Expected Value Calculator

Calculate expected value (mean) of a probability distribution. E(X) = Σ x⋅P(x). Find expected value for discrete distributions, games of chance, and decision analysis.

📊 Statistics / Probability📐 E(X) = Σ xi × P(xi)🔢 Math

Value x1

Probability P(x1)

Value x2

Probability P(x2)

Value x3 (optional)

Probability P(x3)

Please enter valid values.

Formula & Reference

Variable	Symbol	Formula	Units
Expected Value Calculator	—	E(X) = Σ xi × P(xi)	same as outcome units

Step-by-Step Examples

Example 1

Coin Flip Game

Win $10 if heads (p=0.5), lose $5 if tails (p=0.5).

E(X) = 10×0.5 + (-5)×0.5
= 5 - 2.5 = 2.50
Positive expected value: play this game!
In 100 games, expect to win ~$250 total

✓ E(X) = $2.50 per game

Example 2

Lottery Ticket

Win $1000 (p=0.001), Win $10 (p=0.05), Lose $5 (p=0.949). Ticket costs $5.

E(X) = 1000×0.001 + 10×0.05 + (-5)×0.949
= 1.00 + 0.50 - 4.745 = -3.245
Negative expected value — lose $3.25 per ticket on average

✓ E(X) = -$3.25 per ticket (bad bet)

Example 3

Insurance Decision

No insurance: lose $10,000 if disaster (p=0.02), $0 otherwise (p=0.98). Premium = $150.

E(loss without insurance) = 10000×0.02 = $200
E(cost with insurance) = $150 (certain)
Insurance saves $50 in expectation PLUS eliminates catastrophic risk
Expected value slightly favors insurance here

✓ E(uninsured loss) = $200 vs $150 premium

Real-World Applications

🎲

Game Theory

Expected payoffs determine optimal strategies in games and economic decisions.

💰

Finance

Expected return E(r) = Σ(probability of scenario × return in scenario) for investments.

💊

Medical Decision Making

Expected utility theory guides treatment decisions weighing outcomes and their probabilities.

💼

Business Risk Analysis

Expected value of perfect information (EVPI) quantifies what research or data is worth.

Common Mistakes to Avoid

⚠️

Probabilities not summing to 1

All P(xi) must sum to 1.0 for a valid probability distribution. Check your probabilities.

⚠️

Confusing expected value with most likely outcome

E(X) is the long-run average — it may never actually occur. A coin flip has E=0.5 but only gives 0 or 1. Expected value is a theoretical average, not a prediction of one outcome.

⚠️

Ignoring risk despite same expected value

Two choices with E=100: Choice A: always $100. Choice B: $200 with p=0.5 or $0 with p=0.5. Same expected value, very different risk. Risk aversion means people often prefer A.

Frequently Asked Questions

What is expected value? ▾

The probability-weighted average of all possible outcomes. E(X) = Σ xi×P(xi). Represents the long-run average outcome if the experiment is repeated infinitely many times.

What is the law of large numbers? ▾

As the number of trials increases, the sample average converges to the expected value. After 10,000 fair coin flips, the average will be very close to 0.5.

What is variance of a distribution? ▾

Var(X) = Σ P(xi)×(xi-E(X))². Measures spread around the expected value. High variance = more uncertainty. SD = √Var(X).

How does expected value apply to gambling? ▾

Most casino games have negative expected value for the player (house edge). Slot machines: E ≈ -5% to -15%. Blackjack with perfect strategy: E ≈ -0.5%. Positive EV games are exploited until corrected.

What is expected utility? ▾

Extends expected value with a utility function U(x) to model risk preferences. E[U(X)] = Σ P(xi)×U(xi). Risk-averse individuals prefer certain outcomes over equal-expected-value gambles.

Related Math Calculators

Formula Explorer connections

Interpretation: This formula summarizes data, models uncertainty or supports inference about a population or random process. Assumption: The sampling design and distribution assumptions must match the data. Independence, sample size, outliers and measurement quality can materially affect interpretation.

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