Expected Value
Long-run average value of a random variable.
Formula
Variables
Example
Die roll: E = 3.5
How it works
Expected value is the probability-weighted average of all possible outcomes. Each outcome contributes in proportion to how likely it is, so rare results count less and common results count more; over many repeated trials, the average result tends to this weighted mean. It works because repeated long-run outcomes occur in frequencies that match the probabilities, so adding up outcome × probability captures the average payoff or measurement you would expect.
When to use it
Use this when a random variable can take several discrete values and you want its average outcome over many repetitions, such as games of chance, insurance payouts, or scores on a quiz. It is especially useful for comparing strategies or deciding whether a gamble has a positive or negative average payoff.
Worked example, step by step
A game pays \u00a310 if you draw a heart from a standard deck, \u00a35 if you draw a spade, and \u00a32 otherwise. There are 13 hearts and 13 spades in 52 cards, so P(heart) = 13/52 = 1/4 and P(spade) = 13/52 = 1/4. The probability of drawing neither a heart nor a spade is 1 - 1/4 - 1/4 = 1/2. E[X] = (10)(1/4) + (5)(1/4) + (2)(1/2) E[X] = 2.5 + 1.25 + 1.0 E[X] = 4.75 So the expected value of the game is \u00a34.75 per draw.
Common mistakes to avoid
- Forgetting that probabilities must add up to 1 before calculating the average
- Multiplying values by counts instead of by probabilities
- Confusing expected value with the most likely outcome
Did You Know?
Expected value need not be an achievable outcome.
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