The long-run centre of a distribution
The expected value is the probability-weighted average of a random variable's values: multiply each outcome by its probability and add. It is not a prediction of any single trial, a die's expected roll is 3.5, a face that does not exist, but the average of many trials converges to it, which is why casinos, insurers and pension funds can plan around it with confidence.
X takes 0, 1, 2 with probabilities 0.5, 0.3, 0.2: E(X) = 0(0.5) + 1(0.3) + 2(0.2) = 0.7. E(X²) = 0 + 0.3 + 0.8 = 1.1, so Var = 1.1 − 0.49 = 0.61 and SD = 0.78. A game paying this X in pounds costs a fair price of exactly £0.70 per play. Charge more and the house edges ahead by precisely the difference, every game, on average.
Probabilities that do not sum to 1. The calculator checks and warns: a distribution that accounts for only 90% of what can happen produces an "expected value" of a world that does not exist.
Frequently asked questions
Is the expected value the most likely value?
No, the mode is. Expected value is the balance point, and for skewed distributions it can sit far from any likely outcome, dragged by rare large values.
What does a negative expected value mean?
On average, a loss: every lottery ticket ever sold. Whether playing is still worth it depends on the price of the entertainment, not the arithmetic.