Expected Value Calculator
Calculate the expected value (mean) of a probability distribution with up to 20 outcomes
Value-Probability Pairs
| Value (x) | Probability P(x) | Product | Action |
|---|---|---|---|
| - | |||
| - |
• Enter up to 20 value-probability pairs
• Probabilities must be between 0 and 1
• The sum of all probabilities must equal 1.0
• New rows appear automatically when you fill the last row
Expected Value Results
Common Examples
Dice Roll Example
For a standard 6-sided die, each outcome has probability 1/6:
E(X) = 1×(1/6) + 2×(1/6) + 3×(1/6) + 4×(1/6) + 5×(1/6) + 6×(1/6) = 21/6 = 3.5
The expected value of a dice roll is 3.5, meaning on average you'll roll 3.5.
Betting Example
You win $100 with 35% chance, lose $45 with 65% chance:
E(X) = 100×(0.35) + (-45)×(0.65) = 35 - 29.25 = $5.75
Positive expected value means this is a favorable bet in the long run.
Investment Example
Stock returns: +10% (prob 0.4), +5% (prob 0.3), -2% (prob 0.3):
E(X) = 10×(0.4) + 5×(0.3) + (-2)×(0.3) = 4 + 1.5 - 0.6 = 4.9%
Expected return is 4.9%, helping evaluate investment risk vs. reward.
Expected Value Formula
E(X): Expected value
xi: Each possible value
P(xi): Probability of value xi
Σ: Sum over all possible values
Key Properties
Expected value is the theoretical mean
Can be negative, positive, or zero
Probabilities must sum to 1.0
Similar to weighted average calculation
Common Applications
Financial risk assessment
Gaming and gambling analysis
Business decision making
Investment portfolio analysis
Understanding Expected Value
What is Expected Value?
Expected value is the average outcome you would expect if you repeated an experiment many times. It's the theoretical mean of a probability distribution, calculated by multiplying each possible outcome by its probability and summing all products.
Mathematical Foundation
- •Each outcome has an associated probability
- •All probabilities must sum to exactly 1.0
- •Formula: E(X) = Σ(xi × P(xi))
- •Similar to weighted average with probabilities as weights
Practical Applications
Risk Assessment
Evaluate potential losses and gains in financial decisions
Quality Control
Predict average defect rates and production outcomes
Game Theory
Analyze optimal strategies in competitive scenarios