Expected Value Calculator

What this expected value calculator does

Expected value is the average result of a random process, weighted by how likely each result is. Multiply every possible outcome by its probability and add the products, and you have E(X). It tells you what the process pays, costs or scores on average in the long run, which makes it a standard tool for comparing games, tickets, investments in a simplified model and decisions under chance.

The first mode takes a full probability distribution and returns the expected value, the variance and the standard deviation. The second mode is for games and tickets: enter the gross prizes and the cost to play, and it returns the expected net winnings and the cost that would make the game fair. The third mode, which is optional, works out the expected value of a bet from the stake, the odds and your own estimate of the win probability. For a list of raw observations rather than a distribution, use the Mean Median Mode Calculator.

How to enter outcomes and probabilities

Each row holds one outcome and its probability. Add rows with the Add outcome button and remove them with the × button. A few rules keep the distribution valid:

• One row per distinct outcome. This is a probability distribution, not a data set. If a die can land on six faces, there are six rows, not one row per roll.
• Probabilities can be decimals, percentages or fractions. 25% is the same as 0.25, and 1/4 is too. Choose percentages in the entry format menu to type 25 instead of 0.25, or add a % sign to any value.
• Each probability lies between 0 and 1 (0% and 100%). A value such as 25 in decimal format is flagged, with a reminder to type 25% if that is what you meant.
• The probabilities must total 1 (100%). The running total sits under the table, and a total that is off is reported as a problem rather than quietly accepted. Include every outcome, even one with a value of 0, so nothing is missing.
• Repeating decimals need care. Six rows of 0.1667 total 1.0002, which is not 1. Type 1/6 or use more decimal places.

Random variable X

The quantity whose value depends on chance, such as the number on a die, the prize on a ticket or the result of a bet. This calculator handles a discrete X, one with a list of separate possible values.

Outcome x_i

One possible value of X. Enter it in the outcome column. It can be negative or zero.

Probability P(X = x_i)

The chance that X takes the value x_i, from 0 to 1. Together the probabilities form a discrete probability distribution and must total 1.

Expected value E(X)

The probability-weighted average of the outcomes, also written μ and called the mean of the distribution.

Variance and standard deviation

Measures of how far outcomes spread around E(X). They are reported next to the expected value but are separate from it.

Gross prize and net result

The gross prize is the total you receive. The net result is the prize minus what you paid to play, so it is negative when you lose money.

The expected value formula

For a discrete random variable, E(X) is the sum of each outcome multiplied by its probability. The variance measures spread around that average, and the standard deviation is its square root.

• E(X) = Σ x_i × P(X = x_i) (sum over every outcome)
• μ = E(X) (the mean of the distribution)
• Σ P(X = x_i) = 1, and each P(X = x_i) is between 0 and 1
• E(X²) = Σ x_i² × P(X = x_i)
• Var(X) = E(X²) − [E(X)]² = Σ P(X = x_i) × (x_i − μ)²
• Standard deviation = √Var(X)
• Net result = gross prize − cost to play, for every outcome; E(net) = E(prize) − cost
• Fair game: expected net gain = 0, that is, cost = E(prize)
• Betting EV = (win probability × profit if win) − (loss probability × stake)

The two forms of the variance are algebraically identical. The calculator evaluates it from the deviations, which stays accurate when the outcomes are large numbers with small differences between them, and shows the E(X²) − [E(X)]² form in the steps.

Step-by-step example: a fair die

Each face of a fair die has probability 1/6. Multiply each face by 1/6 and add:

So E(X) = 3.5. The same table gives the spread: E(X²) = (1 + 4 + 9 + 16 + 25 + 36) ÷ 6 = 91/6 ≈ 15.1667, the variance is 15.1667 − 3.5² = 2.9167 (exactly 35/12), and the standard deviation is about 1.7078. Choose Fill with an example in the first mode to load this distribution.

What the result means

• It is a long-run average. As the number of independent repetitions grows, the average of the results tends toward E(X). Over a handful of repetitions the average can be far from it.
• It does not predict the next outcome. A fair die never shows 3.5. The expected value does not have to be a possible outcome, and often is not.
• It can be negative. A negative value means the average result is a loss. A positive value means an average gain, and zero means the gains and losses balance.
• It says nothing about spread by itself. Two distributions can share an expected value and differ greatly in how far single results stray from it. See the variance section below.

