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Expected Value Calculator

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About This MicroSim

Ready to crack this nut? This interactive calculator helps you compute the expected value (also called the mean) of a discrete random variable. Enter your own values and probabilities, or use the pre-loaded die roll example to see how the calculation works step-by-step.

The expected value tells you what you'd expect the average outcome to be if you repeated the random process many, many times. It's your best prediction for the long-run average!

How to Use

  1. Click any cell in the table to edit it
  2. Type a number and press Enter to confirm
  3. Tab to move to the next cell
  4. Watch the contribution column update automatically (x * P(X=x))
  5. Check that your probabilities sum to 1 (valid distribution indicator)
  6. See the expected value calculated at the bottom

Preset Buttons

  • Die Roll: Load the standard fair die distribution (values 1-6, each with probability 1/6)
  • Clear All: Start fresh with empty cells
  • Add Row: Add another outcome (up to 8 total)
  • Remove Row: Remove the last row

Key Concepts

  • Expected Value E(X): The weighted average of all possible values, where each value is weighted by its probability
  • Formula: E(X) = Sum of [x * P(X = x)]
  • Valid Distribution: Probabilities must be between 0 and 1, and must sum to exactly 1
  • Long-Run Average: Expected value is what you'd expect the average to be over many trials

Learning Objectives

After using this MicroSim, you'll be able to:

  • Calculate expected value using the formula E(X) = Sum of [x * P(X=x)]
  • Verify that a probability distribution is valid
  • Understand that expected value represents a long-run average
  • Recognize that expected value doesn't have to equal any possible outcome

Example: Insurance Company Profit

Try entering this example:

Value (x) P(X = x)
250 0.999
-99750 0.001

This represents an insurance company's profit from a $250 policy with a 0.1% chance of paying out $100,000. What's the expected profit per policy?

Sylvia Says

"The expected value is like a balance point - it's the center of your probability distribution. For a fair die, E(X) = 3.5, which isn't even a possible roll! That's okay - it's the long-run average, not a prediction for any single roll."

Lesson Plan

Introduction (5 minutes)

Ask: "If you play a game many times, how much would you expect to win on average?" Introduce the idea that we can calculate this precisely using expected value.

Guided Exploration (10 minutes)

  1. Start with the die roll example - verify E(X) = 3.5
  2. Modify probabilities to create an unfair die - how does E(X) change?
  3. Create a simple game: win $10 with P=0.4, lose $5 with P=0.6
  4. Calculate expected value - is this a fair game?

Practice Activities

  1. Insurance Problem: Model an insurance policy and find expected profit
  2. Lottery Problem: Model a lottery ticket (cost $2, win $1M with P=0.000001, etc.)
  3. Design a Fair Game: Create a game where E(X) = 0

Assessment Questions

  1. A game costs $5 to play. You win $20 with probability 0.2 and $0 otherwise. What is E(X) for your net winnings? Should you play?
  2. Why can the expected value of rolling a die be 3.5 when you can't actually roll a 3.5?
  3. If E(X) = 0 for a game, what does that tell you about the game?

Embedding This MicroSim

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<iframe src="https://dmccreary.github.io/statistics-course/sims/expected-value-calculator/main.html" height="552px" scrolling="no"></iframe>

Technical Notes

  • Built with p5.js 1.11.10
  • Uses canvas-based controls for iframe compatibility
  • Keyboard input for editing cells
  • Real-time validation of probability distribution
  • Drawing height: 450px, Control height: 100px

References

  • Chapter 13: Random Variables
  • Concepts: Expected Value, Calculating Expected Value, Probability Distribution, Valid Distribution