# Advanced Lesson: Expected Value

In this "Cultivating Data" lesson, students will learn to calculate expected value to analyze risk.

**STANDARDS**

- High School: CCSS.Math.Content.HSS-MD.B.5b
- Grades 9–12: NCTM Data Analysis and Probability

**OBJECTIVE**

Advanced (beyond grade 8) math students will be able to:

- Calculate and interpret expected value; and
- Explore using the mean and deviation from it to draw conclusions.

**TIME REQUIRED:** 40 minutes, plus additional time for worksheet

**MATERIALS**

- Advanced Worksheet: PDF or whiteboard-friendly
- Worksheet Answer Key (PDF)
- Classroom Poster (PDF)

**ADDITIONAL RESOURCES**

**DIRECTIONS**

1. Pose the following problem to your class: A middle school volleyball team has 20 players. Four of the players are 5'4" (64") tall, four are 5'6" (66"), four are 5'8" (68"), three are 5'10" (70"), and five are 6' (72"). If the coach picked a player at random, how tall would you expect the player to be? Discuss possible solution methods with the class, including finding the mean by adding all the heights and dividing by 20.

2. Indicate that an **expected** **value** calculation is another way to determine the solution. Explain that expected value is a calculation that predicts value by summing the values of all possible outcomes, with each outcome multiplied by the probability of its occurring. Demonstrate how to calculate the expected value of the height of a volleyball player picked at random. First, make a frequency table showing the number of times a specific height is represented on the team.

3. Then, show how to calculate expected value by writing the following on the board:

20%(64") = 12.8"

20%(66") = 13.2"

20%(68") = 13.6"

15%(70") = 10.5"

25%(72") = 18.0"

Total = 68.1" (Expected Value)

First, divide the frequency of each height by the total number of players, resulting in a percentage (for example, 3 players at 70" divided by 20 players = 3/20 = .15 = 15%). Then, multiply the result by each respective height represented on the team (for example, 15%(70") = 10.5". Finally, add the products together to find the expected value.

4. Explain that the total equals the expected value of the height of a player picked at random. Further, explain how expected value is the predicted value of a future event, taking into account each possible outcome multiplied by its probability.

5. Note that in this data set, the expected value is not the height of one specific individual on the team. Indicate that this is not uncommon with statistical measures, e.g., the mean or, in some cases, the median of a group of numbers.

6. Connect expected value to mean as follows with this example: Consider the numbers 3, 3, 4, 5, 5. The mean is 4 [(3 + 3 + 4 + 5 + 5)/5]. Another way to think about this group is that 3 makes up 40% of the group (2/5), 4 makes up 20%, and 5 makes up 40%. So:

7. Give this problem to show the practical use of calculating expected value:

A town carnival includes a game of chance to benefit the local food bank. Each player pays $5 to spin a prize wheel. The wheel is divided into the following segments:

50%: No Prize

25%: $1 Prize

20%: $5 Prize

5%: $50 Prize

Ask: "Will the food bank make money on this game?" [Yes, the expected value of each play to the charity is $1.25 or $5 - (50%(0) + .25($1) + .20($5) + .05($50).] Write this calculation on the board and explain as necessary. Note that the expected value of the total profit for the evening for the food bank is $1.25 times the number of people who play the game.

Note that on any one spin of the wheel, the charity will either win $5 or $4, break even, or lose $45. However, when the outcomes are weighted by the probability of their occurrence, the result is an expected profit of $1.25. The food bank is counting on many people playing the game so that, in total, the actual outcomes of the spins approach their theoretical probabilities.

For more information about theoretical versus experimental probability, see: www.actuarialfoundation.org/probabilitychallenge/lab.

8. Distribute the Advanced Worksheet to students, then review answers with class.

**EXTENSION:** For those students with sufficient background knowledge and a strong passion for mathematics needing an additional challenge in advanced topics like standard deviation, direct them to the self-study program at: www.actuarialfoundation.org/pdf/Probstat_SG_2012.pdf, beginning on page 24.