Compound Events Examples: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Compound Events.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

Compound events are probability events made up of two or more simple events combined using 'and' (both events occur) or 'or' (at least one occurs). For independent 'and' events, multiply probabilities; for 'or' events, add probabilities and subtract any overlap.

Simple event: rolling a 6. Compound event: rolling a 6 AND then flipping heads. For 'and,' multiply probabilities. For 'or,' add them (but subtract overlap if any).

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: Compound Events 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 compound events 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 card is drawn from a deck. What is

P(\text{a face card or a diamond})

Answer

P = \frac{11}{26}

First step

Face cards: 12. Diamonds: 13. Diamond face cards (overlap): 3.

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Example 2

medium

P(A)=0.6

P(B)=0.4

, and

A

and

B

are independent. Find

P(A\text{ and }B)

and

P(A\text{ or }B)

P(A)=0.6, P(B)=0.4, independent — find P(A and B) and P(A or B)

Example 3

hard

Two dice are rolled. What is

P(\text{at least one die shows a 5})

Example 4

challenge

Cards are drawn one at a time without replacement from a standard deck. What is

P(\text{first card is a king and second card is a queen})

Example 5

easy

A bag contains 3 red and 5 blue marbles. You draw one marble, replace it, then draw another. What is the probability of getting red then blue?

Example 6

medium

A bag contains 4 red and 6 blue marbles. You draw two marbles without replacement. What is the probability that both are red?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A coin is flipped and a die is rolled. What is

P(\text{heads and a 4})

Coin flip then die roll — find P(H and 4)

Example 2

easy

A die is rolled. What is

P(\text{rolling a 2 or a 5})

Example 3

easy

Two coins are flipped. What is

P(\text{both heads})

Two coin flips — find P(both heads)

Example 4

easy

A card is drawn from a deck. What is

P(\text{a king or a queen})

Example 5

easy

A spinner lands red with

P=\frac{1}{4}

each spin. Spun twice, what is

P(\text{red both times})

Spinner spun twice — find P(red both times)

Example 6

easy

A die is rolled. What is

P(\text{even or a 3})

Example 7

easy

A coin is flipped twice. What is

P(\text{tails on both flips})

Coin flipped twice — find P(tails on both)

Example 8

easy

A bag has 10 marbles: 4 red, 6 blue. What is

P(\text{red or blue})

Example 9

medium

A card is drawn from a deck. What is

P(\text{a heart or a king})

Hearts and Kings in a 52-card deck — find P(heart or king)

Example 10

medium

A die is rolled twice. What is

P(\text{a 6 on at least one roll})

Example 11

medium

A coin is flipped and a die rolled. What is

P(\text{heads or a 6})

Coin flip and die roll — find P(heads or a 6)

Example 12

medium

Two dice are rolled. What is

P(\text{both even})

Example 13

medium

A spinner is red with

P=0.2

, blue

0.5

, green

0.3

. Spun once, what is

P(\text{red or green})

Example 14

medium

A student passes math with

P=0.8

and passes science with

P=0.7

, independently. What is

P(\text{passes both})

Example 15

medium

A die is rolled twice. What is

P(\text{first roll odd and second roll a 6})

Die rolled twice — find P(first odd and second a 6)

Example 16

medium

A spinner lands red

\frac{2}{5}

or blue

\frac{3}{5}

, spun twice. What is

P(\text{at least one red})

Spinner spun twice — find P(at least one red) via complement: 1 − P(BB)

Example 17

medium

A card is drawn. What is

P(\text{a spade or a face card})

Spades and Face Cards in a 52-card deck — find P(spade or face card)

Example 18

challenge

In a group,

P(\text{coffee})=0.7

P(\text{tea})=0.4

P(\text{both})=0.25

. What is

P(\text{exactly one of the two})

Coffee and Tea preferences — find P(exactly one of the two)

Example 19

challenge

Three coins are flipped. What is

P(\text{at least two heads})

Example 20

challenge

A bag has 5 red and 5 blue. Two drawn without replacement. What is

P(\text{both same color})

Example 21

easy

A die is rolled and a coin is flipped. What is

P(\text{a 3 and tails})

Die rolled and coin flipped — find P(3 and tails)

Example 22

easy

A die is rolled. What is

P(\text{a 1 or an even number})

Example 23

easy

A card is drawn from a 52-card deck. What is

P(\text{a heart or a spade})

Example 24

easy

A bag has 5 red and 5 blue marbles. Two marbles are drawn with replacement. What is

P(\text{red then blue})

Two marbles drawn with replacement — find P(red then blue)

Example 25

easy

A die is rolled. What is

P(\text{less than 3 or greater than 5})

Example 26

easy

Two dice are rolled. What is

P(\text{both show a 1})

Example 27

medium

A coin is flipped 3 times. What is

P(\text{at least one head})

Example 28

medium

A bag has 4 red and 6 blue marbles. Two are drawn without replacement. What is

P(\text{both red})

Example 29

medium

Two dice are rolled. What is

P(\text{sum of 7 or sum of 11})

Example 30

medium

A coin is flipped 4 times. What is

P(\text{exactly 2 heads})

Example 31

medium

Two cards are drawn from a deck without replacement. What is

P(\text{both hearts})

Example 32

medium

A spinner has

P(\text{red})=0.4

. Spun 5 times, what is

P(\text{no red})

Example 33

hard

A bag has 3 red, 4 blue, 5 green marbles. Two are drawn without replacement. What is

P(\text{one red and one blue})

, in either order?

Example 34

hard

Three cards are drawn from a 52-card deck without replacement. What is

P(\text{all three are aces})

Example 35

hard

A coin is flipped 5 times. What is

P(\text{exactly 3 heads})

Example 36

hard

P(A)=0.5

P(B)=0.3

P(A \cap B)=0.1

. Find

P(A \cup B)

P(A)

Example 37

hard

A bag has 3 red, 5 blue, 2 green marbles. One marble is drawn. What is

P(\text{red or green})

Bag of 3 red, 5 blue, 2 green — find P(red or green)

Example 38

challenge

Three dice are rolled. What is

P(\text{at least one shows a 6})

Example 39

medium

A spinner has sections: Red (50%), Blue (30%), Green (20%). You spin twice. What is the probability of getting the same colour both times?

Example 40

hard

A deck has 52 cards. You draw 2 cards without replacement. What is the probability of drawing at least one ace?

← Back to Compound Events Practice Mode

Related Concepts

Basic Probability Sample Space Independent Events Conditional Probability

Background Knowledge

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

probability basicstat sample space