Expected Value Formula

The Formula

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

When to use: 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.

Quick 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.

What This Formula Means

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.

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.

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.

Worked Examples

Example 1

medium
A game costs \2 to play. You roll a fair die: if you roll a 6, you win \10; otherwise, you win nothing. Find the expected value per game.

Solution

  1. 1
    Step 1: P(6) = \frac{1}{6}, winnings = \10 - \2 = \8 net. P(\text{not 6}) = \frac{5}{6}, winnings = \0 - \2 = -\2 net.
  2. 2
    Step 2: E(X) = \frac{1}{6}(8) + \frac{5}{6}(-2) = \frac{8}{6} - \frac{10}{6} = -\frac{2}{6} \approx -\$0.33.
  3. 3
    Step 3: On average, you lose about 33 cents per game.

Answer

E(X) \approx -\$0.33 per game.
Expected value is the long-run average outcome. A negative expected value means the game favours the house — on average, you lose money over many plays.

Example 2

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A raffle sells 200 tickets at \5 each. There is one prize of \500. Find the expected value for a ticket buyer.

Common Mistakes

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

Why This Formula 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.

Frequently Asked Questions

What is the Expected Value formula?

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.

How do you use the Expected Value formula?

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.

Why is the Expected Value formula important in Statistics?

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.

What do students 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 the Expected Value formula?

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

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