Lotteries: How to Find Expected Values for Games of Chance

 Lotteries: How to find the expected value of games of chance

Understanding lotteries & expected values

Megan bought a lottery ticket together with her friends. The ticket requires Megan and her friends to select 4 numbers between 1-10. However, her friends feel that this is not a great deal. They are keen to try another lottery game in which they choose 6 numbers from 1 through 48. They feel like they have a better chance at winning. Each week, they purchase 1 ticket. What ticket should they buy?

It is a common statistical problem: What's my chance of winning the lottery. You will learn all about different lotteries, and how to determine the expected value of each.

First, let's talk about expected value. It is the number expected to succeed in an experiment. Also, how likely are you to win? The formula for expected value (n*P) is the number of trials and the probability of success on each trial. We don't know how likely we are to succeed, but we know how many successful outcomes we have: 1. That is, there is only one winning ticket. We will write our problem as: n *P = 1, and then rearrange our formula to calculate our probability, which is: P=1/n.

If each ticket contained one number and 500 people bought lottery tickets, the problem would be: 1/500, or 0.2%. Yikes! It's a slim chance.

Lotteries are more complicated than that. Let's examine two types of lotteries.

Autoplay Speed Normal Save Timeline

Video Quiz Course

11K views

You can choose 4 lotteries

Megan and her friend played a first game in which she had to choose four numbers between 1-10. In this game, the order of the numbers doesn't matter. Each number can only been chosen once.

To solve this particular probability problem, we can use a combination method. Megan and her friends don't care what numbers they choose. The probability will always remain the same.

Before you can calculate the possible combinations of numbers, it is important to understand the concept combination. Let's assume you were drawing cards using a regular deck. What are your chances of drawing a 6 or more of hearts, 4 of clubs, 7 of diamonds, a 4 and 7 of clubs, or a king of spades from a regular deck of cards? This is the same problem Megan had with her lottery ticket. First, you may be aware that the chance of drawing a 6 of heart is 1 out 52. This is because there is only one six-of-a-kind of heart and 52 total cards. Because you never replace the first card, your next probability is 1 out of 51. You can draw six hearts from any deck regardless of whether you draw the 6 hearts first, 2nd, 3rd or 4th. Also, you must account for all the other cards in the scenario: the 4 clubs, 7 of the diamonds, and the king of spades.

There are so many options it can become confusing. This problem can be solved using the combination formula. Combination formula uses factorials to calculate the possible combinations for all possible outcomes. This is how the combination formula looks:

This formula, which is also called a factorial in mathematics, uses an exclamation punct. Except for combinations, statistics and probability don't use factorials often. You will need a graphing tool to calculate factorial values for large numbers. You can find more information about factorials in our other lessons.  Learn How to Win the Lottery by Using Analysis Based on the Odds