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Why is expected value described as a weighted average?

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sample space for a pair of dice — probability theory Sample space for a pair of dice.

In probability theory, the expected value—also known as the mean—is described as a weighted average because it calculates the long-term average outcome of a random variable by factoring in the likelihood of each possible result.123

Rather than treating every outcome as equally important, the expected value assigns "weights" to each potential value based on its probability of occurring. In the discrete case, these weights are provided by the probability mass function, while in the continuous case, they are provided by the probability density function.23

The calculation is defined as the sum (or integral) of each possible value multiplied by its corresponding probability:13

$E (x) = \sum x_{i} f (x_{i})$

Because the sum of all probabilities must equal 1, this formula effectively balances the values according to their relative frequency. If every outcome were equally likely, the weighted average would simplify to a standard arithmetic mean.23