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{59}{165}[/tex]
C. [tex]\frac{25}{33}[/tex]
D. [tex]\frac{2}{5}[/tex]



Answer :

Let's break down the solution step-by-step:

1. Total number of authors: 165

2. Number of male authors:
[tex]\[ 0.60 \times 165 = 99 \][/tex]

3. Number of authors who write only nonfiction:
[tex]\[ 0.40 \times 165 = 66 \][/tex]

4. Number of male authors who write only nonfiction: 40 (given)

5. Number of female authors who write only nonfiction:
[tex]\[ 66 - 40 = 26 \][/tex]

6. Number of male authors who write fiction:
[tex]\[ 99 - 40 = 59 \][/tex]

7. Number of female authors:
[tex]\[ 165 - 99 = 66 \][/tex]

8. Number of female authors who write fiction:
[tex]\[ 66 - 26 = 40 \][/tex]

Next, we determine the total number of authors who write nonfiction or are men.

9. Total number of authors who write nonfiction: 66 (given by calculation)

10. Total number of authors who are men: 99 (calculated above)

Now, to avoid double-counting the male authors who write only nonfiction, we need to subtract them once from the total of 99 male authors and 66 nonfiction authors:

11. Total number of authors who write nonfiction or are men:
[tex]\[ 66 + 99 - 40 = 125 \][/tex]

12. Probability that a randomly picked book is either written by an author who writes only nonfiction or a male author:
[tex]\[ \frac{125}{165} \][/tex]

Simplify this fraction:

13. [tex]\(\frac{125}{165} = 0.7575757575757576 \approx \frac{25}{33}\)[/tex]

Thus, the probability that a randomly picked book is either written by an author who writes only nonfiction or a male author is:

Answer: C. [tex]\(\frac{25}{33}\)[/tex]