A student is randomly selected from this table. What is the probability that they are a freshman, given that they are a girl?

\begin{tabular}{|c|c|c|c|c|}
\hline \multicolumn{5}{|c|}{ Students on a Team } \\
\hline & Freshman & Sophomore & Junior & Senior \\
\hline Boy & 7 & 9 & 7 & 5 \\
\hline Girl & 5 & 5 & 4 & 2 \\
\hline
\end{tabular}

Calculate [tex]\( P(\text{Freshman} \mid \text{Girl}) \)[/tex] using the formula:
[tex]\[ P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)} \][/tex]

Enter your answer:



Answer :

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].