1. **State the problem:** Determine if events V (Took a vacation) and C (Have cable TV) are independent. 2. **Recall the definition of independence:** Two events V and C are independent if and only if $$P(V \cap C) = P(V) \times P(C)$$ 3. **Given data:** - $P(V) = \frac{111}{166} \approx 0.669$ - $P(V|C) = \frac{97}{135} \approx 0.719$ - Total with cable TV: 135 out of 166, so $$P(C) = \frac{135}{166} \approx 0.813$$ - $P(V \cap C) = \frac{97}{166} \approx 0.584$ 4. **Calculate $P(V) \times P(C)$:** $$P(V) \times P(C) = 0.669 \times 0.813 = 0.543$$ 5. **Compare $P(V \cap C)$ and $P(V) \times P(C)$:** - $P(V \cap C) = 0.584$ - $P(V) \times P(C) = 0.543$ Since $P(V \cap C) \neq P(V) \times P(C)$, the events V and C are **not independent**. 6. **Alternative check using conditional probability:** - If independent, $P(V|C) = P(V)$ - Given $P(V|C) = 0.719$ and $P(V) = 0.669$, since $0.719 \neq 0.669$, events are not independent. **Final answer:** Events V and C are not independent.