Expected Value | Econ Academy

When outcomes are uncertain, we need a way to summarize what we can “expect” on average. The expected value ( $E V$ ) is the probability-weighted average of all possible outcomes. It tells us what we’d get on average if we could repeat the gamble many, many times. Expected value is the foundation of decision-making under uncertainty.

Key Term

Expected Value Formula

For a random variable with outcomes $w_{1}, w_{2}, \dots, w_{n}$ occurring with probabilities $p_{1}, p_{2}, \dots, p_{n}$ :

$E (w) = p_{1} \times w_{1} + p_{2} \times w_{2} + \dots + p_{n} \times w_{n} = \sum_{i} p_{i} \times w_{i}$

Probabilities must sum to 1: $\sum_{i} p_{i} = 1$ . Expected value is a weighted average — outcomes with higher probability count more.

Expected value: the probability-weighted average of all possible outcomes

The expected value is often an outcome that never actually occurs. If you flip a coin for 0 dollars or 100 dollars, $E (w) = 50$ dollars — but you never actually get 50 dollars. Expected value summarizes the center of the probability distribution, not an actual likely outcome. It’s most meaningful for decisions that repeat (like insurance pricing) or as a benchmark for comparing risky options.

Worked Example

You’re offered a gamble: with probability 0.3 you win 1,000 dollars, with probability 0.5 you win 200 dollars, and with probability 0.2 you lose 500 dollars. What is the expected value?

Interactive — Expected Value as a Balance Point

Loading interactive...

Three outcomes sit as weighted circles on a wealth number-line; the blue fulcrum below the line marks the expected value — the point where the system balances.

Check yourself · no marks

Is a gamble with positive expected value always worth taking?

Commit to one first. Guessing and being wrong beats reading the answer cold.

Practice · 1 / 4

No penalty for a wrong answer

Expected value:

A lottery ticket costs 5 dollars. With probability 0.01 you win 200 dollars, otherwise you win nothing. The expected value of this ticket is:

Like what you read?