A national survey of small-business owners was conducted to determine the challe

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 11% 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?

Explanation / Answer

A: owner believes the economy is a challenge for growth, P(A)=0.46

B: owner believes that finding qualified workers is a challenge for growth, P(B)=0.37

P( A and B)=0.11

a) P(A given B)=P(A and B)/P(B) =0.11/0.37=0.29729

b) P(B given A)=P(A and B)/P(A) =0.11/0.46=0.2391

c) P(B given not A)=P(B and not A)/P(not A) =(0.37-0.11)/(1-0.46)=0.4814

d) P(not A and not B) =1- P( A or B)=1-(P(A)+P(B)-P(A and B))=1-(0.46+0.37-0.11)=0.28

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