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).

	fiction	nonfiction	total
hardcover	10	30	40
total	30	90	120

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: Recall independence rule

Two events \( A \) and \( B \) are independent if \( P(A|B) = P(A) \) (or \( P(B|A)=P(B) \)).

Step2: Calculate \( P(NF|PB) \)

\( P(NF|PB)=\frac{\text{Number of PB and NF}}{\text{Number of PB}}=\frac{60}{80}=\frac{3}{4} \).

Step3: Calculate \( P(NF) \)

\( P(NF)=\frac{\text{Total NF}}{\text{Total books}}=\frac{90}{120}=\frac{3}{4} \).

Step4: Compare \( P(NF|PB) \) and \( P(NF) \)

Since \( P(NF|PB)=\frac{3}{4} \) and \( P(NF)=\frac{3}{4} \), they are equal. So the events are independent because \( P(NF|PB) = P(NF) \).

Answer:

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