Written by Dackohn; last reviewed November 2026.
Expected value is the long-run average of a random outcome: multiply each outcome by its probability and add them up. A fair die has E(X) = 3.5. A €5 raffle ticket in the example below has E(X) = −€3.50, so you lose that much per ticket on average.
The formula
E(X) = Σ x × P(x)
A die roll
Each face has probability 1/6, so E(X) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5. You can never roll 3.5. Expected value is not what you expect on one roll, but what the average approaches over many rolls. The law of large numbers is what makes that true.
A raffle
| Outcome | Net gain x | P(x) | x × P(x) |
|---|---|---|---|
| Big prize | €95 | 0.01 | 0.95 |
| Small prize | €5 | 0.05 | 0.25 |
| Nothing | −€5 | 0.94 | −4.70 |
| Total | 1 | −3.50 |
Roulette
On a European wheel, a €1 bet on red wins €1 with probability 18/37 and loses €1 with probability 19/37. E(X) = 18/37 − 19/37 = −1/37 = −€0.027. Any single spin can go either way, but over many spins the casino keeps about 2.7% of what is bet.
Risk: why the mean is not enough
Two choices can have the same expected value and very different risks. A certain €50 and a coin flip for €100 both have E(X) = €50, yet many people prefer the certain money. Variance measures that spread: Var(X) = Σ x² P(x) − E(X)². The raffle has a variance of 102.75, so an SD of €10.14, about three times the size of the expected loss.
Useful rules
- Sums: E(X + Y) = E(X) + E(Y), even when X and Y are dependent.
- Multiples: E(aX + b) = a × E(X) + b.
- Binomial: E(X) = np. For 100 tosses of a fair coin, that gives 50 heads.
- Poisson: E(X) = λ.
Common mistakes
- Probabilities that do not sum to 1. Check first.
- Forgetting the cost, and using prizes instead of net gains.
- Confusing the expected value with the most likely outcome. In the raffle the most likely result is losing €5, not losing €3.50.
Try it yourself
Expected Value Calculator →Enter outcomes and probabilities and see the steps. Binomial Calculator →Probabilities for repeated yes/no trials.Frequently asked questions
What is expected value in simple terms?
The average result you would get if you repeated something many times.
Can expected value be negative?
Yes. It means you lose on average, as with most lottery tickets and casino bets.
Is the expected value what will most likely happen?
No. It is a long-run average, and it may not even be a possible single outcome.
Related reading: Law of large numbers · Bayes’ theorem explained · Beat the Algorithm game
Sources & further reading: NIST/SEMATECH e-Handbook of Statistical Methods · OpenStax Introductory Statistics. How we compute & validate: methodology. Written & reviewed by Dackohn; last reviewed August 2026.