The formula
E(X) = Σ (value × probability)
What it means
Each outcome contributes its value multiplied by the probability of it happening, and the expected value is the sum of all those contributions. It can be a number that is not on the list at all: the expected value of a die is 3.5, and no die ever rolls 3.5.
How to work it out by hand
- Write each outcome with its value and its probability
- Multiply each value by its probability as a decimal
- Add all the products together
- Check that the probabilities add up to a hundred
What is worth knowing
Expected value is a very long-run average and says nothing about a single go. A straight-up roulette bet has an expected value of −0.027 per unit staked, and yet most of the time you lose the whole unit and occasionally you win thirty-five. That gap between the average and what actually happens is the standard deviation, and it is what lets a game with a slightly negative expected value ruin or enrich someone in the short run. Which is why, for decisions that will not be repeated many times — selling a house, changing jobs — expected value is only part of the answer.