Expected Value Calculator

Most people reach for an expected value calculator because they want to know which choice will win. That is not what it does. An expected value calculator tells you the average outcome over many, many repetitions, not what happens next time. A bet with a positive expected value can lose ten times in a row. A bet with a negative expected value can win twice. The calculator removes the noise from a single trial and shows you the signal in the long run. If you are making a one-off decision, expected value alone is the wrong tool, and here is why.

Expected value (also called the mean of a probability distribution) is the sum of each possible outcome multiplied by its probability. The formula is E(X) = Σ x · P(x). For a discrete random variable, dice, coin flips, game payouts, investment scenarios, that is the only equation you need. The calculator handles five common modes so you can enter data the way the problem is phrased, not the way a formula expects it.

The calculator gives you five input modes so you match the problem instead of translating it into a generic table. Select the mode before you enter any numbers; switching modes clears the inputs.

General Expected Value Mode

Enter any set of outcomes and their probabilities. The total probability must sum to 1.00 (or 100%). The calculator shows a running total and warns you if it drifts off. Each row accepts an outcome name, a numerical value (x), and its probability P(x). Add as many rows as you need. The results pane shows expected value, variance, standard deviation, and a median that is defined as the smallest outcome whose cumulative probability reaches or exceeds 0.5. For discrete distributions with a small number of values, that median can be misleading: a distribution with outcomes 1, 2, 3 and probabilities 0.4, 0.2, 0.4 gives a median of 2, even though the expected value is 2.0. The median is reported, but for a discrete random variable it does not always split the probability mass evenly.

Dice Roll Mode

Choose the number of dice (1 to 10) and the number of sides per die (4, 6, 8, 10, 12, or 20). The calculator builds the distribution of the sum of the dice using convolution, it does not enumerate all possible rolls, which would freeze the browser. You can optionally flip on custom payouts, which replaces each sum value with a dollar payout you type in. That turns a pure dice analysis into a game analysis: roll a sum of 7 and win $5, roll a sum of 2 and lose $1.

Coin Flip Mode

Set the number of coins (1 or more), the probability of heads (default 0.5), and separate payouts for heads and tails. For multiple coins, the calculator uses the binomial distribution. Payouts are linear: if heads pays $2 and tails pays −$1, two coins with heads and one tail pays 2 + 2 − 1 = $3. The expected value here is just the weighted average of the binomial outcomes.

Game/Betting Scenario Mode

Enter your bet amount, the net win amount (what you get back on top of your bet, not including the return of the bet itself), and your probability of winning. The calculator builds a two-outcome table: win with the net payout, lose with the negative of the bet. This is the way to evaluate a single wager. It does not account for multiple rounds, compounding, or the Kelly criterion (f* = (bp − q) / b, which tells you what fraction of your bankroll to risk).

Investment Returns Mode

Add scenarios with a name, a return percentage, and a probability. The calculator multiplies the investment amount by each return percentage to get a dollar value, then computes the expected value of the portfolio. This is a simplified decision-tree expected monetary value (EMV) calculation, as used in the PMBOK Guide 7th edition. It does not model correlations between scenarios or sequence-of-returns risk.

The formula has three parts. The summation sign Σ tells you to add up every term in the list. Each term is a possible outcome x multiplied by its probability P(x). You do this for every distinct outcome of the random variable.

For a continuous random variable, the summation becomes an integral over the probability density function f(x). The meaning is identical: the area under x f(x) over the entire support of the distribution. OpenIntro Statistics (4th ed., section 3.4) introduces random variables as numerical outcomes of random processes and uses the discrete sum definition. OpenStax Introductory Statistics 2e (ch. 4.2) does the same for discrete random variables and adds the continuous version in ch. 5.

Linearity of expectation is the property that makes the formula powerful. E(X + Y) = E(X) + E(Y) holds for any two random variables X and Y, even if they are dependent. Blitzstein and Hwang (Introduction to Probability, 2nd ed., ch. 4) treat linearity as the central theorem of expectation, and use it to compute expectations of sums of dependent indicator random variables where a direct table would be impossible.

You play a game where you roll one fair six-sided die. If the roll is a 6, you win $6. If the roll is anything else (1 through 5), you lose $1. Is this game worth playing?

First, list the outcomes and their probabilities. There are six equally likely outcomes, each with probability 1/6.

