Odds and Probability Explained in the Context of a Lottery Game


Last updated on February 22, 2024

Odds and probability are two different terms. They are not mathematically equivalent. Knowing the difference between the two is crucial for you as a lottery player.

Many people use the term odds when they mean probability and vice versa. Perhaps the main reason for the confusion is that they are synonymous.

Unfortunately, these terms are used interchangeably in conversations, internet sites, and published materials. Ronald Wasserstein of the American Statistical Association caught that occasion well in one article in The New York Times.1

But let’s set one thing straight. In Statistical Science, odds and probability are two different mathematical terms.2

So, let’s talk about the differences and why they are so essential to all lottery players like you.

The Definition of Probability

Probability is the measurement of the likelihood of an event’s occurrence.3 The value is expressed between 0 and 1, where 0 indicates impossibility, and 1 means certainty. So, we usually expressed probability in decimal terms.

Some writers prefer to use a simpler word for probability. For instance, we use the more comfortable term “percentage” to represent “chances” as an alternative to decimals.

For example, an event with a 25% chance of occurring is equivalent to 1/4 or 0.25.

P(event A) = 1/4 = 0.25 = 25%

The Definition of Odds

Odds refer to the ratio of two events.4 There are two flavors of which we express odds:

  • Odds against = The number of ways an event does not occur against the number of ways an event does occur.
  • Odds in favor = The number of ways an event occurs against the number of ways an event does not occur

In the context of the lottery, we commonly use the latter expression.

So, let’s say event A will occur 25 times out of 100 total events. Therefore, the odds in favor of A are 25 to 75.

Odds in favor of A = 25 / 75 = 1/3 or 1:3

  • 25 refers to the numbers by which event A will occur.
  • 75 refers to the numbers by which event A will not occur.

As you can see, from a layman’s point of view, odds are the success-to-failure ratio.

Odds equals successes over failures

From the lottery perspective, odds refer to the ratio of the number of ways you get favorable shots to the number of ways you don’t get favorable shots.

The Visual Difference

Let’s say we have four marbles.

4 marbles = 1 red and 3 blue marbles

First, let’s calculate the probability of the red marble:

P(red marble) = 1 / 4 or 25%

So, the probability P(red marble) is described visually in the following way:

One red marble over 4 marbles

So when you put all those marbles in a bag and you are told to pick one with your eyes closed, the probability that you will get the red one is 25%.

Now, let’s calculate the odds:

As I have explained above, odds refer to the success-to-failure ratio.

So, in this particular example, the odds refer to the ratio of red marble occurring successfully to the number of times red marble will fail to occur.

one fourth divided three fourth equals one third


The odds in favor of red marble is 1:3. We express this ratio visually in the following way:

One red marbles over 3 blue marbles

So let’s recap:

Probability of red marble = 1/4 or 25%

Odds in favor of red marble = 1/3 or 1:3

Did you see the difference?

Now, the discrepancy may be too tiny. But when the number of events gets bigger, such as in the lottery, knowing the difference takes a significant role when making decisions

Odds and Probability in the Lottery

All combinations in the lottery have an equal probability of getting drawn because there’s only one way to win the jackpot.

All combinations have the same probability. Because there's only one way to match the winning combination

So does that mean 5-10-15-20-25-30 is equally likely?

Well, yes. That’s because, theoretically:

P(5-10-15-20-25-30) = 1 / All possible combinations

The same calculations apply to the 1-2-3-4-5-6 or 2-4-6-8-10-12 combination. The same formula applies to all combinations.

So many lotto experts advise against playing 1-2-3-4-5-6 because if you happen to win, there’s a possibility that you split your jackpot to a lot of winners.

That’s true.

However, this explanation doesn’t apply in some situations.

For example, are you willing to play the 1-2-10-20-11-22 combination and spend $2 on it?

You’ll probably answer no. That’s because part of you screams that something is not right.

But why do you worry if you stand up firmly and say 1-2-10-20-11-22 possesses the same probability as any other combination?

You see, a strategy based on “gut feeling” doesn’t add up. You’ve got to explain things from a mathematical perspective.

In this article, I propose a way to explain “gut feeling” using the success-to-failure ratio.

Minding Your Success-to-Failure Ratio

Combinations are not created equally. A combination is a composition of numbers. To measure the success-to-failure ratio, composition matters.

Let’s consider analyzing the lottery using odd and even numbers.

There are 4,655,200 ways you can combine 3-odd-3-even combinations. So if you play a 3-odd-3-even combination, 33 in every 100 draws will put you in a 1 to 4.6 million advantage rather than 1 to 14 million.

