Question

a book shelf holds twenty books. eleven are hard - covered, nine of the books are nonfiction, and four of the nonfiction books are hard - covered, as shown in the venn diagram. let event h be a hard - covered book and let event n be a nonfiction book. a book is chosen randomly. complete the steps to find the probability that the book is nonfiction given that it is hard - covered. 1. find the probability that a chosen book is nonfiction and hard - covered. ( p(ncap h)=) 2. find the probability that a chosen book is hard - covered. ( p(h)=) 3. calculate the conditional probability. ( \frac{p(ncap h)}{p(h)}=)

Explanation:

Step1: Find \( P(N\cap H) \)

The number of books that are non - fiction and hard - covered is \( 4 \), and the total number of books is \( 20 \). So \( P(N\cap H)=\frac{4}{20}=\frac{1}{5} \).

Step2: Find \( P(H) \)

The number of hard - covered books is \( 7 + 4=11 \), and the total number of books is \( 20 \). So \( P(H)=\frac{11}{20} \).

Step3: Calculate the conditional probability \( \frac{P(N\cap H)}{P(H)} \)

Substitute the values of \( P(N\cap H) \) and \( P(H) \) into the formula. \( \frac{P(N\cap H)}{P(H)}=\frac{\frac{4}{20}}{\frac{11}{20}}=\frac{4}{11} \) (using the rule \( \frac{a/b}{c/b}=\frac{a}{c}\) for \( b
eq0 \)).

Answer:

1. \( \frac{1}{5} \)
2. \( \frac{11}{20} \)
3. \( \frac{4}{11} \)