Expected Value Calculator
Calculate the expected value of a random variable from outcome-probability pairs. Useful for decision analysis, gambling odds, insurance pricing, and investment evaluation under uncertainty.
This free online expected value calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Value of the first outcome.
Probability of the first outcome (0 to 1).
Value of the second outcome.
Probability of the second outcome (0 to 1).
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Expected Value Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Expected Value Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Expected Value Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
When to Use This Calculator
- •Use the Expected Value Calculator when you need accurate results quickly without the risk of manual computation errors or unit conversion mistakes.
- •Use it to verify calculations made by hand or in spreadsheets — an independent check can catch errors before they lead to costly decisions.
- •Use it to explore how changing input parameters affects the output — a quick way to develop intuition and identify the most influential variables.
- •Use it when collaborating with others to ensure everyone is working from the same numbers and applying the same assumptions.
About This Calculator
The Expected Value Calculator is a free, browser-based calculation tool for engineers, students, and technical professionals. Calculate the expected value of a random variable from outcome-probability pairs. Useful for decision analysis, gambling odds, insurance pricing, and investment evaluation under uncertainty. It implements standard formulas and supports both metric (SI) and imperial unit systems with automatic unit conversion. All calculations are performed instantly in your browser with no data sent to a server. Use this calculator as a quick reference and sanity-check tool during design, analysis, and learning. Always verify results against primary engineering references and applicable standards for any safety-critical application.
About Expected Value Calculator
The expected value calculator computes the long-run average outcome of a random process by weighting each possible outcome by its probability. Expected value is the single most important concept in decision theory under uncertainty. It tells you what outcome to expect on average over many repetitions and is used to evaluate bets, insurance policies, business investments, clinical trial endpoints, and lottery tickets. A positive expected value means the proposition is favorable on average, while a negative one means it is unfavorable. This calculator accepts two outcome-probability pairs and computes their weighted sum.
The Math Behind It
Formula Reference
Expected Value
E(X) = sum of (x_i * P(x_i))
Variables: x_i = value of outcome i; P(x_i) = probability of outcome i
Worked Examples
Example 1: Simple bet evaluation
A bet pays $100 with probability 0.3 and loses $50 with probability 0.7. What is the expected value?
The expected value is -$5 per bet, meaning on average you lose $5 each time. This is an unfavorable bet.
Example 2: Insurance decision
A $200/year policy pays $50,000 for a loss that has 0.5% probability. Is it worth it?
The insurance has a positive expected value of $50 for the buyer in this simplified scenario.
Common Mistakes & Tips
- !Forgetting that probabilities must sum to 1. If they do not, the expected value calculation is invalid.
- !Confusing expected value with the most likely outcome. The expected value may not equal any single possible outcome.
- !Ignoring the sign of outcomes; losses should be entered as negative values.
Related Concepts
Used in These Calculators
Calculators that build on or apply the concepts from this page:
Frequently Asked Questions
Can the expected value be a number that is not a possible outcome?
Yes. For example, the expected value of a fair die roll is 3.5, which is not any face value. Expected value represents the long-run average, not a single trial result.
How does expected value relate to gambling?
Casino games are designed so the house has a positive expected value and the player has a negative one. The house edge is essentially the negative of the player's expected value per unit bet.
What if I have more than two outcomes?
Simply add more x_i * p_i terms. The formula generalizes to any number of outcomes: E(X) = x1*p1 + x2*p2 + ... + xn*pn.