Expected Value Calculator

What Is Expected Value?

The Expected Value (E(X)) is the long-run average outcome of a random variable if an experiment were repeated many times. It is the sum of each possible outcome weighted by its probability. Think of it as the “center of gravity” of a probability distribution — where the distribution would balance if placed on a fulcrum.

Formula: E(X) = Σ [xᵢ × P(xᵢ)]

where xᵢ is each possible outcome and P(xᵢ) is its probability. All probabilities must sum to exactly 1.0.

Variance and Standard Deviation

Expected value alone does not capture risk. Two distributions can share the same expected value but differ enormously in spread. The variance measures that spread:

Var(X) = Σ [P(xᵢ) × (xᵢ - E(X))²]

Standard deviation = √Var(X). A higher standard deviation means outcomes are more unpredictable.

Applications

Gambling and casino games: A casino game with a house edge always has a negative expected value for the player. A standard roulette wheel (American) has E(X) = -$0.053 per $1 bet. Over thousands of spins, the casino is guaranteed to profit — this is the Law of Large Numbers in action.

Insurance: Actuaries calculate the expected value of claims to price premiums. If there is a 1% chance of a $50,000 loss, the expected loss is $500 — and the insurer charges more than that to cover overhead and profit.

Business decisions: A product launch might have a 40% chance of earning $200,000 and a 60% chance of losing $50,000. E(X) = 0.40 × 200,000 + 0.60 × (-50,000) = $80,000 - $30,000 = $50,000. Positive expected value suggests proceeding.

Dice example: Rolling a fair six-sided die: outcomes {1, 2, 3, 4, 5, 6} each with probability 1/6. E(X) = (1+2+3+4+5+6)/6 = 21/6 = 3.5. No single roll gives exactly 3.5, but the average of many rolls converges to 3.5.

Important Note

Expected value guides rational decisions under uncertainty, but it does not eliminate risk. A rational investor considers both expected value and variance (risk/reward tradeoff). All probabilities you enter must sum to 1.0 — this calculator will warn you if they do not.

Where expected value misleads. EV is the right rule only when outcomes can be repeated enough times for averages to settle, and when bad outcomes don’t end the game. Compare two bets: (A) 50% chance of +$10, 50% chance of −$10, with E(X) = 0; and (B) 1% chance of +$1,000,000, 99% chance of −$10,000, with E(X) = +$100. Bet B has the higher expected value, but 99 times out of 100 it costs the player $10,000 — enough to bankrupt most participants before the big win arrives. Real decision-making under uncertainty layers risk metrics on top of EV: variance, value-at-risk, ruin probability, and utility theory all exist because expected value alone hides ruin risk.

The most useful property — linearity. E[aX + bY] = a·E[X] + b·E[Y] holds whether or not X and Y are independent. That is what lets you find the expected sum of two dice as simply 3.5 + 3.5 = 7 without working out the joint distribution. Linearity is the workhorse identity behind almost every quick expected-value calculation in practice.