Expected Value Examples in Math

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 Math.

Concept 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.

Read the full concept explanation →

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 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

A fair die has six equally likely outcomes

\{1,2,3,4,5,6\}

, each with probability

\frac{1}{6}

Full solution

2
Apply the expected value formula: $E(X) = \sum x_i \cdot P(x_i) = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6)$
3
Compute the sum: $\frac{1}{6} \times 21 = \frac{21}{6} = 3.5$

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 \sim \text{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

\$k^2

. 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(\text{sum})

Example 31

medium

A roulette bet on red pays $1 with prob

\frac{18}{38}

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

\frac{1}{2}

4

with prob

\frac{1}{4}

8

with prob

\frac{1}{4}

. 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 \in \{1, 3\}

each with prob

0.5

Y \in \{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(\text{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(\text{payout})

Example 40

challenge

You roll a fair die repeatedly until you get a 6. Find

E(\text{number of rolls})

Example 41

challenge

In a class of 20, each student picks a random month for their birthday. Find

E(\text{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?

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Related Concepts

Probability Mean Variance

Background Knowledge

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

probabilitymean