Let's analyze the given problem step-by-step:
1. Given Probabilities:
- The probability that a person eats dinner, [tex]\( P(\text{dinner}) = 0.8 \)[/tex].
- The probability that a person eats dessert, [tex]\( P(\text{dessert}) = 0.7 \)[/tex].
- The probability that a person eats both dinner and dessert, [tex]\( P(\text{dinner and dessert}) = 0.4 \)[/tex].
2. Check for Independence:
- Two events, A and B, are independent if and only if [tex]\( P(A \text{ and } B) = P(A) \times P(B) \)[/tex].
3. Calculate [tex]\( P(\text{dinner}) \times P(\text{dessert}) \)[/tex]:
[tex]\[
P(\text{dinner}) \times P(\text{dessert}) = 0.8 \times 0.7 = 0.56
\][/tex]
4. Compare [tex]\( P(\text{dinner} \text{ and } \text{dessert}) \)[/tex] with [tex]\( P(\text{dinner}) \times P(\text{dessert}) \)[/tex]:
- Given [tex]\( P(\text{dinner and dessert}) = 0.4 \)[/tex]
- Calculated [tex]\( P(\text{dinner}) \times P(\text{dessert}) = 0.56 \)[/tex]
Since [tex]\( P(\text{dinner and dessert}) = 0.4 \)[/tex] is not equal to [tex]\( P(\text{dinner}) \times P(\text{dessert}) = 0.56 \)[/tex], the events are dependent.
Therefore, the correct statement is:
B. Eating dinner and eating dessert are dependent events because [tex]\( P(\text{dinner}) \times P(\text{dessert}) = 0.8 \cdot 0.7 = 0.56 \)[/tex], which is not equal to [tex]\( P(\text{dinner and dessert}) = 0.4 \)[/tex].
Hence, the correct answer is:
B.
