Practice Dependence (Statistical) in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy

A deck of 52 cards. Find

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

without replacement.

Example 2

easy

A bag has

4

red and

6

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

Example 3

medium

P(A)=0.6

and

P(A\cap B)=0.18

. Find

P(B\mid A)

Example 4

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 5

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 6

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 7

medium

True or false: if

A

and

B

are independent then

P(A \mid B) = P(A)

Example 8

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 9

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 10

easy

True or false: if

P(B \mid A) = P(B)

, then

A

and

B

are independent.

Example 11

easy

True or false: dependence implies causation.

Example 12

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 13

easy

P(A) = 0.4

P(B) = 0.5

, and

P(A \cap B) = 0.20

, are

A

and

B

independent?

Example 14

easy

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

Example 15

medium

Roll a fair die. Let

A

= 'outcome is even' and

B

= 'outcome

\le 3

'. Are

A

and

B

independent?

Example 16

hard

A jar contains

2

red,

3

blue, and

5

green balls. Two are drawn without replacement. Find

P(\text{same color})

Example 17

easy

P(A)=0.3

P(B)=0.5

P(A\cap B)=0.15

, are

A

and

B

independent?

Example 18

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 19

medium

A box has

3

defective and

7

good items. Two are picked without replacement. Find

P(\text{both defective})

Example 20

medium

P(A)=0.4

P(B)=0.5

P(A\cap B)=0.3

. Are

A

and

B

independent? Show the check.

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Related Concepts

Probability Independent Events Conditional Probability Causation