Risk Assessment Slides Calculated Risks
Mastering Expected Value
"If we played this game 1,000 times, what would the average outcome be?"
The Big Idea
Expected Value (E)
The theoretical long-run average of a random variable across many trials.
Interpretation
It's not what will happen once ; it's what happens on average in the long run.
The Formula
\[ E(X) = \sum x \cdot P(x) \]
\(x\) = Outcome
\(P(x)\) = Probability
The Blueprint
01
List Outcomes
Identify all possible results (\(x\)). Don't forget negative values for losses!
02
Find Probabilities
Assign a probability (\(P(x)\)) to each outcome. Sum must equal 1.
03
Multiply & Add
Multiply outcome by probability, then add them all together.
Outcome
$10
Probability
0.20
Product
$2.00
Real World Case
The "Quick-Fix" Warranty
Cost: $50/year | Max Benefit: $600 repair
Risk Analyst View
Scenario Profit/Loss (x) Prob. P(x) x · P(x) No Repair Needed -$50 0.95 -$47.50 Full Repair Paid +$550 0.05 +$27.50 Total Expected Value (EV): -$20.00
"On average, the customer expects to lose $20 per year to have this peace of mind."
Interpreting Results
The "Long Run" Sentence
"If this event is repeated many times, the average of all outcomes will be approximately EV value."
USE THIS EVERY TIME
DON'T SAY:
"I will definitely lose $20 next year."
DO SAY:
"Over many years, I expect to spend an average of $20/year."
Calculated Risks Worksheet Calculated Risks
Expected Value & Decision Analysis
Name:
Date:
Formula
\[ E(X) = \sum x \cdot P(x) \]
Multiply each outcome (x) by its probability (P(x)), then sum them up.
Part 1: The Basics
1. Match the term to its meaning:
A
Expected Value
B
Random Variable
____ A variable whose numerical value is determined by chance.
____ The long-run average outcome of a random process.
Part 2: The Carnival Challenge
In a game called "Spinstar", you pay $2 to spin a wheel.
Outcome (x) Prob. P(x) Product: \(x \cdot P(x)\) +$8.00 0.10 $0.00 0.30 -$2.00 0.60 Total Expected Value:
Interpretation Practice
Fill in the blanks using your result from above:
"If a person plays Spinstar many times, they should expect to lose an average of ________ per spin in the long run."
Part 3: Protection Plans
A electronics store sells a laptop protection plan for $100. If the laptop breaks (5% chance), the store pays $1,200 for a replacement (Net cost to store: -$1,100). If it doesn't break (95% chance), the store keeps the $100 (Net profit: +$100).
Calculate the Expected Value for the Store:
Checklist
Set up table
\(x \cdot P(x)\)
Sum it up
Final Interpretation:
Write a complete sentence describing what this value means for the store.
____________________________________________________________________________________________________
Why doesn't the expected value usually match any of the actual outcomes?
Remember: The mean of a dice roll is 3.5, even though you can't actually roll a 3.5!
Value Check Exit Ticket Value Check
Quick Assessment: Expected Value
Name:
1
The Mystery Box
A raffle sells tickets for $5. One ticket wins a $100 prize (Net: +$95). The probability of winning is 0.02. All other tickets win nothing (Net: -$5).
Calculation Area
x P(x) x·P(x) +$95 0.02 -$5 0.98
The Result
EV = ________________
Interpretation
Complete the sentence:
Over many trials of this raffle, the average outcome per ticket is...
2
Risk Decision
If a game has an expected value of -$0.50, should you play it once if you want to win money? Why or why not?
Self-Reflection
How confident do you feel about interpreting "the long run"?
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Risk Analysis Teacher Guide Intervention Guide
Expected Value & Probability Distributions
Target Group
Tier 2 Small Group (3-5 Students)
Standard
CO HS.S-MD.A.2
Duration
30-45 Minutes
1. The Hook (5 Mins)
Discussion Prompt: "If you play a game where you lose $1 every time, but win $100 once in every 50 games, is that a 'good' game? How do we know?"
"Listen for students who mention 'averaging it out' or 'overall gain.' This is the intuitive seed of expected value."
Key Misconception
Students often think the Expected Value is a "prediction" of what will happen in a single trial. Emphasize that EV is often a number that is impossible to get in one turn (e.g., $0.50 gain).
2. Guided Modeling (15 Mins)
Use the Risk Assessment Slides. Focus on the Table Method:
Visualization
Show how the probability "weights" the outcome. A high probability makes that outcome more influential on the mean.
Net Gain
Ensure students subtract the cost to play from the winnings before calculating. (Example: $10 win - $2 cost = +$8 outcome).
Sentence Frame
Strictly enforce the "In the long run..." phrasing. It bridges the gap between calculation and meaning.
Diagnostic Error Analysis
If Student Does This... It Means They... Try This Intervention... Adds all probabilities to a number > 1 Confused about distribution basics. Use a pie chart visual to show "100% of the possibilities." Forgets negative signs for losses Treats money only as absolute values. Use a number line to show money "leaving" the pocket vs. "entering." Interprets -$0.10 as "I'll lose a dime next turn" Lack of "Long Run" conceptualization. Ask: "Can you actually lose exactly one dime in this game?"
Progress Monitoring
Use the Value Check Exit Ticket. Look for 100% accuracy in table setup and "many trials" keywords in the written response.
80%
Mastery Goal
Answer Key: Calculated Risks Worksheet
Part 2 (Spinstar):
Products: 0.80, 0.00, -1.20
EV = -$0.40
Part 3 (Store):
Products: -55.00, 95.00
EV = +$40.00 (Profit for store)
