Question

Pick any type of public lottery and find the probability of winning using combinations or permutations. Be sure to explain the rules to the lottery( how many numbers chosen, from what set of numbers, power ball numbers etc...) Then also explain how you achieved the probability you did. The lotteries could be from a different state or country as long as it is public

Answer 1

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 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:

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.