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Expected Value — Definition, Formula & Examples

Expected Value
Mean of a Random Variable

A quantity equal to the average result of an experiment after a large number of trials. For example, if a fair 6-sided die is rolled, the expected value of the number rolled is 3.5. This is a correct interpretation even though it is impossible to roll a 3.5 on a 6-sided die. This sort of thing often occurs with expected values.

 

Two tables showing Expected Value formula: E=P1x1+P2x2+…+Pnxn, with die example yielding 3.5.

 

See also

Weighted average

Key Formula

E(X)=∑i=1nxi⋅P(xi)E(X) = \sum_{i=1}^{n} x_i \cdot P(x_i)
Where:
  • E(X)E(X) = The expected value of random variable X
  • xix_i = The i-th possible outcome
  • P(xi)P(x_i) = The probability of the i-th outcome occurring
  • nn = The total number of possible outcomes

Worked Example

Problem: A game costs 5toplay.Youdrawonecardfromastandard52−carddeck.IfyoudrawanAce,youwin5 to play. You draw one card from a standard 52-card deck. If you draw an Ace, you win20. If you draw a King, you win $10. For any other card, you win nothing. What is the expected value of your net profit per game?
Step 1: Identify each outcome and its net profit (winnings minus the $5 cost).
Ace: 20−5=$15King: 10−5=$5Other: 0−5=−$5\text{Ace: } 20 - 5 = \$15 \quad \text{King: } 10 - 5 = \$5 \quad \text{Other: } 0 - 5 = -\$5
Step 2: Find the probability of each outcome. There are 4 Aces, 4 Kings, and 44 other cards in a 52-card deck.
P(Ace)=452P(King)=452P(Other)=4452P(\text{Ace}) = \frac{4}{52} \qquad P(\text{King}) = \frac{4}{52} \qquad P(\text{Other}) = \frac{44}{52}
Step 3: Multiply each outcome by its probability and sum the results.
E(X)=15⋅452+5⋅452+(−5)⋅4452E(X) = 15 \cdot \frac{4}{52} + 5 \cdot \frac{4}{52} + (-5) \cdot \frac{44}{52}
Step 4: Compute each term and add them together.
E(X)=6052+2052+−22052=−14052≈−2.69E(X) = \frac{60}{52} + \frac{20}{52} + \frac{-220}{52} = \frac{-140}{52} \approx -2.69
Answer: The expected value of your net profit is approximately −2.69pergame.Onaverage,youloseabout2.69 per game. On average, you lose about2.69 each time you play.

Another Example

Problem: Find the expected value when rolling a fair 6-sided die.
Step 1: Each face (1 through 6) has an equal probability of 1/6.
P(xi)=16 for each xi∈{1,2,3,4,5,6}P(x_i) = \frac{1}{6} \text{ for each } x_i \in \{1, 2, 3, 4, 5, 6\}
Step 2: Multiply each outcome by its probability and sum.
E(X)=1⋅16+2⋅16+3⋅16+4⋅16+5⋅16+6⋅16E(X) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6}
Step 3: Simplify by adding the numerators.
E(X)=1+2+3+4+5+66=216=3.5E(X) = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5
Answer: The expected value of a fair die roll is 3.5. You can never roll a 3.5, but over many rolls, the average will converge to this number.

Frequently Asked Questions

Can the expected value be a number that is impossible to actually get?
Yes. The expected value is an average over many trials, not a prediction for a single trial. For a fair die, the expected value is 3.5 even though no single roll can produce 3.5. This is perfectly normal and does not mean the calculation is wrong.
How is expected value different from the most likely outcome?
The most likely outcome (the mode) is the single result with the highest probability. The expected value is the probability-weighted average of all outcomes. For example, if you have a 90% chance of winning 0anda100 and a 10% chance of winning100, the most likely outcome is 0,buttheexpectedvalueis0, but the expected value is10.

Expected Value vs. Arithmetic Mean (Average)

The arithmetic mean is computed from data you have already collected — you add up observed values and divide by the count. The expected value is computed from probabilities before any experiment happens — you weight each possible outcome by how likely it is. When you repeat an experiment many times, the arithmetic mean of your results approaches the expected value. This connection is known as the Law of Large Numbers.

Why It Matters

Expected value is the foundation of decision-making under uncertainty. Insurance companies use it to set premiums, investors use it to compare financial strategies, and game designers use it to balance rewards. Whenever you need to judge whether a risky choice is 'worth it' on average, expected value is the tool you reach for.

Common Mistakes

Mistake: Forgetting to account for unequal probabilities and simply averaging the outcomes.
Correction: Each outcome must be multiplied by its own probability. Only when all outcomes are equally likely does dividing by the number of outcomes give the correct expected value.
Mistake: Interpreting the expected value as the result you will get on any single trial.
Correction: Expected value describes the long-run average. In any single trial, the actual result may be far from the expected value. Think of it as what you'd expect 'on average' over hundreds of repetitions.

Related Terms