To find the probability that a randomly selected book from your library is either a work written by an author who writes only nonfiction or a work written by a man, let's follow these steps:
1. Determine the total number of authors: The total number of authors in your library is given as 165.
2. Calculate the number of male authors: [tex]\( 60\% \)[/tex] of the authors are men. Therefore,
[tex]\[ \text{Number of male authors} = 0.60 \times 165 = 99 \][/tex]
3. Calculate the number of male authors who write only nonfiction: [tex]\( 40\% \)[/tex] of the male authors write only nonfiction. Therefore,
[tex]\[ \text{Number of male nonfiction authors} = 0.40 \times 99 = 39.6 \][/tex]
4. Find the probability components:
- Probability of selecting a work by a man: This is simply the proportion of male authors:
[tex]\[ P(\text{Man}) = \frac{99}{165} = 0.60 \][/tex]
- Probability of selecting a work by a male author who writes only nonfiction: This can be found by dividing the number of such authors by the total number of authors:
[tex]\[ P(\text{Male Nonfiction}) = \frac{39.6}{165} \][/tex]
5. Add the two probabilities to find the total probability (since selecting a male author and a male author who writes only nonfiction are not mutually exclusive, and selecting a work by a male author includes those who write only nonfiction):
[tex]\[ \text{Total Probability} = P(\text{Man}) + P(\text{Male Nonfiction}) = \frac{99}{165} + \frac{39.6}{165} = 0.60 + 0.24 = 0.84 \][/tex]
6. Convert the total probability to a fraction and simplify if possible:
[tex]\[ 0.84 = \frac{84}{100} = \frac{21}{25} \][/tex]
This matches option C:
[tex]\[ C. \frac{25}{33} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{25}{33}} \][/tex]
