Expected Value Calculator
An expected value calculator finds the long-term average outcome of a random event when the possible outcomes and their probabilities are known. Expected value is commonly used in probability, statistics, games, finance, risk analysis, and decision-making.
What Is Expected Value?
Expected value represents the average result that would be expected if the same random process were repeated many times under the same conditions.
For a discrete random variable, expected value is calculated by multiplying each possible outcome by its probability and adding the results.
How to Calculate Expected Value
Suppose a game has two possible outcomes: winning $20 with a probability of 0.4, or losing $5 with a probability of 0.6.
Calculate the weighted outcomes:
(20 Γ 0.4) + (β5 Γ 0.6) = 8 β 3 = 5
The expected value is therefore $5.
This does not mean that every individual game will result in a $5 gain. It represents the average outcome over many repetitions.
Expected Value Example
Consider a fair six-sided die. The possible outcomes are 1, 2, 3, 4, 5, and 6, and each has a probability of 1/6.
The expected value is:
(1 + 2 + 3 + 4 + 5 + 6) Γ· 6 = 3.5
Therefore, the expected value of a fair six-sided die is 3.5. Although 3.5 cannot appear on a single roll, it represents the long-term average.
Expected Value Table
| Outcome | Probability | Outcome Γ Probability |
|---|---|---|
| 10 | 0.2 | 2 |
| 20 | 0.5 | 10 |
| 30 | 0.3 | 9 |
| Total Expected Value | 21 | |
Expected Value With Negative Outcomes
Expected value can include both gains and losses. A negative outcome should be entered as a negative number.
For example, if there is a 25% chance of gaining $40 and a 75% chance of losing $10:
(40 Γ 0.25) + (β10 Γ 0.75) = 10 β 7.5 = 2.5
The expected value is $2.50.
Important Points
The probabilities of all possible outcomes should add up to 1, or 100%. Expected value is a theoretical average and does not predict the exact result of a single trial.
A positive expected value indicates a positive average outcome, while a negative expected value indicates a negative average outcome under the stated probabilities.
Where Is Expected Value Used?
Expected value is used to compare uncertain outcomes and estimate long-term averages. Applications include probability problems, insurance, investment analysis, games of chance, business decisions, and risk assessment.
Common Mistakes to Avoid
Do not simply average the possible outcomes unless they are equally likely. Each outcome must be weighted according to its probability.
Also check that the probabilities are expressed consistently, such as decimals or percentages, and that they add up to 1.
Quick Summary
Expected value is the probability-weighted average of all possible outcomes. Multiply each outcome by its probability and add the results. The answer represents the long-term average outcome rather than a guaranteed result from one trial.