To determine whether events A (cloudy days) and B (rainy days) are independent or dependent, we can use the definition of independent events:
Events A and B are independent if and only if: \[ P(A \cap B) = P(A) \cdot P(B) \]
Where:
- \( P(A \cap B) \) is the probability that both events A and B occur (i.e., a day that is both cloudy and rainy).
- \( P(A) \) is the probability of event A occurring (i.e., a cloudy day).
- \( P(B) \) is the probability of event B occurring (i.e., a rainy day).
From the information provided, we have the following:
- Total days = 60
- Cloudy days (Event A) = 24
- Rainy days (Event B) = 15
- Cloudy and rainy days (Event A and B) = 10
Now we can calculate the probabilities:
- \( P(A) = \frac{24}{60} = 0.4 \)
- \( P(B) = \frac{15}{60} = 0.25 \)
- \( P(A \cap B) = \frac{10}{60} \approx 0.1667 \)
Now we check if \( P(A \cap B) = P(A) \cdot P(B) \):
Calculating \( P(A) \cdot P(B) \): \[ P(A) \cdot P(B) = 0.4 \cdot 0.25 = 0.1 \]
Comparing:
- \( P(A \cap B) = 0.1667 \)
- \( P(A) \cdot P(B) = 0.1 \)
Since \( P(A \cap B) \neq P(A) \cdot P(B) \), this indicates that events A and B are dependent.
Thus, the correct response is:
dependent, because P(A and B) ≠ P(A)⋅P(B)