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 the 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 that a student is earning an A given that the student is a woman. - \( P(A \cap W) \) is the probability that a student is both earning an A and is a woman. - \( P(W) \) is the probability that a student is a woman. From the information provided: - 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, \( A \cap W = 7 \) Now we can calculate \( P(A \cap W) \) and \( P(W) \): 1. **Calculate \( P(A \cap W) \)**: \[ P(A \cap W) = \frac{\text{Number of women earning an A}}{\text{Total number of students}} = \frac{7}{40} \] 2. **Calculate \( P(W) \)**: \[ P(W) = \frac{\text{Number of women}}{\text{Total number of students}} = \frac{22}{40} = \frac{11}{20} \] Now we can substitute these values into the conditional probability formula: \[ P(A | W) = \frac{P(A \cap W)}{P(W)} = \frac{\frac{7}{40}}{\frac{11}{20}} \] To divide by a fraction, we multiply by its reciprocal: \[ P(A | W) = \frac{7}{40} \times \frac{20}{11} = \frac{7 \times 20}{40 \times 11} = \frac{140}{440} = \frac{7}{22} \] Thus, the probability that a randomly selected student is earning an A given that the student is a woman is: \[ \boxed{\frac{7}{22}} \]