The weather report says that in the following 60 days, 24 days will be cloudy, 15 days will be rainy, and 10 days will be cloudy and rainy. Let event A represent cloudy days and event B represent rainy days. Are the events A and B independent or dependent, and why?

Answer 1

To determine whether events A (cloudy days) and B (rainy days) are independent or dependent, we need to analyze the probabilities of these events and their intersection.

1. Define the events:

   • Let \( P(A) \) be the probability of a cloudy day.
   • Let \( P(B) \) be the probability of a rainy day.
   • Let \( P(A \cap B) \) be the probability of a day being both cloudy and rainy.

2. Calculate the probabilities:

   • Total days = 60.
   • Days that are cloudy (A) = 24.
   • Days that are rainy (B) = 15.
   • Days that are both cloudy and rainy (A ∩ B) = 10.

From this data, we can calculate:

• \( P(A) = \frac{24}{60} = 0.4 \)
• \( P(B) = \frac{15}{60} = 0.25 \)
• \( P(A \cap B) = \frac{10}{60} = \frac{1}{6} \approx 0.1667 \)

3. Check for independence: Events A and B are independent if: \[ P(A \cap B) = P(A) \times P(B) \]

Calculate \( P(A) \times P(B) \): \[ P(A) \times P(B) = 0.4 \times 0.25 = 0.1 \]

Now compare \( P(A \cap B) \) with \( P(A) \times P(B) \): \[ P(A \cap B) = \frac{1}{6} \approx 0.1667 \quad \text{and} \quad P(A) \times P(B) = 0.1 \]

Since \( P(A \cap B) \neq P(A) \times P(B) \), the two events are not independent.

Conclusion: Events A (cloudy days) and B (rainy days) are dependent. The occurrence of one affects the likelihood of the occurrence of the other, as evidenced by the difference between \( P(A \cap B) \) and \( P(A) \times P(B) \).