Expected Value Formula
The expected value of a random variable is the probability-weighted average of all possible outcomes — the long-run mean over many repetitions.
The Formula
When to use: Expected value is what you would "expect" on average after very many trials — not the most likely single outcome, but the center of the distribution.
Quick Example
Notation
What This Formula Means
The expected value of a random variable is the probability-weighted average of all possible outcomes — the long-run mean over many repetitions.
Expected value is what you would "expect" on average after very many trials — not the most likely single outcome, but the center of the distribution.
Formal View
Worked Examples
Example 1
easyAnswer
First step
Full solution
- 2 Apply the expected value formula:
- 3 Compute the sum:
Example 2
mediumExample 3
mediumCommon Mistakes
- Averaging the outcome values without weighting by probability — multiply each value by its chance first.
- Letting the probabilities not sum to 1 — a complete set of outcomes must have probabilities adding to 1.
- Treating as the result of one trial — it is the long-run average, which need not be an achievable single outcome.
Why This Formula Matters
Expected value turns uncertainty into a single decision number — it is how you judge whether a lottery, insurance policy, or business gamble is worth it, and it generalizes the mean to weighted outcomes. The trap is treating it as a prediction of one trial rather than a long-run average. Recognizing it by "Am I weighting each outcome by its probability and summing to get a long-run average?" — rather than by familiar numbers — is what lets a student tell it apart from mean (of data) and mode / most likely outcome and probability in a mixed problem set.
Frequently Asked Questions
What is the Expected Value formula?
The expected value of a random variable is the probability-weighted average of all possible outcomes — the long-run mean over many repetitions.
How do you use the Expected Value formula?
Expected value is what you would "expect" on average after very many trials — not the most likely single outcome, but the center of the distribution.
What do the symbols mean in the Expected Value formula?
or denotes the expected value of random variable
Why is the Expected Value formula important in Math?
Expected value turns uncertainty into a single decision number — it is how you judge whether a lottery, insurance policy, or business gamble is worth it, and it generalizes the mean to weighted outcomes. The trap is treating it as a prediction of one trial rather than a long-run average. Recognizing it by "Am I weighting each outcome by its probability and summing to get a long-run average?" — rather than by familiar numbers — is what lets a student tell it apart from mean (of data) and mode / most likely outcome and probability in a mixed problem set.
What do students get wrong about Expected Value?
The procedure for expected value is the easy part; the trap is averaging the outcome values without weighting by probability. Asking "Am I weighting each outcome by its probability and summing to get a long-run average?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
What should I learn before the Expected Value formula?
Before studying the Expected Value formula, you should understand: probability, mean.