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ACT Math Hard
⏱ 12 min 📊 Hard ⭐ Premium

Expected Value

The expected value E(X) is the long-run average outcome of a random variable: Multiply each outcome by its probability, then add. Example: A fair die roll…

Before you read
Answer first — see what you already know.

A spinner has: 50% chance of $2, 30% chance of $5, 20% chance of $0. What is the expected value?

Check your answer
Answer:

B

Before you read
Answer first — see what you already know.

A lottery ticket costs $5. You win $100 with probability 0.02, $10 with probability 0.1, and nothing otherwise. What is the expected profit?

Check your answer
Answer:

A

Theory

Expected Value

The expected value E(X)E(X) is the long-run average outcome of a random variable:
E(X)=∑xi⋅P(xi)E(X) = \sum x_i \cdot P(x_i)

Multiply each outcome by its probability, then add.

Expected Value CalculationOutcomeValue ()ProbabilityValue × ProbWin1000.220Lose-100.8-8E(X) =12
E(X) = (100)(0.2) + (-10)(0.8) = 20 - 8 = $12

Example: A fair die roll. E=1(16)+2(16)+...+6(16)=216=3.5E = 1(\frac{1}{6}) + 2(\frac{1}{6}) + ... + 6(\frac{1}{6}) = \frac{21}{6} = 3.5

You expect an average of 3.5 per roll over many rolls.

20-8Win $100Lose $10
Each outcome's contribution to expected value
Example 1

A game: win $10 with probability 0.2, lose $3 with probability 0.8. What is the expected value?

E=10(0.2)+(−3)(0.8)=2−2.4=−0.4E = 10(0.2) + (-3)(0.8) = 2 - 2.4 = -0.4

−$0.40-\$0.40 (expected loss per game)
Tip

ACT Pro Tip

Expected value questions are essentially weighted averages. Multiply each outcome by its probability and sum. If the expected value is negative, you expect to lose money on average — the game favors the house.

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