Question

Wing Foot is a particular shoe franchise knows that its stores will not show a profit unless they gross over $940,000 per year. Let A be the event that a new store grosses over $940,000 its first year. Let B be the event that a store grosses over $940,00 its second year. The franchise has an administrative policy of closing a new store if it does not show a profit in either of the first 2 years. The accounting office at the franchise provided the following information: 67% of all the franchise stores show a profit the first year; 69% of all the franchise stores show a profit the second year (this includes stores that did not show a profit the first year); however, 85% of the franchise stores that showed a profit the first year also showed a profit the second year. Compute the following. (Use 2 decimal places.)(a) P(A).(b) P(B).(c) P(B | A).(d) P(A and B).(e) P(A or B).(f) What is the probability that a new Wing Foot store will not be closed after 2 years?What is the probability that a new Wing Foot store will be closed after 2 years?

Answer 1

Answer:

(a) 0.67         (b) 0.69

(c) 0.85         (d) 0.8254

(e) 0.5146     (f) 0.8254, 0.4854

Step-by-step explanation:

The events are:

A = A new store grosses over $940,000 its first year.

B =  A new store grosses over $940,000 its second year.

Given:

P (A) = 0.67, P (B) = 0.69 and P (B | A) = 0.85

Also, the franchise has an administrative policy of closing a new store if it does not show a profit in either of the first 2 years.

(a)

The probability that a new store grosses over $940,000 its first year is:

P (A) = 0.67.

(b)

The probability that a new store grosses over $940,000 its second year is:

P (B) = 0.69.

(c)

The probability that a store that showed a profit the first year also showed a profit the second year is:

P (B | A) = 0.85.

(d)

The probability that a store showed profit in both the first and second year is:

$P(A\cap B)=\frac{P(B|A)P(A)}{P(B)} =\frac{0.85\times0.67}{0.69} =0.8254$

Thus, the value of P (A and B) is 0.8254.

(e)

The probability that a store showed profit in either the first or the second year is:

$P(A\cup B)=P(A) + P(B) - P(A\cap B)=0.67+0.69-0.8254=0.5146$

Thus, the value of P (A or B) is 0.5146.

(f)

A store will be closed if it does not shows the profit in the first 2 years.

The probability that the Wing foot store will not be closed after 2 years is:

P (Wing foot will show profit for both 2 years) = P (A and B) = 0.8254.

Thus, the probability that the Wing foot store will not be closed after 2 years is 0.8254.

The probability that the Wing foot store will be closed after 2 years is:

P (Wing foot will not show profit for any of the 2 years) = P (neither A nor B)

                                                                                           $=P(A^{c}\cup B^{c})\\=1-P(A\cup B)\\=1-0.5146\\=0.4854$

Thus, the probability that a new store will be closed after 2 years is 0.4854.