Expected Value

The expected value E(X) is the long-run average outcome of a random variable: [formula] Multiply each outcome by its probability, then add. Example: A fair die roll. E = 1(frac{1}{6}) + 2(frac{1}{6}) + ... + 6(frac{1}{6}) = frac{21}{6} = 3.5 You expect an average of 3.5 per roll over many rolls.

Example: A game: win 10 with probability 0.2, lose 3 with probability 0.8. What is the expected value?

Expected value questions are essentially weighted averages. Multiply each outcome by its probability and sum. If the expected value is negative, you expect to lose money on average — the game favors the house.