To determine if events A and B are independent, we need to clarify what each event represents and compute their probabilities.
Event A: P(yellow) - the probability of selecting a yellow shape. Event B: P(yellow | triangle) - the probability of selecting a yellow shape given that it is a triangle.
First, let's calculate the probabilities:
-
P(yellow) (which is event A):
- Total number of shapes = 80
- Total number of yellow shapes = 48
- Therefore, \( P(A) = \frac{48}{80} = 0.6 \).
-
P(yellow | triangle) (which is event B):
- Total number of triangles = 50
- Total number of yellow triangles = 30
- Therefore, \( P(B) = P(yellow | triangle) = \frac{30}{50} = 0.6 \).
Now, for two events to be independent, the following must hold true: \[ P(A) = P(B) \text{ or } P(A) = P(B | A) \]
However, we have:
- \( P(A) = 0.6 \)
- \( P(B) = P(yellow | triangle) = 0.6 \)
Since both of these probabilities are equal (0.6), the groups of shapes that they consider—general yellow shapes (event A) and yellow triangles (event B)—do not reflect that one affects the other regarding independence in these specific probabilities. However, we have to check the conditional probabilities or how one affects the other as independent events.
Using the correct evaluation of dependency:
- P(yellow and triangle) = The probability of picking a yellow shape that is also a triangle:
- This would be \( \frac{30}{80} = 0.375 \)
Then:
- To check for independence:
- \( P(A) \cdot P(B) = 0.6 \cdot 0.6 = 0.36 \)
Since \( P(yellow) \cdot P(yellow | triangle) \neq P(yellow \text{ and } triangle) \), this means they are not independent.
Therefore, the answer to whether events A and B are independent is: No, because both events do not include yellow.
However, if this option were listed incorrectly, they are indeed dependent because the fits of these probabilities do not match up as required for independence. The closest option that matches correctly is saying they do not reflect independence conditions.