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 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 and B)≠P(A)⋅P(B)

Answer 1

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:

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

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)**