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)≠P(B)P(B|A) \neq P(B), meaning the events share information.

Knowing AA happened tells you something about BB—they're connected.

Showing a random 20 of 50 problems.

Example 1

easy
A deck of 52 cards. Find P(drawing two aces in a row)P(\text{drawing two aces in a row}) without replacement.

Example 2

easy
A bag has 44 red and 66 green balls. Draw one, replace it, draw another. Are the draws dependent?

Example 3

medium
P(A)=0.6P(A)=0.6 and P(A∩B)=0.18P(A\cap B)=0.18. Find P(B∣A)P(B\mid A).

Example 4

medium
If AA and BB are independent with P(A)=0.3P(A)=0.3 and P(B)=0.5P(B)=0.5, find P(A∩B)P(A\cap B) and P(B∣A)P(B\mid A).

Example 5

hard
Disease test: P(disease)=0.05P(\text{disease}) = 0.05. Test positive given disease: P(+∣D)=0.90P(+|D) = 0.90. Test positive given no disease: P(+∣Dc)=0.10P(+|D^c) = 0.10. Find P(D∩+)P(D \cap +) and P(Dc∩+)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(red)P(\text{red}).

Example 7

medium
True or false: if AA and BB are independent then P(A∣B)=P(A)P(A \mid B) = P(A).

Example 8

hard
A test has sensitivity P(+∣D)=0.95P(+\mid D) = 0.95 and specificity P(−∣Dc)=0.90P(-\mid D^c) = 0.90, with disease prevalence P(D)=0.02P(D) = 0.02. Find P(D∣+)P(D \mid +).

Example 9

hard
Verify whether smoking and lung cancer are dependent using the following: P(cancer)=0.06P(\text{cancer}) = 0.06, P(cancer∣smoker)=0.15P(\text{cancer}|\text{smoker}) = 0.15. What does this tell us about the relationship?

Example 10

easy
True or false: if P(B∣A)=P(B)P(B \mid A) = P(B), then AA and BB are independent.

Example 11

easy
True or false: dependence implies causation.

Example 12

easy
For independent events, P(A∩B)=P(A)×P(B)P(A\cap B)=P(A)\times P(B). If P(A)=0.5P(A)=0.5, P(B)=0.2P(B)=0.2 and they are independent, find P(A∩B)P(A\cap B).

Example 13

easy
If P(A)=0.4P(A) = 0.4, P(B)=0.5P(B) = 0.5, and P(A∩B)=0.20P(A \cap B) = 0.20, are AA and BB 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 AA = 'outcome is even' and BB = 'outcome ≤3\le 3'. Are AA and BB independent?

Example 16

hard
A jar contains 22 red, 33 blue, and 55 green balls. Two are drawn without replacement. Find P(same color)P(\text{same color}).

Example 17

easy
If P(A)=0.3P(A)=0.3, P(B)=0.5P(B)=0.5, P(A∩B)=0.15P(A\cap B)=0.15, are AA and BB independent?

Example 18

medium
A card is drawn. Event AA: it is red. Event BB: it is a heart. Are AA and BB independent? Use P(B∣A)P(B\mid A) vs P(B)P(B).

Example 19

medium
A box has 33 defective and 77 good items. Two are picked without replacement. Find P(both defective)P(\text{both defective}).

Example 20

medium
P(A)=0.4P(A)=0.4, P(B)=0.5P(B)=0.5, P(A∩B)=0.3P(A\cap B)=0.3. Are AA and BB independent? Show the check.
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