To find the probability that a student is involved in sports given that they are a senior, we follow these steps:
1. Determine the given probabilities:
- Probability that a student is a senior: [tex]\( P(\text{senior}) \)[/tex]
- Probability that a student is a senior and involved in sports: [tex]\( P(\text{senior and sports}) \)[/tex]
From the provided table, we find:
- Total number of seniors: 35
- Number of seniors involved in sports: 25
- Total number of students: 100
2. Calculate the probabilities:
- [tex]\( P(\text{senior}) = \frac{\text{Number of seniors}}{\text{Total number of students}} = \frac{35}{100} = 0.35 \)[/tex]
- [tex]\( P(\text{senior and sports}) = \frac{\text{Number of seniors involved in sports}}{\text{Total number of students}} = \frac{25}{100} = 0.25 \)[/tex]
3. Find the conditional probability:
The probability that a student is involved in sports given that they are a senior is given by the formula:
[tex]\[
P(\text{sports} \mid \text{senior}) = \frac{P(\text{senior and sports})}{P(\text{senior})}
\][/tex]
Plugging in the values we calculated:
[tex]\[
P(\text{sports} \mid \text{senior}) = \frac{0.25}{0.35} = 0.7142857142857143
\][/tex]
To convert this probability to a percentage:
[tex]\[
P(\text{sports} \mid \text{senior}) \times 100 = 0.7142857142857143 \times 100 = 71.42857142857143\%
\][/tex]
So, the probability that a student is involved in sports given that they are a senior is approximately [tex]\( 71.43\% \)[/tex].
