To determine if events A and B are independent, we can use the definition of independence of events. Two events A and B are independent if:
\[ P(A | B) = P(A) \]
Where:
- \( P(A) \) is the probability of event A occurring.
- \( P(B) \) is the probability of event B occurring.
- \( P(A | B) \) is the probability of event A occurring given that event B has occurred.
First, we calculate \( P(A) \) and \( P(B) \).
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Calculate P(A) (Probability of yellow):
- Total yellow = 48
- Total observations = 80
- So, \( P(A) = \frac{\text{Total yellow}}{\text{Total observations}} = \frac{48}{80} = 0.6 \).
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Calculate P(B) (Probability of yellow given triangle):
- Total triangles = 50
- Yellow triangles = 30
- So, \( P(B) = P(\text{yellow | triangle}) = \frac{\text{Yellow triangles}}{\text{Total triangles}} = \frac{30}{50} = 0.6 \).
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Calculate P(A | B) (Probability of yellow given triangle):
- As calculated above, since event B considers only triangles and the probability of selecting a yellow triangle is already calculated as \( P(B) = 0.6 \).
Now we need to check if \( P(A | B) = P(A) \).
Since we have \( P(A | B) \) calculated from the triangle row and it results in the same value as \( P(A) \), the values of both are:
- \( P(A) = 0.6 \)
- \( P(A | B) = 0.6 \)
Thus, we can conclude:
Yes, because \( P(A | B) = P(A) \).
So, the correct response would be:
Yes, because \( P(A | B) = P(A) \).