Question

4. Suppose that we randomly select one American Adult. Let A be the event that the individuals annual income $100,000 and let B be the event that the individual has at least a bachelors degree.

a. Without knowing any of the actual probabilities involved, would you expect the events A and B to be independent or not? Clearly explain in a few words.

According to a Census Bureau, P(A)= 0.20, P(B)= 0.35, P(A ∩ B)= 0.14

b. What is the probability that the individual selected does not have a bachelors degree?

d. What is the probability that the individual neither has a bachelors degree nor has income over $100,000

f. Given that the individual does not have a bachelors degree, what is the probability that he or she has income over $100,000?

h. Are the events A and B disjoint? clearly explain in a few words

Answer 1

a)A and B cannot be independent because intuitively speaking, annual income depends on the degrees a person possesses, higher the degree, higher will be the income and vice versa.
So, even if the probabilities are not specified, we may say that A and B are not independent.

b) Required probability = 1 – P(B) = 1 – 0.35 = 0.65

d) Required probability = P(A U B)^c = 1 – P(A U B) = 1 – {P(A) + P(B) – P(A ∩ B)} = 1 – (0.2 + 0.35 – 0.14)
= 0.59

f) Required probability = P(A|B^c) = P(A ∩ B^c) / P(B^c) = [P(A) – P(A ∩ B)] / [1 – P(B)] = (0.2-0.14)/(1-0.35)
= 0.0923

h) No, A and B are not disjoint because P(A ∩ B) = 0.14 > 0.[ A and B are called disjoint if P(A∩B)=0]