- Calculate the probability of picking a reference book first: \(P(R_1) = \frac{2}{17}\).
- Identify the remaining total after one pick: \(16\) books.
- Calculate the conditional probability of picking a nonfiction book second: \(P(N_2 | R_1) = \frac{5}{16}\).
- Multiply the probabilities to get the joint probability: \(\frac{2}{17} \times \frac{5}{16} = \boxed{\frac{5}{136}}\).
Explanation
- Analyze the collection of books and the experiment structure
Nadia has a bookshelf with a collection of books: \(10\) fiction books, \(2\) reference books, and \(5\) nonfiction books. The total number of books is \(T = 10 + 2 + 5 = 17\). We are interested in the probability of a two-step event: picking a reference book first, and then picking a nonfiction book second, without replacing the first book. Since the first book is not replaced, the total number of books for the second pick is \(17 - 1 = 16\).
- Calculate the probability of picking a reference book first
First, we calculate the probability of the first event, \(P(R_1)\), which is picking a reference book from the initial collection. There are \(2\) reference books out of \(17\) total books:
\[P(R_1) = \frac{2}{17}\]
- Calculate the conditional probability of picking a nonfiction book second
After one reference book is removed, the remaining number of books on the shelf is \(16\). The number of nonfiction books remains \(5\), as the book removed was a reference book. The conditional probability of picking a nonfiction book given that the first book was a reference book is:
\[P(N_2 | R_1) = \frac{5}{16}\]
- Calculate and simplify the final probability
To find the probability of both events occurring in sequence, we use the multiplication rule for dependent events:
\[P(R_1 \cap N_2) = P(R_1) \times P(N_2 | R_1) = \frac{2}{17} \times \frac{5}{16}\]
Performing the multiplication:
\[\frac{2 \times 5}{17 \times 16} = \frac{10}{272}\]
We simplify this fraction by dividing both the numerator and the denominator by \(2\):
\[\frac{10 \div 2}{272 \div 2} = \frac{5}{136}\]
Thus, the probability of picking a reference book and then a nonfiction book is \(\frac{5}{136}\).
Examples
This probability calculation is useful in inventory management or library science. For example, if you are looking to understand the likelihood of a specific sequence of book types being borrowed by a patron, you can use this method to determine the probability of those events occurring sequentially, which helps in predicting demand patterns for different sections of a collection. If you have 17 books and want the probability of two specific categories being picked in a certain order, the calculation \(\frac{n_1}{T} \times \frac{n_2}{T-1}\) is the standard approach.