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+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⋅47100 Start Fraction 26 over 100 End Fraction times Start Fraction 47 over 100 End Fraction 26100

1 answer

To solve for \( P(\text{brown} | \text{black}) \), we want to find the conditional probability of selecting a brown pair of shoes given that we have selected a black pair of shoes.

According to the definition of conditional probability:

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]

In this case, let's define:

  • Let \( A \) be the event of selecting a brown pair of shoes.
  • Let \( B \) be the event of selecting a black pair of shoes.

Since brown shoes and black shoes are mutually exclusive (you cannot have a pair that is both brown and black), the intersection \( A \cap B \) is empty, meaning \( P(A \cap B) = 0 \). Therefore, the probability \( P(brown | black) \) is:

\[ P(brown | black) = \frac{0}{P(black)} = 0 \]

In a more direct interpretation:

  1. The number of brown pairs is 26.
  2. The number of black pairs is 47.
  3. The total number of pairs is 100.

Since there are no pairs that are both brown and black, \( P(brown | black) = 0 \).

However, if we are asked how to calculate \( P(\text{brown} | \text{black}) \) in a way that does not assume they are mutually exclusive and instead consider probabilities within the context of pairs selected, the correct response should indicate the conditionality based on a total number of pairs.

But given the context of the question and the definitions, the most accurate and straightforward response remains:

\[ \boxed{0} \]