# Lottery probability formula

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We will get through this together. Updated: March 31, References. Everyone's heard comparisons between the odds of winning the lottery and the odds of other unlikely events, like getting struck by lightning. It's true, the odds of winning the jackpot on a game like Powerball or another pick-6 lottery game are incredibly low.

UNDERSTANDING LOTTERY MATHEMATICS

But just how low are they? And how many times would you have to play to have a better chance of winning? These answers can be found down to the exact odds with some simple calculations. To calculate your odds of winning the lottery, use the formula: factorial of n over factorial of r times factorial of n minus r, where n is the total number of possible numbers and r is the number of numbers chosen. For example, if you're playing a lottery where you can choose 2 numbers from a pool of 5 numbers, the formula would be: factorial of 5 over factorial of 2 times factorial of 3, which equals over To learn how to calculate other kinds of lotto odds, keep reading!

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Calculating Other Lotto Odds. Tips and Warnings. Related Articles.

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Article Summary. Method 1 of Understand the calculations involved. To find the odds of winning any lottery, divide the number of winning lottery numbers by the total number of possible lottery numbers. If the numbers are chosen from a set and the order of the numbers doesn't matter, use the formula n!

In the formula, n stands for the total number of possible numbers and r stands for the number of numbers chosen.The lottery is a random game, but mathematically, you can predict it to an extent.

Probability theory can, indeed, pose as an effective guide in predicting how the lottery behaves. Then, along with lottery prediction, you can pick a better number combination with the best shot at the lottery. Firstly, it is important that I introduce some basic mathematical concepts to you. The two lottery formulas are essential to understanding the odds of winning in various lotteries in the world. In Powerball, winning the jackpot requires matching the extra ball, in this case, the total combination becomes:.

Considering the astronomical odds of winning, you likely think it is impossible to win with any lottery numbers you can think of. In the Mega Millions, the odds of winning are 1 in million and the odds for US Powerball are a staggering 1 in million. Of course, in probability, if you do not play the lottery, your chances of winning reduce to zero or impossible. So this is the problem with playing the lottery in a classical approach. Players look at the odds based on the individual combination. You must learn how to pick lottery numbers from a different perspective. Renato Gianella is a Brazilian mathematician who published a study called the Geometry of Chance in Fortunately, the formula to use is the same in any lottery. We simply change the variable depending on the format. To apply math in the lottery; first, we get the probability of each pattern.

## How To Win The Lottery According To Math

According to The Law Of Large Numbersthe average of the results obtained from a large number of trials should be close to the expected value and will tend to become closer as more trials are performed to infinity. Therefore, to prove that our recommendation of playing the lottery based on patterns is correct, our estimation should be close to the actual results, given a sufficiently large number of data from actual lottery draws.

Probability will always be a means to put a solid number on our lottery prediction scientifically and mathematically.

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Please notice the closeness between the estimated values and the actual occurrences. There are patterns in Australian Saturday Lotto. Pattern 1 is estimated to occur about 17 times in draws. The actual lottery results show Therefore, this proves that combination patterns exhibit different probabilities which are instrumental in lottery prediction.

In Euromillions, pattern, and are estimated that they will never appear. The actual draw indeed shows no occurrence of the said patterns. As a player of Euromillions, you should know how to pick lotto numbers based on the right pattern? There are worst combination patterns in US Powerball. You are probably one of them. To have higher chances of winning the U. Powerball, you should focus your playing strategy on six best patterns. The problem, the majority of lottery players never know they are falling into lottery numbers that will never appear in an Irish lottery. I can almost guarantee, you are probably one of these players. The above table shows that patterns and will not appear. If you were to play the UK lottery, you will not want to waste your money with these kinds of patterns.

Will you? UK Lotto has only one best pattern. You must stop picking lottery numbers based on the bad combination.If you like this topic, please consider buying the entire e-book.

Strategy: It depends on how many numbers are in the game. You need to figure out the number of possible combinations in the game. You can use the COMBIN function as follows to figure out the number of possible combinations for games in which you choose 6 of 40, 44, 48, and so on numbers:. If your state lottery game requires you to select 6 numbers out of 40, then the odds against you winning are 3.

