Conditional Probability Examples in Statistics

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

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

Concept Recap

Conditional probability is the probability that one event happens given that another event has already happened. It narrows the sample space to the cases where the given condition is true.

Once you know event B happened, you no longer look at every outcome. You only look at the part of the sample space where B is true, then ask how much of that smaller space also satisfies A.

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: Conditional Probability 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 conditional probability 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?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Conditional Probability Mistakes

Worked Examples

Example 1

medium

In a study, 60% of subjects exercise, 40% follow a diet plan, and 25% do both. Find the probability a randomly sampled subject follows the diet plan given they exercise.

60% exercise, 40% follow diet plan, 25% do both

Answer

\frac{5}{12}

First step

Identify

P(D\cap E)=0.25

and

P(E)=0.60

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Example 2

medium

A two-way table of 500 customers shows 200 buy Brand A and 80 buy both Brand A and Brand B. Find

P(\text{Brand B} \mid \text{Brand A})

Example 3

medium

A survey shows

P(\text{owns car})=0.7

P(\text{owns bike})=0.4

P(\text{owns both})=0.3

. Find

P(\text{owns bike} \mid \text{owns car})

P(car)=0.7, P(bike)=0.4, P(both)=0.3 — find P(bike | car)

Example 4

hard

Two cards are drawn without replacement from a 52-card deck — a common simple-random-sample model. Find

P(\text{2nd is king} \mid \text{1st is king})

Two cards without replacement from a standard 52-card deck

Example 5

hard

From 800 sampled adults, 320 own pets and 60% of pet owners read newspapers, while 30% of non-pet-owners do. Find

P(\text{newspaper reader} \mid \text{adult sampled})

Example 6

challenge

In a random sample, 1% of subjects have a rare allele. A genetic test has

99\%

sensitivity and

98\%

specificity. If a randomly sampled subject tests positive, find

P(\text{allele} \mid \text{positive})

Example 7

easy

A card known to be a spade is drawn. What is

P(\text{queen} \mid \text{spade})

Example 8

medium

In a school,

60\%

of students play sports and

25\%

play sports AND music. What is

P(\text{music}\mid \text{sports})

Example 9

medium

A test detects a disease in

95\%

of sick people and in

10\%

of healthy people. If

4\%

of the population has the disease, find

P(\text{positive test})

Example 10

medium

In a survey of

200

adults:

120

drink coffee,

80

drink tea, and

50

drink both. Given an adult drinks coffee, find

P(\text{they also drink tea})

200 adults: 120 drink coffee, 80 drink tea, 50 drink both

Example 11

medium

A family has two children. Given at least one is a girl, what is

P(\text{both girls})

? (Assume each child equally likely boy/girl, independent.)

Example 12

medium

In a class,

40\%

are seniors and

60\%

of seniors take calculus. What is

P(\text{senior and takes calculus})

Example 13

hard

1\%

of a population has a disease. A test is

99\%

accurate (true positive and true negative rates both

99\%

). Given a positive test, find

P(\text{disease})

Example 14

hard

Bag

A

has

3

red and

7

white. Bag

B

has

6

red and

4

white. A bag is chosen at random and a ball drawn. Given the ball is red, find

P(\text{bag }A)

Example 15

hard

Three coins are flipped. Given at least two heads, find

P(\text{all heads})

Example 16

hard

In a town,

30\%

of cars are red.

20\%

of red cars and

5\%

of non-red cars have been in an accident. Given a car has been in an accident, find

P(\text{red})

Example 17

challenge

Monty Hall:

3

doors, one car. You pick door

1

. Monty (who knows the car) opens a goat door from

\{2,3\}

. Find

P(\text{car behind }1 \mid \text{Monty opens a goat})

assuming Monty always reveals a goat.

Practice Problems

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

Example 1

easy

A die is rolled and is known to be even. What is the probability it is a 2, given it is even?

