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

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

[tex]\[
P(\text{Sophomore} \mid \text{Boy}) = \underline{[?]}
\][/tex]

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



Answer :

GinnyAnswer
To determine the probability that a randomly selected student is a sophomore given that they are a boy, we need to follow these steps:

1. Identify the number of sophomore boys:
- From the table, we see that the number of sophomore boys is 9.

2. Calculate the total number of boys:
- We sum up the number of boys in each year:
- Freshman boys: 7
- Sophomore boys: 9
- Junior boys: 7
- Senior boys: 5
- Total number of boys = 7 (Freshman) + 9 (Sophomore) + 7 (Junior) + 5 (Senior) = 28

3. Use the conditional probability formula:
- The probability of being a sophomore given that the student is a boy is given by the ratio of the number of sophomore boys to the total number of boys:
[tex]\[ P(\text{Sophomore} \mid \text{Boy}) = \frac{\text{Number of Sophomore Boys}}{\text{Total Number of Boys}} \][/tex]

4. Substitute the values and compute the probability:
- [tex]\[ P(\text{Sophomore} \mid \text{Boy}) = \frac{9}{28} \][/tex]
- Simplifying this fraction, we get:
[tex]\[ P(\text{Sophomore} \mid \text{Boy}) = 0.32142857142857145 \][/tex]

Hence, the probability that a randomly selected student is a sophomore given that they are a boy is approximately 0.3214, or about 32.14%.

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