball 5/69 (Odd-Even Analysis)
US Powerball 5/69 (Low-High Analysis)
Irish Lottery 6/47 (Odd-Even Analysis)
Irish Lottery 6/47 (Low-High Analysis)
EuroMillions 5/50 (Odd-Even Analysis)
EuroMillions 5/50 (Low-High Analysis)
Individual Numbers Don’t Matter; Combinatorial Composition Does
We have discussed how the balls behave under a truly random draw. And we effectively show how combinatorial composition matters using odd/even and low/high numbers. However, there’s more to the lottery composition than meets the eye.
For example, the 1–2–3–4–5–6 line is considered best under odd/even analysis. But we know that line is one of the worst under low/high analysis since all numbers are low. According to probability theory, a lottery draw tends to pick numbers across the number field. Therefore, it’s rare that you get a winning combination composed of purely low numbers.
Therefore, there’s an unacceptable contradiction when we deal with two separate analyses.
We must combine the two analyses into one combinatorial analysis using Lotterycodex to avoid contradiction. (See The Winning Lottery Formula Based on Combinatorics and Probability Theory)
There’s no such thing as lucky or cursed numbers, and predicting the winning numbers is impossible. That’s because each lotto draw is as random as nature can be.
But as we saw from our analysis of both mathematical theory and real-world results, some patterns show up much more frequently than others. For instance, the 3-odd-3-even pattern is over 34 times more likely to show up than the 6-even pattern.
Leveraging the advantage of combinatorial patterns is the key to getting the best shot possible when playing a lotto game. You can’t guarantee success, but this way, you’ll have a far better ratio of success to failure.
Please don’t put all your hope in the lottery to better your life. A lottery game is an entertainment, so just bet the money you can afford to lose.
