Answer :
To determine 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, we will follow these steps.
1. Calculate the number of male authors:
We know that 60% of the authors are men.
[tex]\[ \text{Number of male authors} = 0.60 \times 165 = 99 \][/tex]
2. Calculate the number of authors who write only nonfiction works:
We know that 40% of the authors write only nonfiction.
[tex]\[ \text{Number of non-fiction authors} = 0.40 \times 165 = 66 \][/tex]
3. Given data on the number of male authors who write only nonfiction:
We are given that 40 of the male authors write only nonfiction.
4. Determine the union of the two sets (non-fiction works and works by men):
We need to use the principle of inclusion and exclusion to avoid double-counting the male authors who write only nonfiction.
[tex]\[ \text{Number of authors who either write only nonfiction or are men} = (\text{Number of male authors}) + (\text{Number of non-fiction authors}) - (\text{Number of male authors who write only nonfiction}) \][/tex]
[tex]\[ = 99 + 66 - 40 = 125 \][/tex]
5. Calculate the probability:
The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.
[tex]\[ \text{Probability} = \frac{\text{Number of authors who either write only nonfiction or are men}}{\text{Total number of authors}} = \frac{125}{165} \][/tex]
6. Simplify the fraction:
Simplify [tex]\(\frac{125}{165}\)[/tex] to its lowest terms. Determining the GCD (Greatest Common Divisor) of 125 and 165 to simplify the fraction:
[tex]\[ \frac{125}{165} = \frac{25}{33} \][/tex]
Hence, 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 is [tex]\(\boxed{\frac{25}{33}}\)[/tex].
1. Calculate the number of male authors:
We know that 60% of the authors are men.
[tex]\[ \text{Number of male authors} = 0.60 \times 165 = 99 \][/tex]
2. Calculate the number of authors who write only nonfiction works:
We know that 40% of the authors write only nonfiction.
[tex]\[ \text{Number of non-fiction authors} = 0.40 \times 165 = 66 \][/tex]
3. Given data on the number of male authors who write only nonfiction:
We are given that 40 of the male authors write only nonfiction.
4. Determine the union of the two sets (non-fiction works and works by men):
We need to use the principle of inclusion and exclusion to avoid double-counting the male authors who write only nonfiction.
[tex]\[ \text{Number of authors who either write only nonfiction or are men} = (\text{Number of male authors}) + (\text{Number of non-fiction authors}) - (\text{Number of male authors who write only nonfiction}) \][/tex]
[tex]\[ = 99 + 66 - 40 = 125 \][/tex]
5. Calculate the probability:
The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.
[tex]\[ \text{Probability} = \frac{\text{Number of authors who either write only nonfiction or are men}}{\text{Total number of authors}} = \frac{125}{165} \][/tex]
6. Simplify the fraction:
Simplify [tex]\(\frac{125}{165}\)[/tex] to its lowest terms. Determining the GCD (Greatest Common Divisor) of 125 and 165 to simplify the fraction:
[tex]\[ \frac{125}{165} = \frac{25}{33} \][/tex]
Hence, 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 is [tex]\(\boxed{\frac{25}{33}}\)[/tex].
