A reliable method
For a discrete distribution, multiply each value by its probability and add. For a binomial count, verify a fixed number of independent trials, two outcomes per trial and a constant success probability.
The common trap
An expected value need not be one of the possible individual outcomes. A mean count of 2.4 successes is meaningful over repeated samples even though no individual sample has 2.4 successes.
Make the connection
For a binomial count use mean np and probability choose(n,k)p^k(1−p)^(n−k). Do not apply this model when the trials strongly affect one another.
Weight outcomes instead of just listing them
Suppose X is 0, 2 or 4 with probabilities .25, .50 and .25. E(X)=2, while E(X²)=6 and Var(X)=6−2²=2. Equal numerical spacing is not a reason to ignore probabilities. A distribution with the same three values but different probabilities can have a different mean.
Count arrangements in a binomial event
For four fair independent tosses, exactly one head can appear in four positions. Each complete sequence has probability 1/16, so the event has probability 4/16. The value 1/16 counts only one arrangement; 15/16 counts at least one head. The event definition determines which sequences to include.
Transfer check
A process has ten trials and success probability .3. The expected number of successes is 3, but that does not guarantee exactly three in each run. Check fixed n, binary outcomes, independence and a constant probability before using a binomial model; sampling a large fraction without replacement can violate independence.
X takes values 0, 1, and 2, with probabilities 1/4, 1/2, and 1/4. What is E(X)?
- Multiply each value by its probability.
- The weighted sum is 0 + 1/2 + 2/4.
- E(X) = 1.
Compare the other answer choices
Use the largest possible outcome as the expectation.
Square each probability before weighting.
Compute E(X²) rather than E(X).
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Statistics 2026–27 course framework