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?

Question

(1 point)
Responses

dependent, because P(A and B)≠P(A)⋅P(B)dependent, because P(A and B)≠P(A)⋅P(B)

dependent, because P(A)=P(B) dependent, because cap p times cap A is equal to cap p times cap b

independent, because P(A and B)=P(A)⋅P(B) independent, because P(A and B)=P(A)⋅P(B)

dependent, because P(A)≠P(B)dependent, because P(A)≠P(B)

Answer 1

To determine whether events \( A \) (cloudy days) and \( B \) (rainy days) are independent or dependent, we need to check whether the probability of both events happening, \( P(A \cap B) \), is equal to the product of their individual probabilities, \( P(A) \cdot P(B) \).

Given:

• Total days = 60
• Cloudy days (event \( A \)) = 24
• Rainy days (event \( B \)) = 15
• Days that are both cloudy and rainy (event \( A \cap B \)) = 10

First, we calculate the probabilities:

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

Now we check if \( P(A \cap B) = P(A) \cdot P(B) \):

\[ P(A) \cdot P(B) = 0.4 \cdot 0.25 = 0.1 \]

Since \( P(A \cap B) \approx 0.1667 \) is not equal to \( P(A) \cdot P(B) = 0.1 \), we conclude that the events are dependent.

The correct response is:

**dependent, because ( P(A \cap B) \neq P(A) \cdot P(B) **