Expected Value Calculator

Expected value, written E[X], is the long-run average payoff of a random outcome. You multiply each possible value by its probability and add the products: E[X] = sum of (xi * pi). It is not the value you expect on any single trial; it is the value you would converge to if you repeated the experiment forever. A bet that pays $10 on heads and loses $5 on tails has E[X] = 0.5 * 10 + 0.5 * (-5) = $2.50, even though every individual flip pays out $10 or -$5 and never $2.50.

Because the outcomes you list have to cover every possible result of the experiment exactly once. If your probabilities sum to less than 1, you have forgotten an outcome; if they sum to more than 1, you have double-counted something. The probabilities must sum to 1 within a tiny floating-point tolerance before the calculator returns a result. It does not calculate or normalize an incomplete distribution.

Variance measures how spread out the outcomes are around E[X]. The formula is Var(X) = sum of pi * (xi - E[X])^2: take each outcome, subtract the mean, square the gap, weight by probability, and add. Standard deviation is the square root of variance, in the same units as your outcomes, so it is easier to interpret. A coin flip paying $10 / -$5 has E[X] = 2.5, variance = 0.5 * (10 - 2.5)^2 + 0.5 * (-5 - 2.5)^2 = 56.25, and standard deviation = 7.5. Big standard deviation means the bet is volatile around its mean; small standard deviation means it is close to deterministic.

A positive E[X] means the modeled average net payoff is positive. Under repeated independent trials with the stated distribution, the sample average tends toward that value, but no finite run is guaranteed to make money. Variance, dependence, transaction costs, changing probabilities, and the risk of ruin can all change a real decision.

No. Expected value treats a 50% chance of doubling your savings and a 50% chance of losing them the same as a guaranteed small profit, as long as the average matches. In real life, going broke once kills your ability to keep playing, so a bet with positive E[X] but catastrophic downside can still be irrational. This is why investors use risk-adjusted measures (Sharpe ratio, drawdown), why insurance exists at all (people accept negative E[X] to avoid ruin), and why the Kelly criterion sizes bets by both E[X] and volatility rather than by E[X] alone. Expected value tells you the direction of the bet; variance, standard deviation, and your bankroll tell you how much to put on the table.