Asked by AAA
Use the table to answer the 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.
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
There are no human answers yet.
Answered by
GPT-4o mini
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 \( P(B) \).
1. **Calculate P(A)** (Probability of yellow):
- Total yellow = 48
- Total observations = 80
- So, \( P(A) = \frac{\text{Total yellow}}{\text{Total observations}} = \frac{48}{80} = 0.6 \).
2. **Calculate P(B)** (Probability of yellow given triangle):
- Total triangles = 50
- Yellow triangles = 30
- So, \( P(B) = P(\text{yellow | triangle}) = \frac{\text{Yellow triangles}}{\text{Total triangles}} = \frac{30}{50} = 0.6 \).
3. **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 \( P(A | B) = P(A) \).**
\[ P(A | B) = P(A) \]
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 \( P(B) \).
1. **Calculate P(A)** (Probability of yellow):
- Total yellow = 48
- Total observations = 80
- So, \( P(A) = \frac{\text{Total yellow}}{\text{Total observations}} = \frac{48}{80} = 0.6 \).
2. **Calculate P(B)** (Probability of yellow given triangle):
- Total triangles = 50
- Yellow triangles = 30
- So, \( P(B) = P(\text{yellow | triangle}) = \frac{\text{Yellow triangles}}{\text{Total triangles}} = \frac{30}{50} = 0.6 \).
3. **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 \( P(A | B) = P(A) \).**
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