What the expected value tells you
The expected value of a discrete random variable X, written E(X) or μ, is the probability-weighted average of every outcome it can take. If X can take values x₁, x₂, …, xₙ with probabilities p₁, p₂, …, pₙ, then:
E(X) = Σ xᵢ · pᵢ = x₁p₁ + x₂p₂ + … + xₙpₙ
You multiply each outcome by how likely it is, then add up the products. Because it is an average over the whole distribution, the expected value is a single number that summarizes the "center of mass" of the outcomes. It is what you would converge to if you repeated the experiment many, many times and averaged the results — this is the meaning made precise by the law of large numbers.
Why the expected value is often not a possible outcome
A common misconception is that the expected value must be one of the outcomes. It usually is not. Roll a fair six-sided die: the outcomes are 1 through 6, each with probability 1/6, so E(X) = (1+2+3+4+5+6)/6 = 21/6 = 3.5. You can never roll a 3.5, yet 3.5 is the long-run average roll. The expected value is a property of the distribution, not a prediction of any single trial.
Probabilities must sum to 1
For a valid probability distribution, every pᵢ is between 0 and 1 and the probabilities sum to exactly 1 (100%). If your probabilities do not add to 1, either an outcome is missing or a value is mistyped, and the resulting E(X) will be biased. This calculator reports the total probability alongside the answer so you can check it at a glance. If you only have relative weights rather than true probabilities, divide each weight by the sum of all weights first so they normalize to 1.
Common reference values
- Fair coin, heads = 1, tails = 0: E(X) = 0.5.
- Fair six-sided die: E(X) = 3.5. A fair 20-sided die gives 10.5, and in general a fair die with faces 1…n has E(X) = (n+1)/2.
- Sum of two fair dice: E(X) = 7 (each die averages 3.5, and expectation adds).
- Single European roulette number (pays 35:1): stake $1, win $35 with probability 1/37 and lose $1 with probability 36/37, so E(net) = 35·(1/37) − 1·(36/37) = −1/37 ≈ −$0.027 per $1 bet.
- Bernoulli trial (success = 1 with probability p): E(X) = p.
Expected value of a bet or gamble
A frequent use is deciding whether a wager or investment is worth taking. Set each outcome to the net gain or loss (winnings minus stake), weight by probability, and sum. A positive expected value means the bet favors you on average; a negative one means it favors the house. Nearly every casino game has a negative expected value for the player by design — that negative number is the house edge.