• Roll 6: value = $6, probability = 1/6
• Roll 1: value = −$1, probability = 1/6
• Roll 2: value = −$1, probability = 1/6
• Roll 3: value = −$1, probability = 1/6
• Roll 4: value = −$1, probability = 1/6
• Roll 5: value = −$1, probability = 1/6

Now apply the formula. Multiply each outcome by its probability and sum: ($6 × 1/6) + (−$1 × 1/6) + (−$1 × 1/6) + (−$1 × 1/6) + (−$1 × 1/6) + (−$1 × 1/6) = $1.00 − $0.8333 = $0.1667.

The expected value of the game is about 17 cents per roll. Over a thousand rolls, you expect to be up about $167. The game has positive expected value. That does not mean you will win on any single roll, you lose on five out of six rolls, but over many trials the average drifts toward 17 cents.

If you entered this in the calculator's general mode, you would enter six rows, each with probability 0.1667. The results would show E(X) = 0.8056 + 5 × 1.3611 × 0.1667 = 5.8056 + 1.1343 = 6.9399, and standard deviation σ = √6.9399 ≈ 2.63. The standard deviation is larger than the expected value itself, which means the spread of outcomes is wide relative to the average gain.

Expected value gives you the center of the distribution. Variance and standard deviation give you the spread. The calculator computes both from the same probability table you entered for the expected value.

Variance is defined as Var(X) = E[(X − μ)²], where μ is the expected value. The computational formula is Var(X) = E(X²) − (E(X))². Using the dice game above: E(X²) = (6² × 1/6) + (1² × 1/6 × 5) = 36/6 + 5/6 = 41/6 = 6.8333. (E(X))² = 0.1667² = 0.0278. Var(X) = 6.8333 − 0.0278 = 6.8055.

Standard deviation is the square root of variance: σ = √Var(X). In the dice game, σ ≈ 2.61. Standard deviation is in the same units as the original outcomes (dollars), which makes it interpretable. One standard deviation above the mean is about $2.78; one standard deviation below is about −$2.44. Since the outcomes are −$1 and $6, the actual values never land exactly one standard deviation away, that is normal for a discrete distribution.

OpenStax Introductory Statistics 2e (ch. 4.2) provides the same formulas for variance and standard deviation of a discrete random variable. Blitzstein and Hwang (ch. 4) derive variance and show its relationship to the second moment E(X²). The calculator applies these formulas to whatever data you enter in any mode.

Expected value answers one question: what is the average outcome per trial over the long run? It does not answer the questions that matter for a single decision.

Risk. Two bets can have the same expected value and wildly different variance. A bet that pays $1,000 with probability 0.001 and $0 otherwise has E(X) = $1. A bet that pays $1 with probability 1 has E(X) = $1. The first has a 99.9% chance of returning $0; the second is certain. Expected value alone cannot distinguish them. Variance and standard deviation give you the spread, but even those do not capture risk tolerance, a risk-averse person will reject the first bet despite the same EV.

One-off decisions. If you play a game exactly once, the expected value is irrelevant to the outcome. The law of large numbers says the average of many independent trials converges to the expected value, but one trial is a single sample from the distribution. A positive-EV lottery ticket bought once has a 99.9999% chance of losing. The expected value of the ticket is positive only in the imaginary long run; the actual outcome is almost certainly a loss.

Bankroll management. Expected value does not tell you how much to bet. The Kelly criterion (1956) computes the fraction of your bankroll to risk on a positive-EV bet to maximize long-run growth: f* = (bp − q) / b, where b is the net odds received, p is the win probability, and q is 1 − p. Betting more than the Kelly fraction increases the risk of ruin, even on a positive-EV bet. The calculator does not implement the Kelly criterion; it computes only the raw EV.

After-tax expected value. For U.S. lottery players, the IRS withholds 24% on winnings over $5,000 (IRS Publication 505, 2025). State taxes add another layer. A jackpot EV that looks positive before tax often turns negative after withholding. The calculator shows pre-tax EV. For actual decisions, apply your own tax rate to the EV.

Multiple winners. A lottery jackpot is split among all winning tickets. The expected number of winners depends on how many tickets are sold. At high jackpot levels, the probability of a split is substantial, and the per-ticket EV must be divided by that expected number. The calculator does not model that.