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How to Calculate Lottery Probability

Updated on December 13, 2016
Source

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:

No. of Matching Nos.
Prize Money (in Php)
Prize Money (in $)
6
minimum of 30 million
~700,000
5
150,000
~3,500
4
2,000
~47
3
150
~4

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 Permutations, where order is important. | Source
This is the formula for Combinations, where order is not important.
This is the formula for Combinations, where order is not important. | Source

, 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:

Source

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:

Source

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:

Source

If we write this in a more general form, we get:

Source

, 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)
0
C(6,0) * C(49,6)/C(55,6)
0.48237
2.07308
1
C(6,1) * C(49,5)/C(55,6)
0.39466
2.53777
2
C(6,2) * C(49,4)/C(55,6)
0.10963
9.12158
3
C(6,3) * C(49,3)/C(55,6)
0.01271
78.67367
4
C(6,4) * C(49,2)/C(55,6)
0.00060
1643.40561
5
C(6,5) * C(49,1)/C(55,6)
0.00001
98604.33673
6
C(6,6) * C(49,0)/C(55,6)
0.00000003
28989675

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!

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    • dezalyx profile image
      Author

      dezalyx 3 days ago from Philippines

      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.).

    • profile image

      jazer 4 days ago

      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.

    • profile image

      Kgomotso Comfort 8 weeks ago

      I just want know how can I calculate the probabilities of 6/49

    • dezalyx profile image
      Author

      dezalyx 3 months ago from Philippines

      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.

    • profile image

      Edgard 3 months ago

      Dear dezalix,

      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]

      Samples Average

      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

      Regards

      Edgard

    • dezalyx profile image
      Author

      dezalyx 3 months ago from Philippines

      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.

    • profile image

      Edgard 3 months ago

      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 profile image
      Author

      dezalyx 3 months ago from Philippines

      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]

    • profile image

      Edgard 3 months ago

      ¿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 profile image

      Steve Ho Ong 18 months ago from Dumaguete City

      This article is very clear because it is a step by step procedure.