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Year 12 · Statistics & Probability

Expected value

Expected value is the long-run average of a random quantity: each value times its probability, added up.

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How to do it

  1. Multiply each outcome by its probability.
  2. Add the products.
  3. For a game, subtract the cost to play.

Worked example

A random variable X has P(X=1) = 1/9, P(X=2) = 2/9, P(X=3) = 3/9, P(X=4) = 3/9. Find E(X) to two decimal places.
E(X) = Σ x·P(x) = 1×1/9 + 2×2/9 + 3×3/9 + 4×3/9 = 2.89.
Answer: 2.89

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A negative expected value means the game loses money on average.

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