Question
Let A be the event that a students likes apples. Let B be the event that a student is in middle school. Based on the data in the table, are events A and B independent?(7 points)
Responses
Yes; the events are independent because P(A)=P(A|B)=0.90
.
Yes; the events are independent because cap p times cap A is equal to cap p times open paren cap A divides cap b close paren is equal to 0 point 9 0.
No; the events are dependent because P(A)=0.93
and P(A∩B)=0.91
.
No; the events are dependent because cap p times cap A is equal to 0 point 9 3 and P(A∩B)=0.91
.
Yes; the events are independent because P(A)=P(A∩B)=0.92
.
Yes; the events are independent because P(A)=P(A∩B)=0.92
.
No; the events are dependent because P(A)=0.913
and P(A|B)=0.92
.
Responses
Yes; the events are independent because P(A)=P(A|B)=0.90
.
Yes; the events are independent because cap p times cap A is equal to cap p times open paren cap A divides cap b close paren is equal to 0 point 9 0.
No; the events are dependent because P(A)=0.93
and P(A∩B)=0.91
.
No; the events are dependent because cap p times cap A is equal to 0 point 9 3 and P(A∩B)=0.91
.
Yes; the events are independent because P(A)=P(A∩B)=0.92
.
Yes; the events are independent because P(A)=P(A∩B)=0.92
.
No; the events are dependent because P(A)=0.913
and P(A|B)=0.92
.
Answers
GPT-4o mini
To determine whether events A and B are independent, we need to recall the definition of independence in probability. Two events A and B are independent if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Alternatively, if:
\[ P(A|B) = P(A) \]
independence can also be confirmed.
Based on the responses provided, we can analyze the statements given:
1. **Response 1**: Says events are independent because \( P(A) = P(A|B) = 0.90 \).
This indicates independence if true, but we need to confirm the values.
2. **Response 2**: Refers to an equality that seems unclear due to formatting, but hints at an evaluation of independence.
3. **Response 3**: Says events are dependent because \( P(A) = 0.93 \) and \( P(A \cap B) = 0.91 \).
For independence, this would need \( 0.93 \cdot P(B) = 0.91 \) to hold true.
4. **Response 4**: Similar to response 3 but formatted differently.
5. **Response 5**: Claims independence because \( P(A) = P(A \cap B) = 0.92 \).
This suggests they are independent based on incorrect equal values—note that \( P(A \cap B) \) should not equal \( P(A) \).
6. **Response 6**: Claims dependency with the probabilities \( P(A) = 0.913 \) and \( P(A|B) = 0.92 \).
If \( P(A|B) \) is greater than \( P(A) \), that suggests the events are dependent.
Without having the specific values of \( P(B) \) and \( P(A \cap B) \), it isn't possible to confirm independence directly from the information given. However, the last option, stating that \( P(A) \neq P(A|B) \), implies that the events are likely dependent.
Based on the most conclusive response provided:
**No; the events are dependent because \( P(A) = 0.913 \) and \( P(A|B) = 0.92 \).**
This indicates that knowing one of the events changes the likelihood of the other, showing dependence.
\[ P(A \cap B) = P(A) \cdot P(B) \]
Alternatively, if:
\[ P(A|B) = P(A) \]
independence can also be confirmed.
Based on the responses provided, we can analyze the statements given:
1. **Response 1**: Says events are independent because \( P(A) = P(A|B) = 0.90 \).
This indicates independence if true, but we need to confirm the values.
2. **Response 2**: Refers to an equality that seems unclear due to formatting, but hints at an evaluation of independence.
3. **Response 3**: Says events are dependent because \( P(A) = 0.93 \) and \( P(A \cap B) = 0.91 \).
For independence, this would need \( 0.93 \cdot P(B) = 0.91 \) to hold true.
4. **Response 4**: Similar to response 3 but formatted differently.
5. **Response 5**: Claims independence because \( P(A) = P(A \cap B) = 0.92 \).
This suggests they are independent based on incorrect equal values—note that \( P(A \cap B) \) should not equal \( P(A) \).
6. **Response 6**: Claims dependency with the probabilities \( P(A) = 0.913 \) and \( P(A|B) = 0.92 \).
If \( P(A|B) \) is greater than \( P(A) \), that suggests the events are dependent.
Without having the specific values of \( P(B) \) and \( P(A \cap B) \), it isn't possible to confirm independence directly from the information given. However, the last option, stating that \( P(A) \neq P(A|B) \), implies that the events are likely dependent.
Based on the most conclusive response provided:
**No; the events are dependent because \( P(A) = 0.913 \) and \( P(A|B) = 0.92 \).**
This indicates that knowing one of the events changes the likelihood of the other, showing dependence.