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Chapter 4: Problem 12

Selecting a Book At a used-book sale, 100 books are adult books and 160 are children鈥檚 books. Of the adult books, 70 are nonfiction while 60 of the children鈥檚 books are nonfiction. If a book is selected at random, find the probability that it is a. Fiction b. Not a children鈥檚 nonfiction book c. An adult book or a children鈥檚 nonfiction book

Short Answer

Expert verified
a. \(\frac{1}{2}\), b. \(\frac{10}{13}\), c. \(\frac{8}{13}\)

Step by step solution

01

Determine Total Books

First, we calculate the total number of books available. Add the number of adult books (100) and children's books (160):\[100 + 160 = 260\]So, there are 260 books in total.
02

Calculate Fiction Books

We are given that there are 70 nonfiction adult books and 60 nonfiction children's books. First, find the total number of nonfiction books:\[70 + 60 = 130\]Then, subtract the nonfiction books from the total books to find the fiction books:\[260 - 130 = 130\]Thus, there are 130 fiction books.
03

Calculate Probability of a Fiction Book

Use the formula for probability, which is the number of favorable outcomes divided by the total number of possible outcomes. Here, the number of fiction books is the favorable outcome:\[P(\text{Fiction}) = \frac{130}{260} = \frac{1}{2}\]
04

Calculate Probability of Not a Children's Nonfiction Book

First, find how many children's nonfiction books there are, which is 60. The books that are not children's nonfiction are the total books minus the children's nonfiction:\[260 - 60 = 200\]Then, calculate the probability:\[P(\text{Not a children's nonfiction}) = \frac{200}{260} = \frac{10}{13}\]
05

Calculate Probability of an Adult Book or a Children's Nonfiction Book

For this, we use the formula for probability of either event A or event B:\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]where:- \(P(A)\) is the probability of selecting an adult book, \(P(A) = \frac{100}{260}\)- \(P(B)\) is the probability of selecting a children鈥檚 nonfiction book, \(P(B) = \frac{60}{260}\)- \(P(A \cap B)\) is zero because a book cannot be an adult book and a children's nonfiction book at the same time.Calculate:\[P(A \cup B) = \frac{100}{260} + \frac{60}{260} = \frac{160}{260} = \frac{8}{13}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nonfiction
Nonfiction books are those that are based on factual information. They are not made-up stories, but rather, they deal with real events and actual figures. Nonfiction can include a wide range of genres:
  • Biographies
  • History books
  • Documentaries
  • Self-help guides
In the context of this exercise, we have two categories of nonfiction books: adult nonfiction and children's nonfiction. Understanding what's included in nonfiction can help students distinguish between different types of books when calculating probabilities.
Fiction
Fiction books are based on imaginative narration. Unlike nonfiction, their content is not constrained to factual accuracy. Fiction includes genres like:
  • Novels
  • Short stories
  • Fantasies
  • Science fiction
In the given problem, fiction books are calculated by subtracting the nonfiction books from the total number of books. This allows us to determine the number of fiction books which helps in computing various probabilities.
Children's Books
Children's books are specifically written and marketed for younger readers. They often contain illustrations and simpler text to engage this audience.
In the problem, we categorize children's books into two: nonfiction and fiction. It's worth noting:
  • The total children's books include both fiction and nonfiction.
  • There are 60 children's nonfiction books.
  • Understanding these categories is crucial for probability calculation, as different scenarios affect these groups differently.
Adult Books
Adult books are aimed at mature readers and can include a wide array of genres, just like children's books; they are divided into fiction and nonfiction categories.
This exercise details that out of 100 adult books:
  • 70 are nonfiction.
  • Understanding the adult book category helps when determining probabilities involving adult fiction and nonfiction books.
Probability Formula
Probability is a key concept in this problem. It's the measure of the likelihood of a particular event happening. The basic probability formula is:\[P( ext{event}) = \frac{ ext{Number of favorable outcomes}}{ ext{Total number of possible outcomes}}\]This formula is applicable to each scenario in the exercise:
  • For the fiction books, it uses the formula to determine what fraction of the total books are fiction.
  • For non-children's nonfiction, the formula helps us determine how likely we will pick a book that is not a children's nonfiction book from the lot.
  • In calculating the probability of selecting an adult book or children's nonfiction, it involves understanding the union of events, which broadens the application of the probability formula.
Grasping the probability formula is essential to solve the exercise effectively, ensuring we can clearly determine the chances of picking books from different categories.

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