For a game with 44 numbers, the odds are 7 million to 1. This payoff is only slightly in your favor. Here, the sequence in which the balls are drawn is not relevant.

If you had a game in which you had to match both the numbers and the order in which they were drawn, you would want to use the PERMUT function to find the number of permutations of drawing 6 numbers in sequence out of Additional Details: Since the first edition of this book, two multi-state lotteries have become popular in the United States. These require the player to match five numbers from one pool of numbers and then one number from a separate pool of numbers. This means you have to win two drawings to win the jackpot.

Multiply the combinations from the first drawing with the combinations from the second drawing. Here are the calculations for Mega Millions and PowerBall lotteries. As you can see, lotteries are a tax on people who can't use Excel. The Formulas, Functions and Visual Basic procedures on this web site are provided "as is" and we do not guarantee that they can be used in all situations.

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You can earn a commission for sales leads that you send to us by joining our affiliate program. Excel Book Excerpts. All rights reserved.In our quest to find the best way of winning lottery prizes, we hear about past winners who claim that they used a formula to predict lottery numbers. According to some past lottery winners mathematical formulas based on probability and statistical analysis are capable of predicting the outcome of random events.

As the lottery is a game where numbers are drawn randomly, then applying mathematics these type of formulas may be just what smart lottery players need to do. Although computers can do really amazing things these days at their core they are designed to make use of a set of rules for computing calculations or solving problems.

### How to Win the Lottery (and Win Sooner According to Math)

This has led its users to always make use of this tool each time any sort of calculation needs to be made. There are very simple ways to use mathematics to help you win the lottery. The challenge which every lottery player comes up against is the huge odds stacked against him or her. The odds that are stacked against you are astronomical when you play in the traditional and common way. Since the game of lottery is a game that is based on probability and not just pure random selection though randomness is involvedMath can be used to predict how the numbers in the game will perform to some extent.

Truth be told, no lottery player who wants to win wants to depend on a stroke of luck alone to win. The thrill of dreaming about what one can do with the huge winnings is too strong to leave it to chance when an offer on how to narrow the odds is presented. Contrary to the belief that each number in a lottery draw has the same number of chances of getting picked. Therefore the way to ensure you have a better chance of having a number in your selection match a number that is picked in the winning lottery draw you must play more numbers than you normally would?

The probability of an event happening is rather dependent on its division by the number of possible outcomes. For instance, in a lottery game with 45 balls, the probability of picking an odd number is greater than that of an even number. This is simply because of the fact that there are more odd numbers in the draw than there are even numbers. Lottery players can leverage this knowledge when picking numbers.

Although this is just a simple example it does highlight the fact that the lottery is not just about pure chance and luck — there is some mathematical laws involved. It is important to take this into consideration in order to avoid selecting numbers with a lower probability of getting drawn like going for hot numbers and avoiding cold numbers.

If you are like most of us who do not have the flair for Math, depending on its use for selecting the lottery numbers to play offers a tricky proposition.Dez has been a mathematician since grade school and has a master's degree in Applied Mathematics. As a mathematician, I have never purchased a lottery ticket. I find the odds depressing and have never had luck in winning anything from these kinds of games. This hub is all about calculating lottery probability or odds.

There will be two different cases discussed in the hub: the probability of winning the game with all six numbers matching, and the probability of having n numbers matching. It is always important to find out the rules of any game before participating in it. It is also possible to win some money if you are able to match three, four, or five numbers of the winning combination.

Note that the order of the winning combination here does not matter. Before we start with the calculations, I would like to talk about Permutations and Combinations. This is one of the basic concepts you learn in Probability Theory. The main difference being that permutations consider order to be important, while in combinations, order isn't important.

In a lottery ticket, permutation should be used if the numbers in your ticket have to match the order of the draw for the winning string of numbers.

The next formulas only apply for numbers without repetition. This means that if the number x is drawn, it cannot be drawn again. If the number drawn from the set is returned before the next draw, then that has repetition. Note that based on the formulas given, C n,k is always less than or equal to P n,k. You will see later on why it is important to make this distinction for calculating lottery odds or probability. Because order is not important, we will use the formula for combination:.

