How to Calculate Lotto Odds
Last Updated: June 1, 2020 References
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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. 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.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…
How to Calculate Lottery Probability
About the Author
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. In order to make it more relevant to me, I decided to base it on the Grandlotto 6/55, the lottery game with the biggest prize money here in the Philippines. 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.
Rules of the Lottery Game
It is always important to find out the rules of any game before participating in it. For the Grandlotto 6/55, in order to win the jackpot prize, you have to match six numbers from a pool of 55 numbers ranging from 1-55. The initial payout is a minimum of P20 (or around $0.47). 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.
Here is a table for the prizes you can obtain:
minimum of 30 million
Some Probability Concepts
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. In the Grandlotto 6/55, order is not important because so long as you have the winning set of numbers, you can win the prize.
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.
This is the formula for Permutations, where order is important.
This is the formula for Combinations, where order is not important.
, where n! = n * (n – 1) * (n – 2) * . * 3 * 2 * 1.
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.
How to Calculate Lottery Probability for 6 Matching Numbers
So now that we know the basic concepts of permutations and combinations, let us go back to the example of Grandlotto 6/55. For the game, n = 55, the total number of possible choices. k = 6, the number of choices we can make. 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: 0.0000000344 or 0.00000344%. 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 0.0000000000479 or 0.00000000479%.
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.
How to Calculate Lottery Probability with Less Than 6 Matching Numbers
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. Because 6 are winning numbers, we have 55 – 6 = 49 balls to choose the losing numbers from. 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:
If we write this in a more general form, we get:
, where n = total number of balls in the set, k = total number of balls in the winning combination for the jackpot prize, and x = total numbers of balls matching the winning set of numbers.
If we use this formula to calculate the probability (and the odds) of winning the Grandlotto 6/55 with only x matching numbers, we get the following:
How to Choose the Winning Numbers in Lottery
As you can see from the math in this hub, the probability of winning the lottery is the same for any 6-number combination available in the Grandlotto 6/55 game. This is also applicable for other lottery games out there.
As I was researching for this hub, I came across links that said never choose numbers that are sequential, like from 1-6 or some such nonsense. There is no such secret to winning the lottery! Each number is as equally likely to come up in the draw as the next number.
If you are willing to face the very little probability of winning the lottery, I say go choose any number you want. You can base it on your birthdays, special days, anniversaries, lucky numbers, etc. Just remember that with great risk comes great reward!
Brandon on February 13, 2020:
In a game of lotto, balls are numbered 1 through to 44. They are placed in a barrel and six balls are drawn without replacement. The balls are of the same size and are equally likely to be drawn. The first five balls drawn out are numbered 34, 2, 15, 29 and 42. what is the probability that the next ball draw out will be number 26?
Rocco on July 16, 2019:
I want to determine some odds. There are 4 different variables. Column one has a 1 in 9 chance of getting what you want, column 2 has a 1 in 8 chance of getting what you want, column 3 has a 1 in 5 chance of getting what you want, and column 4 has a 1 in 5 chance of getting what you want. How do I determine the odds of getting what I want from each column in one random roll/event?
iegsaan on April 19, 2019:
i just feel this person is trying to make lottery predictions to difficult because she hasn’t figured out all lotteries strategies across the globe. any game of chance whether it be lotto , dice, cards its all a con. if you go into a casino and you good at memorizing cards they kick your ass out saying you cheating but bottom line any betting house crook the games. lotteries are the worse. think of it like this the lottery is a business and to make a profit it needs to eat your money. if they played fairly then we would have a lot more winners and lottery boards would go bankrupt . that is why they need time before the draw to run through all the tickets and choose the numbers with the least amount to payout. so predicting numbers is easy try to crook a crook much more difficult. funny how government allow this kind of white collar crimes but end up sending other people to jail for the exact same things.
Zack on January 29, 2019:
I am looking at a lottery game here in Brazil that says something abouts the odds that sounds “odd” to me. Pick 50 numbers out of 100. They say the odds of getting all 20 drawn numbers correct is the same odds of getting zero numbers correct (1:11,372,635). This doesn’t sound possible. What do you think?
Andy on January 17, 2019:
In-fact even in close proximity they did well enough.This hub is all about calculating lottery probability or odds. In order to make it relevant, I decided to base it on the Grandlotto 6/55, the lottery game with the biggest prize money here in the Philippines. There will be two different cases in the hub: the probability of winning the game with all six numbers matching, and the probability of having n numbers matching. ]]>