Teacher Guide: Expected Value
Learn how to calculate the average outcome of a random event over many trials.
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Class quiz
10 questions on Probability Distributions. Students join with a name, you see everyone's score.
For Teachers
- Define expected value as the long-run average of a random experiment
- Calculate expected value using the formula
- Interpret expected value in real-world contexts (gambling, insurance, investments)
- Distinguish between expected value and actual outcomes
- Use expected value to make informed decisions under uncertainty
- • Understanding of basic probability (outcomes, events, probability of an event)
- • Ability to work with fractions and decimals
- • Familiarity with weighted averages (helpful but not required)
- 1. Why do casinos always make money in the long run, even though some players win?
- 2. If a lottery has negative expected value, why do people still play?
- 3. How might insurance companies use expected value to set their prices?
- 4. Would you take a bet with positive expected value if there was a chance of losing everything?
Expected value tells you what will happen next
Higher expected value is always better
Outcomes with probability 0 don't matter
For Struggling Students:
- • Start with equally likely outcomes (fair die, fair coin)
- • Use physical dice to demonstrate averaging over many rolls
- • Focus on the concept before introducing the formula
- • Use simple whole-number payoffs initially
For On-Level Students:
- • Calculate expected value for games with unequal probabilities
- • Determine if games are fair, favor the player, or favor the house
- • Work with both positive and negative outcomes (wins and losses)
For Advanced Students:
- • Analyze real casino games (roulette, blackjack basic strategy)
- • Explore variance and standard deviation alongside expected value
- • Investigate the law of large numbers through simulation
- • Calculate expected value for multi-stage games
- S-MD.A.2 (CCSS.MATH.CONTENT.HSS.MD.A.2)
Calculate the expected value of a random variable; interpret it as the mean of the probability distribution
- S-MD.B.5 (CCSS.MATH.CONTENT.HSS.MD.B.5)
Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values
- S-MD.B.6 (CCSS.MATH.CONTENT.HSS.MD.B.6)
Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator)
- visualInteractive Dice Simulator
Roll dice many times and watch the average converge to expected value
- activityCasino Game Analysis
Calculate expected value for different casino games
- worksheetIs This Game Fair?
Analyze games and determine if they favor the player or house
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Definition
Worked Examples
What is the expected value when rolling a fair six-sided die?
List all outcomes and their probabilities
Outcomes: . Each has probability → 6 equally likely outcomes
Apply the expected value formula
→ Sum of weighted outcomes
Simplify the calculation
→
Answer: . On average, you expect to roll 3.5 (even though you can never actually roll 3.5)!
Common Mistakes
Forgetting that expected value can be a non-integer (like 3.5 on a die)
Why it's wrong: Expected value is a theoretical average, not an actual outcome. You cannot roll 3.5, but over many rolls, your average will approach 3.5.
Correct: Understand that expected value represents the long-run average, not a specific outcome you will get.
Using equal probabilities when outcomes are not equally likely
Why it's wrong: A biased coin or weighted die has different probabilities for each outcome.
Correct: Always check if outcomes are equally likely. Use the actual probability for each outcome.
Forgetting to subtract costs when calculating expected profit
Why it's wrong: Winning 10 euros is not profit if you paid 15 euros to play!
Correct:
Adding probabilities instead of multiplying outcome by probability
Why it's wrong: The formula is , not .
Correct: Each outcome must be multiplied by its probability, then sum all the products.
Why It Matters
- Casino games: Casinos set payouts so that the expected value favors the house
- Insurance: Companies calculate expected payouts to set premium prices
- Investments: Investors compare expected returns to make decisions
- Game theory: Expected value helps determine optimal strategies
- Quality control: Manufacturers predict average defect rates
Real World Applications
Casino Games and House Edge
Casinos design games so that the expected value always favors the house. In roulette, the house edge means players lose about 2.7% (European) or 5.3% (American) of their bets on average.
Example:
In European roulette, betting 10 euros on red gives: euros. You lose 27 cents on average per 10-euro bet.
A game pays 35 times your bet if you guess the exact number on a roulette wheel (37 numbers total). You bet 1 euro.
What is the expected value of this bet?
Step 1: Write the mathematical expression
Calculate:
Investment Decisions
Investors use expected value to compare different investment options by weighing potential returns against their probabilities.
Example:
Stock A has a 60% chance of gaining 20% and a 40% chance of losing 10%. expected return.
Investment B has a 30% chance of gaining 50% and a 70% chance of gaining 5%.
What is the expected return of Investment B?
Step 1: Write the mathematical expression
Calculate:
Game Show Strategy
Game show contestants use expected value to decide whether to keep a guaranteed prize or risk it for a chance at something bigger.
Example:
You have won 1000 euros. You can keep it or flip a coin: heads wins 2500 euros, tails wins nothing. euros. Mathematically, you should flip!
You won 500 euros. You can keep it OR spin a wheel: 25% chance of 3000 euros, 75% chance of 0 euros.
What should you do based on expected value?
Step 1: Write the mathematical expression
Calculate:
Key Takeaways
- 1Expected value is the long-run average of a random experiment repeated many times
- 2Formula: (sum of each outcome times its probability)
- 3Expected value can be a number you cannot actually get (like 3.5 on a die)
- 4Positive expected value means you gain on average; negative means you lose
- 5To find expected profit, subtract costs from expected winnings
Frequently Asked Questions
Why is the expected value of a die 3.5 if I can never roll 3.5?
How is expected value different from the mode or median?
Can expected value be negative?
Should I always choose the option with highest expected value?
Glossary
- Expected value
- The long-run average outcome of a random experiment, calculated as the sum of each outcome times its probability. Notation:
- Random variable
- A variable whose value depends on the outcome of a random event
- Probability distribution
- A list of all possible outcomes and their probabilities
- Weighted average
- An average where some values contribute more than others, based on their weights (here, probabilities)
- House edge
- The casino's expected profit as a percentage of each bet