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 covered. ( p(h)=\frac{11}{20})
3. calculate the conditional probability. ( \frac{p(ncap h)}{p(h)}=\frac{4}{11})

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 \(P(N\cap H)=\frac{4}{20}\) (using the formula \(P(A)=\frac{n(A)}{n(S)}\), where \(n(A)\) is the number of elements in event \(A\) and \(n(S)\) is the number of elements in the sample space).

Step2: Find \(P(H)\)

The number of hard - covered books is \(11\), and the total number of books is \(20\). So \(P(H)=\frac{11}{20}\) (using the formula \(P(A)=\frac{n(A)}{n(S)}\)).

Step3: Calculate the conditional probability

The formula for conditional probability is \(P(N|H)=\frac{P(N\cap H)}{P(H)}\). Substitute \(P(N\cap H)=\frac{4}{20}\) and \(P(H)=\frac{11}{20}\) into the formula: \(\frac{P(N\cap H)}{P(H)}=\frac{\frac{4}{20}}{\frac{11}{20}}=\frac{4}{11}\) (using the rule \(\frac{\frac{a}{b}}{\frac{c}{b}}=\frac{a}{c}\) for \(b
eq0\)).

Answer:

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