In a class of 26 students, 5 are female and 12 have an A in the class. There are 3 students who are female and have an A in the class. What is the probability that a student who has an A is a female?

To find the probability that a student who has an A is a female, we can use the concept of conditional probability. Let's denote the following events:

• $F$: The event that a student is female.
• $A$: The event that a student has an A in the class.

We are given the following information:

• The total number of students in the class is 26.
• The number of female students is 5.
• The number of students who have an A is 12.
• The number of students who are both female and have an A is 3.

We need to find $P(F|A)$, the probability that a student is female given that the student has an A. This can be calculated using the formula for conditional probability:

$P(F|A) = \frac{P(F \cap A)}{P(A)}$

Where:

• $P(F \cap A)$ is the probability that a student is both female and has an A.
• $P(A)$ is the probability that a student has an A.

First, we calculate $P(F \cap A)$:

$P(F \cap A) = \frac{\text{Number of students who are both female and have an A}}{\text{Total number of students}} = \frac{3}{26}$

Next, we calculate $P(A)$:

$P(A) = \frac{\text{Number of students who have an A}}{\text{Total number of students}} = \frac{12}{26}$

Now, we can find $P(F|A)$:

$P(F|A) = \frac{P(F \cap A)}{P(A)} = \frac{\frac{3}{26}}{\frac{12}{26}} = \frac{3}{12} = \frac{1}{4}$

Therefore, the probability that a student who has an A is a female is $\frac{1}{4}$ or 0.25.