Expected Value
Expected value is the long-run average outcome of a random variable, calculated by weighting each possible value by its probability.
Formula
E(X) = \sum (\text{each value} \times \text{its probability})
Definition
Expected value is the average result you would get if you repeated an experiment many, many times, what you "expect" on average in the long run. The expected value $E(X) = \sum x_i P(X=x_i)$ for a discrete random variable is the probability-weighted average of all possible values, used in decision theory, gambling, insurance, and economics to compare choices with uncertain outcomes. Formally, $E[X] = \int x f(x)\,dx$ for continuous $X$, with properties including linearity $E[aX+bY] = aE[X]+bE[Y]$ and, for independent $X,Y$, $E[XY] = E[X]E[Y]$; in decision theory, expected utility maximization generalizes expected value by applying a utility function $u$: $E[u(X)]$ instead of $E[X]$.
Example
In a game where you roll a die and win that many dollars, $E(\text{win}) = 1(1/6) + 2(1/6) + \ldots + 6(1/6) = 21/6 = \$3.50$ on average per roll, even though you cannot actually roll $3.5$. A lottery ticket costing $\$2$ that pays $\$100$ with $P=0.01$, $\$10$ with $P=0.05$, and $\$0$ with $P=0.94$ has $E(\text{payout}) = \$1.50$, so the expected profit is $-\$0.50$: on average you lose $50$ cents per ticket. The St. Petersburg paradox illustrates a game with infinite expected value, $\sum_{n=1}^{\infty} (1/2^n)2^n = \infty$, yet most people would not pay more than $\$20$-$\$30$ to play, showing the need for expected utility rather than expected value.
Key Insight
Expected value might not be a value that can actually occur, but it is still the long-run average; if a game has a negative expected value, you lose money in the long run regardless of short-term luck, which is why casinos design all games with negative expected value for players. Jensen's inequality states that for a concave utility function $u$ (risk aversion), $E[u(X)] \le u(E[X])$, meaning risk-averse decision-makers prefer a certain gain of $E[X]$ over the lottery $X$ even when both have the same expected value, the mathematical foundation of insurance markets.