Therefore, your success-to-failure ratio is:

Odds in favor of 3-odd-3-even combinations is equal to 4,655,200 over 9,328,616 or 1 is to 2

In comparison, 6-even combinations will give you the odds of 1 to 134,595, but be aware that this advantage will happen only once in 100 draws.

The odds of 6-even combinations is equal to 1 every 103 draws

That means if you play 2-4-6-8-10-12, then expect that a favorable opportunity to win the jackpot only comes around every 104 draws.

As a lotto player, you don’t want to spend your money and wait for 104 draws to get one favorable shot. Do you?

Your first duty as a lotto player is to get more favorable shots in fewer attempts.

Your goal is to win the lottery, and the first thing you should know when playing a lottery game is to know your success-to-failure ratio.5 You cannot change the underlying probability and you cannot beat the lottery’s odds, but as a lotto player, you have the power to make informed choices.6

Odds and Probability as a Guide

Lotterycodex exists to help you make informed choices. We’re talking about a certain mathematical advantage.

As a lotto player, I don’t think you will spend money on a combination that will only go down the drain for most of the draws. For example, a ticket with 1-2-3-4-5-6 on it belongs to a group with less favorable shots and more opportunity to lose.

So, while all combinations exhibit the same probability, combinations are not created equally. There exist combinatorial groups with varying success-to-failure ratios.

Lotterycodex suggests that all lotto players use these combinatorial groups as guides to get the best shot possible.

Numbers don’t lie. And that’s how combinatorial mathematics and probability theory can help improve your skill for picking better combinations.

In lotterycodex, I use probability to forecast the likely outcome of the lottery. And I do that from the context of combinatorial groups based on the law of large numbers.7

So, to set things straight, we cannot predict the “exact” combination. No one can do that. Some people take it from a different context. And that’s how the confusion starts.

All in all, thanks to probability theory. Thanks to the great mathematicians who invented all these mathematical tools for us to enjoy.8

Probability theory in the lottery - in a lotto 649 game, players are advised to play the 3-odds-3-even patterns because of its high probability of occurring in a draw.

A mathematical strategy is impossible if odds and probability don’t exist.

I invite you to check out the Lotterycodex calculator to help you make informed choices when playing your favorite lottery game.

Questions and Answers

What is the difference between odds and probability in a lottery context?

The difference between odds and probability is fundamental for strategic playing in a lottery context. Probability is the likelihood of an event occurring, expressed as a number between 0 and 1. In contrast, odds compare the likelihood of an event happening to it not happening, expressed as a success-to-failure ratio. This success-to-failure ratio is an important guide for all lottery players to make informed choices.

How is the success-to-failure ratio expressed in the lottery?

In the lottery, odds are typically expressed as a ratio of favorable and unfavorable outcomes. For example, if a lottery has one favorable event and 999,999 unfavorable events, the S/F ratio can be 1 to 999,999. This expression clearly compares the number of ways you get favorable shots against the number of ways you don’t get favorable shots.

Do all lottery combinations have the same probability of winning?

Yes, in a typical lottery game, all combinations have the same probability of winning. This is because the lottery is a random draw, and each combination is unique and equally likely to be drawn as the winning combination. The concept of equal probability for all combinations is central to the fairness and unpredictability of lottery games.

What influences a lottery player’s success-to-failure ratio?

The composition of a combination influences a lottery player’s success-to-failure ratio. Combinatorial groups have varying compositions with varying success-to-failure ratios. Players may use this mathematical information to make informed decisions when selecting numbers, giving them more favorable shots to winning the lottery.

What does Lotterycodex suggest to improve the lottery playing strategy?

Lotterycodex suggests avoiding statistical analysis and utilizing combinatorial mathematics and probability theory to analyze how lottery balls behave in a random lottery draw. This approach involves choosing combinatorial groups with favorable success-to-failure ratios. This method doesn’t guarantee winning but aims to optimize the selection process for a more informed and strategic approach to playing the lottery.

Can we predict exact winning lottery combinations?

No, we cannot predict exact winning lottery combinations. Lottery games are designed to be random, and each combination has an equal chance of being drawn. While mathematical strategies can potentially improve your number selection process, they do not enable the prediction of specific winning combinations.

What is the significance of combinatorial groups in lottery strategies?

Combinatorial groups in lottery strategies are significant because they categorize combinations into different groups based on their varying success-to-failure ratio. This classification allows players to make informed choices and play the lottery with the best shot possible.

Additional Resources

  1. Know the Difference    []
  2. Statistics — Probability vs. Odds    []
  3. Probability    []
  4. Odds    []
  5. The Lotto Secret: Three Math Strategies for Winning Revealed    []
  6. How to Win the Lottery According to Math    []
  7. The Winning Lottery Formula Using Math    []
  8. List of Important Mathematicians – Timeline    []

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