Games and tickets: gross payout versus net gain

Money outcomes can mean two different things, so be clear which one you are entering. A gross prize is the total you receive. A net result is what you end up with after paying to play. In the second mode you enter gross prizes and a cost, and the calculator subtracts the cost on every outcome. A $2 ticket that returns a $50 prize gives a net gain of $48, not $50, and a ticket that pays nothing gives a net result of −$2. Subtracting the cost only from the losing outcome, or not at all, is the usual mistake. If your values are already net gains and losses, use the first mode instead.

Take a $2 ticket that pays $50 with probability 0.02, $10 with probability 0.08 and nothing with probability 0.90 (illustrative figures):

The expected prize is 50 × 0.02 + 10 × 0.08 + 0 × 0.90 = $1.80, and the expected net result is $1.80 − $2 = −$0.20 per ticket. The game is fair only if the ticket costs the expected prize, here $1.80. A fair game does not mean winning and losing are equally likely. It means the probability-weighted gains and losses cancel.

Variance and standard deviation: the spread around the mean

Expected value is the centre of a distribution. Variance and standard deviation describe how widely outcomes spread around it, and they are shown separately so that a good average is not mistaken for a safe one. Suppose game A pays $0 or $10 with equal probability, and game B always pays $5. Both have an expected value of $5. Game A has a variance of 25 and a standard deviation of $5. Game B has a standard deviation of 0. Same average, very different experience.

The ticket example above has a standard deviation of about $7.40 on an expected result of −$0.20, so single plays swing far more than the average suggests. Subtracting a fixed cost moves the mean but leaves the variance unchanged. For the spread of a list of raw values, the Variance Calculator and Standard Deviation Calculator are the right tools.

Betting expected value (optional)

The third mode applies the same idea to a single bet with two outcomes: it wins, or it loses. The expected value is (win probability × profit if it wins) − (loss probability × stake). Enter the stake, your own estimate of the win probability, and the odds as American odds, decimal odds or a profit amount. Profit and total payout are different things. Profit is what you gain beyond getting your stake back, and the total payout is stake plus profit. American odds of +150 pay a profit of 150 on a stake of 100, and −110 needs a stake of 110 to win 100. Decimal odds of 2.50 pay 250 in total on a stake of 100, which is a profit of 150.

The odds imply a break-even win probability of stake ÷ (stake + profit), which is 110/210 ≈ 52.38% here. An estimate above that gives a positive expected value, and an estimate below it gives a negative one. This is where the caution matters: the result is only as good as the probability you supply. The calculator cannot know the true chance, a positive expected value does not guarantee a profit on any single bet or over a short run, and the probability implied by a price usually includes the bookmaker's margin. This is arithmetic for learning and checking numbers. It is not betting or financial advice, and gambling carries a real risk of loss.

Input checks and edge cases

The calculator stops and explains the problem, rather than showing a misleading number, when:

• an outcome or probability is missing from a half-filled row, or is not a number;
• a probability is negative or above 1 (above 100%);
• the probabilities do not total 1, in which case it shows the total and how far off it is;
• the cost to play is negative, or the stake is not above zero;
• American odds fall between −100 and +100, or lack a sign, or decimal odds are not above 1.

Some unusual cases are accepted. A single outcome with probability 1 is valid and has a variance of 0. A probability of 0 is allowed and contributes nothing. Outcome values can repeat, and they are handled as separate rows. If the probabilities total 1 only to within rounding (about one part in a million), the result uses the numbers exactly as typed and says so.

Assumptions and limitations

• The calculator handles discrete outcomes with known probabilities, up to 50 outcomes. Expected values of continuous distributions need integration and are not covered here. For the normal curve, see the Normal Distribution Calculator.
• The long-run reading assumes the repetitions are independent and the probabilities stay the same each time.
• Expected value ignores risk, the amount of money you can afford to lose and how much you value outcomes. Read it together with the standard deviation.
• The expected cell counts in a chi-square test are a different calculation. For those, use the Chi-Square Calculator.
• This calculator provides a mathematical result from the values you enter. It is an informational and educational tool, not professional, financial or statistical advice. For decisions that matter, consult a qualified professional.

Formula sources

The expected value, variance, fair-game and net-winnings formulas, and the odds conventions, were checked against these independent references:

• OpenStax, Introductory Statistics, 4.2 Mean or Expected Value and Standard Deviation
• Hossein Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, 3.2.4 Variance (the computational formula Var(X) = E(X²) − [E(X)]²)
• LibreTexts, College Mathematics for Everyday Life, 3.4 Expected Value and Law of Large Numbers (fair games and net winnings)
• Covers, Odds Converter (American and decimal odds conversions and implied probability)

Frequently asked questions