To find the probability that a student selected at random is a freshman given that they are a girl, we need to follow these steps:
1. Identify the given data:
[tex]\[
\begin{array}{c|c|c|c|c}
\text{} & \text{Freshman} & \text{Sophomore} & \text{Junior} & \text{Senior} \\
\hline
\text{Boy} & 7 & 9 & 7 & 5 \\
\text{Girl} & 5 & 5 & 4 & 2 \\
\end{array}
\][/tex]
2. Determine the total number of girls:
To calculate the total number of girls, sum up the number of girls in each category (Freshman, Sophomore, Junior, Senior):
[tex]\[
5 \text{ (Freshman girls)} + 5 \text{ (Sophomore girls)} + 4 \text{ (Junior girls)} + 2 \text{ (Senior girls)} = 16 \text{ girls}
\][/tex]
3. Calculate the probability:
According to the formula [tex]\(P(\text{Freshman} \mid \text{Girl}) = \frac{P(\text{Girl and Freshman})}{P(\text{Girl})}\)[/tex], we need to find the probability that the student is both a girl and a freshman, divided by the probability that they are a girl.
- The number of freshman girls is [tex]\(5\)[/tex].
- The total number of girls is [tex]\(16\)[/tex].
The probability that a student is a freshman given that they are a girl is:
[tex]\[
P(\text{Freshman} \mid \text{Girl}) = \frac{5}{16}
\][/tex]
4. Simplify the fraction if necessary and convert to a decimal:
[tex]\[
\frac{5}{16} = 0.3125
\][/tex]
So, the probability that a student is a freshman given that they are a girl is [tex]\(0.3125\)[/tex] or [tex]\(31.25\% \)[/tex].
