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 article 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: 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:
|No. of Matching Nos.||Prize Money (in Php)||Prize Money (in $)|
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.
, 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:
|x matches||Calculation||Probability||Odds (1/Probability)|
C(6,0) * C(49,6)/C(55,6)
C(6,1) * C(49,5)/C(55,6)
C(6,2) * C(49,4)/C(55,6)
C(6,3) * C(49,3)/C(55,6)
C(6,4) * C(49,2)/C(55,6)
C(6,5) * C(49,1)/C(55,6)
C(6,6) * C(49,0)/C(55,6)
How to Choose the Winning Numbers in Lottery
As you can see from the math in this article, 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 this topic, 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.
Andy on January 17, 2019:
if any of you had used a combi like this you would of done ok
01-06-12-13-54-55 it's all comes don't to brilliance!!!
Andy Martin on January 17, 2019:
Play the lottery hotpicks because the odds are 800/1 but only use 3 number sequences. The possibilities should prove an overall win return.
dean martin on January 17, 2019:
The lottery itself is designed from patterns then created into numbers also restricting and removing numbers also... reducing your odds even further.
So mathematics can't be applied but in someway can help with sequences or possibilities. It is not even a 50/50 odds simply because we can't use a 50.50 odds system if numbers are removed.
paul smith on January 17, 2019:
You can't use mathematics to win the lottery simple because the lottery removes numbers. The most simplest way to calculate your odds are 1/59 then 1/58 1/57 1/56 1/55 1/54
Gabriel on January 08, 2019:
According to quantum theory and the theory of everything, we are actually able to calculate the exact lottery winning numbers of a particular day with using the theory of time and space, if we know the exact day, time and location that the numbers are withdraw from the machine. nanosecond is 10 -9 s; one picosecond is 10 -12 s. So this means if our time are exactly picosecond accurate with the lottery machine we can predict the numbers. Which would be near impossible but not impossible.
Brigid on January 06, 2019:
@ Patel Ali how can permutations help with a pick 4 draw ?? Have you tried it and has it worked for you?
Patel ali on December 23, 2018:
4 digits number quite easy to predict for draw..
Example... 9269 =2057(permutation)
Are u agree with me?
roby on October 21, 2018:
what is the formula of choosing 2 numbers where these 2 number will appear in the 6 draw numbers of a lottery
Ralph on October 07, 2018:
No, probability will not be able to predict any future draw, patterns should be searched in sequences, try Tesla...
anna on August 24, 2018:
The Powerball lottery is decided every Wednesday and Saturday night by drawing five white balls out of a drum with 69 balls and one red ball out of a drum with 26 red balls. The Powerball jackpot is won by matching all five white balls in any order and the red Powerball. Each ticket costs $2.
a. [2 pts] How many possible different Powerball tickets can be purchased?
b. [1 pts] How many possible different winning Powerball tickets are there?
Sean Castleton on August 06, 2018:
Unfortanetly, the Law of Probability simply goes out the window if a lottery system is rigged, which many of them are.
Nardo on March 26, 2018:
Probability is one part of the whole thing which is true by itself. However there are also many factors(known and many unknown) which produces lottery results that happen to produce patterns, which may help in striking like few matching numbers but still ultra difficult in winning 6 matching numbers. It's still a combination of luck, for now i conclude it's combination of probability+luck (L.O.A or whatever u wanna call it)...cheers n good luck hehe
dezalyx (author) from Philippines on January 15, 2018:
From the definition of classical probability, every statistical outcome will contain elements that are equally likely to happen. This means that if you roll a 6-sided dice, it is equally probable that the outcome will be 1, 2, 3, 4, 5, or 6. Or in the case of flipping a coin, the probability of heads will be equal to the probability of tails. No magic coins, no loaded dice, all equally probable to happen.
When calculating the lottery probability in this article, this assumption is already used. This is usually the case when calculating probabilities theoretically. Having no affiliation to companies who hold the lottery, it would be difficult for me to assume that certain balls are more likely to surface because they are loaded.
In my opinion, historical data does not have any bearing on lottery outcomes. Statistically speaking, working on other probability approaches will not work because of this. Just because the number N appears frequently in lottery drawings, does not mean that the ball is more likely to surface than the number N+1 in the next drawing.
Subs on January 14, 2018:
Can the classical probability theory be used here? And please give an example
Subs on January 14, 2018:
What other probability approaches can be used?
dezalyx (author) from Philippines on September 01, 2017:
Hi, Odds. Thanks for commenting.
I did not say choosing one's birthday would contribute to one's luck. I just said you can choose any number you want if you're willing to face the fact that there's only a very small chance that you can win the jackpot prize. Heck, you could choose 1, 2, 3, 4, 5, 6 if you wanted to. The basis of this article is that the probability of each ball getting drawn is the same as any other ball in the lottery.
While I kept getting comments on how likely certain numbers will get picked over other numbers, mathematically speaking, there is no such thing as having weighted balls for lottery drawings.
While you might not have studied mathematics, you certainly were on point when you said that you should never try your luck in gambling. That's actually a mathematician's saying. Never gamble, because the house always wins.
ODDS on August 30, 2017:
I didn't study mathematics, but you were wrong when you mentioned that the numbers of the player's birthday would contribute to his luck, rather it would add up to the improbability, because certainly the player would not live enough to wait his birthday's numbers to come up!
And yeah you certainly can top up your chances even if mathematically every number may pop up randomly.
