Practice Expected Value in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The expected value of a random variable is the probability-weighted average of all possible outcomes — the long-run mean over many repetitions.

Expected value is what you would "expect" on average after very many trials — not the most likely single outcome, but the center of the distribution.

Showing a random 20 of 50 problems.

Example 1

medium
A weighted die shows 66 with probability 0.50.5 and each of 11–55 with probability 0.10.1. Find E(X)E(X).

Example 2

hard
Roll a fair die. Let XX = number of distinct prime factors of the outcome. Find E(X)E(X).

Example 3

medium
A raffle sells 100 tickets at \$2 each; one ticket wins \$150. What is the expected value of buying one ticket (net)?

Example 4

easy
What is the expected value of one roll of a fair 6-sided die?

Example 5

medium
A discrete variable: X=10X=10 with prob 0.30.3, X=20X=20 with prob 0.50.5, X=30X=30 with prob 0.20.2. Find E(X)E(X).

Example 6

medium
Draw one card. You win $4 for a face card (J/Q/K) else lose $1. Find E(X)E(X).

Example 7

medium
XX has E(X)=5E(X) = 5. Find E(2X−7)E(2X - 7).

Example 8

medium
A raffle has 100100 tickets. One ticket wins $500\$500 and two tickets win $50\$50 each. Each ticket costs $10\$10. Find the expected net gain per ticket.

Example 9

challenge
In a class of 20, each student picks a random month for their birthday. Find E(number of January birthdays)E(\text{number of January birthdays}).

Example 10

challenge
You pay $1 to play. A die is rolled; you win $n\$n where nn is the face. After all costs, what is your expected net gain per play?

Example 11

easy
A bet pays $6 with prob 13\frac{1}{3} and $0 otherwise. Find E(X)E(X).

Example 12

hard
A binomial X∌Binomial(10,0.3)X \sim \text{Binomial}(10, 0.3). Find E(X)E(X).

Example 13

easy
A game pays $10 with probability 0.20.2 and $0 otherwise. Find the expected payout.

Example 14

medium
You draw one card from a deck. You win \$13 for an ace, else lose \$1. Find the expected gain.

Example 15

medium
A spinner gives 00 points with prob 0.40.4, 55 points with prob 0.40.4, and 1010 points with prob 0.20.2. Find E(X)E(X).

Example 16

hard
Two independent variables: X∈{1,3}X \in \{1, 3\} each with prob 0.50.5, Y∈{0,2}Y \in \{0, 2\} each with prob 0.50.5. Find E(XY)E(XY).

Example 17

easy
A random variable takes value 00 with probability 11. What is its expected value?

Example 18

medium
A game costs $3 to play and pays $10 with probability 0.250.25, else nothing. Find the expected net gain.

Example 19

challenge
You roll a fair die repeatedly until you get a 6. Find E(number of rolls)E(\text{number of rolls}).

Example 20

medium
Two fair dice are rolled. Find E(sum)E(\text{sum}).
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Related Concepts

ProbabilityMeanVariance