To find the probability that a randomly selected student is a freshman given that they are a girl, we need to follow these steps:
1. Determine the Total Number of Girls:
First, we sum the number of girls in each class (Freshman, Sophomore, Junior, Senior).
[tex]\[
\text{Total number of girls} = 5 + 5 + 4 + 2 = 16
\][/tex]
2. Identify the Number of Freshman Girls:
From the table, we see that the number of freshman girls is 5.
3. Calculate the Conditional Probability:
To find the probability that a student is a freshman given that they are a girl, we use the formula:
[tex]\[
P(\text{Freshman} \mid \text{Girl}) = \frac{P(\text{Freshman and Girl})}{P(\text{Girl})}
\][/tex]
Here, [tex]\( P(\text{Freshman and Girl}) \)[/tex] corresponds to the number of freshman girls, and [tex]\( P(\text{Girl}) \)[/tex] corresponds to the total number of girls.
4. Substitute the Values:
[tex]\[
P(\text{Freshman} \mid \text{Girl}) = \frac{\text{Number of Freshman Girls}}{\text{Total Number of Girls}} = \frac{5}{16}
\][/tex]
5. Simplify the Fraction:
After division, we get:
[tex]\[
P(\text{Freshman} \mid \text{Girl}) = 0.3125
\][/tex]
Therefore, the probability that a randomly selected student is a freshman given that they are a girl is [tex]\( 0.3125 \)[/tex].
