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.

Step-by-step explanation:

Hello!

Be two events A and B of the sample space S. A and B are independent when the ocurrence of A doesn't affect the probability of occurrence of B.

Symbolically:

P(B/A)= P(B) -or- P(A/B)= P(A)

If A and B are not independent then P(B/A) ≠ P(B) -or- P(A/B)≠ P(A)

P(A∩B)= P(A)*P(A/B)

a.

A: "the day had precipitation" ⇒ P(A)= 0.32

B: "the day was a weekday" ⇒ P(B)= 0.25

P(A∩B)= 0.08

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:

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

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

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

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

I hope it helps!

1e+27

step-by-step explanation:

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.