# Lotteries: How to Find Expected Values for Games of Chance

Lotteries — Finding the Expected Values of Chance Games

Understanding Lotteries: Expected Values

Megan will be buying a ticket for the lottery with her friends. Megan is asked by her friends to pick four numbers between 1–10. The ticket asks her and her friends to pick 4 numbers between 1 and 10. They’d like to pick another lottery game that allows them to pick 6 numbers, from 1 to 48. They feel that they have a better chance to win. Each week, they buy one ticket. They need to decide which ticket they should buy.

It’s a common statistical question: What are my chances at winning the lottery? This lesson will explain the differences between lotteries and show you how to calculate the expected value.

Let’s first discuss the expected value. This refers to the number of positive outcomes that can be expected from an experiment. This is how you can determine your chances of winning. The formula to calculate expected value is n x P. Here, n refers to the number of trials. P indicates the likelihood of success for each trial. In this instance, the probability for success is unknown, but the number and successful outcomes are known: 1. It means that there is only a single winning ticket. This means that we need to write our problem in the following format: n*P = 1 and then rewrite our formula to find our probability. It is P=1/n.

If each ticket had only 1 number and 500 people bought a ticket, our problem might look like this: 1.500 or 0.2%. Yikes! This is a small chance.

Lotteries are much more complex than this. Let’s now look at two types.

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Megan played the first game with her friends, in which she had four numbers to choose from between 1 and 10. You can choose any number once, regardless of their order.

A combination formula can be used to solve this particular probability issue. Megan and her friends may choose different numbers, but it doesn’t really matter. The probability will always be the exact same.

To calculate the many combinations of numbers possible, we need to first understand the concept called combinations. Let’s imagine that you were playing with regular cards. What are the chances that you draw a 6th of hearts, a 4th of clubs and 7th of diamonds? This is similar to Megan’s lottery ticket problem. First, it is likely that you know that drawing 6 hearts is a one in 52 chance. This is because there are only six hearts and 52 cards. You cannot replace that first card so the probability of you drawing that second card will change to 1 out 51. There are many probabilities that you will draw six of the same cards from the deck, regardless of whether the first, second or third card is drawn. In this scenario, you also need to account the following cards: 4 of clubs (7 of diamonds), and a King of Spades.

The many options can make it very confusing. We will use the combination formula to solve this problem. The combination formula is a probabilistic formula that uses factorials for the number of possible combinations in each experiment. This is how the combination equation looks like:

You might notice that the formula uses an exclamation, also known as factorials in mathematics. Probability and statistics don’t usually use factorials, except for the case of combinations. A graphing calculator is required to determine the factorial value for large numbers. Our other lessons provide more information on factorials. **Learn How to Win the Lottery by Using Analysis Based on the Odds**