Question

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: 69% of all the franchise stores show a profit the first year; 77% of all the franchise stores show a profit the second year (this includes stores that did not show a profit the first year); however, 89% of the franchise stores that showed a profit the first year also showed a profit the second year. Compute the following. (Enter your answers to four 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 store will not be closed after 2 years?

Answer 1

Answer:

(a) 0.69      (d) 0.798

(b) 0.77       (e) 0.662

(c) 0.77        (f) 0.338

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) = 0.69, P (B) = 0.77 and P (B | A) = 0.89

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) = 0.69.

(b)

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

P (B) = 0.77.

(c)

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

P (B | A) = 0.89

(d)

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

$P (A\ and\ B)=\frac{P(B|A)P(A)}{P(B)}=\frac{0.89\times0.69}{0.77}=0.798$

Thus, the value of P (A and B) is 0.798.

(e)

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

$P(A\ or\ B)=P(A)+P(B)-P(A\ and\ B)=0.69+0.77-0.798=0.662$

Thus, the value of P (A or B) is 0.662.

(f)

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

Compute the value of $P(A^{c}\ or\ B^{c})$ as follows:

$P(A^{c}\ or\ B^{c})=1-P(A\ or\ B)=1-0.662=0.338$

Thus, the probability that a new store will not be closed after 2 years is 0.338.