These are the odds or the total number of possible combinations for any 6-digit number to win the game. To find the probability, just divide 1 by the number above, and you will get: 0. See what I mean by depressing odds? So what if we're talking about a different lottery game where order does matter. We will now use the permutation formula to get the following:. Compare these two results and you will see that the odds for getting the winning combination where order matters has 3 additional zero's!

It's going from about 28 million:1 odds to 20 billion:1 odds! The probability of winning for this case is 1 divided by the odds which equals to 0. As you can see, because the permutation is always greater than or equal to the combination, the probability of winning a game where order matters is always less than or equal to the probability of winning a game where order does not matter.

Because the risk is greater for games where order is required, this implies that the reward must also be higher. Because you can also win prizes if you have less than 6 matching numbers, this section will show you how to calculate the probability if there are x matches to the winning set of numbers. First, we need to find the number of way to choose x winning numbers from the set and multiply it by the number of ways to choose the losing numbers for the remaining 6-x numbers.

Consider the number of ways to choose x winning numbers. Because there are only 6 possible winning numbers, in essence, we are only choosing x from a pool of 6. And so, because order does not matter, we get C 6, x.

Next, we consider the number of ways to choose the remaining 6-x balls from the pool of losing numbers. So, the number of possibilities for choosing a losing ball can be obtained from C 49, 6 - x. Again, order does not matter here. So, in order to calculate the probability of winning with x matching numbers out of a possible 6, we need to divide the outcome from the previous two paragraphs by the total number of possibilities to win with all 6 matching numbers.

We get:.In this article, I will show you how you can use Math to increase your chances of winning. And we will prove everything using actual data from actual lottery draws.

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. The traditional school of thought tells us that each lottery combination has the same chances of getting picked in a draw. 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.

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. 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.

In mathematics, probability is the likelihood of one or more events happening divided by the number of possible outcomes. 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 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. For now, you can simply pick an answer and later, we will find out if you are correct. 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:.

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. If we follow the same formula with the rest of the patterns, we will be able to arrive at the completed table below:. Going back to the question above, which of these combinations do you think have better chances of getting 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.

## Formula To Predict Lottery Numbers

In the following tables, I will be comparing Expected frequencies with Observed frequencies.Lottery mathematics is used to calculate probabilities of winning or losing a lottery game.

It is based heavily on combinatoricsparticularly the twelvefold way and combinations without replacement. If the six numbers on a ticket match the numbers drawn by the lottery, the ticket holder is a jackpot winner— regardless of the order of the numbers. The probability of this happening is 1 in 13, The chance of winning can be demonstrated as follows: The first number drawn has a 1 in 49 chance of matching. When the draw comes to the second number, there are now only 48 balls left in the bag, because the balls are drawn without replacement.

So there is now a 1 in 48 chance of predicting this number. Thus for each of the 49 ways of choosing the first number there are 48 different ways of choosing the second. However; the order of the 6 numbers is not significant. That is, if a ticket has the numbers 1, 2, 3, 4, 5, and 6, it wins as long as all the numbers 1 through 6 are drawn, no matter what order they come out in.

Dividing 10, by gives 13, also written as 49! This yields a final formula of.

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The range of possible combinations for a given lottery can be referred to as the "number space". The numerator equates to the number of ways to select the winning numbers multiplied by the number of ways to select the losing numbers.

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The total number of combinations giving that result is, as stated above, the first number multiplied by the second. The generalisation of this formula is called the hypergeometric distribution. When a bonus number is included, the adjusted odds are: . Many lotteries have a Powerball or "bonus ball". If the powerball is drawn from a pool of numbers different from the main lottery, the odds are multiplied by the number of powerballs.

For example, in the 6 from 49 lottery, given 10 powerball numbers, then the odds of getting a score of 3 and the powerball would be 1 in Another example of such a game is Mega Millionsalbeit with different jackpot odds. Where more than 1 powerball is drawn from a separate pool of balls to the main lottery for example, in the EuroMillions gamethe odds of the different possible powerball matching scores are calculated using the method shown in the " other scores " section above in other words, the powerballs are like a mini-lottery in their own rightand then multiplied by the odds of achieving the required main-lottery score. 