If i were to buy a ticket, i would consider not selecting numbers that are altogether evens/odds, divisible by 5 altogether etc. If the order does not matter, you will get more chances!
But you know what, never try your luck in gambling, it's made to steal your money and you end up depressed.
dezalyx (author) from Philippines on August 13, 2017:
Hi, jazer. I've been receiving a ton of comments lately asking for what the winning number will be for the next lottery draw, but the article is just about the probability of winning the lottery. I don't know about the specifics of the composition of lotto balls, since it probably differs for every company that makes them, but looking for trends when there is no factual basis will just result in you losing money. And given the odds of winning the lottery, you would be better off investing that money in something else with a higher chance of getting a profit. While I agree with the idea that maybe the composition of the balls would get it drawn more than the others, you can never bank on that, since it could be changed without you ever knowing about it (repainting the balls numbers, changing the balls entirely, having different sets of balls for different drawings, etc.).
jazer on August 13, 2017:
Probability is probability. We cannot predict the future. But for the purpose of conversation, some people used they called a "trend". A history of past result. From trend you may "guess" which number can be with higher probability. Is this true in nature? Actually, this is only my opinion. My reasoning with the trend is somewhat related to the composition of the lotto balls e.g. weight, etc, that's why some numbers were drawn more that the others. What is your opinion on this? Thanks.
Kgomotso Comfort on June 19, 2017:
I just want know how can I calculate the probabilities of 6/49
dezalyx (author) from Philippines on May 08, 2017:
Edgard, it seems to me that your question is still about the gambler's fallacy. Just because a certain number comes out approximately 6 times out of x number of drawings will not change the probability of the number being drawn again in the future.
Imagine changing the game into a simple heads or tails coin flip. The probability of getting either outcome is 1/2. Just because you get 6 heads in a row does not mean the next result would be a tail. Even if you get 100 heads in a row, the probability of getting either outcome is still going to be 1/2.
There won't be any probability equations to predict what you are looking for. If there is, the people who have discovered such equations would already be taking advantage of the lottery system.
The only thing you can do with the study is plot historical graphs on how many times a certain number has come out previously and marvel at how some numbers seem to appear more than others.
Edgard on May 05, 2017:
We really appreciate your time and answers
We are aware that there will always be an average for any set of numbers, also we can find a trend to explain any series of numbers, but these work outs are not a guaranty that we will get really meaningful results.
Our previous questions came out from the following fact:
We made a sequential review of winning numbers of 260 draws of a 6/45 lottery, to find out the number of times (frequency) that each of the 45 balls was drawn; we started with a 160 draws sample, and increased the sample with 10 draws data each time, until we got the last sample of 260 draws; with every sample of draws, we worked out the rate = [total draws of the sample / frequency of each ball in the sample] for every ball, then we worked out the average of this 45 ratios; the result clearly showed a steady trend for this average, which started at 6.63, for the 160 draws sample, to 6.55 for the 260 draws sample; of course we could work with more sample, but we just decided to stop at 260 draws sample. We initially thought this ratio means the average number of draws we expect to wait for a ball to be
drawn; but, what would be the right mining of this average (6.55), if any?, why the average has a steady trend?, and is there any probability equation to predict this average?
ratio ni = [N° draws/Freq ni]
i = [1, 45]
N° Draws of ratios Min ratio Max ratio
260 6.55 4.73 8.97
250 6.56 4.63 8.93
240 6.57 4.44 8.57
230 6.59 4.26 8.85
220 6.58 4.31 8.46
210 6.59 4.29 9.13
200 6.61 4.17 9.09
190 6.61 4.13 8.64
180 6.62 3.91 9.00
170 6.62 3.95 8.95
160 6.63 3.81 8.89
dezalyx (author) from Philippines on May 04, 2017:
No. The law of large numbers indicate that given a large number of opportunities, we should expect a certain event to happen again, no matter how unlikely it may be. For example, a set of 6 lottery numbers can get picked 2 weeks in a row. But, just because it happened, doesn't mean you should go running to buy the same exact set of numbers in the hopes of getting lucky.
Each lottery picking is still independent of each other. So the probability of picking the correct set of numbers will remain the same for each drawing. Saying that one number has the chance to be drawn every specific number of games is succumbing to gambler's fallacy (mistaken belief that just because something happens more frequently than normal during some period, it will happen less frequently in the future).
Odds represent the ratio that allow you to calculate how much money you will win on your bet in comparison to your stake. So your $10 ticket can become $23.67 if one number gets drawn. That is why you will notice that the odds keep increasing as you get more matches, meaning you get more money the more matches you get.
I hope this answers your question, Edgard.
Edgard on May 04, 2017:
Dear dezalyx, Thanks again,
Regarding the same game 6/45:
As the probability of one match is 0.42417, then Odds (1/Probability) will be 2.3678. According to probability theory and the law of large numbers, is it right to say, at least theoretically, that every 2.3678 games we should expect one match; in other words, one number has the chance to be drawn every 2.3678 games?
dezalyx (author) from Philippines on May 03, 2017:
Hi, Edgar. Just replace the numbers in the formula (from 55 to 45, 49 to 39) then solve for the answer to your question. [0.4241]
Edgard on May 03, 2017:
¿What would be the probability of one single ball to be part of the winner combination in a game where we have to pick a 6 number combination out of 45 numbers?.
Thanks in advance for your answer!
Steve Ho Ong from Dumaguete City on February 04, 2016:
This article is very clear because it is a step by step procedure.