Question

A national survey of small-business owners was conducted to determine the challenges for growth for their businesses. The top challenge, selected by 46% of the small-business owners, was the economy. A close second was finding qualified workers (37%). Suppose 12% of the small-business owners selected both the economy and finding qualified workers as challenges for growth. A small-business owner is randomly selected.
a. What is the probability that the owner believes the economy is a challenge for growth if the owner believes that finding qualified workers is a challenge for growth?
b. What is the probability that the owner believes that finding qualified workers is a challenge for growth if the owner believes that the economy is a challenge for growth?
c. Given that the owner does not select the economy as a challenge for growth, what is the probability that the owner believes that finding qualified workers is a challenge for growth?
d. What is the probability that the owner believes neither that the economy is a challenge for growth nor that finding qualified workers is a challenge for growth?

Answer 1

Answer:

A) 12/37

B) 12/46

C) 25/54

D) 29/100

Step-by-step explanation:

Let event A = small-business owner selected economy,

event B = small-business owner selected finding qualified workers,

Thus, probabilities;

P(A)=46/100

P(B)=37/100

P(A∩B)=12/100

A) P(A ∩ B)/P(B) = (12/100)/(37/100) = 12/37

B) P(A∩B)/P(A) = (12/100)/(46/100)= 12/46

C) P(B∩A')= P(B) − P(A∩B) =

37/100 - 12/100 =25/100

P(B∩A')/P(A') =

(25/100)/(1-46/100) = 25/54

D) Using inclusion-exclusion principle;

P(A∪B) = P(A)+P(B)−P(A∩B)=

46/100 + 37/100 - 12/100 = 71/100

Hence, the probability that the owner believes neither that the economy is a challenge for growth nor that finding qualified workers is a challenge for growth is;

P( A' ∩ B') = 1 − P(A∪B)= 1− (71/100)= 29/100