Example 2

easy

A card is known to be a face card. What is

P(\text{it is a king} \mid \text{face card})

Example 3

easy

A die shows a number greater than 3. What is

P(\text{it is a 5} \mid >3)

Example 4

easy

Using

P(A\mid B)=\frac{P(A\cap B)}{P(B)}

, find

P(A\mid B)

P(A\cap B)=0.2

and

P(B)=0.5

Example 5

easy

A card is known to be red. What is

P(\text{it is a heart} \mid \text{red})

Example 6

easy

In a class, 12 students play sports and 4 of those also play music. What is

P(\text{music} \mid \text{sports})

12 students play sports; 4 of those also play music

Example 7

easy

A two-coin flip is known to have at least one head. The outcomes are HH, HT, TH. What is

P(HH \mid \text{at least one head})

Example 8

easy

P(A\cap B)=\frac{1}{6}

and

P(B)=\frac{1}{2}

, find

P(A\mid B)

Example 9

medium

A die is rolled. Given the result is odd, what is

P(\text{it is greater than 2})

Example 10

medium

In a survey,

P(\text{owns a dog})=0.4

and

P(\text{owns a dog and a cat})=0.1

. What is

P(\text{cat} \mid \text{dog})

Example 11

medium

A bag has 3 red and 2 blue. Two drawn without replacement. Given the first is red, what is

P(\text{second is red})

Bag: 3 red, 2 blue — draw two without replacement

Example 12

medium

Of 50 people, 30 like tea, 20 like coffee, and 10 like both. What is

P(\text{coffee} \mid \text{tea})

50 people surveyed: 30 like tea, 20 like coffee, 10 like both

Example 13

medium

Two dice are rolled. Given the sum is 8, what is

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

Example 14

medium

A test is 95% accurate. If

P(\text{disease and positive})=0.0095

and

P(\text{positive})=0.0595

, find

P(\text{disease} \mid \text{positive})

(round to 2 decimals).

Example 15

medium

A die is rolled. Let A = even, B = greater than 3. Find

P(A\mid B)

Example 16

medium

A bag has 4 white and 1 black. Two drawn without replacement. Given the first is white, what is

P(\text{second is black})

Bag: 4 white, 1 black — draw two without replacement

Example 17

medium

In a class of 40, 24 study Spanish, 18 study French, and 8 study both. What is

P(\text{French} \mid \text{Spanish})

Class of 40: 24 study Spanish, 18 study French, 8 study both

Example 18

challenge

A box has 4 red and 6 green. Two drawn without replacement. Given at least one is red, what is

P(\text{both red})

Example 19

challenge

Two dice are rolled. Given that at least one die shows a 4, what is

P(\text{the sum is 7})

Example 20

challenge

A family has two children. Given at least one is a boy, what is

P(\text{both are boys})

? (Assume BB, BG, GB, GG equally likely.)

Example 21

easy

In a sample of 200 adults, 80 exercise weekly and 50 of those 80 also eat vegetarian. Find

P(\text{vegetarian} \mid \text{exercises})

200 adults: 80 exercise weekly, 50 of those are also vegetarian

Example 22

easy

A two-way table from a survey shows: of 300 voters, 180 prefer Party A and 90 of those also support Policy X. Find

P(\text{Policy X} \mid \text{Party A})

Example 23

easy

In a sample of 500 students, 200 take statistics, and 60 of those also take chemistry. Find

P(\text{chemistry} \mid \text{statistics})

Example 24

easy

From a batch of 1000 widgets, 40 are defective. Of those 40 defective widgets, 25 came from Machine A. Find

P(\text{Machine A} \mid \text{defective})

Example 25

medium

A survey of 400 commuters found 250 take the bus, 150 take the train, and 80 take both. Find

P(\text{train} \mid \text{bus})

400 commuters: 250 take bus, 150 take train, 80 take both

Example 26

medium

Out of 1000 sampled emails, 200 are spam, and 180 of those spam emails contain the word 'free'. Find

P(\text{contains 'free'} \mid \text{spam})

Example 27

medium

A randomly sampled patient tests positive. Given

P(\text{disease})=0.02

, sensitivity

0.99

, and

P(\text{positive})=0.0492

, find

P(\text{disease} \mid \text{positive})

Example 28

medium

A sample of 300 customers: 120 own a smartphone, 60 own a tablet, 30 own both. Find

P(\text{smartphone} \mid \text{tablet})

300 customers: 120 own smartphone, 60 own tablet, 30 own both

Example 29

medium

In a clinical trial, 50% of subjects get the treatment, and 30% of all subjects both get the treatment and respond. Find

