To determine the probability that a student participates in sports given that they are a senior, we'll use the concept of conditional probability. Let's break down the steps to solve this problem systematically:
1. Understand the problem: We need the probability that a student participates in sports given that they are a senior, which is denoted \(P(\text{sports} \mid \text{senior})\).
2. Define the probabilities given:
- The probability that a student is a senior, \(P(\text{senior})\).
- The probability that a student is both in sports and a senior, \(P(\text{sports and senior})\).
3. Extract the required values from the table:
- The total number of students is 100.
- The number of seniors is 35.
- The number of seniors who participate in sports is 25.
4. Calculate the individual probabilities:
- \(P(\text{senior})\) is calculated as the number of seniors divided by the total number of students:
[tex]\[
P(\text{senior}) = \frac{35}{100} = 0.35
\][/tex]
- \(P(\text{sports and senior})\) is calculated as the number of seniors who participate in sports divided by the total number of students:
[tex]\[
P(\text{sports and senior}) = \frac{25}{100} = 0.25
\][/tex]
5. Apply the formula for conditional probability:
The formula for conditional probability is given by:
[tex]\[
P(\text{sports} \mid \text{senior}) = \frac{P(\text{sports and senior})}{P(\text{senior})}
\][/tex]
Plugging in the values we have:
[tex]\[
P(\text{sports} \mid \text{senior}) = \frac{0.25}{0.35}
\][/tex]
6. Perform the division:
[tex]\[
P(\text{sports} \mid \text{senior}) = \frac{0.25}{0.35} \approx 0.7143
\][/tex]
7. Convert to a percentage and round to the nearest whole number:
[tex]\[
P(\text{sports} \mid \text{senior}) \times 100\% \approx 71.43\%
\][/tex]
Rounded to the nearest whole number, this is 71%.
Answer: The probability that a student is in sports given that they are a senior is approximately 71%.
