Select the correct answer.

Your personal library contains books written by 165 authors. [tex]60\%[/tex] of the authors are men. [tex]40\%[/tex] of the authors write only nonfiction works. Also, 40 of the male authors write only nonfiction works. Assuming that your library has only one book by each author, what is the probability that a book picked at random is either a work written by an author who writes only nonfiction or a work written by a man?

A. [tex]\frac{3}{5}[/tex]
B. [tex]\frac{58}{165}[/tex]
C. [tex]\frac{25}{33}[/tex]
D. [tex]\frac{2}{5}[/tex]



Answer :

To solve the problem, we will analyze the information step-by-step:

1. Total Number of Authors: There are 165 authors.

2. Percentage of Male Authors: [tex]\(60\%\)[/tex] of the authors are men.
- Number of male authors:
[tex]\[ \text{Number of male authors} = 165 \times 0.60 = 99 \][/tex]

3. Percentage of Authors Who Write Only Nonfiction: [tex]\(40\%\)[/tex] of the authors write only nonfiction.
- Number of nonfiction authors:
[tex]\[ \text{Number of nonfiction authors} = 165 \times 0.40 = 66 \][/tex]

4. Number of Male Authors Who Write Only Nonfiction:
- Given that 40 male authors write only nonfiction.

5. Number of Male Authors Who Write Other Works:
- The total number of male authors is 99, and 40 write only nonfiction. Therefore, the number of male authors who write other works is:
[tex]\[ \text{Number of male authors who write other works} = 99 - 40 = 59 \][/tex]

6. Number of Nonfiction Authors Who Are Not Male:
- There are 66 nonfiction authors in total. Out of these, 40 are male.
[tex]\[ \text{Number of nonfiction authors who are not male} = 66 - 40 = 26 \][/tex]

_We are interested in finding the probability that a book picked at random is either a work written by an author who writes only nonfiction or a work written by a man._

7. Number of Favorable Cases:
- Books written by male authors who write only nonfiction: 40
- Books written by male authors who write other works: 59
- Books written by nonfiction authors who are not male: 26
The number of favorable cases is:
[tex]\[ \text{Total favorable cases} = 40 + 59 + 26 = 125 \][/tex]

8. Calculate the Probability:
- The probability that a randomly selected book is either written by an author who writes only nonfiction or written by a man is the number of favorable cases divided by the total number of authors:
[tex]\[ \text{Probability} = \frac{\text{Total favorable cases}}{\text{Total number of authors}} = \frac{125}{165} \][/tex]

9. Simplify the Fraction:
[tex]\[ \frac{125}{165} \approx 0.7576 \quad \text{(a recurring decimal which is also approximately)} \quad \frac{25}{33} \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{\frac{25}{33}} \][/tex]

Therefore, the correct answer is C.