From the following list of hypothetical situations, identify those that are independent events using the given probabilities of each event and the formal definition of independence.

a. In a yearly weather survey of a city, 32% of the days had precipitation, 25% of the days were weekend days, and 8% of the days had precipitation and were weekend days
b. In a voter study, 25% of the voters had a low income, 45% of the voters were registered Democrats, and 15% of voters had a low income and were registered Democrats.
c. In a study of color blindness, 10% of the individuals were color-blind, 52% of the individuals were male, and 8% of the individuals were color-blind and were male.
d. In a college course survey, 20% of students received an A in chemistry, 25% of students received an A in biology, and 5% of students received an A in both chemistry and biology.

To check if both events are independent you have to calculate the conditional probability between them and compare it with the probability of the event alone:

[tex]P(B/A)= \frac{P(AnB)}{P(A)} = \frac{0.08}{0.32}= 0.25[/tex]

Then P(B/A)= P(B) ⇒ A and B are independent.

b.

A: "the voter had low income" ⇒ P(A)= 0.25

B: "the voter is a registered Democrat" ⇒ P(B)= 0.45

P(A∩B)= 0.15

[tex]P(B/A)= \frac{P(AnB)}{P(A)} = \frac{0.15}{0.25} = 0.6[/tex]

P(B/A)≠P(B) ⇒ A and B are not independent.

c.

A: "the individual is color-blind" ⇒ P(A)= 0.10

B: "the individual is male" ⇒ P(B)= 0.52

P(A∩B)= 0.08

[tex]P(A/B)= \frac{P(AnB)}{P(B)}= \frac{0.08}{0.52}= 0.15[/tex]

P(A/B)≠P(A) ⇒ A and B are not independent.

d.

A:"The student received an A in Chemistry" ⇒ P(A)= 0.20

B:"The student received an A in Biology" ⇒ P(B)= 0.25

P(A∩B)= 0.05

[tex]P(A/B)= \frac{P(AnB)}{P(B)} = \frac{0.05}{0.25}= 0.2[/tex]

P(A/B)= P(A) ⇒ A and B are independent.

I hope it helps!