Expected value: averages over many repetitions
What expected value means
The expected value of a random quantity is the probability-weighted average of its possible values. It is the long-run average over many independent repetitions, not a promise about any single repetition. It need not even be a value the quantity can actually take.
E[X] = Σ x × P(X = x)Example: a scoring exercise
Imagine a classroom exercise where each student draws one card and records 'points' for a lab report: 0 points for a number card, 1 for a face card, 3 for an ace. Probabilities: 36/52 for 0, 12/52 for 1, 4/52 for 3. Expected points = 0 × 36/52 + 1 × 12/52 + 3 × 4/52 = 24/52 ≈ 0.46.
The score is merely a numeric label used to practise the calculation. Nothing is staked on it and no actual reward depends on it.
Linearity of expectation
Expected values add, even when the pieces are dependent. In a five-card draw, each card has a 13/52 chance of being a heart, so the expected number of hearts is 5 × 13/52 = 1.25 without ever computing a distribution. This shortcut solves many problems that would otherwise need long case analyses.
Beyond cards
- Waiting time: if a bus comes on average every 10 minutes at random, the expected wait for a randomly arriving passenger can be calculated from the timetable distribution.
- Quality control: if 2% of items are faulty, the expected number of faulty items in a batch of 500 is 10, yet batches with 7 or 14 are entirely ordinary.
- Variance and standard deviation describe how far single results typically stray from the expected value.
Check your understanding
Expected number of face cards in a four-card draw?
4 × 12/52 ≈ 0.923.
A fair six-sided die is rolled. What is E[X]?
(1+2+3+4+5+6)/6 = 3.5, a value the die can never show.
Would the simulated average of 100 draws exactly equal the expected value?
Almost never. It will be close, and closer on average as the number of draws grows.