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Teacher Guide: Expected Value

Learn how to calculate the average outcome of a random event over many trials.

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Probability Distributions. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define expected value as the long-run average of a random experiment
  • Calculate expected value using the formula E(X)=∑xi⋅P(xi)E(X) = \sum x_i \cdot P(x_i)
  • 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
Prerequisites
  • • 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)
Discussion Starters
  • 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?
Common Misconceptions

Expected value tells you what will happen next

Remediation: Emphasize that expected value is about long-run averages, not individual outcomes. Use dice rolling simulation to show variation in short runs but convergence over many trials.

Higher expected value is always better

Remediation: Introduce the concept of risk. A 50% chance of winning 1 million vs guaranteed 400,000 euros has higher expected value, but most people prefer the certainty. Discuss risk aversion.

Outcomes with probability 0 don't matter

Remediation: Technically correct, but students sometimes assign 0 probability to unlikely events. Rare events still contribute to expected value.

Differentiation Ideas

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
Standards Alignment
  • 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)

Lesson Resources
  • 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.

Definition

The expected value (also called the mean or expectation) of a random variable is the long-run average value you expect to get if you repeat an experiment many times.
For a discrete random variable XX with outcomes x1,x2,…,xnx_1, x_2, \ldots, x_n and corresponding probabilities P(x1),P(x2),…,P(xn)P(x_1), P(x_2), \ldots, P(x_n):
E(X)=x1⋅P(x1)+x2⋅P(x2)+⋯+xn⋅P(xn)E(X) = x_1 \cdot P(x_1) + x_2 \cdot P(x_2) + \cdots + x_n \cdot P(x_n)
Or more compactly:
E(X)=∑i=1nxi⋅P(xi)E(X) = \sum_{i=1}^{n} x_i \cdot P(x_i)
Key insight: Expected value is a weighted average where each outcome is weighted by its probability.

Worked Examples

What is the expected value when rolling a fair six-sided die?
1

List all outcomes and their probabilities

Outcomes: 1,2,3,4,5,61, 2, 3, 4, 5, 6. Each has probability 16\frac{1}{6} → 6 equally likely outcomes
2

Apply the expected value formula

E(X)=1⋅16+2⋅16+3⋅16+4⋅16+5⋅16+6⋅16E(X) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6} → Sum of weighted outcomes
3

Simplify the calculation

E(X)=16(1+2+3+4+5+6)=216E(X) = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6) = \frac{21}{6} → 216=3.5\frac{21}{6} = 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: E(profit)=E(winnings)−costE(\text{profit}) = E(\text{winnings}) - \text{cost}

Adding probabilities instead of multiplying outcome by probability

Why it's wrong: The formula is x⋅P(x)x \cdot P(x), not x+P(x)x + P(x).

Correct: Each outcome must be multiplied by its probability, then sum all the products.

Why It Matters

Expected value is one of the most powerful concepts in probability and statistics:
  • 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
Understanding expected value helps you make rational decisions when outcomes are uncertain!

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: E=10⋅1837+(−10)⋅1937≈−0.27E = 10 \cdot \frac{18}{37} + (-10) \cdot \frac{19}{37} \approx -0.27 euros. You lose 27 cents on average per 10-euro bet.

1Try It Yourself

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: E=35⋅137+(−1)⋅3637E = 35 \cdot \frac{1}{37} + (-1) \cdot \frac{36}{37}

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%. E(return)=0.20⋅0.60+(−0.10)⋅0.40=0.08=8%E(\text{return}) = 0.20 \cdot 0.60 + (-0.10) \cdot 0.40 = 0.08 = 8\% expected return.

2Try It Yourself

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: E=0.50⋅0.30+0.05⋅0.70E = 0.50 \cdot 0.30 + 0.05 \cdot 0.70

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. E(flip)=2500⋅0.5+0⋅0.5=1250E(\text{flip}) = 2500 \cdot 0.5 + 0 \cdot 0.5 = 1250 euros. Mathematically, you should flip!

3Try It Yourself

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: E(spin)=3000⋅0.25+0⋅0.75E(\text{spin}) = 3000 \cdot 0.25 + 0 \cdot 0.75

Key Takeaways

  • 1Expected value E(X)E(X) is the long-run average of a random experiment repeated many times
  • 2Formula: E(X)=∑xi⋅P(xi)E(X) = \sum x_i \cdot P(x_i) (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?

Expected value is the average over many trials, not a single outcome. If you roll a die 1000 times and average all results, you'll get very close to 3.5. It's a theoretical center of the distribution.

How is expected value different from the mode or median?

Expected value (mean) weights outcomes by probability. The mode is the most likely outcome, and the median is the middle value. For a fair die: mean = 3.5, mode = none (all equally likely), median = 3.5.

Can expected value be negative?

Yes! If losses outweigh gains on average, the expected value is negative. Most gambling games have negative expected value for players, which is why casinos are profitable.

Should I always choose the option with highest expected value?

Not necessarily. Expected value ignores risk tolerance. Would you bet your entire savings on a coin flip that doubles or loses it all? The expected value is positive, but most people wouldn't take that risk.

Glossary

Expected value
The long-run average outcome of a random experiment, calculated as the sum of each outcome times its probability. Notation: E(X)E(X)
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

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