## Expected Value

#### Aligned To Common Core Standard:

**High School Statistics** - HSS-MD.A.2

**What is the Expected Value in Statistics?** - When we are exploring the possible outcomes to an event through an experiment or data that we have collected common sense tells us of an anticipated result. To make it as simplistic as possible, consider flipping a penny. There are two possible outcomes. Putting the Chaos Theory to the side, we would anticipate a fifty percent result of landing on a heads or tails result. In statistics we would refer to that anticipated result as the expected value. This is the mean or weighted average of all the possible variables. It is also referred to as the probability distribution. The basic formula that you see is compilation of the product of the probability (P(x)) of the event multiplied by the number of time the event occurs (n): (P(x) n). As the situation gets more complex, so does the formula that you use. This topic will start by exploring the most basic form of this calculation and advance to more complex applications of this skill.
This series of lessons and worksheets has students discover the best guess value of the probability-weighted average of all variables that are present in the problems.

### Printable Worksheets And Lessons

- Jake and the Roulette Wheel Step-by-step Lesson- See if you can figure out Jake's chance of success.
- Guided Lesson - Flipping a coin, the cultural show lottery, and the cost of picking chocolates.
- Guided Lesson Explanation - The probability distributions that are shown here are very commonly used on tests.
- Practice Worksheet - Sorry for the repetitive nature of these problems.
- Matching Worksheet - Match the random variables to the questions that created them.

#### Homework Sheets

These are fun. Who doesn't want to calculate their winnings?

- Homework 1 - Mike bets $2 on one of the numbers and wins $80 when the wheel lands on his number. If the wheel stops at another number, he would have lost his money.
- Homework 2 - See if you can gauge the value of one ticket.
- Homework 3 - Five thousand tickets are sold at $25 each for a charity raffle. Tickets are to be drawn at random and cash prizes are to be awarded as follows: 1 prize of $200, 4 prizes of $160, and 6 prizes of $90. What is the expected value of this raffle if you buy 1 ticket?

#### Practice Worksheets

When you actually run the numbers for some of these lottery questions, you wonder why people even buy lottery tickets.

- Practice 1 - An alphabet wheel has 26 equally likely spaced slots numbered A to Z. Sherry bets $2 on one of the numbers and wins $50. You earn $50 if it lands on your letter; otherwise you lose your bet. The random variable X assigns $50 to the wheel landing on letter chosen and -$2 to having the wheel land on any other letter. What is expected value of your winnings?
- Practice 2 - The random variable X assigns $ 100 to the wheel landing on number chosen and -$95 to having the wheel land on any other number. What is the expected value of your winnings?

#### Math Skill Quizzes

I went heavy on the spinning wheel question because you will see a lot of that on most exams.

- Quiz 1 - A hundred tickets are sold for a circus at the cost of $20 each. Some tickets have cash prizes as a part of promotional campaign. One prize of $500, three prizes of $ 250, and five prizes of $100. What is the expected value if you buy 1 ticket?
- Quiz 2 - An experiment consists of choosing 1 of 6 cookies labeled 1 through 6 in a packet at random. Let the random variable X be the cookie that turns up. What is the expected value of X?

### How to Find the Expected Value?

Expected value is a term that we come across very frequently in our daily lives and even in mathematics. Expected value refers to the result you can expect from a particular action. Students have to understand the expected value even in probability and statistics where it refers the mean value of the theoretical values that we get after repeating an experiment several times. It is the measure of central tendency. In case of normal probability distribution, the outcome plurality will be close to the expected value. The primary purpose of this value is to summarize all the information a variable hold. In case of discrete random variable, we can calculate the expected value by multiplying each outcome with its probability. The scaling of random variable effects the expected value. The basic expected value formula is; P(x) × n. Here P(x) is the probability of an outcome and n is number of times an event or experiment is taking place. The formula for calculating expected value of binomial random variable is; P(x)×X Here P(x) is the probability of success and X is the number of trials.