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%.
