Expected value is the probability-weighted average of all possible outcomes: what you would get per trial, on average, if you could repeat a gamble, game or decision indefinitely. It is the starting point for judging bets, pricing insurance, valuing projects and checking whether a game is fair. Enter the outcomes and their probabilities (fractions like 1/6 and percentages like 25% are accepted) to get the expected value, the variance and standard deviation, and the expected total over many repetitions.
How to use the expected value calculator
- List the possible outcomes in the first box. Use net values, so a loss is negative.
- List the matching probabilities in the same order, as decimals, fractions or percentages.
- If your second list holds counts or weights rather than probabilities, tick Rescale.
- Optionally enter a number of repetitions to see the expected total and its typical spread.
- Read E(X) on the tape; the table shows each outcome’s contribution and the chart shows the distribution.
Expected value formula
For a discrete random variable X taking values x with probabilities P(x):
The spread around that mean is measured by
Over n independent repetitions, the total has mean nμ and standard deviation √n × σ.
Worked examples
A dice game. You pay $4 to roll a die and win the face value in dollars. Your net result is −3, −2, −1, 0, 1 or 2, each with probability 1/6.
- E(X) = (−3 − 2 − 1 + 0 + 1 + 2) ÷ 6 = −3 ÷ 6 = −$0.50 per game.
- E(X²) = (9 + 4 + 1 + 0 + 1 + 4) ÷ 6 = 3.1667, so Var(X) = 3.1667 − 0.25 = 2.9167 and σ = $1.71.
- Over 100 games you expect to lose about $50, give or take $17 (one standard deviation, √100 × 1.71).
The game would be fair at a $3.50 entry fee, the average face value.
A raffle. A $2 ticket has a 1% chance of a $5 prize (net +$3), a 0.1% chance of a $500 prize (net +$498), and otherwise loses $2. Entering outcomes −2, 3, 498 with probabilities 0.989, 0.01, 0.001 gives E(X) = −1.978 + 0.03 + 0.498 = −$1.45 per ticket. About 72% of the ticket price goes, on average, to the organizer.
Using expected value for decisions
Compare options on the same scale
To choose between alternatives, compute the expected value of each with consistent units, usually net profit. A sales strategy that earns $40,000 with probability 0.6 and loses $10,000 otherwise has E = 0.6 × 40,000 + 0.4 × (−10,000) = $20,000, which can be weighed against a safer option.
Risk is not in the average
Expected value treats a sure $10 and a long-shot lottery as equivalent if their averages match. Most people and businesses are risk-averse for large stakes, which is why insurance sells despite its negative expected value. Look at the standard deviation and at the probability of a loss (P(X > 0) is shown on the tape) before deciding.
The law of large numbers
The average result converges to E(X) only over many independent repetitions. With a handful of plays, the spread dominates. That is why casinos, which run millions of bets, can rely on a small negative expected value for players, while an individual gambler can still walk away ahead.
Related calculations
A weighted average of values uses the same arithmetic with weights instead of probabilities; see the weighted average calculator. For the expected number of successes in repeated yes-or-no trials (simply np), and the full distribution around it, use the binomial probability calculator.
Frequently asked questions
What does expected value mean in plain English?
It is the average result per trial if you could repeat the situation many times. A bet with an expected value of −$0.50 does not cost exactly 50 cents each time; you win or lose various amounts, but over hundreds of plays you lose about 50 cents per play on average.
Can the expected value be an outcome that never happens?
Yes. A fair die has an expected value of 3.5, which no single roll can produce. The expected value is a long-run average, not a prediction of any one result.
What makes a game fair?
A game is fair when its expected value is zero: neither side gains on average. Casino games and lotteries have negative expected values for players, which is how the house profits. Insurance also has a negative expected value for buyers, who accept it in exchange for protection against large losses.
Why does the variance matter if I know the expected value?
Two choices with the same expected value can carry very different risk. A guaranteed $10 and a 1% chance at $1,000 both have an expected value of $10, but the second has a standard deviation of about $99. The variance tells you how far individual results typically stray from the average.
My probabilities do not add up to 1. What should I do?
A complete probability distribution must sum to exactly 1. Check for a missing outcome or a typo. If you entered frequencies or weights (such as counts of how often each outcome occurred), tick the rescale option and the calculator converts them to probabilities.