Practice Conditional Probability 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 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.

Showing a random 20 of 54 problems.

Example 1

medium

A family has

3

children. Given the eldest is a girl,

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

Assume independent and equally likely.

Example 2

medium

70\%

of customers want coffee; among them,

30\%

also want pastry.

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

Example 3

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 4

medium

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

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

Example 5

easy

P(B) = 0.5

and

P(A|B) = 0.6

. Find

P(A \cap B)

Example 6

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 7

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 8

challenge

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

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

Example 9

easy

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

. What must be true of

P(B)

for this to be defined?

Example 10

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 11

easy

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

P(\text{second is heads})

Example 12

medium

P(A)=0.6

P(B)=0.5

P(A\cap B)=0.3

. Are

A,B

independent? Answer

1

for yes.

Example 13

easy

A

and

B

are independent and

P(A) = 0.4

, find

P(A|B)

Example 14

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})

Example 15

medium

A bag has

5

red,

3

green,

2

yellow. Two drawn without replacement.

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

Example 16

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 17

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 18

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 19

hard

Three coins flipped. Given at least two heads,

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

Example 20

easy

P(A\cap B)=0.2

and

P(B)=0.5

. Find

P(A|B)

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Related Concepts

Probability Independent Events