Expected Value Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Expected Value.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

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

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.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

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.

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.

Worked Examples

Example 1

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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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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A spinner has outcomes: \1 (prob 0.5), \3 (prob 0.3), \$10 (prob 0.2). Find the expected value.

Example 2

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A game costs \4 to play. You win \20 with probability 0.1, \5 with probability 0.3, and \0 otherwise. Find the expected net value of one play.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

probability basicweighted average