Expected Value
Calculate the expected value (mean) of a probability distribution.
Key formula for statistics and decision-making.
The Formula
E(X) = Σ xᵢ × P(xᵢ)
The expected value is the long-run average of a random variable over many trials. It tells you what outcome to expect "on average" when an experiment is repeated many times.
Variables
| Symbol | Meaning |
|---|---|
| E(X) | Expected value of random variable X |
| xᵢ | Each possible outcome |
| P(xᵢ) | Probability of each outcome |
| Σ | Sum over all possible outcomes |
Example 1
Expected value of a fair six-sided die
E(X) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6)
= (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21/6
= 3.5
Example 2
A lottery ticket costs $2. You win $100 with probability 0.01 and $0 otherwise.
E(X) = 100(0.01) + 0(0.99) = 1.00
Expected profit = 1.00 - 2.00 (cost)
= -$1.00 (you lose $1 on average per ticket)
When to Use It
Use the expected value formula when:
- Evaluating whether a gamble or investment is worth taking
- Calculating the mean of a probability distribution
- Making decisions under uncertainty (decision theory)
- Analyzing insurance premiums, game strategies, or business projections