Practice Expected Value in Math

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 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.

Showing a random 20 of 50 problems.

Example 1

medium

A weighted die shows

6

with probability

0.5

and each of

1

–

5

with probability

0.1

. Find

E(X)

Example 2

hard

Roll a fair die. Let

X

= number of distinct prime factors of the outcome. Find

E(X)

Example 3

medium

A raffle sells 100 tickets at $2 each; one ticket wins $150. What is the expected value of buying one ticket (net)?

Example 4

easy

What is the expected value of one roll of a fair 6-sided die?

Example 5

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 6

medium

Draw one card. You win $4 for a face card (J/Q/K) else lose $1. Find

E(X)

Example 7

medium

X

has

E(X) = 5

. Find

E(2X - 7)

Example 8

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 9

challenge

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

E(\text{number of January birthdays})

Example 10

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?

Example 11

easy

A bet pays $6 with prob

\frac{1}{3}

and $0 otherwise. Find

E(X)

Example 12

hard

A binomial

X \sim \text{Binomial}(10, 0.3)

. Find

E(X)

Example 13

easy

A game pays $10 with probability

0.2

and $0 otherwise. Find the expected payout.

Example 14

medium

You draw one card from a deck. You win $13 for an ace, else lose $1. Find the expected gain.

Example 15

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 16

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 17

easy

A random variable takes value

0

with probability

1

. What is its expected value?

Example 18

medium

A game costs $3 to play and pays $10 with probability

0.25

, else nothing. Find the expected net gain.

Example 19

challenge

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

E(\text{number of rolls})

Example 20

medium

Two fair dice are rolled. Find

E(\text{sum})

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Related Concepts

Probability Mean Variance