Conditional Probability Examples in Math

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 Math.

Concept Recap

The conditional probability $P(A|B)$ is the probability of event $A$ occurring given that event $B$ has already occurred.

If I know $B$ happened, what's the chance of $A$ ? Updates probability with new info.

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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: $P(A|B)$ is the chance of $A$ once you restrict attention to only the cases where $B$ already happened.

Common stuck point: The procedure for conditional probability is the easy part; the trap is dividing by the whole sample space. Asking "Has some information already been revealed that shrinks the set of possible outcomes?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Has some information already been revealed that shrinks the set of possible outcomes?

Worked Examples

Example 1

medium

In a class of

30

students,

18

play soccer,

12

play basketball, and

6

play both. If a student plays soccer, what is the probability they also play basketball?

Answer

P(B \mid S) = \frac{1}{3}

First step

We need

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

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Example 2

hard

A test for a disease is

95\%

accurate (true positive rate) with a

3\%

false positive rate. If

1\%

of the population has the disease, what is the probability a person who tests positive actually has the disease?

Example 3

medium

200

surveyed,

120

own a dog and

80

own a cat;

50

own both.

P(\text{cat}|\text{dog})=?

Example 4

medium

Show why

P(A|B) \ne P(B|A)

in general by example:

P(A)=0.5,P(B)=0.1,P(A\cap B)=0.05

. Find both.

Example 5

hard

Among

1000

patients,

100

have disease

D

. A test has

90\%

sensitivity and

80\%

specificity. Of those testing positive, how many actually have

D

Example 6

challenge

Monty Hall:

3

doors,

1

prize. You pick door

1

. Monty reveals a goat behind door

3

. If you switch to door

2

P(\text{win})=?

Example 7

challenge

A pair of dice is rolled until the sum is 7 or 11. Find

P(\text{the stopping sum is 11})

Practice Problems

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

Example 1

medium

Two cards are drawn without replacement from a standard deck. Given that the first card is a king, what is the probability the second card is also a king?

Example 2

hard

A jar contains

6

red marbles and

4

blue marbles. Two marbles are drawn without replacement. Given that at least one marble is blue, what is the probability both marbles are blue?

Example 3

easy

A bag has 10 marbles, 4 of which are red.

P(\text{red})=?

Example 4

easy

Of 20 students, 12 like math and 5 like both math and art.

P(\text{art}|\text{math})=?

Example 5

easy

P(A\cap B)=0.2

and

P(B)=0.5

. Find

P(A|B)

Example 6

easy

A die is rolled. Given the result is even,

P(\text{it is }6)=?

Example 7

easy

A card is drawn from 52. Given it is a heart,

P(\text{it is the ace})=?

Example 8

easy

P(B)=0.4

P(A|B)=0.5

. Find

P(A\cap B)

Example 9

easy

In a class,

P(\text{passes}|\text{studied})=0.9

. If 20 studied, how many are expected to pass?

Example 10

easy

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

. What must be true of

P(B)

for this to be defined?

Example 11

medium

A box has 5 red and 3 blue. Two are drawn without replacement.

P(\text{2nd red}|\text{1st red})=?

Example 12

medium

60% of emails are spam; 80% of spam contains 'free'.

P(\text{spam and 'free'})=?

Example 13

medium

A test is 95% accurate.

P(\text{positive}|\text{disease})=0.95

P(\text{disease})=0.02

. Find

P(\text{disease and positive})

Example 14

medium

Table: of 100 people, 30 smoke; 18 of smokers have cough.

P(\text{cough}|\text{smoke})=?

Example 15

medium

Two dice rolled. Given the sum is 7,

P(\text{one die shows }3)=?

Example 16

medium

P(A)=0.3

P(B)=0.5

P(A\cap B)=0.15

. Find

P(A|B)

and state if

A,B

independent.

Example 17

medium

A jar: 7 green, 3 yellow. Draw two without replacement.

