Every lottery combination has an equal probability of being drawn. In a fair game, that applies to every ticket regardless of who holds it. If a lottery has 13,983,816 possible combinations, each one carries a 1 in 13,983,816 chance per draw.
That’s a fact. And it’s the source of a lot of confusion.
A reader once sent me this question, and it cuts right to the heart of the matter:
Hi Edvin,
Your reply to this one question will answer all my other questions. All combinations in the lottery have an equal probability of getting drawn. But you said combinations are not created equally. Since both statements cannot be true simultaneously, which statement is true?
Both are true. They just describe different things.
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Probability and odds are related but not the same thing
All combinations have equal probability.1 That does not change. What differs between combinations is their combinatorial composition, which is a separate mathematical property altogether.
To sort this out, you need to understand that probability and odds are two distinct concepts with two different equations.2
Probability = A / B
Odds = A / (B – A)
Where A is the number of favorable combinations and B is the total number of possible combinations. This expresses odds in favor — the ratio of favorable outcomes to unfavorable ones.
Probability measures how likely a single outcome is across all possibilities. Odds compare favorable outcomes to unfavorable ones. They are related, but they do not measure the same thing.
Because all combinations share equal probability in any single draw, the chance that any particular ticket gets picked cannot be changed. For a single specific combination, A = 1, making the probability equal for every combination.
For a combinatorial composition group, A equals the total number of combinations in that group — and that count differs across groups.
What combinatorial composition actually means
Combinations with equal probability can still differ in how they are built. A combination made entirely of even numbers and a combination with a balanced mix of odd and even numbers both carry the same single-draw probability. But their combinatorial compositions are different, and that difference shows up when examining how often each type appears across thousands of draws.3
When combinations are grouped by their composition, some groups contain far more combinations than others. A combinatorial composition group with more combinations occupying the total outcome space will appear more often over a large number of draws. That is how probability and counting work together.3
In Lotterycodex, this ratio of favorable to unfavorable outcomes within a combinatorial group is called the frequency ratio. It describes long-run statistical behavior, not what will happen in the next draw.
I use “frequency ratio” rather than “odds” deliberately. In everyday usage, “odds” tends to be read as odds of winning or losing. That framing does not apply when the measurement is about how often a combinatorial composition appears across many draws. “Frequency ratio” carries the same mathematical structure as the classical odds formula but frames it as a long-run statistical description, not a prediction and not a win/loss comparison.
Equal probability tells you every combination has the same shot in a single draw. Combinatorial composition tells you how different groups of combinations are distributed across the entire outcome space. Both statements are true. They answer different questions.
What combinatorial analysis shows about player hesitation
Many players know all combinations have equal probability, and yet many still avoid combinations like 2-4-6-8-10-12 or 10-20-30-40-50. Combinatorial analysis offers a mathematical explanation for that observation, even though it does not validate or explain the hesitation itself as rational or irrational.
Those combinations are entirely valid. They carry the same single-draw probability as any other. Combinatorial analysis shows they belong to groups that occupy a smaller share of the total outcome space. Over many draws, those groups appear less often. Not because randomness avoids them. Just because there are fewer combinations built that way.
What the math actually describes
No selection method changes the equal probability each combination carries into a draw. Mathematics describes how combinations are distributed across the outcome space, not how to control or predict draw results.
I built the Lotterycodex framework in 2017 to separate combinations into groups based on their combinatorial compositions and to calculate the frequency ratio for each group. The goal was to build a descriptive model, not a predictive one. Frequency ratios describe long-run statistical behavior under probability theory. They do not forecast what any specific draw will produce.
Equal probability is the starting point for understanding the lottery honestly. Once that is clear, the mathematical question becomes: given that all combinations share equal probability, do different combinatorial compositions occupy different portions of the total outcome space? The math says yes. That is a description of how the outcome space is structured. It says nothing about which combination will be drawn next.
Understand Lottery Games Using Math-Based and Data-Driven Analysis
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References