the two - way table shows the distribution of book style to genre. miguel claims that given that the book is…

Explanation:

Step1: Calculate $P(NF)$

The total number of books is $120$, and the number of non - fiction books is $90$. So $P(NF)=\frac{90}{120}=\frac{3}{4}$.

Step2: Calculate $P(NF|PB)$

The number of paperback books ($PB$) is $80$, and the number of paperback non - fiction books is $60$. By the formula for conditional probability $P(NF|PB)=\frac{P(NF\cap PB)}{P(PB)}$, and since $P(NF\cap PB)=\frac{60}{120}$ and $P(PB)=\frac{80}{120}$, then $P(NF|PB)=\frac{60/120}{80/120}=\frac{60}{80}=\frac{3}{4}$.

Step3: Check independence

Two events $A$ and $B$ are independent if $P(A|B) = P(A)$. Here, $A$ is the event of a book being non - fiction ($NF$) and $B$ is the event of a book being paperback ($PB$). Since $P(NF|PB)=\frac{3}{4}$ and $P(NF)=\frac{3}{4}$, the two events are independent.

Answer:

Yes, the two events are independent because $P(NF|PB)=P(NF)$.