Question
Color
Shape Red Yellow TOTAL
Circle 12 18 30
Triangle 20 30 50
Total 32 48 80
Let event A
represent the P(yellow)
and let event B
represent the P(yellow|triangle)
. Are they independent? Explain.
(1 point)
Responses
Yes, because P(A|B)=P(A)
.
Yes, because upper P left parenthesis upper A vertical bar upper B right parenthesis equals upper P left parenthesis upper A right parenthesis .
Yes, because P(A)=P(B)
.
Yes, because upper P left parenthesis upper A right parenthesis equals upper P left parenthesis upper B right parenthesis .
No, because both events do not include triangles.
No, because both events do not include triangles.
No, because both events do not include yellow.
Answers
GPT-4o mini
To determine whether events A (P(yellow)) and B (P(yellow|triangle)) are independent, we need to review the definition of independence. Two events A and B are independent if:
\[ P(A \cap B) = P(A) \times P(B) \]
1. **Calculate P(A):** The total probability of yellow (event A).
- Total yellow balls = 48
- Total balls = 80
\[
P(A) = \frac{Total \, yellow}{Total \, balls} = \frac{48}{80} = 0.6
\]
2. **Calculate P(B):** The probability of yellow given that the shape is a triangle (event B).
- Yellow triangles = 30
- Total triangles = 50
\[
P(B) = P(yellow | triangle) = \frac{30}{50} = 0.6
\]
3. **Calculate P(A ∩ B):** The probability that a ball is yellow and a triangle.
Since "B" is about given triangles, we need to find how many yellow triangles there are, which is just yellow triangles out of total yellow.
\[
P(A \cap B) = P(Yellow \, and \, Triangle) = \frac{30}{80} = 0.375
\]
4. **Check independence:**
- Now we compare \( P(A) \times P(B) \) and \( P(A \cap B) \):
\[
P(A) \times P(B) = 0.6 \times 0.6 = 0.36
\]
Since
\[
P(A \cap B) = 0.375 \neq 0.36 = P(A) \times P(B)
\]
events A and B are not independent.
### Conclusion:
The correct response to whether events A and B are independent is:
**No, because both events do not include triangles.**
However, this statement isn't entirely accurate, as the real reason for dependency is calculated probabilities not aligning. It may lead you to choose a different option, but none match the reasoning perfectly based on the provided responses. If you're explaining independence based on the calculation, the best response would be to imply that the probability of yellow instances in triangles indicates dependence even if it doesn't entirely include the triangles or yellow directly in two events.
\[ P(A \cap B) = P(A) \times P(B) \]
1. **Calculate P(A):** The total probability of yellow (event A).
- Total yellow balls = 48
- Total balls = 80
\[
P(A) = \frac{Total \, yellow}{Total \, balls} = \frac{48}{80} = 0.6
\]
2. **Calculate P(B):** The probability of yellow given that the shape is a triangle (event B).
- Yellow triangles = 30
- Total triangles = 50
\[
P(B) = P(yellow | triangle) = \frac{30}{50} = 0.6
\]
3. **Calculate P(A ∩ B):** The probability that a ball is yellow and a triangle.
Since "B" is about given triangles, we need to find how many yellow triangles there are, which is just yellow triangles out of total yellow.
\[
P(A \cap B) = P(Yellow \, and \, Triangle) = \frac{30}{80} = 0.375
\]
4. **Check independence:**
- Now we compare \( P(A) \times P(B) \) and \( P(A \cap B) \):
\[
P(A) \times P(B) = 0.6 \times 0.6 = 0.36
\]
Since
\[
P(A \cap B) = 0.375 \neq 0.36 = P(A) \times P(B)
\]
events A and B are not independent.
### Conclusion:
The correct response to whether events A and B are independent is:
**No, because both events do not include triangles.**
However, this statement isn't entirely accurate, as the real reason for dependency is calculated probabilities not aligning. It may lead you to choose a different option, but none match the reasoning perfectly based on the provided responses. If you're explaining independence based on the calculation, the best response would be to imply that the probability of yellow instances in triangles indicates dependence even if it doesn't entirely include the triangles or yellow directly in two events.