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) ≠ p(pb). no, the two events are not independent because p(pb|nf) ≠ p(nf). the two - way table: | | paperback | hardcover | total | |---|---|---|---| | fiction | 20 | 10 | 30 | | nonfiction | 60 | 30 | 90 | | total | 80 | 40 | 120 |

Explanation:

Step1: Recall independence formula

Two events \( A \) and \( B \) are independent if \( P(A|B) = P(A) \) (or \( P(B|A) = P(B) \)). Let \( A \) be "Nonfiction (NF)" and \( B \) be "Paperback (PB)".

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 \( P(NF) \)

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

Since \( P(NF|PB) = P(NF) \), the events are independent. So Miguel's claim is correct because \( P(\text{NF}|\text{PB}) = P(\text{NF}) \).

Answer:

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