P(\text{both green})=?

Example 18

medium

In a deck,

P(\text{king}|\text{face card})=?

(face cards: J, Q, K).

Example 19

medium

P(\text{rain})=0.3

P(\text{traffic}|\text{rain})=0.7

P(\text{traffic}|\text{no rain})=0.2

. Find

P(\text{traffic})

Example 20

challenge

A test:

P(+|D)=0.9

P(+|D')=0.1

P(D)=0.05

. Find

P(D|+)

(Bayes).

Example 21

challenge

Three urns equally likely. Urn contents:

P(\text{red})=0.2,0.5,0.8

. A red is drawn.

P(\text{urn 3}|\text{red})=?

Example 22

challenge

Family has two children. Given at least one is a boy,

P(\text{both boys})=?

Example 23

easy

P(A \cap B) = 0.12

and

P(B) = 0.3

. Find

P(A|B)

Example 24

easy

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

P(\text{it is }3)

Example 25

easy

A card is drawn from

52

. Given it is a face card (J, Q, K), what is

P(\text{queen})

Example 26

easy

A coin is flipped twice. Given the first flip is heads, what is

P(\text{second is heads})

Example 27

easy

A class has

25

students;

10

play soccer and

4

of those also play tennis.

P(\text{tennis}|\text{soccer})=?

Example 28

easy

P(B) = 0.5

and

P(A|B) = 0.6

. Find

P(A \cap B)

Example 29

medium

Two dice are rolled. Given the sum is

8

, what is

P(\text{a die shows }5)

Example 30

medium

A box has

4

red and

6

blue balls. Two are drawn without replacement. Given the first is blue,

P(\text{second is blue})=?

Example 31

medium

70\%

of customers want coffee; among them,

30\%

also want pastry.

P(\text{coffee and pastry})=?

Example 32

medium

A two-way table:

40

students total.

24

took the bus,

16

walked. Among bus riders,

18

ate breakfast.

P(\text{breakfast}|\text{bus})=?

Example 33

medium

P(A)=0.6

P(B)=0.5

P(A\cap B)=0.3

. Are

A,B

independent? Answer

1

for yes.

Example 34

medium

A bag has

5

red,

3

green,

2

yellow. Two drawn without replacement.

P(\text{second is red}|\text{first is green})=?

Example 35

medium

P(\text{snow})=0.2

P(\text{accident}|\text{snow})=0.4

P(\text{accident}|\text{no snow})=0.05

P(\text{accident})=?

Example 36

medium

A family has

3

children. Given the eldest is a girl,

P(\text{all three are girls})=?

Assume independent and equally likely.

Example 37

hard

P(\text{rare})=0.01

P(+|\text{rare})=0.99

P(+|\text{not rare})=0.05

. By Bayes' rule,

P(\text{rare}|+)=?

Round to

3

decimals.

Example 38

hard

Three coins flipped. Given at least two heads,

P(\text{all three heads})=?

Example 39

hard

Two dice rolled. Given at least one die shows

6

P(\text{sum is }10)=?

Example 40

hard

Two urns: Urn 1 has

3

red,

1

blue; Urn 2 has

1

red,

3

blue. An urn is chosen at random, then a ball. Given the ball is red,

P(\text{Urn 1})=?

Example 41

challenge

A family has

2

children. Given that one of them is a boy born on a Tuesday,

P(\text{both are boys})=?

(Independent, equally likely days and sexes.)

Example 42

hard

A die is rolled twice. Given the sum is at least 9, find

P(\text{first roll is 6})

Example 43

challenge

A family has three children. Given at least one boy, find

P(\text{at least two boys})

. Assume each child independently is a boy with probability

1/2

Example 44

challenge

A randomly chosen integer from 1 to 1000 is divisible by 5. Find

P(\text{also divisible by 6}\mid\text{divisible by 5})

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Related Concepts

Probability Independent Events

Background Knowledge

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

probabilityindependent events