Expected Value Examples in Statistics

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.

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How to Use These Examples

  • 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 NN has P(N=k)=(1/2)kP(N=k)=(1/2)^k for k=1,2,…k=1,2,\dots. Find E(N)E(N).

Answer

22

First step

1
This is geometric with p=0.5p=0.5.

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Example 2

hard
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 0.50.5 and $0 with probability 0.50.5. 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 0.010.01 and $0 otherwise. Find the expected winnings.

Example 4

easy
A random variable takes value 2 with probability 0.30.3 and 5 with probability 0.70.7. Find E(X)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-\$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)=2.5E(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 0.30.3, $0 otherwise. Find the expected net value per play.

Example 10

medium
A random variable: P(0)=0.2P(0)=0.2, P(1)=0.5P(1)=0.5, P(2)=0.3P(2)=0.3. Find E(X)E(X).

Example 11

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

Example 14

medium
A carnival game pays $5, $2, or $0 with probabilities 0.10.1, 0.30.3, 0.60.6. Find the expected payout.

Example 15

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

Example 16

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

Example 17

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 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 0.40.4 and $1 with probability 0.60.6. If E(X)=$3E(X)=\$3, find xx.

Example 20

challenge
A game offers: −$1-\$1 with probability $0.5,$0withprobability$0.3, \$0 with probability \$0.3, and $kwithprobability$0.2 with probability \$0.2. Find kk so the game is fair (E(X)=0E(X)=0).

Example 21

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 22

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

Example 23

easy
A random variable takes values −2-2 and 55 with equal probability. Find E(X)E(X).

Example 24

easy
A game pays $20 with probability 0.10.1, $5 with probability 0.40.4, and $0 otherwise. Find EE(winnings).

Example 25

easy
A coin is flipped; you win $1 on heads and lose $1 on tails. Find EE(net per flip).

Example 26

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

Example 27

medium
A raffle sells 500500 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)=0.2P(0)=0.1, P(2)=0.4, P(5)=0.3, P(10)=0.2. Find E(X)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 EE(correct).

Example 31

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 32

medium
You pay $3 to play a game that pays $10 with probability 0.20.2 and $0 otherwise. Find EE(net per play).

Example 33

hard
A box has 3 red and 2 blue balls. You draw 2 without replacement. Let XX be the number of red drawn. Find E(X)E(X).

Example 34

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 35

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 36

hard
A die is rolled until a 66 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)E(\text{number of HH adjacent pairs in the sequence}).

Example 38

hard
A continuous random variable has PDF f(x)=2xf(x)=2x for 0≤x≤10 \le x \le 1. Find E(X)E(X).

Example 39

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 40

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 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.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

probability basicweighted average