Question

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 miguel’s claim correct? 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)
eq p(pb) ). no, the two events are not independent because ( p(pb|nf)
eq p(nf) ).

Explanation:

Step1: Calculate \(P(\text{PB}|\text{NF})\)

By the formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), for \(P(\text{PB}|\text{NF})\), \(P(\text{PB}\cap\text{NF})=\frac{60}{120}\), \(P(\text{NF})=\frac{90}{120}\). Then \(P(\text{PB}|\text{NF})=\frac{P(\text{PB}\cap\text{NF})}{P(\text{NF})}=\frac{\frac{60}{120}}{\frac{90}{120}}=\frac{60}{90}=\frac{2}{3}\)

Step2: Calculate \(P(\text{PB})\)

\(P(\text{PB})=\frac{80}{120}=\frac{2}{3}\)

Step3: Check independence

Two events \(A\) and \(B\) are independent if \(P(A|B) = P(A)\). Here \(A=\text{PB}\), \(B = \text{NF}\). \(P(\text{PB}|\text{NF})=\frac{2}{3}\), \(P(\text{PB})=\frac{80}{120}=\frac{2}{3}\). Also, \(P(\text{NF}|\text{PB})=\frac{P(\text{PB}\cap\text{NF})}{P(\text{PB})}=\frac{\frac{60}{120}}{\frac{80}{120}}=\frac{60}{80}=\frac{3}{4}\), \(P(\text{NF})=\frac{90}{120}=\frac{3}{4}\)

Answer:

Yes, the two events are independent because \(P(\text{NF}|\text{PB}) = P(\text{NF})\).