Expected Value

Probability Theory
definition

Grade 9-12

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

Definition

The expected value of a random variable is the long-run average outcome of a random process, calculated as the weighted sum of each possible outcome times its probability. It represents what you would earn or lose on average per trial if the process were repeated infinitely many times.

💡 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.

Formula

E(X) = \sum x \cdot P(x)

🌟 Why It Matters

Expected value is the mathematical foundation of rational decision-making under uncertainty. It is used in gambling odds, insurance premium pricing, stock portfolio valuation, and game theory strategy.

💭 Hint When Stuck

To calculate expected value, list every possible outcome and its probability. Multiply each outcome by its probability, then add up all the products: E(X) = \sum x_i \cdot P(x_i). Check that your probabilities sum to 1 before computing. The result tells you the average outcome per trial over the long run.

Formal View

For a discrete random variable X with outcomes x_1, x_2, \ldots, x_k and probabilities p_1, p_2, \ldots, p_k, the expected value is E[X] = \sum_{i=1}^{k} x_i \, p_i. For a continuous random variable with density f(x), E[X] = \int_{-\infty}^{\infty} x \, f(x) \, dx.

🚧 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 expected value of a random variable is the long-run average outcome of a random process, calculated as the weighted sum of each possible outcome times its probability. It represents what you would earn or lose on average per trial if the process were repeated infinitely many times.

What is the Expected Value formula?

E(X) = \sum x \cdot P(x)

When do you use Expected Value?

To calculate expected value, list every possible outcome and its probability. Multiply each outcome by its probability, then add up all the products: E(X) = \sum x_i \cdot P(x_i). Check that your probabilities sum to 1 before computing. The result tells you the average outcome per trial over the long run.

How Expected Value Connects to Other Ideas

To understand expected value, you should first be comfortable with probability basic and weighted average. Once you have a solid grasp of expected value, you can move on to standard deviation intro.

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