Question

analyzing a claim
the two - way table shows the distribution of book style to genre. miguel claims that given that the book is paperback (pb) does not affect the outcome that the book is nonfiction (nf). is miguels claim correct?
fiction nonfiction total
paperback 20 60 80
hardcover 10 30 40
total 30 90 120
yes, the two events are independent because p(nf|pb)=p(nf).
yes, the two events are independent because p(pb|nf)=p(nf).
no, the two events are not independent because p(pb|nf)≠p(pb).
no, the two events are not independent because p(pb|nf)≠p(nf).

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 $P(NF|PB)=P(NF)$.