Question

Question
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?(1 point)
Responses

dependent, because P(A)=P(B)
dependent, because upper P left parenthesis upper A right parenthesis equals upper P left parenthesis upper B right parenthesis

dependent, because P(A and B)≠P(A)⋅P(B)
dependent, because upper P left parenthesis upper A and upper B right parenthesis does not equal upper P left parenthesis upper A right parenthesis times upper P left parenthesis upper B right parenthesis

independent, because P(A and B)=P(A)⋅P(B)
independent, because upper P left parenthesis upper A and upper B right parenthesis equals upper P left parenthesis upper A right parenthesis times upper P left parenthesis upper B right parenthesis

dependent, because P(A)≠P(B)
dependent, because upper P left parenthesis upper A right parenthesis does not equal upper P left parenthesis upper B right parenthesis
Remaining Attempts : 3
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Answer 1

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

1. **Calculate the probabilities:**

- Total days: \( 60 \)
- Days that are cloudy (Event \( A \)): \( 24 \)
- Days that are rainy (Event \( B \)): \( 15 \)
- Days that are both cloudy and rainy (Event \( A \cap B \)): \( 10 \)

The probabilities of events \( A \) and \( B \) are:
\[
P(A) = \frac{24}{60} = \frac{2}{5}
\]
\[
P(B) = \frac{15}{60} = \frac{1}{4}
\]
\[
P(A \cap B) = \frac{10}{60} = \frac{1}{6}
\]

2. **Check independence:**

Events \( A \) and \( B \) are independent if:
\[
P(A \cap B) = P(A) \times P(B)
\]

Calculate \( P(A) \times P(B) \):
\[
P(A) \times P(B) = \left( \frac{2}{5} \right) \times \left( \frac{1}{4} \right) = \frac{2}{20} = \frac{1}{10}
\]

Now compare \( P(A \cap B) \) with \( P(A) \times P(B) \):
- \( P(A \cap B) = \frac{1}{6} \)
- \( P(A) \times P(B) = \frac{1}{10} \)

Since \( \frac{1}{6} \neq \frac{1}{10} \), we have:
\[
P(A \cap B) \neq P(A) \times P(B)
\]

Therefore, the events \( A \) and \( B \) are **dependent** because \( P(A \cap B) \neq P(A) \cdot P(B) \).

So the correct response is:
- dependent, because \( P(A \cap B) \neq P(A) \cdot P(B) \)