Question

a book publishing company surveyed 450 customers about their favorite genre and preferred season. the results are displayed in the table.

	winter	spring	summer	fall
romance	14	61	24	36
sci-fi	23	2	8	12
non - fiction	28	22	16	24

is preferring winter independent of favoring sci - fi books? justify mathematically.

• yes, because 0.051 ≠ (0.311)(0.100)
• yes, because 0.031 = (0.311)(0.100)
• no, because 0.051 ≠ (0.311)(0.100)
• no, because 0.031 = (0.311)(0.100)

Explanation:

Step1: Calculate total Sci-Fi

Sum Sci-Fi across seasons: $23 + 2 + 8 + 12 = 45$

Step2: Calculate total Winter

Sum Winter across genres: $75 + 14 + 23 + 28 = 140$

Step3: Find joint count (Sci-Fi & Winter)

From table: $23$

Step4: Compute probabilities

$P(\text{Sci-Fi}) = \frac{45}{450} = 0.100$, $P(\text{Winter}) = \frac{140}{450} \approx 0.311$, $P(\text{Sci-Fi and Winter}) = \frac{23}{450} \approx 0.051$

Step5: Check independence

Independence requires $P(A \cap B) = P(A)P(B)$. $0.051
eq (0.311)(0.100) \approx 0.031$, so not independent.

Answer:

No, because 0.051 ≠ (0.311)(0.100)