Probability of No History Books Being Chosen When Arranging Books on a Shelf
Ana has a collection of 3 distinct math books, 4 distinct science books, and 5 distinct history books. She needs to arrange 6 of these books on a shelf. We want to find the probability that none of the books on the shelf are history books.
1. Determine the Total Number of Books
Ana has:
3 distinct math books 4 distinct science books 5 distinct history booksThe total number of books is:
3 4 5 12
2. Calculate the Total Ways to Choose 6 Books from 12
We use the combination formula to find the number of ways to choose 6 books from 12:
binom{12}{6} frac{12!}{6! cdot 12-6!} frac{12!}{6! cdot 6!} 924
3. Calculate the Number of Ways to Choose 6 Books with No History Books
We only consider the math and science books (no history books) when choosing the 6 books:
3 distinct math books 4 distinct science booksThe total number of non-history books is:
3 4 7
We use the combination formula again to find the number of ways to choose 6 non-history books from these 7:
binom{7}{6} frac{7!}{6! cdot 7-6!} 7
4. Calculate the Probability
The probability that no history book is on the shelf can be calculated as the ratio of the number of favorable outcomes to the total outcomes:
Ptext{no history books} frac{text{Number of ways to choose 6 non-history books}}{text{Total ways to choose 6 books}} frac{7}{924}
5. Simplify the Probability
Now, we simplify the probability:
Ptext{no history books} frac{1}{132}
Therefore, the probability that no history book is on the shelf is frac{1}{132}.
Conclusion
This article provides a detailed explanation and calculations to find the probability of no history book being chosen when 6 books are arranged from a collection of 3 math, 4 science, and 5 history books on a shelf. The key steps include determining the total number of books, calculating the total number of ways to choose 6 books from the 12 available, calculating the number of ways to choose 6 books without any history books, and simplifying the probability calculation.
Understanding these steps can help in solving similar problems in probability and combinatorics and are valuable for SEO related to math and probability.