Expected Value

Probability Theory
definition

Grade 9-12

The long-run average outcome of a random process, calculated as the sum of each outcome times its probability. Expected value is the foundation of rational decision-making under uncertainty.

Definition

The long-run average outcome of a random process, calculated as the sum of each outcome times its probability.

💡 Intuition

If you played a game forever, expected value is your average result per play. Positive EV = profitable long-term. Negative EV = you'll lose over time. It's the mathematical way to evaluate risky decisions.

🎯 Core Idea

Expected value is the long-run average outcome per trial, calculated by summing each outcome multiplied by its probability. It guides rational decisions under uncertainty.

Example

Lottery: \1 ticket, 1/1000 chance of \500. EV = \frac{1}{1000}(500) + \frac{999}{1000}(-1) = 0.50 - 0.999 = -0.499 You lose ~50¢ per ticket on average.

🌟 Why It Matters

Expected value is the foundation of rational decision-making under uncertainty. It's used in gambling, insurance, investment, and game theory.

🚧 Common Stuck Point

Students confuse the expected value with the most likely outcome. Expected value is a long-run average; it may not even be a possible single outcome.

⚠️ Common Mistakes

  • Confusing EV with most likely outcome
  • Ignoring EV for emotional decisions
  • Not accounting for all outcomes

Frequently Asked Questions

What is Expected Value in Statistics?

The long-run average outcome of a random process, calculated as the sum of each outcome times its probability.

Why is Expected Value important?

Expected value is the foundation of rational decision-making under uncertainty. It's used in gambling, insurance, investment, and game theory.

What do students usually get wrong about Expected Value?

Students confuse the expected value with the most likely outcome. Expected value is a long-run average; it may not even be a possible single outcome.

What should I learn before Expected Value?

Before studying Expected Value, you should understand: probability basic, weighted average.

How Expected Value Connects to Other Ideas

To understand expected value, you should first be comfortable with probability basic and weighted average.

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