To solve this problem, we need to determine how many books the bookstore needs to rent on average per hour to cover the running cost of \[tex]$6 per hour. Here are the steps:
1. Calculate the expected number of books rented per hour using the given distribution:
\[
\text{Expected number of books} = \sum (\text{Number of books} \times \text{Probability})
\]
Given the distribution:
\[
E(X) = 0 \times \frac{1}{16} + 1 \times \frac{4}{16} + 2 \times \frac{6}{16} + 3 \times \frac{4}{16} + 4 \times \frac{1}{16}
\]
\[
E(X) = 0 + \frac{4}{16} + \frac{12}{16} + \frac{12}{16} + \frac{4}{16}
\]
\[
E(X) = \frac{4}{16} + \frac{12}{16} + \frac{12}{16} + \frac{4}{16} = \frac{32}{16} = 2.0
\]
Therefore, the expected number of books rented per hour is 2.
2. Calculate the expected income per hour:
Since each book is rented for \$[/tex]2,
[tex]\[
\text{Expected Income} = 2 \times \text{Expected number of books}
\][/tex]
[tex]\[
\text{Expected Income} = 2 \times 2 = 4.0
\][/tex]
So, the expected income is \[tex]$4 per hour.
3. Determine how many books need to be rented to break even:
The bookstore's running cost is \$[/tex]6 per hour, and renting a book brings in \$2. To break even, the income from renting books must equal the running cost.
[tex]\[
\text{Break-even books} = \frac{\text{Running cost per hour}}{\text{Rental cost per book}}
\][/tex]
[tex]\[
\text{Break-even books} = \frac{6}{2} = 3.0
\][/tex]
Thus, the bookstore needs to rent 3 books each hour, on average, to break even. The correct answer is:
[tex]\[
3
\][/tex]
