Initial Bookshelf Composition

Oct 13, 2025 by wbrum.org 115 views

Introduction

When dealing with probability, it's essential to understand the concept of conditional probability, which is the probability of an event occurring given that another event has occurred. In this scenario, we have a bookshelf containing a mix of fiction and nonfiction books. We are given that 5 books are removed from the bookshelf, with 1 of them being nonfiction. Our goal is to determine the probability that the next book removed will be fiction.

Initial Bookshelf Composition

Initially, the bookshelf contains 12 fiction books and 15 nonfiction books, making a total of 27 books. The probability of selecting a fiction book from the bookshelf is the number of fiction books divided by the total number of books, which is 12/27 or approximately 0.4444.

Removal of 5 Books

When 5 books are removed from the bookshelf, the total number of books decreases to 22. Out of these 5 removed books, 1 is nonfiction, leaving 4 fiction books removed. This means that the number of fiction books remaining on the bookshelf is 12 - 4 = 8, and the number of nonfiction books remaining is 15 - 1 = 14.

Conditional Probability

To determine the probability that the next book removed will be fiction, we need to consider the conditional probability. The probability of selecting a fiction book given that 4 fiction books have already been removed is the number of remaining fiction books divided by the total number of remaining books. This can be calculated as 8/22 or approximately 0.3636.

Probability of Removing a Fiction Book

However, we are interested in the probability that the next book removed will be fiction, given that one more book is removed. To calculate this, we need to consider the total number of books remaining, which is 21 (22 - 1). The number of fiction books remaining is still 8, and the number of nonfiction books remaining is 14.

Calculating the Probability

The probability of removing a fiction book from the remaining 21 books can be calculated as the number of fiction books divided by the total number of remaining books, which is 8/21 or approximately 0.3809.

Conclusion

In conclusion, the probability that the next book removed from the bookshelf will be fiction, given that one more book is removed, is approximately 0.3809 or 38.09%. This is a relatively high probability, indicating that the likelihood of removing a fiction book is still significant, even after removing 5 books from the bookshelf.

Example Use Case

This problem can be applied to real-world scenarios, such as a library or a bookstore. For instance, if a librarian or a bookstore owner wants to remove a certain number of books from the shelves, they can use this probability to determine the likelihood of removing a fiction or nonfiction book.

Mathematical Formulation

Mathematically, this problem can be formulated as follows:

Let F be the event that the next book removed is fiction, and let N be the event that the next book removed is nonfiction. We want to find P(F|N1), the probability of F given that 1 nonfiction book has already been removed.

Using Bayes' theorem, we can calculate P(F|N1) as follows:

P(F|N1) = P(F ∩ N1) / P(N1)

where P(F ∩ N1) is the probability of F and N1 occurring together, and P(N1) is the probability of N1 occurring.

Simplifying the Formulation

We can simplify the formulation by using the fact that P(F ∩ N1) = P(F) * P(N1|F), where P(F) is the probability of F occurring, and P(N1|F) is the probability of N1 occurring given that F has occurred.

Substituting this into the previous equation, we get:

P(F|N1) = P(F) * P(N1|F) / P(N1)

Calculating the Probabilities

We can calculate the probabilities as follows:

P(F) = 12/27 = 0.4444

P(N1|F) = 1/12 = 0.0833

P(N1) = 1/22 = 0.0455

Substituting the Probabilities

Substituting these probabilities into the previous equation, we get:

P(F|N1) = 0.4444 * 0.0833 / 0.0455

Simplifying the Equation

Simplifying the equation, we get:

P(F|N1) = 0.3809

Conclusion

Q: What is the probability of removing a fiction book from a mixed bookshelf?

A: The probability of removing a fiction book from a mixed bookshelf is approximately 0.4444, which is the number of fiction books divided by the total number of books.

Q: How does the removal of 5 books affect the probability of removing a fiction book?

A: When 5 books are removed from the bookshelf, the total number of books decreases to 22. Out of these 5 removed books, 1 is nonfiction, leaving 4 fiction books removed. This means that the number of fiction books remaining on the bookshelf is 12 - 4 = 8, and the number of nonfiction books remaining is 15 - 1 = 14.

Q: What is the probability of removing a fiction book given that 4 fiction books have already been removed?

A: The probability of selecting a fiction book given that 4 fiction books have already been removed is the number of remaining fiction books divided by the total number of remaining books, which is 8/22 or approximately 0.3636.

Q: What is the probability of removing a fiction book from the remaining 21 books?

A: The probability of removing a fiction book from the remaining 21 books can be calculated as the number of fiction books divided by the total number of remaining books, which is 8/21 or approximately 0.3809.

Q: How does the probability of removing a fiction book change when one more book is removed?

A: The probability of removing a fiction book when one more book is removed is still approximately 0.3809 or 38.09%. This is a relatively high probability, indicating that the likelihood of removing a fiction book is still significant, even after removing 5 books from the bookshelf.

Q: Can this problem be applied to real-world scenarios?

A: Yes, this problem can be applied to real-world scenarios, such as a library or a bookstore. For instance, if a librarian or a bookstore owner wants to remove a certain number of books from the shelves, they can use this probability to determine the likelihood of removing a fiction or nonfiction book.

Q: What is the mathematical formulation of this problem?

A: Mathematically, this problem can be formulated as follows:

Using Bayes' theorem, we can calculate P(F|N1) as follows:

P(F|N1) = P(F ∩ N1) / P(N1)

where P(F ∩ N1) is the probability of F and N1 occurring together, and P(N1) is the probability of N1 occurring.

Q: How can I simplify the mathematical formulation of this problem?

A: We can simplify the formulation by using the fact that P(F ∩ N1) = P(F) * P(N1|F), where P(F) is the probability of F occurring, and P(N1|F) is the probability of N1 occurring given that F has occurred.

Substituting this into the previous equation, we get:

P(F|N1) = P(F) * P(N1|F) / P(N1)

Q: What are the probabilities used in the mathematical formulation?

A: We can calculate the probabilities as follows:

P(F) = 12/27 = 0.4444

P(N1|F) = 1/12 = 0.0833

P(N1) = 1/22 = 0.0455

Q: How can I substitute the probabilities into the mathematical formulation?

A: Substituting these probabilities into the previous equation, we get:

P(F|N1) = 0.4444 * 0.0833 / 0.0455

Q: What is the final answer to the mathematical formulation?

A: Simplifying the equation, we get:

P(F|N1) = 0.3809

Q: What is the conclusion of this problem?

A: In conclusion, the probability that the next book removed from the bookshelf will be fiction, given that one more book is removed, is approximately 0.3809 or 38.09%. This is a relatively high probability, indicating that the likelihood of removing a fiction book is still significant, even after removing 5 books from the bookshelf.