Question

Arthur Andersen Enterprise Group/National Small Business United,Washington,
conducted a national survey of small-business owners to determine the challenges
for growth for their businesses. The top challenge, selected by 46% of the
smallbusiness owners, was the economy. A close second was finding qualified workers
(37%). Suppose 15% 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) 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?
(b) 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

Let event A = small-business owner selected economy,

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

probabilities

$$P(A)=46/100, P(B)=37/100,P(A\cap B)=15/100.$$

b) Using inclusion-exclusion principle

$$P(A\cup B) = P(A)+P(B)-P(A\cap B)=$$

$$46/100+37/100-15/100 = 68/100$$

then  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

$$P(\bar{A}\cap \bar{B})= 1-P(A\cup B)=1 - 68/100 = 32/100.$$

a) Probability

$$P(B\cap \bar{A})=P(B)-P(A\cap B)=$$

$$37/100 - 15/100 = 22/100.$$

Conditional probability that the owner believes that finding qualified workers is a

challenge for growth, given that the owner does not select the economy as a

challenge for growth is

$$P(B|\bar{A})=P(B\cap\bar{A})/P(\bar{A})=$$

$$(22/100)/(1-46/100) = 22/54 = 11/27.$$

Answer: a) 11/27, b) 32/100.