About Expected Value Calculator
Expected value is the mathematical heartbeat of probability, finance, insurance, and decision science. It tells you the long-run average outcome of a random scenario if it were repeated thousands of times. Our Expected Value Calculator turns a list of outcomes and their probabilities into a single weighted average, so you can compare bets, projects, or risks on equal footing. Suppose an investment has a 30% chance of returning $50, a 50% chance of returning $20, and a 20% chance of losing $10. The expected value is $23, a number that helps you decide whether the opportunity fits your risk tolerance. You will learn why probabilities must sum to 1 for a complete distribution, how weighting by probability changes the ordinary average, and how expected value underpins everything from lottery ticket design to clinical trial planning.
How It Works
The calculator parses your comma-separated outcomes and probabilities into parallel lists. It multiplies each outcome by its matching probability, then adds all those products together. It also sums the probabilities separately and reports that total, which helps you catch incomplete or malformed distributions.
Formula & Calculation Logic
The expected value formula is E(X) = Σ [xᵢ · P(xᵢ)], where xᵢ is each outcome and P(xᵢ) is its probability. For a valid probability distribution, Σ P(xᵢ) = 1. If the probabilities do not sum to 1, the result is still a weighted sum, but it is not a true expected value.
Step-by-Step Guide
- Step 1: List all possible outcomes separated by commas.
- Step 2: List the matching probabilities separated by commas.
- Step 3: Click Calculate to multiply each outcome by its probability.
- Step 4: Review the Expected Value and Probability Sum to validate your distribution.
Example Calculations
- Scenario 1: A raffle ticket pays $100 with probability 0.01, $10 with probability 0.19, and $0 with probability 0.80. Expected value = 100(0.01) + 10(0.19) + 0(0.80) = $2.90.
- Scenario 2: A project yields $40,000 with probability 0.6 and loses $10,000 with probability 0.4. Expected value = 40,000(0.6) + (-10,000)(0.4) = $20,000.
Common Use Cases
- Comparing investment opportunities and project forecasts under uncertainty.
- Evaluating casino games, raffles, and lottery tickets.
- Pricing insurance policies and estimating actuarial risk.
- Teaching probability and statistics concepts with concrete numbers.
Pro Tips
- Always check that probabilities sum to 1 for a complete distribution.
- Express losses as negative outcomes so the expected value reflects net gain or loss.
- Use expected value alongside variance for a fuller risk picture.
- Break complex decisions into clear, mutually exclusive outcomes before entering them.
Common Mistakes to Avoid
- Forgetting to convert percentages to decimals, such as entering 50 instead of 0.5.
- Listing a different number of outcomes than probabilities.
- Using probabilities that sum to more than 1 without normalizing.
- Ignoring negative signs on losing outcomes.
Why Use This Tool?
- Turns uncertain scenarios into a single comparable number.
- Catches distribution errors by reporting the probability sum.
- Supports financial, academic, and gaming use cases.
- Works alongside binomial, Poisson, and weighted-average tools.