Start with the recap, study the fully worked examples, then use the practice problems to
check your understanding of Expected Value.
This page combines explanation, solved examples, and follow-up practice so you can move
from recognition to confident problem-solving in Statistics.
Concept 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.
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 starts by naming the possible outcomes and the event rule before assigning or combining probabilities.
Common stuck point:Students often know a procedure related to expected value but skip the recognition step: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition? That leads to a calculation or graph that looks reasonable but answers a different question.
Sense of Study hint:Ask: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?
Worked Examples
Example 1
medium
A coin is flipped until heads appears. The number of flips N has P(N=k)=(1/2)k for k=1,2,…. Find E(N).
Answer
2
First step
1
This is geometric with p=0.5.
See the full worked solution + why-it-works coaching
Setup·Key insight·Why it works·Common pitfall·Connection
You draw cards one at a time without replacement from a standard 52-card deck until you draw an ace. What is the expected number of draws?
Example 3
medium
A game costs $2 to play. You roll a fair die: if you roll a 6, you win $10; otherwise, you win nothing. Find the expected value per game.
Example 4
medium
A raffle sells 200 tickets at $5 each. There is one prize of $500. Find the expected value for a ticket buyer.
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
A game pays $10 with probability 0.5 and $0 with probability 0.5. Find the expected value.
Example 2
easy
A fair die is rolled. Find the expected value of the result.
Example 3
easy
A raffle ticket wins $100 with probability 0.01 and $0 otherwise. Find the expected winnings.
Example 4
easy
A random variable takes value 2 with probability 0.3 and 5 with probability 0.7. Find E(X).
Example 5
easy
A coin flip pays $3 on heads and costs $1 on tails (fair coin). Find the expected value.
Example 6
easy
A lottery has expected value −$0.40 per $1 ticket. Is it a good long-term bet?
Example 7
easy
A spinner pays $4, $0, $0, $0 on four equally likely regions. Find the expected payout.
Example 8
easy
If E(X)=2.5 per play, what total is expected over 8 plays?
Example 9
medium
A game costs $2 to play and pays $5 with probability 0.3, $0 otherwise. Find the expected net value per play.
Example 10
medium
A random variable: P(0)=0.2, P(1)=0.5, P(2)=0.3. Find E(X).
Example 11
medium
An insurance policy collects $300 and pays a $10000 claim with probability 0.02. Find the company's expected profit per policy.
Example 12
medium
A die game pays the face value in dollars but costs $4 to play. Find the expected net value.
Example 13
medium
A bet wins $8 with probability 0.25, loses $2 with probability 0.75. Find E(X) and state if the bet is favorable.
Example 14
medium
A carnival game pays $5, $2, or $0 with probabilities 0.1, 0.3, 0.6. Find the expected payout.
Example 15
medium
A fair coin is flipped twice; X is the number of heads. Find E(X).
Example 16
medium
What entry fee makes a game fair if it pays $10 with probability 0.3 and $0 otherwise?
Example 17
medium
A random variable: P(−5)=0.2, P(0)=0.5, P(10)=0.3. Find E(X).
Example 18
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 19
challenge
A spinner pays $x with probability 0.4 and $1 with probability 0.6. If E(X)=$3, find x.
Example 20
challenge
A game offers: −$1 with probability $0.5,$0withprobability$0.3, and $kwithprobability$0.2. Find k so the game is fair (E(X)=0).
Example 21
easy
A random variable X takes the value 4 with probability 0.25 and 8 with probability 0.75. Find E(X).
Example 22
easy
A fair 4-sided die labeled {1,2,3,4} is rolled. Find E(X).
Example 23
easy
A random variable takes values −2 and 5 with equal probability. Find E(X).
Example 24
easy
A game pays $20 with probability 0.1, $5 with probability 0.4, and $0 otherwise. Find E(winnings).
Example 25
easy
A coin is flipped; you win $1 on heads and lose $1 on tails. Find E(net per flip).
Example 26
easy
A spinner has equal regions paying $1, $2, $3, $4, $5. Find E(payoff).
Example 27
medium
A raffle sells 500 tickets at $2 each. The prize is $300. Find the expected net value of one ticket.
Example 28
medium
A discrete distribution: P(0)=0.1,P(2)=0.4,P(5)=0.3,P(10)=0.2. Find E(X).
Example 29
medium
Two fair dice are rolled. Find the expected value of the sum.
Example 30
medium
A test has 4 multiple-choice questions, each with 5 options. Random guessing: find E(correct).
Example 31
medium
A weighted die: faces {1,2,3,4,5,6} with P(6)=0.3 and other faces share the remaining 0.7 equally. Find E(X).
Example 32
medium
You pay $3 to play a game that pays $10 with probability 0.2 and $0 otherwise. Find E(net per play).
Example 33
hard
A box has 3 red and 2 blue balls. You draw 2 without replacement. Let X be the number of red drawn. Find E(X).
Example 34
hard
An insurance policy pays $50{,}000 if a claim occurs (probability 0.003) and $0 otherwise. What annual premium gives the insurer an expected profit of $50?
Example 35
hard
In a roulette bet on a single number (p=1/38), payoff is 35:1 (win $35, lose $1). Find E(net per $1 bet).
Example 36
hard
A die is rolled until a 6 appears. Find the expected number of rolls.
Example 37
hard
A fair coin is flipped 100 times. By indicator variables, find E(number of HH adjacent pairs in the sequence).
Example 38
hard
A continuous random variable has PDF f(x)=2x for 0≤x≤1. Find E(X).
Example 39
hard
A test pays $10 for each correct answer, −$3 for each wrong (no penalty if blank). Each question has 4 choices. If you guess on every question, what is E(score per question)?
Example 40
challenge
In the St. Petersburg paradox, a coin is flipped until tails; if tails appears on flip n, payoff is 2n. Find E(payoff).
Example 41
medium
A spinner has outcomes: $1 (prob 0.5), $3 (prob 0.3), $10 (prob 0.2). Find the expected value.
Example 42
medium
A game costs $4 to play. You win $20 with probability 0.1, $5 with probability 0.3, and $0 otherwise. Find the expected net value of one play.