analyzing a claim\nthe two - way table shows the distribution of book style to genre. miguel claims that…

Explanation:

Step1: Recall the definition of independent events

Two events $A$ and $B$ are independent if $P(A|B)=P(A)$. Here, let event $A$ be the book is non - fiction ($NF$) and event $B$ be the book is paperback ($PB$). We need to check if $P(NF|PB)=P(NF)$. The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In terms of the table values, $P(NF|PB)=\frac{n(NF\cap PB)}{n(PB)}$, where $n(NF\cap PB)$ is the number of non - fiction paperbacks and $n(PB)$ is the number of paperbacks. Also, $P(NF)=\frac{n(NF)}{n(Total)}$.

Step2: Calculate $P(NF|PB)$

$n(NF\cap PB) = 60$, $n(PB)=80$. So $P(NF|PB)=\frac{60}{80}=\frac{3}{4}$.

Step3: Calculate $P(NF)$

$n(NF)=90$, $n(Total) = 120$. So $P(NF)=\frac{90}{120}=\frac{3}{4}$. Since $P(NF|PB)=P(NF)$, the two events are independent.

Answer:

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