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How to Calculate Lottery Probability

Updated on December 13, 2016
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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:

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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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