Select the correct answer.

Your personal library contains books written by 165 authors. 60% of the authors are men. 40% of the authors write only nonfiction works. Additionally, 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. \(\frac{3}{5}\)
B. \(\frac{58}{165}\)
C. \(\frac{25}{33}\)
D. [tex]\(\frac{2}{5}\)[/tex]



Answer :

To find the correct probability, let's break down the problem step-by-step:

1. Total Authors: The total number of authors is 165.

2. Percentage of Men: 60% of the authors are men. This means the number of male authors is:
[tex]\[ 165 \times 0.60 = 99 \text{ men} \][/tex]

3. Percentage of Nonfiction Authors: 40% of the authors write only nonfiction. This means the number of nonfiction authors is:
[tex]\[ 165 \times 0.40 = 66 \text{ nonfiction authors} \][/tex]

4. Male Nonfiction Authors: We know that 40 male authors write only nonfiction.

5. Female Authors: Since there are 99 male authors, the number of female authors is:
[tex]\[ 165 - 99 = 66 \text{ women} \][/tex]

6. Female Nonfiction Authors: Out of 66 nonfiction authors, 40 are men. So, the number of nonfiction authors who are women is:
[tex]\[ 66 - 40 = 26 \text{ female nonfiction authors} \][/tex]

7. Male Fiction Authors: The number of men who write fiction only is:
[tex]\[ 99 - 40 = 59 \text{ male fiction authors} \][/tex]

8. Female Fiction Authors: The number of women who write fiction only is:
[tex]\[ 66 - 26 = 40 \text{ female fiction authors} \][/tex]

9. Nonfiction or Male Authors: To find out how many authors write either only nonfiction or are men, we sum up:
- The number of nonfiction authors (men and women): 66
- The number of male fiction authors: 59
So, the total number is:
[tex]\[ 66 + 59 = 125 \text{ authors} \][/tex]

10. Probability: The probability that a book picked at random is either a work by an author who writes only nonfiction or by a male author is:
[tex]\[ \frac{125}{165} \][/tex]

To simplify the fraction \(\frac{125}{165}\), we can divide the numerator and the denominator by their greatest common divisor, which is 5:
[tex]\[ \frac{125 \div 5}{165 \div 5} = \frac{25}{33} \][/tex]

Thus, the correct answer is:

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