Dependence (Statistical) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Dependence (Statistical).

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

Concept Recap

Two events are statistically dependent when knowing one event occurred changes the probability of the other — formally, $P(B|A) \neq P(B)$ , meaning the events share information.

Knowing $A$ happened tells you something about $B$ —they're connected.

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: Two events are dependent when $P(B|A)\neq P(B)$ — learning $A$ happened shifts $B$ 's probability.

Common stuck point: The procedure for dependence (statistical) is the easy part; the trap is multiplying $P(A)\times P(B)$ for dependent events. Asking "Does knowing the first event occurred change the probability of the second?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does knowing the first event occurred change the probability of the second?

Worked Examples

Example 1

medium

A bag has 5 red and 3 blue balls. Two balls are drawn without replacement. Find

P(\text{both red})

using the multiplication rule for dependent events.

Answer

P(\text{both red}) = \frac{5}{14} \approx 0.357

First step

Event A = first ball is red:

P(A) = \frac{5}{8}

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

hard

Disease test:

P(\text{disease}) = 0.05

. Test positive given disease:

P(+|D) = 0.90

. Test positive given no disease:

P(+|D^c) = 0.10

. Find

P(D \cap +)

and

P(D^c \cap +)

Example 3

medium

A box has

3

defective and

7

good items. Two are picked without replacement. Find

P(\text{both defective})

Example 4

medium

A bag has

5

red and

5

blue marbles. Without replacement, find

P(\text{red then blue})

Example 5

hard

P(A) = 0.5

P(B \mid A) = 0.6

P(B \mid A^c) = 0.2

. Find

P(B)

using the law of total probability.

Example 6

challenge

Roll two dice. Are the events 'sum is 7' and 'first die is 3' independent?

Practice Problems

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

Example 1

easy

A deck of 52 cards. Find

P(\text{drawing two aces in a row})

without replacement.

Example 2

hard

Verify whether smoking and lung cancer are dependent using the following:

P(\text{cancer}) = 0.06

P(\text{cancer}|\text{smoker}) = 0.15

. What does this tell us about the relationship?

Example 3

easy

P(B)=0.4

but

P(B\mid A)=0.4

as well, are

A

and

B

dependent?

Example 4

easy

P(B\mid A)=0.7

and

P(B)=0.4

, are

A

and

B

dependent?

Example 5

easy

For independent events,

P(A\cap B)=P(A)\times P(B)

. If

P(A)=0.5

P(B)=0.2

and they are independent, find

P(A\cap B)

Example 6

easy

Are successive flips of a fair coin dependent or independent?

Example 7

easy

Drawing two cards WITHOUT replacement: is the second draw dependent on the first?

Example 8

easy

Rain and carrying umbrellas are statistically dependent. Does rain CAUSE umbrellas to exist?

Example 9

easy

For dependent events, the joint probability is

P(A)\times P(B\mid A)

. If

P(A)=0.5

and

P(B\mid A)=0.6

, find

P(A\cap B)

Example 10

easy

P(A)=0.3

P(B)=0.5

P(A\cap B)=0.15

, are

A

and

B

independent?

Example 11

medium

P(A)=0.4

P(B)=0.5

P(A\cap B)=0.3

. Are

A

and

B

independent? Show the check.

Example 12

medium

P(A)=0.6

and

P(A\cap B)=0.18

. Find

P(B\mid A)

Example 13

medium

A bag has 5 red and 5 blue marbles. Drawing two without replacement, find

P(\text{both red})

Example 14

medium

Knowing a person owns a raincoat raises the probability they own an umbrella. Is this dependence, causation, or both, and why?

Example 15

medium

A

and

B

are independent with

P(A)=0.5

P(B)=0.5

, find

P(A\cup B)

Example 16

medium

Why is using

P(A)\times P(B)

wrong for two dependent draws from a small bag?

Example 17

medium

A card is drawn. Event

A

: it is red. Event

B

: it is a heart. Are

A

and

B

independent? Use

P(B\mid A)

P(B)

Example 18

medium

Two fair dice are rolled. Event

A

: first die is even. Event

B

: second die is 5. Are

A

and

B

independent?

Example 19

medium

A

and

B

are independent with

P(A)=0.3

and

P(B)=0.5

, find

P(A\cap B)

and

P(B\mid A)

Example 20

challenge

P(A)=0.7

P(B)=0.6

P(A\cup B)=0.88

. Determine

P(A\cap B)

and whether

A,B

are independent.

Example 21

challenge

A jar has 3 defective and 7 good items. Two are inspected without replacement. Find

P(\text{at least one defective})

Example 22

challenge

Given

P(A)=0.5

P(B\mid A)=0.8

P(B\mid A^c)=0.2

, find

P(B)

using the law of total probability.

Example 23

easy

P(A) = 0.4

P(B) = 0.5

, and

P(A \cap B) = 0.20

, are

A

and

B

independent?

Example 24

easy

P(A) = 0.5

and

P(B \mid A) = 0.7

. Find

P(A \cap B)

Example 25

easy

A bag has

4

red and

6

green balls. Draw one, replace it, draw another. Are the draws dependent?

Example 26

easy

P(A) = 0.6

P(A \cap B) = 0.24

. Find

P(B \mid A)

Example 27

medium

P(A) = 0.3

P(B) = 0.4

P(A \cup B) = 0.58

. Find

P(A \cap B)

and decide independence.

Example 28

medium

Roll a fair die. Let

A

= 'outcome is even' and

B

= 'outcome

\le 3

'. Are

A

and

B

independent?

Example 29

medium

Draw two cards without replacement from a

52

-card deck. Find

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

Example 30

medium

In a study,

P(\text{exercise}) = 0.4

and

P(\text{exercise} \mid \text{healthy}) = 0.7

. Are exercise and being healthy dependent?

Example 31

medium

P(A \cap B) = 0.18

P(A) = 0.6

P(B) = 0.3

. Are

A

and

B

independent?

Example 32

medium

Roll a fair die twice. Let

A

= 'first roll is

6

' and

B

= 'sum is

10

'. Find

P(A \cap B)

P(A)

P(B)

, and decide dependence.

Example 33

medium

In a population,

P(\text{smoker}) = 0.2

and

P(\text{smoker} \mid \text{cancer}) = 0.5

. Are smoking and cancer statistically dependent?

Example 34

hard

A test has sensitivity

P(+\mid D) = 0.95

and specificity

P(-\mid D^c) = 0.90

, with disease prevalence

P(D) = 0.02

. Find

P(D \mid +)

Example 35

hard

A jar contains

2

red,

3

blue, and

5

green balls. Two are drawn without replacement. Find

P(\text{same color})

Example 36

hard

A factory has two machines:

M_1

produces

60\%

of parts with

2\%

defect rate;

M_2

produces

40\%

with

5\%

defect rate. Find

P(M_1 \mid \text{defective})

Example 37

hard

Three urns. Urn 1: 2 red, 3 blue. Urn 2: 4 red, 2 blue. Urn 3: 1 red, 5 blue. Pick an urn uniformly, then a ball. Find

P(\text{red})

Example 38

hard

A

and

B

are independent with

P(A) = 0.4

P(B) = 0.5

. Find

P(A \cup B)

and

P(A \cap B^c)

Example 39

challenge

A spam filter flags

80\%

of spam and

5\%

of non-spam. Prior

P(\text{spam}) = 0.4

. Find

P(\text{spam} \mid \text{flagged})

← Back to Dependence (Statistical) Formula Explained Practice Mode

Related Concepts

Probability Independent Events Conditional Probability Causation

Background Knowledge

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

probabilityindependent events