Expected Value
The predicted average of a random variable.
This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.
Core idea
Overview
The expected value represents the long-term average outcome of a random variable across many repetitions of an experiment. In its simplest form, it is calculated by multiplying the probability of a specific event occurring by the numerical value or payoff associated with that event.
When to use: Use this formula when assessing risk, making financial projections, or evaluating games of chance where outcomes are uncertain. It assumes that the probability and potential payoff are known and that the process can be reasonably averaged over time.
Why it matters: It allows decision-makers to quantify uncertainty and compare different choices on a level playing field. It serves as the mathematical foundation for modern insurance underwriting, investment portfolio management, and statistical decision theory.
Symbols
Variables
E = Expected Value, P = Probability, V = Value of Event
Walkthrough
Derivation
Understanding Expected Value (Frequency)
Expected frequency estimates how many times an outcome will occur over many repeated trials.
- The probability stays constant.
- Trials are independent.
Use the expected frequency formula:
Multiply the number of trials n by the probability of the event P(A).
Example calculation:
Rolling a fair die 100 times, you would expect about 17 threes on average.
Note: Expected value is not guaranteed; real results vary due to randomness.
Result
Source: Standard curriculum — GCSE Mathematics (Probability)
Free formulas
Rearrangements
Solve for
Make P the subject
To make P, Probability, the subject of the Expected Value formula, divide both sides by V, the Value of Event.
Difficulty: 2/5
Solve for
Make V the subject
To make V the subject of the Expected Value formula, divide both sides by P.
Difficulty: 2/5
Solve for
Make E the subject in the Expected Value Formula
This problem demonstrates how to express the Expected Value formula with simplified multiplication notation, where E is already the subject.
Difficulty: 2/5
The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.
Visual intuition
Graph
The graph is a straight line passing through the origin with a slope equal to V, showing that the expected value increases at a constant rate as the probability increases. For a student of data, this means that small probability values result in a low expected value, while large probability values lead to a higher expected value. The most important feature is the linear relationship, which means that doubling the probability will always double the expected value.
Graph type: linear
Why it behaves this way
Intuition
Imagine a weighted average, where each possible outcome is a point on a number line, and its probability acts as a weight, pulling the overall average towards the more likely or impactful outcomes.
Free study cues
Insight
Canonical usage
The expected value (E) will have the same units as the value or payoff (V), as probability (P) is dimensionless.
Common confusion
A common mistake is incorrectly assigning units to probability (P) or failing to ensure that the expected value (E) carries the appropriate units from the value or payoff (V).
Unit systems
One free problem
Practice Problem
A marketing analyst determines there is a 15% chance that a specific campaign will generate 50,000 dollars in revenue. Calculate the expected value of this specific outcome.
Solve for:
Hint: Multiply the decimal probability by the total dollar amount.
The full worked solution stays in the interactive walkthrough.
Where it shows up
Real-World Context
Lottery win expectation.
Study smarter
Tips
- Ensure probabilities are expressed as decimals between 0 and 1.
- Treat financial losses or costs as negative values for V.
- Remember that the expected value itself may not be a possible single outcome of the experiment.
Avoid these traps
Common Mistakes
- Confusing with most likely outcome.
Common questions
Frequently Asked Questions
Expected frequency estimates how many times an outcome will occur over many repeated trials.
Use this formula when assessing risk, making financial projections, or evaluating games of chance where outcomes are uncertain. It assumes that the probability and potential payoff are known and that the process can be reasonably averaged over time.
It allows decision-makers to quantify uncertainty and compare different choices on a level playing field. It serves as the mathematical foundation for modern insurance underwriting, investment portfolio management, and statistical decision theory.
Confusing with most likely outcome.
Lottery win expectation.
Ensure probabilities are expressed as decimals between 0 and 1. Treat financial losses or costs as negative values for V. Remember that the expected value itself may not be a possible single outcome of the experiment.
References
Sources
- Wikipedia: Expected value
- Ross, A First Course in Probability
- Montgomery and Runger, Applied Statistics and Probability for Engineers
- Britannica: Expected Value
- Ross, S. M. (2014). Introduction to Probability and Statistics for Engineers and Scientists. Academic Press.
- Standard curriculum — GCSE Mathematics (Probability)