In a class of 40 students, 22 are women, 10 are earning an \( A \), and 7 are women that are earning an \( A \). If a student is randomly selected from the class, find the probability that th student is earning an \( A \) given that the student is a woman.

To find the probability that a student is earning an A given that the student is a woman, we can use the formula for conditional probability: \[ P(A|W) = \frac{P(A \cap W)}{P(W)} \] Where: - \( P(A|W) \) is the probability of a student earning an A given that the student is a woman. - \( P(A \cap W) \) is the probability of a student being both a woman and earning an A. - \( P(W) \) is the probability of a student being a woman. We are given: - Total number of students \( N = 40 \) - Number of women \( W = 22 \) - Number of students earning an A \( A = 10 \) - Number of women earning an A \( W \cap A = 7 \) First, we calculate \( P(W \cap A) \), which is the probability of a student being both a woman and earning an A. This is simply the number of women earning an A divided by the total number of students: \[ P(W \cap A) = \frac{W \cap A}{N} = \frac{7}{40} \] Next, we calculate \( P(W) \), which is the probability of a student being a woman: \[ P(W) = \frac{W}{N} = \frac{22}{40} \] Now we can calculate \( P(A|W) \): \[ P(A|W) = \frac{P(W \cap A)}{P(W)} = \frac{\frac{7}{40}}{\frac{22}{40}} \] To simplify this, we can cancel out the common factor of 40 in the numerator and denominator: \[ P(A|W) = \frac{7}{22} \] So, the probability that a student is earning an A given that the student is a woman is \( \frac{7}{22} \).