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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{58}{165}[/tex]
C. [tex]\frac{25}{33}[/tex]
D. [tex]\frac{2}{5}[/tex]

Sagot :

GinnyAnswer
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].
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