Practice Expected Value in Statistics

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 long-run average outcome of a random process, calculated as the weighted sum of each possible outcome times its probability. It represents what you would earn or lose on average per trial if the process were repeated infinitely many times.

If you played a game forever, expected value is your average result per play. Positive EV = profitable long-term. Negative EV = you'll lose over time. It's the mathematical way to evaluate risky decisions.

Showing a random 20 of 50 problems.

Example 1

challenge
A game: roll a die; if you roll a 6 you win \$12, otherwise you lose \$2. Find the expected value and decide whether to play.

Example 2

medium
A test has 4 multiple-choice questions, each with 5 options. Random guessing: find EE(correct).

Example 3

easy
A spinner has equal regions paying $1, $2, $3, $4, $5. Find EE(payoff).

Example 4

easy
A random variable XX takes the value 44 with probability 0.250.25 and 88 with probability 0.750.75. Find E(X)E(X).

Example 5

easy
What does E(X)E(X) represent in plain words?

Example 6

medium
An insurance policy collects $300 and pays a $10000 claim with probability 0.020.02. Find the company's expected profit per policy.

Example 7

medium
Two fair dice are rolled. Find the expected value of the sum.

Example 8

medium
What entry fee makes a game fair if it pays $10 with probability 0.30.3 and $0 otherwise?

Example 9

challenge
In the St. Petersburg paradox, a coin is flipped until tails; if tails appears on flip nn, payoff is 2n2^n. Find EE(payoff).

Example 10

easy
If E(X)=2.5E(X)=2.5 per play, what total is expected over 8 plays?

Example 11

hard
An insurance policy pays $50{,}000 if a claim occurs (probability 0.0030.003) and $0 otherwise. What annual premium gives the insurer an expected profit of $50?

Example 12

easy
A fair 4-sided die labeled {1,2,3,4}\{1,2,3,4\} is rolled. Find E(X)E(X).

Example 13

medium
A weighted die: faces {1,2,3,4,5,6}\{1,2,3,4,5,6\} with P(6)=0.3P(6)=0.3 and other faces share the remaining 0.70.7 equally. Find E(X)E(X).

Example 14

medium
A spinner has outcomes: \$1 (prob 0.5), \$3 (prob 0.3), \$10 (prob 0.2). Find the expected value.

Example 15

medium
A fair coin is flipped twice; XX is the number of heads. Find E(X)E(X).

Example 16

medium
A raffle sells 200 tickets at \$5 each. There is one prize of \$500. Find the expected value for a ticket buyer.

Example 17

hard
In a roulette bet on a single number (p=1/38p=1/38), payoff is 35:135:1 (win $35, lose $1). Find EE(net per $1 bet).

Example 18

hard
A test pays $10 for each correct answer, −$3-\$3 for each wrong (no penalty if blank). Each question has 4 choices. If you guess on every question, what is EE(score per question)?

Example 19

medium
A random variable: P(−5)=0.2P(-5)=0.2, P(0)=0.5P(0)=0.5, P(10)=0.3P(10)=0.3. Find E(X)E(X).

Example 20

medium
A bet wins $8 with probability 0.250.25, loses $2 with probability 0.750.75. Find E(X)E(X) and state if the bet is favorable.
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