Intervention Roadmap Teacher Guide Intervention Roadmap
Expected Value Essentials • Teacher Facilitation Guide
Lesson Focus
This Tier 2 intervention is designed for students who struggle with multi-step probability calculations. We move from the concrete (a physical game) to the representational (a probability table) to the abstract (the expected value formula).
Standard
CO HS.S-MD.A.2: Calculate and interpret the expected value of a random variable.
Prerequisite
Identifying outcomes and calculating simple probabilities for a discrete random variable.
Facilitation Steps
1
The Hook (5 mins)
"If you played this game 100 times, how much money would you expect to have in the end?"
Use Slide 2 to introduce the "Lucky Spinner." Allow students to guess before teaching any math. This builds conceptual buy-in.
2
The Table Method (10 mins)
Distribute the Expected Value Blueprint . Guide students through filling out the first table. Emphasize that the product column (\(x \cdot P(x)\)) represents the "weighted contribution" of each outcome.
3
Guided Practice (15 mins)
Work through Scenario 2 on the slides. Have students use their Blueprint worksheet to follow along. Stop after each multiplication to check for accuracy.
Common Misconceptions
Outcome vs. Average: Students may think the expected value must be one of the possible outcomes (e.g., if you can win $1 or $5, the EV might be $2.50, even though you can't win exactly $2.50).Probability Errors: Ensure probabilities in the table sum to 1. If they don't, the EV will be incorrect.Interpretation: Students often think EV is what will happen on the next turn. Remind them it is the long-run average over many trials.
Progress Monitoring
Use the Progress Probe at the end of the session. Check for:
Objective Met
Needs More Support
Calculates product \(x \cdot P(x)\) correctly and sums them to find EV. Correctly uses "per play" or "average" in interpretation.
Struggles to convert probabilities to decimals or fractions. Confuses the value of \(x\) with the count of outcomes.
Value Venture Slides Value Venture
Mastering Expected Value
Small Group Intervention
The Lucky Spinner
Imagine a spinner with 4 equal sections:
Win $10 (1 section)Lose $5 (3 sections)
If you play this 100 times, will you be rich or broke?
What is Expected Value?
The Mathematical Mean
Expected Value (\(E\)) is the long-run average of a random variable.
The Long Run
It doesn't tell you what will happen once . It tells you what happens if you play thousands of times .
The Step-by-Step Table
Outcome (\(x\)) Probability \(P(x)\) Product \(x \cdot P(x)\) Win $10 \(1/4\) (0.25) \(2.50\) Lose $5 (\(-5\)) \(3/4\) (0.75) \(-3.75\) Sum (Expected Value): \(-\$1.25\)
"On average, you lose $1.25 every time you spin."
Wait... How can I lose $1.25?
You can't lose exactly $1.25 on a single spin. You either win $10 or lose $5.
Expected Value is a Theoretical Mean.
If you played 100 times, you would expect to be down $125 total.
Scenario 2: The Dice Duel
Roll one standard 6-sided die:
Roll a 6 Win $20
Roll a 4 or 5 Win $5
Roll a 1, 2, or 3 Lose $10
Let's build the table together on your worksheet!
Expected Value Blueprint Worksheet Expected Value Blueprint
Statistics Intervention • Game Theory Lab
Name:
Date:
The Strategy
To find the Expected Value (\(E\)) , we multiply each possible outcome (\(x\)) by its probability (\(P(x)\)) , then add those products together.
\[ E(X) = \sum x \cdot P(X) \]
1
The Lucky Spinner
A spinner has 4 equal sections: one Win $10 section and three Lose $5 sections. Fill in the table below.
Outcome (\(x\)) Probability \(P(x)\) Calculation: \(x \cdot P(x)\) Win $10 \(1/4\) (or 0.25) \(10 \times 0.25 = 2.50\) Lose $5 (\(-5\)) \(3/4\) (or 0.75) Multiply these... Sum of products (Expected Value):
Interpret the result:
In the long run, I can expect to...
2
Solo Challenge: The Marble Bag
A bag contains 10 marbles: 1 Gold, 4 Silver, and 5 Blue. It costs $5 to play . If you draw a Gold, you win $20. If you draw a Silver, you win $5. Blue wins nothing.
Tip: Don't forget to subtract the $5 cost to play from each "win" to find the true outcome \(x\)!
Outcome (\(x\)) Probability \(P(x)\) Product: \(x \cdot P(x)\) Final Expected Value:
Final Question:
Is this a fair game? (Fair means the Expected Value is $0). Explain your reasoning.
Progress Probe Exit Ticket Progress Probe
Targeted Assessment: Expected Value
Student Name
Date
1
Calculate
A game has the following probability distribution. Complete the table and find the Expected Value .
Outcome (\(x\)) Probability \(P(x)\) \(x \cdot P(x)\) Lose $10 (\(-10\)) 0.60 Win $20 (\(20\)) 0.40
Expected Value (\(E\)):
$
2
Interpret
In your own words, explain what your answer to Problem 1 means if a player plays this game 500 times.
Teacher Scoring
Calculated Correctly
Interpreted Correctly
Ready for Extension
