Answer :
Certainly! Let's break down the problem step by step to find 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.
### Step 1: Determine the Total Number of Authors
The total number of authors in the library is given as 165.
### Step 2: Calculate the Number of Male Authors
We know that 60% of the authors are men.
[tex]\[ \text{Number of male authors} = 165 \times \frac{60}{100} = 99 \][/tex]
### Step 3: Calculate the Number of Nonfiction Authors
We also know that 40% of the authors write only nonfiction.
[tex]\[ \text{Number of nonfiction authors} = 165 \times \frac{40}{100} = 66 \][/tex]
### Step 4: Calculate the Number of Male Nonfiction Authors
It is given that 40 of the male authors write only nonfiction.
### Step 5: Calculate the Number of Male Non-Nonfiction Authors
[tex]\[ \text{Number of male non-nonfiction authors} = \text{Number of male authors} - \text{Number of male nonfiction authors} = 99 - 40 = 59 \][/tex]
### Step 6: Calculate the Total Number of Non-Nonfiction Authors
[tex]\[ \text{Total number of non-nonfiction authors} = \text{Total authors} - \text{Nonfiction authors} = 165 - 66 = 99 \][/tex]
### Step 7: Calculate the Number of Female Non-Nonfiction Authors
[tex]\[ \text{Number of female non-nonfiction authors} = \text{Total non-nonfiction authors} - \text{Male non-nonfiction authors} = 99 - 59 = 40 \][/tex]
### Step 8: Calculate the Number of Books Written by Male Authors or Nonfiction Authors
To determine the number of books that are either written by male authors or nonfiction authors, we exclude the non-male and non-nonfiction authors (i.e., female fiction authors) from the total.
[tex]\[ \text{Books by male or nonfiction authors} = \text{Total authors} - \text{Female non-nonfiction authors} = 165 - 40 = 125 \][/tex]
### Step 9: Calculate the Probability
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 can be found by dividing the number of favourable outcomes by the total number of outcomes.
[tex]\[ \text{Probability} = \frac{\text{Books by male or nonfiction authors}}{\text{Total authors}} = \frac{125}{165} \][/tex]
Simplifying the fraction:
[tex]\[ \frac{125}{165} = \frac{25}{33} \][/tex]
### Conclusion
Therefore, 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]\(\frac{25}{33}\)[/tex].
The correct answer is:
[tex]\[ \boxed{\frac{25}{33}} \][/tex]
Thus, the correct choice is:
[tex]\[ \boxed{\text{C}} \][/tex]
### Step 1: Determine the Total Number of Authors
The total number of authors in the library is given as 165.
### Step 2: Calculate the Number of Male Authors
We know that 60% of the authors are men.
[tex]\[ \text{Number of male authors} = 165 \times \frac{60}{100} = 99 \][/tex]
### Step 3: Calculate the Number of Nonfiction Authors
We also know that 40% of the authors write only nonfiction.
[tex]\[ \text{Number of nonfiction authors} = 165 \times \frac{40}{100} = 66 \][/tex]
### Step 4: Calculate the Number of Male Nonfiction Authors
It is given that 40 of the male authors write only nonfiction.
### Step 5: Calculate the Number of Male Non-Nonfiction Authors
[tex]\[ \text{Number of male non-nonfiction authors} = \text{Number of male authors} - \text{Number of male nonfiction authors} = 99 - 40 = 59 \][/tex]
### Step 6: Calculate the Total Number of Non-Nonfiction Authors
[tex]\[ \text{Total number of non-nonfiction authors} = \text{Total authors} - \text{Nonfiction authors} = 165 - 66 = 99 \][/tex]
### Step 7: Calculate the Number of Female Non-Nonfiction Authors
[tex]\[ \text{Number of female non-nonfiction authors} = \text{Total non-nonfiction authors} - \text{Male non-nonfiction authors} = 99 - 59 = 40 \][/tex]
### Step 8: Calculate the Number of Books Written by Male Authors or Nonfiction Authors
To determine the number of books that are either written by male authors or nonfiction authors, we exclude the non-male and non-nonfiction authors (i.e., female fiction authors) from the total.
[tex]\[ \text{Books by male or nonfiction authors} = \text{Total authors} - \text{Female non-nonfiction authors} = 165 - 40 = 125 \][/tex]
### Step 9: Calculate the Probability
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 can be found by dividing the number of favourable outcomes by the total number of outcomes.
[tex]\[ \text{Probability} = \frac{\text{Books by male or nonfiction authors}}{\text{Total authors}} = \frac{125}{165} \][/tex]
Simplifying the fraction:
[tex]\[ \frac{125}{165} = \frac{25}{33} \][/tex]
### Conclusion
Therefore, 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]\(\frac{25}{33}\)[/tex].
The correct answer is:
[tex]\[ \boxed{\frac{25}{33}} \][/tex]
Thus, the correct choice is:
[tex]\[ \boxed{\text{C}} \][/tex]
