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)?
A.
100
20
⋅
100
47
B.
100
26
⋅
99
47
C.
100
26
+
99
47
D.
100
26
A.
100
20
⋅
100
47
B.
100
26
⋅
99
47
C.
100
26
+
99
47
D.
100
26
Answers
Answered by
GPT-4o mini
To find the probability \( P(\text{brown} \mid \text{black}) \), we need to use the conditional probability formula:
\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]
In this scenario:
- \( A \) is the event of selecting a brown pair of shoes.
- \( B \) is the event of selecting a black pair of shoes.
However, since brown and black shoes cannot be paired simultaneously (one cannot choose both a brown pair and a black pair at the same time), we recognize that \( P(A \cap B) = 0 \) since there are no shoes that are both brown and black.
Thus, we cannot calculate a meaningful probability \( P(brown \mid black) \) using standard definitions because \( P(brown \mid black) \) is not a quantifiable component within the possibilities of the shoe pairs.
In the context of the answer options:
- **Option A, B, C, and D** do not have the potential to compute \( P(brown \mid black) \) correctly under the defined formula due to the lack of overlap between brown and black shoes.
Therefore, the answer is that none of the options provided correctly represent the conditional probability because it is technically irrelevant in this context (the event probabilities are zero).
In other words, \( P(brown \mid black) = 0 \) due to the absence of any overlapping pairs.
If you need to answer differently based on assumptions or specifics concerning definitions, please clarify, as in probability theory \( P(A \mid B) \) cannot yield a result when \( P(A \cap B) = 0 \).
\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]
In this scenario:
- \( A \) is the event of selecting a brown pair of shoes.
- \( B \) is the event of selecting a black pair of shoes.
However, since brown and black shoes cannot be paired simultaneously (one cannot choose both a brown pair and a black pair at the same time), we recognize that \( P(A \cap B) = 0 \) since there are no shoes that are both brown and black.
Thus, we cannot calculate a meaningful probability \( P(brown \mid black) \) using standard definitions because \( P(brown \mid black) \) is not a quantifiable component within the possibilities of the shoe pairs.
In the context of the answer options:
- **Option A, B, C, and D** do not have the potential to compute \( P(brown \mid black) \) correctly under the defined formula due to the lack of overlap between brown and black shoes.
Therefore, the answer is that none of the options provided correctly represent the conditional probability because it is technically irrelevant in this context (the event probabilities are zero).
In other words, \( P(brown \mid black) = 0 \) due to the absence of any overlapping pairs.
If you need to answer differently based on assumptions or specifics concerning definitions, please clarify, as in probability theory \( P(A \mid B) \) cannot yield a result when \( P(A \cap B) = 0 \).