Addition Rule Examples: 47 Problems with Answers

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

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 addition rule finds the probability that at least one of two events occurs. It adds the probabilities of the two events and then subtracts any overlap so the shared outcomes are not counted twice.

If you want “A or B,” start by adding A and B. Then fix the double-counting by removing the part that belongs to both events.

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: Addition Rule 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 addition rule 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

easy

P(A)=0.5

P(B)=0.5

P(A\cap B)=0.5

. Find

P(A\cup B)

and interpret.

Answer

0.5

First step

P(A\cup B)=0.5+0.5-0.5=0.5

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Example 2

medium

Two dice are rolled. Find

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

Example 3

hard

Among 120 students, 60 take Math, 45 take Physics, 50 take Chemistry; 20 take both Math and Physics, 25 Math and Chemistry, 15 Physics and Chemistry, and 10 take all three. How many take at least one subject?

Practice Problems

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

Example 1

easy

Find

P(A\cup B)

P(A)=0.3

P(B)=0.4

, and

P(A\cap B)=0.1

Use P(A∪B) = P(A) + P(B) − P(A∩B)

Example 2

easy

Events

A

and

B

are mutually exclusive with

P(A)=0.2

P(B)=0.5

. Find

P(A\cup B)

Mutually exclusive: the circles do not overlap

Example 3

easy

A die is rolled. Find

P(\text{even or }6)

Example 4

easy

From a deck, find

P(\text{heart or spade})

Example 5

easy

P(A)=0.6

P(B)=0.5

P(A\cup B)=0.8

. Find

P(A\cap B)

Find P(A∩B) using the addition rule

Example 6

easy

A card is drawn. Find

P(\text{king or queen})

Example 7

easy

P(A)=0.45

P(B)=0.35

, disjoint. Find

P(A\cup B)

Disjoint events: P(A∪B) = P(A) + P(B)

Example 8

easy

A die is rolled. Find

P(\text{less than 3 or greater than 4})

Example 9

medium

From a deck, find

P(\text{heart or face card})

. There are 13 hearts, 12 face cards, 3 of which are hearts.

Card counts out of 52

Example 10

medium

In a class,

60\%

like pizza,

50\%

like tacos, and

30\%

like both. Find the percent who like at least one.

Shaded region = 'at least one'

Example 11

medium

Using the same class (

60\%

pizza,

50\%

tacos,

30\%

both), find the percent who like neither.

Outside = 'like neither'; the four regions must sum to 1

Example 12

medium

A number is chosen from 1 to 20. Find

P(\text{multiple of 3 or multiple of 5})

Example 13

medium

P(A)=0.5

P(B)=0.4

. If

A

and

B

are independent, find

P(A\cup B)

Example 14

medium

A spinner has regions A (

P=0.4

), B (

P=0.35

), C (

P=0.25

), all disjoint. Find

P(A\text{ or }B)

Example 15

medium

Of 100 students, 40 take French, 30 take Spanish, 10 take both. How many take at least one language?

Shaded union = students taking at least one language

Example 16

medium

P(A)=0.7

P(A\cap B)=0.2

P(A\cup B)=0.9

. Find

P(B)

Use the addition rule to recover P(B)

Example 17

medium

A weather model says

P(\text{rain})=0.4

P(\text{wind})=0.5

P(\text{rain and wind})=0.3

. Find

P(\text{rain or wind})

Shaded union = P(rain or wind)

Example 18

challenge

Three events

A,B,C

are pairwise disjoint with probabilities

0.2,0.3,0.4

. Find

P(A\cup B\cup C)

and the probability of none.

Example 19

challenge

Numbers 1 to 30: find

P(\text{divisible by 2, 3, or 5})

using inclusion-exclusion.

Example 20

challenge

P(A)=0.5

P(B)=0.6

. Find the smallest possible

P(A\cup B)

Example 21

easy

Find

P(A\cup B)

P(A)=0.25

P(B)=0.55

, and

P(A\cap B)=0.15

Read off P(A∪B) = aOnly + intersection + bOnly

Example 22

easy

A die is rolled. Find

P(\text{multiple of 2 or multiple of 3})

Example 23

easy

From a standard 52-card deck, find

P(\text{ace or club})

Example 24

easy

P(A)=0.65

P(B)=0.45

P(A\cup B)=0.85

. Find

P(A\cap B)

Find P(A∩B) = P(A) + P(B) − P(A∪B)

Example 25

easy

A card is drawn from a deck. Find

P(\text{jack or 10})

Example 26

medium

From a deck, find

P(\text{red or king})

Example 27

medium

In a survey,

70\%

own a phone,

40\%

own a laptop, and

30\%

own both. Find the percent who own at least one device.

Shaded union = owns at least one device

Example 28

medium

Using the same survey (

70\%

phone,

40\%

laptop,

30\%

both), find the percent who own neither.

Outside = owns neither device

Example 29

medium

A number is chosen from 1 to 40. Find

P(\text{multiple of 4 or multiple of 6})

Example 30

medium

P(A)=0.6

P(B)=0.3

. If

A

and

B

are independent, find

P(A\cup B)

Example 31

medium

A spinner has four disjoint regions with probabilities

0.20, 0.30, 0.10, 0.40

. Find

P(\text{first or third region})

Example 32

medium

Of 200 students, 80 play soccer, 70 play basketball, and 25 play both. How many play at least one of the two sports?

Shaded union = students who play at least one sport

Example 33

medium

A forecast gives

P(\text{rain})=0.35

P(\text{snow})=0.20

P(\text{rain and snow})=0.05

. Find

P(\text{neither rain nor snow})

Outside = P(neither rain nor snow)

Example 34

medium

From integers 1 to 50, find

P(\text{prime or even})

. (Note: 2 is even and prime.)

Example 35

medium

P(A)=0.4

P(B)=0.5

P(A\cup B)=0.75

. Are

A

and

B

independent?

Example 36

medium

Two dice are rolled. Find

P(\text{sum}\le 4\text{ or both dice show the same number})

Example 37

hard

In a town,

P(\text{owns car})=0.78

P(\text{owns bike})=0.42

, and

P(\text{owns both})=0.30

. A person is chosen at random. Find

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

Exactly one = aOnly + bOnly

Example 38

hard

Given

P(A)=0.5

and

P(B)=0.3

, find the largest and smallest possible values of

P(A\cup B)

Example 39

hard

From a deck, find

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

Example 40

hard

Suppose

P(A)=0.45

P(B)=0.55

, and

A,B

are independent. Find

P(\text{exactly one of }A,B)

Example 41

hard

Numbers 1 to 100: find the count divisible by 4 or 6 or 9, using inclusion-exclusion.

Example 42

challenge

Events

A,B,C

satisfy

P(A)=P(B)=P(C)=0.5

, pairwise independent, and

P(A\cap B\cap C)=0.1

. Find

P(A\cup B\cup C)

Example 43

challenge

P(A)=0.6

P(B)=0.7

. Find the smallest possible value of

P(A\cap B)

Smallest possible P(A∩B) when A and B together must fill the whole space

Example 44

challenge

A coin is flipped 3 times. Find

P(\text{at least one head})

two ways and confirm they agree.

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Related Concepts

Compound Events Sample Space Conditional Probability

Background Knowledge

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

compound eventsstat sample space