Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:Expected value is each outcome's value weighted by its probability, summed — the average you'd settle on over many, many trials.
Common stuck point:The procedure for expected value is the easy part; the trap is averaging the outcome values without weighting by probability. Asking "Am I weighting each outcome by its probability and summing to get a long-run average?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Am I weighting each outcome by its probability and summing to get a long-run average?
Worked Examples
Example 1
easy
A fair six-sided die is rolled. What is the expected value of the outcome?
Answer
E(X)=3.5
First step
1
A fair die has six equally likely outcomes {1,2,3,4,5,6}, each with probability 61.
Full solution
2
Apply the expected value formula: E(X)=∑xi⋅P(xi)=61(1+2+3+4+5+6)
3
Compute the sum: 61×21=621=3.5
The expected value is the long-run average outcome. Note that 3.5 is not a possible outcome of a single roll, but it is the average over many rolls.
Example 2
medium
A game costs $5 to play. You win $20 with probability 0.2 and $0 otherwise. What is the expected profit?
Example 3
medium
Flip 3 fair coins. Let X be the number of heads. Find E(X).
Example 4
hard
A binomial X∼Binomial(10,0.3). Find E(X).
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
medium
A raffle has 100 tickets. One ticket wins $500 and two tickets win $50 each. Each ticket costs $10. Find the expected net gain per ticket.
Example 2
medium
A prize wheel pays $0, $2, $5, and $20 with probabilities 0.50, 0.30, 0.15, and 0.05, respectively. What is the expected payout per spin?
Example 3
easy
A fair coin pays $1 for heads and $0 for tails. What is the expected payout?
Example 4
easy
What is the expected value of one roll of a fair 6-sided die?
Example 5
easy
A game pays $10 with probability 0.2 and $0 otherwise. Find the expected payout.
Example 6
easy
A spinner gives 2 points half the time and 4 points half the time. Expected points?
Example 7
easy
A random variable takes value 0 with probability 1. What is its expected value?
Example 8
easy
A bet: win $5 with probability 0.5, lose $5 with probability 0.5. Expected gain?
Example 9
easy
Outcomes 1,2,3 have probabilities 0.5,0.3,0.2. Find the expected value.
Example 10
easy
A lottery ticket wins $100 with probability 0.01, else $0. Expected value?
Example 11
medium
A game costs $3 to play and pays $10 with probability 0.25, else nothing. Find the expected net gain.
Example 12
medium
A die pays its face value in dollars, except a 6 pays $0. Find the expected payout.
Example 13
medium
A raffle sells 100 tickets at $2 each; one ticket wins $150. What is the expected value of buying one ticket (net)?
Example 14
medium
A discrete variable: X=10 with prob 0.3, X=20 with prob 0.5, X=30 with prob 0.2. Find E(X).
Example 15
medium
You draw one card from a deck. You win $13 for an ace, else lose $1. Find the expected gain.
Example 16
medium
Two independent fair coins are flipped. You win $2 per head. Find the expected winnings.
Example 17
medium
A variable X has E(X)=4. Find E(3X+2).
Example 18
medium
A weighted die shows 6 with probability 0.5 and each of 1–5 with probability 0.1. Find E(X).
Example 19
medium
A spinner gives 0 points with prob 0.4, 5 points with prob 0.4, and 10 points with prob 0.2. Find E(X).
Example 20
challenge
A bag has 3 red and 2 blue balls. You draw 2 without replacement and win $5 per red drawn. Find the expected winnings.
Example 21
challenge
A game: roll a die; if it shows k, you win $k2. Find the expected winnings.
Example 22
challenge
Two players each roll a fair die; you win $1 if your roll is strictly higher. Find your expected winnings.
Example 23
easy
A spinner lands on 1 with prob 0.6 and 5 with prob 0.4. Find E(X).
Example 24
easy
A fair 4-sided die shows 1,2,3,4. Find E(X).
Example 25
easy
Toss a fair coin: $3 for heads, lose $3 for tails. Find E(X).
Example 26
easy
A weighted coin gives heads with prob 0.8. Payouts: heads $10, tails $0. Find E(X).
Example 27
easy
X=100 with prob 0.05, else 0. Find E(X).
Example 28
easy
Insurance pays $1000 with prob 0.01, else $0. The premium is $15. Find the company's expected gain per policy.
Example 29
medium
A carnival game costs $2 to play and pays $10 with prob 0.15. Find expected net gain.
Example 30
medium
Two fair dice are rolled. Find E(sum).
Example 31
medium
A roulette bet on red pays $1 with prob 3818 and loses $1 otherwise. Find E(X).
Example 32
medium
X has E(X)=5. Find E(2X−7).
Example 33
medium
Draw one card. You win $4 for a face card (J/Q/K) else lose $1. Find E(X).
Example 34
medium
X takes value 0 with prob 21, 4 with prob 41, 8 with prob 41. Find E(X).
Example 35
medium
A weighted die has P(6)=0.4 and the rest equally likely. Find E(X).
Example 36
hard
Two independent variables: X∈{1,3} each with prob 0.5, Y∈{0,2} each with prob 0.5. Find E(XY).
Example 37
hard
Roll a fair die. Let X = number of distinct prime factors of the outcome. Find E(X).
Example 38
hard
A bag has 4 chips labeled 1,2,3,4. Two are drawn without replacement. Find E(sum).
Example 39
hard
A game pays $n if you roll a fair die and get n, but only if n is even (otherwise $0). Find E(payout).
Example 40
challenge
You roll a fair die repeatedly until you get a 6. Find E(number of rolls).
Example 41
challenge
In a class of 20, each student picks a random month for their birthday. Find E(number of January birthdays).
Example 42
challenge
You pay $1 to play. A die is rolled; you win $n where n is the face. After all costs, what is your expected net gain per play?