Expected Value Calculator

An expected value calculator turns a list of outcomes and their probabilities into one number: the long-run average result. The formula is E(X) = Σ x P(x). For a fair six-sided die every face has probability 1/6, so the expected value is 21/6 = 3.5, with a variance of 2.9167 and a standard deviation of about 1.7078. Enter up to 12 outcomes above and the mean, variance and spread update as you type.

Three sister tools cover the neighboring questions. The Probability Calculator finds the probabilities in the first place (single events, A and B, A or B, at least once), the Standard Deviation Calculator works from a raw list of observed data rather than a probability model, and the Average Calculator gives the plain mean, median and mode when every value carries equal weight. Use this page when each outcome has its own probability attached and you want the weighted average plus its risk.

How expected value is calculated

Expected value is a probability-weighted average. Every outcome contributes its own value multiplied by how often it happens:

The sum runs over every outcome the random variable can take. Two conditions make the result valid: each probability must sit between 0 and 1, and all of them together must total exactly 1. If your probabilities add to 0.95 you have forgotten an outcome; if they add to 1.4 you have double counted one. The calculator checks the total on every keystroke and offers to rescale the rows proportionally when it is off.

Worked example: a $5 raffle ticket

A club sells raffle tickets for $5 each. One ticket in a thousand wins the $1,000 grand prize, eight in a thousand win $100, and fifty in a thousand win $10. Everything else wins nothing. Work in net terms, subtracting the $5 you paid from every prize, so the grand prize becomes $995, the second prize $95, the small prize $5, and a losing ticket is −$5. The probability of losing is 1 − 0.001 − 0.008 − 0.05 = 0.941.

The expected value is −$2.70 per ticket. Buy 20 tickets and the average loss is $54; buy 100 and it is $270. The variance is 1,079.71 and the standard deviation is √1,079.71 = $32.86, far larger than the mean itself. That gap is the whole character of a raffle: the typical result is a small loss, the rare result is a large win, and the average sits close to the small loss because the win almost never happens. Note also that the gross expected prize is only $2.30 per ticket, which is what the $5 price is set against.

Variance and standard deviation

The expected value alone hides how wild the outcomes are. Variance measures the probability-weighted average squared distance from the mean:

Both forms give the same answer, and the calculator reports E(X²) so you can verify the shortcut yourself. Standard deviation is simply the square root of the variance, which puts the spread back into the same units as the outcomes: dollars, points, hours. That is why the standard deviation, not the variance, is the number people quote when they talk about risk.

Worked example: the fair die, checked two ways

Take the six faces of a fair die, each with probability 1/6. The mean is (1 + 2 + 3 + 4 + 5 + 6)/6 = 21/6 = 3.5. For the variance, square each face first: (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6 = 15.1667. That is E(X²). Subtract the square of the mean, 3.5² = 12.25, and you get a variance of 2.9167, exactly 35/12. The standard deviation is √2.9167 = 1.7078. Working the long way, the squared distances from 3.5 are 6.25, 2.25 and 0.25 for the pairs 1 and 6, 2 and 5, 3 and 4, each with weight 1/6, and (2 × 6.25 + 2 × 2.25 + 2 × 0.25)/6 = 17.5/6 = 2.9167. The two routes agree, which is the standard way to catch an arithmetic slip.

Expected value and spread of common chance devices

These are the distributions that show up in most homework problems, computed with the formulas above. Every die is assumed fair, so each face carries probability 1/n.

The pattern is worth remembering: the mean of an n-sided die is (n + 1)/2 and its variance is (n² − 1)/12. The sum of two dice keeps a mean of 7 but its variance is exactly double one die, because variances of independent variables add while standard deviations do not. Load the Sum of two dice example above to see all eleven outcomes with their weights 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 out of 36.

Same expected value, very different risk

Expected value on its own is never enough to choose between options. These four payoffs all have an expected value of exactly $10,000, yet almost nobody is indifferent between them:

The mean says these are equivalent. The standard deviation column says they are not remotely equivalent, ranging from $0 to $30,000 of typical deviation. This is exactly why the calculator reports the spread next to the mean rather than hiding it: a decision made on expected value alone quietly assumes you can absorb the worst row without changing your life.

How to use this calculator

1. Pick a starting point: load one of the examples (fair die, coin flip, sum of two dice, raffle ticket, business decision) or start typing over the default rows.
2. Choose what the second column holds: pick Probability when your numbers already add to 1, or Weight when you have counts, frequencies or odds ratios that should be normalized for you.
3. Say whether the outcomes are dollars: switching to the dollar format prints results with a currency symbol and two decimals, which is what you want for payoffs, costs and prizes.
4. Enter each outcome and its probability: type probabilities however you like, as 0.25, 25% or 1/4. Outcome values may be negative for losses.
5. Add or remove rows: up to 12 outcomes are supported; a distribution needs at least two to be interesting.
6. Read the total: if the probabilities do not land on 1 you get a warning and a one-click rescale, so a typo never silently corrupts the mean.
7. Read the result: the headline is E(X). Below it sit the variance, the standard deviation, E(X²) and the most likely single outcome, then a row-by-row table showing exactly where the mean and the variance came from.