P(\text{respond} \mid \text{treatment})

Clinical trial: 50% get treatment, 30% of all subjects both treated and respond

Example 30

medium

A sample yields

P(A)=0.4

P(B)=0.5

P(A\cup B)=0.7

. Find

P(B \mid A)

Example 31

medium

Quality-control data: 5% of items are defective, 2% are both defective and from Shift 1, and 40% are from Shift 1. Find

P(\text{defective} \mid \text{Shift 1})

Example 32

medium

From sampled data,

P(A)=0.30

P(B \mid A)=0.40

. Find

P(A\cap B)

Example 33

medium

In a random sample of 600 voters, 360 favor Candidate X. Among the 250 women sampled, 175 favor X. Find

P(\text{favors X} \mid \text{woman})

Example 34

medium

From historical data,

P(\text{rain})=0.30

P(\text{rain and commute delay})=0.18

. Find

P(\text{commute delay} \mid \text{rain})

Example 35

hard

A disease has prevalence

0.5\%

. A test has sensitivity

98\%

and specificity

95\%

. Find

P(\text{disease} \mid \text{positive})

Example 36

hard

In a population,

P(\text{college})=0.40

P(\text{employed})=0.85

P(\text{college and employed})=0.36

. Find

P(\text{college} \mid \text{employed})

and compare with

P(\text{college})

Example 37

hard

A factory's data show

P(\text{defect} \mid \text{Machine A})=0.02

P(\text{defect} \mid \text{Machine B})=0.05

. Machines A and B produce 70% and 30% of items. Find

P(\text{Machine A} \mid \text{defect})

Example 38

hard

Survey data: 25% of teens use Platform A only, 35% use Platform B only, 30% use both, 10% use neither. Find

P(\text{uses A} \mid \text{uses B})

Example 39

hard

A sample of 1200 households finds 300 with broadband only, 200 with mobile only, 600 with both, 100 with neither. Find

P(\text{broadband} \mid \text{mobile})

1200 households: 300 broadband only, 200 mobile only, 600 both, 100 neither

Example 40

hard

From study data:

P(\text{snoring})=0.40

P(\text{heart disease})=0.10

P(\text{snoring and heart disease})=0.06

. Are snoring and heart disease independent? Justify with conditional probability.

Example 41

easy

A die is rolled. Given the result is less than

5

, what is

P(\text{it is a }1)

Example 42

easy

A card is drawn and known to be black. What is

P(\text{spade}\mid \text{black})

Example 43

easy

In a bag of

10

marbles,

6

are red and

4

are blue. You see a marble that is red. What is

P(\text{red}\mid \text{red})

Example 44

easy

A spinner has

8

equal sectors numbered

1

–

8

. Given the result is even, find

P(\text{it is }6)

Example 45

easy

Two dice are rolled. Given the sum is

4

, what is

P(\text{first die is }1)

Example 46

medium

Two dice are rolled. Given the first die is

3

, what is

P(\text{sum is }7)

Example 47

medium

From a deck, two cards are drawn without replacement. Given the first is a heart, what is

P(\text{second is a heart})

Example 48

medium

A box has

5

red,

3

blue, and

2

green balls. One ball is drawn. Given the ball is not blue, find

P(\text{green})

Example 49

medium

Two dice are rolled. Given the sum is even, find

P(\text{sum} = 8)

Example 50

hard

A bag has

4

red and

6

blue. Two drawn without replacement. Given at least one is red, find

P(\text{both red})

Example 51

hard

P(A) = 0.5

P(B) = 0.3

A

and

B

independent. Find

P(A\mid A\cup B)

Example 52

hard

A jar has

5

red and

5

blue. Draw

3

without replacement. Given the first is red, find

P(\text{exactly }2\text{ reds in the }3\text{ draws})

Example 53

challenge

A family has

3

children. Given that at least one is a boy born on Tuesday, find

P(\text{at least }2\text{ boys})

. (Each child independently boy/girl with prob

1/2

, and day of week uniform.)

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Related Concepts

Sample Space Compound Events Two-Way Tables Independent Events Multiplication Rule

Background Knowledge

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

stat sample spacecompound eventstwo way tables