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

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).

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).

Answer

Explanation:

Step1: Recall the independence formula

Two events (A) and (B) are independent if (P(A|B)=P(A)). Here, let (A) be the event that the book is non - fiction ((NF)) and (B) be the event that the book is paperback ((PB)). We need to check if (P(NF|PB)=P(NF)).

Step2: Calculate (P(NF))

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

Step3: Calculate (P(NF|PB))

The formula for conditional probability is (P(NF|PB)=\frac{P(NF\cap PB)}{P(PB)}). (P(NF\cap PB)=\frac{60}{120}) and (P(PB)=\frac{80}{120}). Then (P(NF|PB)=\frac{\frac{60}{120}}{\frac{80}{120}}=\frac{60}{80}=\frac{3}{4}).

Answer:

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