Expected Value
Long-run average.
Key idea
Long-run average.
In depth
E[X] = Σ x·P(x) is the long-run average. It is a weighted mean, with probabilities as weights. It need not be a value X can actually take.
Worked example
A die: E = (1 + 2 + ... + 6)/6 = 3.5. A game paying 10 with probability 0.1 and 0 otherwise has expected payout 1.
Formula
Real-world uses
- Deciding whether a lottery ticket or an insurance policy is a fair deal in the long run
Common mistake
Treating the expected value as the most likely outcome.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Expected Value?
Long-run average. It is a topic at the advanced level in Probability & Statistics.
What is the formula for Expected Value?
The key formula is E[X] = Σ x P(x).
Where is Expected Value used in real life?
Real-world examples: Deciding whether a lottery ticket or an insurance policy is a fair deal in the long run.
What should I learn before Expected Value?
Make sure you are comfortable with Probability Distributions, the previous topic in Probability & Statistics.
What is a common mistake with Expected Value?
Treating the expected value as the most likely outcome.
How can I practise Expected Value?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Expected Value?
Move on to Variance of a Random Variable next.
Related advanced topics in Probability & Statistics
- Quartiles and IQR – Quartiles split data into four parts; IQR = Q3 − Q1.
- Box Plots – Five-number summary visualised.
- Outliers – Values beyond 1.5 IQR from the quartiles.
- Variance – Average squared deviation from the mean.
- Standard Deviation – Square root of variance; typical spread.
- Sample vs Population Variance – Divide by n−1 for sample variance.