What expected value means
The expected value is the long-run average of a random outcome: multiply each outcome by its probability and add them up, E(X) = Σ x × P(x). See expected value explained.
Worked example: a raffle
A €5 ticket has a 1% chance of a net €95 win, a 5% chance of a net €5 win and a 94% chance of losing €5. E(X) = 95 × 0.01 + 5 × 0.05 − 5 × 0.94 = 0.95 + 0.25 − 4.70 = −€3.50. On average you lose €3.50 per ticket, and the standard deviation of €10.14 shows how much single tickets vary.
Variance and standard deviation
Var(X) = Σ x² P(x) − E(X)². Two bets can have the same expected value with very different risk. A fair die has E(X) = 3.5 with SD 1.71, and 3.5 is not even a possible roll.
For specific distributions, use the binomial or Poisson calculators.
Frequently asked questions
How do you calculate expected value?
Multiply each outcome by its probability and add the results.
Can an expected value be impossible?
Yes. A die roll has an expected value of 3.5, which can never occur. It is a long-run average, not a prediction.
What does a negative expected value mean?
On average you lose per play. Most lotteries and casino games have a negative expected value for the player.