To determine if events A and B are independent, we need to use the definitions of probabilities involved in these events.
Event A: \( P(\text{yellow}) \)
From the table, the total probability of yellow (across all shapes) is:
\[ P(A) = \frac{\text{Total yellow}}{\text{Overall total}} = \frac{48}{80} = 0.6 \]
Event B: \( P(\text{yellow} | \text{triangle}) \)
To find this conditional probability, we look at the number of yellow triangles and the total number of triangles:
\[ P(B) = P(\text{yellow} | \text{triangle}) = \frac{\text{Number of yellow triangles}}{\text{Total triangles}} = \frac{30}{50} = 0.6 \]
To check for independence, we need to see if:
\[ P(A|B) = P(A) \]
This means we need to find \( P(A|B) \):
Since event A does not depend directly on whether the shape is a triangle or not, we calculate \( P(A|B) \):
Event A occurs if we have any shape, and since we only consider yellow shapes in this case, we don't have a new calculation that changes depending on being a triangle.
Instead, it would be calculated as:
\[ P(A \cap B) = P(\text{yellow and triangle}) = \frac{\text{Number of yellow triangles}}{\text{Total}} = \frac{30}{80} = 0.375 \]
Now:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.375}{0.6} = 0.625 \quad \text{(Not equal to } P(A) \text{)} \]
Now, since \( P(A|B) \neq P(A) \), the two events A and B are not independent.
The correct response here is:
No, because \( P(A|B) \neq P(A) \).