Expected Value
The long-run average value of a random variable.
Formula
Variables
Example
A fair die: E[X] = (1+2+3+4+5+6)/6 = 3.5.
How it works
Expected value is the weighted average of all possible outcomes, where each outcome is multiplied by how likely it is to occur. It works because outcomes that happen more often should contribute more to the long-run average than rare outcomes. If you repeat the same random process many times, the average result tends to settle near this value.
When to use it
Use this formula whenever you want the long-run average result of a random process, such as games of chance, insurance payouts, prize draws, or decision-making under uncertainty. It is especially useful for comparing options when each outcome has a different payoff and probability.
Worked example, step by step
A game pays $0 with probability 0.70, $5 with probability 0.20, and $20 with probability 0.10. E[X] = (0)(0.70) + (5)(0.20) + (20)(0.10) E[X] = 0 + 1 + 2 E[X] = 3 So the expected value of the game is $3 per play.
Common mistakes to avoid
- Averaging the outcomes without multiplying by their probabilities
- Forgetting to include outcomes with zero payoff, which can still affect the average
- Using percentages as if they were probabilities without converting them to decimals
Did You Know?
Expected value is the long-run average — and it's why the house always wins in the long run.
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More in Statistics & Probability
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BasicThe average of a data set.
Median
BasicThe middle value separating the higher and lower halves.
Variance
IntermediateAverage squared deviation from the mean.
Standard Deviation
IntermediateSquare root of the variance; typical spread.
Z-Score
IntermediateNumber of standard deviations a value is from the mean.
Bayes' Theorem
IntermediateUpdates a probability given new evidence.