Who this calculator is for

• Statistics students checking homework on discrete random variables, where the mean, variance and standard deviation are the three answers every problem asks for.
• Business analysts valuing a project with several scenarios, each carrying its own probability and payoff.
• Insurance and warranty shoppers comparing a fixed premium against an uncertain repair or claim.
• Game designers and players pricing a die roll, a loot table or a card draw in points rather than dollars.
• Anyone facing a gamble who wants to see the average and the spread before deciding, instead of only the headline prize.

Second worked example: is the extended warranty worth it?

A retailer offers a $120 extended warranty on an appliance. From the failure rate you estimate a 12% chance of a repair that would cost you $600 out of pocket, and an 88% chance of no repair at all. Without the warranty your cost is a random variable: $600 with probability 0.12, $0 with probability 0.88. Its expected value is 0.12 × 600 = $72, and its standard deviation is $194.98. With the warranty your cost is a certainty: $120, with a standard deviation of $0.

So the warranty costs $48 more than the risk it removes. On expected value alone you decline it. The break-even failure probability is 120 ÷ 600 = 0.20, so the warranty only pays for itself if you believe the appliance fails at least 20% of the time. What the extra $48 actually buys is the removal of a $600 shock, and whether that is worth $48 depends on whether an unexpected $600 would hurt. That is the honest way to read every expected value comparison: the mean tells you the price of the risk, and the standard deviation tells you why someone might pay more than that price to avoid it.

Key terms explained

• Random variable: a quantity whose value depends on chance, written X. Here it is discrete, meaning it takes a countable list of values.
• Probability distribution: the full list of outcomes with their probabilities. It is complete only when the probabilities total 1.
• Expected value E(X): the probability-weighted mean, also written μ. It is the long-run average, not a prediction of one trial.
• Variance Var(X): the weighted average squared distance from the mean, in squared units.
• Standard deviation: the square root of the variance, in the original units, and the usual measure of risk or spread.
• Mode: the single most likely outcome. It often differs from the expected value.
• Fair game: a bet whose expected value is exactly 0. Anything below 0 favors the other side.
• Law of large numbers: the principle that the average of many independent trials converges on the expected value as the number of trials grows.

What changes the result the most

• Probabilities on the extreme outcomes: moving weight onto the largest or smallest value shifts the mean fastest and inflates the variance dramatically.
• How far apart the outcomes are: the mean depends on the values, but the variance depends on their squared distances, so doubling the spread quadruples the variance.
• Whether you work in gross or net terms: a $1,000 prize on a $5 ticket is $995 net. Mixing the two conventions is the most common source of a wrong answer.
• A missing outcome: if the probabilities total less than 1, the missing case usually has value 0 or a small loss, and adding it always pulls the mean down.
• Rounding the probabilities: entering 0.17 six times instead of 1/6 changes the total to 1.02 and distorts every result. Fractions avoid this entirely.

Practical tips

• Always list the "nothing happens" case. It is the easiest outcome to forget and it usually carries most of the probability.
• Use weight mode for counts. Eleven sums of two dice with weights 1 through 6 and back down are far easier to type than eleven fractions over 36.
• Sanity-check with the shortcut. Compare the variance in the result card against E(X²) − E(X)² using the value shown. If they disagree, a row is wrong.
• Scale the mean to your horizon. Expected value per trial times the number of trials gives the expected total: 1,000 die rolls average 3,500 pips, and 10,000 appliances with a 12% failure rate cost $720,000 in repairs.
• Compare like with like. Two options are only comparable on expected value if both are measured over the same period and in the same units.
• Look at the worst row, not just the mean. If the largest loss would be unrecoverable, expected value is the wrong decision rule no matter how favorable it looks.

Limitations and assumptions

• It handles discrete random variables only. A continuous distribution replaces the sum with an integral and needs a different tool.
• Your probabilities are taken as given. If they are estimates, the expected value inherits every bit of that uncertainty, and no amount of decimal places makes a guessed probability more accurate.
• It assumes the outcomes are mutually exclusive and exhaustive, so exactly one of them happens on each trial.
• Expected value describes the long run. Over a handful of trials the realized average can sit far from it, which is exactly what the standard deviation quantifies.
• It uses the population variance formula, weighted by probability. It does not apply the n − 1 sample correction, which belongs to data you have observed rather than a model you have specified.
• It does not model utility. In real decisions the value of a dollar is not constant, which is why people insure against losses with a negative expected value.

How it compares to related calculators

This page answers "what is the average outcome, and how much does it vary?" when each outcome has a probability. If your question is different, a sister tool fits better:

• To find the probabilities themselves, use the Probability Calculator.
• To measure spread in a list of observed numbers with no probabilities attached, use the Standard Deviation Calculator.
• For an unweighted mean, median and mode, use the Average Calculator.
• To count the arrangements behind a probability, use the Permutation and Combination Calculator.
• To convert a probability into a percentage or back, use the Percentage Calculator.

Sources

Every number on this page comes from deterministic mathematics rather than an external data source, so there is nothing to cite for the results themselves. Specifically:

• The expected value of a discrete random variable is defined as E(X) = Σ x P(x), and the variance as Var(X) = Σ P(x)(x − E(X))², with the identity Var(X) = E(X²) − E(X)². These are exact definitions, not measured quantities.
• The dice and coin figures follow from those definitions on a uniform distribution over n equally likely faces, giving mean (n + 1)/2 and variance (n² − 1)/12.
• The raffle, warranty and business scenarios are illustrative examples with stated probabilities. They are worked arithmetic, not claims about any real product, prize pool or failure rate.