Question
Mr. Jay has 100 pairs of shoes in his closet. He has 26 brown pairs, 12 navy pairs, 47 black pairs, and 15 white pairs. What is P(brown|black)
?(1 point)
Responses
26100⋅47100
Start Fraction 26 over 100 End Fraction times Start Fraction 47 over 100 End Fraction
26100+4799
Start Fraction 26 over 100 End Fraction plus Start Fraction 47 over 99 End Fraction
26100⋅4799
Start Fraction 26 over 100 End Fraction times Start Fraction 47 over 99 End Fraction
26100
?(1 point)
Responses
26100⋅47100
Start Fraction 26 over 100 End Fraction times Start Fraction 47 over 100 End Fraction
26100+4799
Start Fraction 26 over 100 End Fraction plus Start Fraction 47 over 99 End Fraction
26100⋅4799
Start Fraction 26 over 100 End Fraction times Start Fraction 47 over 99 End Fraction
26100
Answers
GPT-4o mini
To find \( P(\text{brown} | \text{black}) \), we are looking for the conditional probability of having a brown pair given that the selected pair is black.
The formula for conditional probability is given by:
\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]
In this case, let:
- \( A \) be the event of choosing a brown pair.
- \( B \) be the event of choosing a black pair.
Since there are no brown pairs that are also black (as they are distinct categories), the intersection \( P(A \cap B) \) is 0.
Instead, we will focus on the context of counting the total number of pairs.
From the problem, we know:
- Total shoes = 100
- Brown pairs = 26
- Black pairs = 47
Thus, the probability \( P(B) \) of selecting a black pair is:
\[
P(B) = \frac{47}{100}
\]
However, since having a brown pair given that we have selected a black one does not make sense (since they are distinct categories), we reconsider that conditional probability \( P(\text{brown} | \text{black}) \) just inspects the base rates and not the intersection.
The probability of getting a brown pair when we know we are getting a black pair is still effectively 0 since there are no brown-black overlaps.
Therefore:
\[
P(\text{brown|black}) = 0
\]
None of the provided response options seem to reflect this scenario accurately, including possible misinterpretations. Please clarify if there is a different context intended.
The formula for conditional probability is given by:
\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]
In this case, let:
- \( A \) be the event of choosing a brown pair.
- \( B \) be the event of choosing a black pair.
Since there are no brown pairs that are also black (as they are distinct categories), the intersection \( P(A \cap B) \) is 0.
Instead, we will focus on the context of counting the total number of pairs.
From the problem, we know:
- Total shoes = 100
- Brown pairs = 26
- Black pairs = 47
Thus, the probability \( P(B) \) of selecting a black pair is:
\[
P(B) = \frac{47}{100}
\]
However, since having a brown pair given that we have selected a black one does not make sense (since they are distinct categories), we reconsider that conditional probability \( P(\text{brown} | \text{black}) \) just inspects the base rates and not the intersection.
The probability of getting a brown pair when we know we are getting a black pair is still effectively 0 since there are no brown-black overlaps.
Therefore:
\[
P(\text{brown|black}) = 0
\]
None of the provided response options seem to reflect this scenario accurately, including possible misinterpretations. Please clarify if there is a different context intended.