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

\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}

[tex]\[
P(\text{Girl} \mid \text{Senior}) = [?]
\][/tex]
[tex]\[
P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)}
\][/tex]



Answer :

To find the probability that the student selected is a girl given that they are a senior, we are looking for [tex]\( P(\text{Girl} \mid \text{Senior}) \)[/tex].

We can start by identifying the relevant information from the table:

1. The total number of seniors.
2. The number of girl seniors.

From the table, the number of seniors can be calculated as follows:
- Number of boy seniors: 5
- Number of girl seniors: 2

Thus, the total number of seniors:
[tex]\[ 5 \text{ (boy seniors)} + 2 \text{ (girl seniors)} = 7 \text{ seniors} \][/tex]

Next, we focus on the number of girl seniors which is given as:
[tex]\[ 2 \text{ girl seniors} \][/tex]

The probability that the student selected is a girl given that they are senior is calculated by:
[tex]\[ P(\text{Girl} \mid \text{Senior}) = \frac{\text{Number of girl seniors}}{\text{Total number of seniors}} \][/tex]

Substituting the identified values:
[tex]\[ P(\text{Girl} \mid \text{Senior}) = \frac{2}{7} \][/tex]

To summarize, the probability that a randomly selected student is a girl given that they are a senior is:
[tex]\[ \frac{2}{7} \approx 0.2857 \][/tex]

Thus, the probability is approximately [tex]\( 0.2857 \)[/tex] or [tex]\( 28.57\% \)[/tex].